[
  {
    "id": "30-60-90-triangle",
    "ja": {
      "term": "30°, 60°, 90° の直角三角形",
      "reading": "さんじゅうどろくじゅうどきゅうじゅうどのちょっかくさんかくけい",
      "alt": [
        "30-60-90 の三角形",
        "半正三角形"
      ]
    },
    "en": {
      "term": "30-60-90 triangle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "3 つの角が 30°, 60°, 90° の直角三角形。辺の比は短い順に 1 : √3 : 2 で、正三角形を高さで半分にした形。",
    "definition_en": "A right triangle whose angles measure 30°, 60°, and 90°; it is half of an equilateral triangle. With shortest side x, the longer leg is x√3 and the hypotenuse is 2x.",
    "latex": "x : x\\sqrt{3} : 2x",
    "spoken_en": "x to x root three to two x",
    "examples": [
      {
        "en": "In a 30-60-90 triangle the hypotenuse is twice the short leg, so if the short leg is 4, the hypotenuse is 8.",
        "ja": "30°, 60°, 90° の直角三角形では斜辺は短い辺の 2 倍だから、短い辺が 4 なら斜辺は 8 だ。",
        "register": "spoken"
      },
      {
        "en": "In a 30-60-90 triangle, the hypotenuse is 12. Find the lengths of the two legs.",
        "ja": "30°, 60°, 90° の直角三角形で、斜辺の長さが 12 である。直角をはさむ 2 辺の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "角の大きさで呼ぶ（1 語の名前は用例コーパスと参照に出てこない）。CK-12 Geometry 4.43 は 30-60-90 triangle と呼び、辺の比を 30-60-90 Theorem と名づける。読みは thirty-sixty-ninety triangle（本プロジェクトの読みの案内）。",
      "日本語版 Wikipedia「三角定規」「直角三角形」は、三角定規を直角二等辺三角形と半正三角形の 2 枚 1 組と書く。CK-12 Geometry 4.43 は辺を短い方から x, x√3, 2x と書く。√3 がつくのは 60° の向かいの長い方の leg で、斜辺ではない。",
      "30° の向かいが短い leg（short leg）、60° の向かいが長い leg（long leg）。取り違えると x と x√3 が入れかわる。"
    ],
    "related": [
      "special-right-triangles",
      "equilateral-triangle",
      "45-45-90-triangle",
      "legs",
      "hypotenuse"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.43 30-60-90 Right Triangles"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角定規」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%AE%9A%E8%A6%8F",
        "note": "直角二等辺三角形と半正三角形の 2 枚 1 組"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "30-60-90 triangle": 3
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 30-60-90 triangle（CK-12 Geometry 4.43 30-60-90 Right Triangles・CK-12 Geometry 4.41 Special Right Triangles and Ratios・CK-12 Geometry 5.17 Area and Perimeter of Rhombuses and Kites・計 14 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "45-45-90-triangle",
    "ja": {
      "term": "直角二等辺三角形",
      "reading": "ちょっかくにとうへんさんかくけい",
      "alt": [
        "45°, 45°, 90° の三角形"
      ]
    },
    "en": {
      "term": "isosceles right triangle",
      "alt": [
        "45-45-90 triangle",
        "right isosceles triangle"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直角をはさむ 2 辺の長さが等しい直角三角形。角は 45°, 45°, 90° で、辺の比は 1 : 1 : √2。",
    "definition_en": "A right triangle whose two legs are equal in length, so its angles are 45°, 45°, and 90°. Its sides are in the ratio 1 : 1 : √2.",
    "latex": "1 : 1 : \\sqrt{2}",
    "spoken_en": "one to one to the square root of two",
    "examples": [
      {
        "en": "Cut a square along a diagonal and you get a 45-45-90 triangle, so the hypotenuse is a leg times root 2.",
        "ja": "正方形を対角線で切ると、45°, 45°, 90° の三角形ができる。だから斜辺は直角をはさむ辺の √2 倍だ。",
        "register": "spoken"
      },
      {
        "en": "The hypotenuse of an isosceles right triangle is 10. Find the length of each leg.",
        "ja": "直角二等辺三角形の斜辺の長さが 10 である。直角をはさむ辺の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "CK-12 Geometry 4.42 は isosceles right triangle を用語として定義し、節の名前と glossary では 45-45-90 triangle を使い、辺の比を 45-45-90 Theorem と名づける。IM Geometry 4.2 Half a Square も isosceles right triangle と呼び（right isosceles triangle とも）、正方形を対角線で半分にした形として扱う。",
      "45-45-90 triangle の数字は forty-five, forty-five, ninety と読む（本プロジェクトの読みの案内）。",
      "斜辺から直角をはさむ辺を求めるときは √2 で割る（10 ÷ √2 = 5√2）。CK-12 4.42 は分母を有理化した形で書き、IM Geometry 4.2 は 12/√2 のまま書く（エントリ rationalizing-the-denominator）。"
    ],
    "related": [
      "special-right-triangles",
      "isosceles-triangle",
      "legs",
      "square-shape",
      "30-60-90-triangle"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 4.2 Half a Square"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.42 45-45-90 Right Triangles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "45-45-90 triangle": 2,
        "isosceles right triangle": 3
      },
      "written": {
        "right isosceles triangle": 1,
        "isosceles right triangle": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは IM の呼び方 isosceles right triangle（Geometry 4.2 Half a Square・Grade 8 6.1 Organizing Data・Grade 8 1.8 Rotation Patterns・計 9 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "aa-similarity",
    "ja": {
      "term": "2 組の角がそれぞれ等しい",
      "reading": "にくみのかくがそれぞれひとしい",
      "alt": [
        "AA 相似"
      ]
    },
    "en": {
      "term": "angle-angle triangle similarity theorem",
      "alt": [
        "AA similarity",
        "AA similarity postulate",
        "angle-angle similarity"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 つの三角形で、2 組の角がそれぞれ等しければ相似であるという相似条件。内角の和が 180° なので、残りの角も等しくなる。",
    "definition_en": "A triangle similarity criterion: two triangles are similar whenever two angles of one are congruent to two angles of the other. The third pair then agrees too, since the three angles of any triangle total 180°.",
    "latex": "\\angle A = \\angle D,\\ \\angle B = \\angle E \\implies \\triangle ABC \\sim \\triangle DEF",
    "spoken_en": "if angle A equals angle D and angle B equals angle E, then triangle ABC is similar to triangle DEF",
    "examples": [
      {
        "en": "They share angle C, and they both have a right angle, so the triangles are similar by AA.",
        "ja": "∠C は共通で、どちらにも直角があるから、2 組の角がそれぞれ等しく、この 2 つの三角形は相似だ。",
        "register": "spoken"
      },
      {
        "en": "In the figure, DE ∥ BC. Prove that △ADE ~ △ABC.",
        "ja": "図で DE ∥ BC のとき、△ADE ∽ △ABC であることを証明せよ。",
        "register": "written"
      },
      {
        "en": "Two angles of one triangle are congruent to two angles of the other, so the triangles are similar by the Angle-Angle Triangle Similarity Theorem.",
        "ja": "2 つの三角形は 2 組の角がそれぞれ等しいので、相似である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "similar by AA",
        "ja": "2 組の角がそれぞれ等しいので相似"
      }
    ],
    "pitfalls": [
      "AA は相似条件で、合同条件ではない。角がすべて等しくても大きさは決まらない（AAA でも合同にはならない）。",
      "相似の記号は、中学校学習指導要領解説（〔用語・記号〕）では ∽、IM Geometry・CK-12 Geometry では ∼（△ABC ∼ △DEF）。",
      "IM Geometry は Angle-Angle Triangle Similarity Theorem と呼び（angle-angle triangle similarity 16 件、angle-angle similarity 1 件）、CK-12 Geometry は AA Similarity Postulate と呼ぶ（7.7 AA Similarity、11 件。AAA Similarity Theorem とも）。用例コーパスには出てこない。"
    ],
    "related": [
      "similar-triangles",
      "scale-factor",
      "sss-congruence",
      "similarity-transformation"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "3.9 Conditions for Triangle Similarity ほか（16 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "7.7 AA Similarity（AA Similarity Postulate）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形〔用語・記号〕: ∽"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）〔用語・記号〕",
        "note": "中3 B 図形: ∽"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 angle-angle triangle similarity theorem（Geometry 3.9 Conditions for Triangle Similarity・Geometry 3.13 Using the Pythagorean Theorem and Similarity・Geometry 3.16 Bank Shot・計 16 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "aas-congruence",
    "ja": {
      "term": "2 組の角とその間にない 1 辺がそれぞれ等しい",
      "reading": "にくみのかくとそのあいだにないいっぺんがそれぞれひとしい",
      "alt": [
        "AAS"
      ]
    },
    "en": {
      "term": "AAS",
      "alt": [
        "angle-angle-side"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「2 組の角とその間にない 1 辺がそれぞれ等しい」は学習指導要領解説に無い（日本語版 Wikipedia「図形の合同」は AAS をほぼ同じ言い方で挙げる）。日本の三角形の合同条件には無い（1 辺とその両端の角が等しい、に帰着する）。",
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの三角形で、2 組の角と、その 2 角にはさまれていない 1 組の辺がそれぞれ等しければ合同であるという合同条件。内角の和から残りの角も等しくなるので、ASA に帰着する。",
    "definition_en": "A triangle congruence criterion: if two angles and a non-included side of one triangle are congruent to the corresponding angles and side of another, the triangles are congruent. It follows from ASA, because the third angles must also be equal.",
    "latex": "\\angle A = \\angle D,\\ \\angle B = \\angle E,\\ BC = EF \\implies \\triangle ABC \\cong \\triangle DEF",
    "spoken_en": "if angle A equals angle D, angle B equals angle E, and BC equals EF, then triangle ABC is congruent to triangle DEF",
    "examples": [
      {
        "en": "The side isn't between the two angles, so it's not ASA, it's AAS. That still works, because the third angles have to match too.",
        "ja": "辺が 2 つの角の間にないから ASA ではなく AAS だ。それでも合同になる。残りの角も必ず等しくなるからだ。",
        "register": "spoken"
      },
      {
        "en": "In △ABC and △DEF, ∠A = ∠D, ∠B = ∠E, and BC = EF. Explain why the two triangles are congruent.",
        "ja": "△ABC と △DEF で ∠A = ∠D, ∠B = ∠E, BC = EF である。この 2 つの三角形が合同である理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "by AAS",
        "ja": "2 組の角とその間にない 1 辺がそれぞれ等しいから"
      }
    ],
    "pitfalls": [
      "AAS と SSA は別物。2 辺とその間にない角（SSA）は合同条件にならない（正弦定理の ambiguous case で、三角形が 2 通りできることがある）。",
      "日本の合同条件にはこの形がないので、日本の答案では残りの角も等しいことを示してから「1 組の辺とその両端の角がそれぞれ等しい」を使う。",
      "OpenStax Algebra and Trigonometry の angle-angle-side は正弦定理で解く問題の場合分けの名前で、合同条件としては用例コーパスにも IM にも出てこない。合同の証明では by AAS と言う。"
    ],
    "related": [
      "asa-congruence",
      "sss-congruence",
      "sas-congruence",
      "law-of-sines",
      "ambiguous-case"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「図形の合同」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%B3%E5%BD%A2%E3%81%AE%E5%90%88%E5%90%8C",
        "note": "AAS を 2 組の角とその間にない 1 組の辺が等しいと説明する"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元 2 の前の修正」の 4）。en.term は AAS、angle-angle-side は en.alt。mapping near。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "absolute-convergence",
    "ja": {
      "term": "絶対収束",
      "reading": "ぜったいしゅうそく",
      "alt": []
    },
    "en": {
      "term": "absolute convergence",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "各項の絶対値をとった級数 Σ|aₙ| が収束すること。絶対収束する級数は収束する。",
    "definition_en": "Convergence of the series of absolute values Σ|aₙ|; an absolutely convergent series converges.",
    "latex": "\\sum |a_n| \\text{ converges}",
    "spoken_en": null,
    "examples": [
      {
        "en": "If it converges absolutely, it converges — you can drop the signs and check.",
        "ja": "絶対収束すれば収束する。符号を外して調べればいい。",
        "register": "spoken"
      },
      {
        "en": "Determine whether Σ (−1)ⁿ/n² converges absolutely, converges conditionally, or diverges.",
        "ja": "Σ (−1)ⁿ/n² が絶対収束するか、条件収束するか、発散するかを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞の形 converges absolutely のほうが多い（書き言葉では名詞の absolute convergence より多い）。形容詞は absolutely convergent。AP の CED は topic 10.9 で absolutely convergent と converges absolutely を使う。"
    ],
    "related": [
      "conditional-convergence",
      "ratio-test",
      "alternating-series-test"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "絶対収束",
        "en": "Absolute convergence"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.9（absolutely convergent）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "absolute convergence": 1
      },
      "written": {
        "absolute convergence": 14
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "absolute-extrema",
    "ja": {
      "term": "最大値・最小値",
      "reading": "さいだいち・さいしょうち",
      "alt": [
        "最大・最小",
        "絶対極値"
      ]
    },
    "en": {
      "term": "absolute extrema",
      "alt": [],
      "variants": [
        {
          "term": "absolute maximum and minimum",
          "register": "written",
          "note": "書き言葉の次点。話し言葉には出てこない。書き言葉は OpenStax Calculus と OpenStax Algebra and Trigonometry が半々。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "ja.alt の「絶対極値」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "区間全体（または定義域全体）での最大値と最小値。",
    "definition_en": "The largest and smallest values a function takes on its whole domain or on a given interval.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "On a closed interval, check the critical points and the endpoints to find the absolute extrema.",
        "ja": "閉区間では、臨界点と端点を調べて最大値・最小値を求める。",
        "register": "spoken"
      },
      {
        "en": "On [0, 2], find the absolute maximum value and the absolute minimum value of f(x) = x³ − 3x.",
        "ja": "区間 [0, 2] における f(x) = x³ − 3x の最大値と最小値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "区間全体の最大・最小には absolute（または global）を付け、極大・極小（local ／ relative）と区別する。"
    ],
    "related": [
      "extreme-value-theorem",
      "local-extremum",
      "critical-point"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 5（absolute extrema）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「絶対極値」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "absolute extrema": 3
      },
      "written": {
        "absolute extrema": 31,
        "absolute maximum and minimum": 12
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "absolute-value-equation",
    "ja": {
      "term": "絶対値を含む方程式",
      "reading": "ぜったいちをふくむほうていしき",
      "alt": [
        "絶対値方程式"
      ]
    },
    "en": {
      "term": "absolute value equation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "未知数を絶対値記号の中に含む方程式。中身の符号で場合分けして解く。",
    "definition_en": "An equation with the unknown inside absolute value bars, solved by splitting into cases by the sign of the inside.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For |x − 3| = 5, split it into two cases: x − 3 = 5 or x − 3 = −5.",
        "ja": "|x − 3| = 5 は x − 3 = 5 または x − 3 = −5 の 2 つに分ける。",
        "register": "spoken"
      },
      {
        "en": "Solve the absolute value equation |2x + 1| = 7.",
        "ja": "方程式 |2x + 1| = 7 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは書き言葉（OpenStax Algebra and Trigonometry・OpenStax Intermediate Algebra）に多い。話し言葉にはほとんど出てこない。"
    ],
    "related": [
      "solve",
      "absolute-value",
      "absolute-value-inequality",
      "split-into-cases"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "absolute value equation": 2
      },
      "written": {
        "absolute value equation": 40
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "absolute-value-function",
    "ja": {
      "term": "絶対値関数",
      "reading": "ぜったいちかんすう",
      "alt": []
    },
    "en": {
      "term": "absolute value function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1"
      ]
    },
    "definition_ja": "f(x) = |x| と、それを平行移動・拡大した f(x) = a|x − h| + k の形の関数。グラフは V 字形で、頂点は (h, k)。",
    "definition_en": "The function f(x) = |x| and its transformations of the form f(x) = a|x − h| + k. Its graph is V-shaped with the vertex at (h, k).",
    "latex": "f(x) = |x|",
    "spoken_en": "f of x equals the absolute value of x",
    "examples": [
      {
        "en": "The graph of an absolute value function makes a V, and the point at the bottom of the V is the vertex.",
        "ja": "絶対値関数のグラフは V の字になって、V のいちばん下の点が頂点だよ。",
        "register": "spoken"
      },
      {
        "en": "Graph the absolute value function f(x) = |x − 2| + 1 and identify its vertex.",
        "ja": "絶対値関数 f(x) = |x − 2| + 1 のグラフをかき、頂点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では |x| を the absolute value of x と読む。f(x) = |x − 2| は f of x equals the absolute value of x minus 2。",
      "頂点 (h, k) の h は、絶対値の中の符号と逆になる。f(x) = |x + 3| − 1 の頂点は (−3, −1)。",
      "a < 0 のとき、グラフは上下が逆の V 字（頂点が最も高い点）になる。"
    ],
    "related": [
      "absolute-value",
      "piecewise-function",
      "vertex",
      "transformations-of-functions",
      "absolute-value-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "absolute value function": 27
      },
      "written": {
        "absolute value function": 72
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "absolute-value-inequality",
    "ja": {
      "term": "絶対値を含む不等式",
      "reading": "ぜったいちをふくむふとうしき",
      "alt": []
    },
    "en": {
      "term": "absolute value inequality",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "definition_ja": "|x − 2| < 3 のように、絶対値の中に未知数を含む不等式。a > 0 のとき、|X| < a は −a < X < a、|X| > a は X < −a または X > a として解く。",
    "definition_en": "An inequality with the variable inside absolute value bars, such as |x − 2| < 3. For a > 0, |X| < a means −a < X < a, and |X| > a means X < −a or X > a.",
    "latex": "|x| < a \\iff -a < x < a \\quad (a > 0)",
    "spoken_en": "the absolute value of x is less than a if and only if x is between negative a and a",
    "examples": [
      {
        "en": "With an absolute value inequality, think distance: |x − 3| > 2 means x is more than 2 units away from 3.",
        "ja": "絶対値を含む不等式は距離で考えよう。|x − 3| > 2 は、x が 3 から 2 より遠くにあるということだ。",
        "register": "spoken"
      },
      {
        "en": "Solve the absolute value inequality |2x − 1| ≤ 5.",
        "ja": "絶対値を含む不等式 |2x − 1| ≦ 5 を解け。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "solve an absolute value inequality",
        "ja": "絶対値を含む不等式を解く"
      }
    ],
    "pitfalls": [
      "|X| > a の答えは X < −a または X > a（英語では or）。「かつ」の範囲にまとめない。",
      "|x − 1| + |x − 3| < 4 のように絶対値が 2 つ以上あるときは、中身の符号で場合分けする。"
    ],
    "related": [
      "absolute-value-equation",
      "linear-inequality",
      "case-analysis",
      "split-into-cases",
      "number-line"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "absolute value inequality": 3
      },
      "written": {
        "absolute value inequality": 26
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "absolute-value",
    "ja": {
      "term": "絶対値",
      "reading": "ぜったいち",
      "alt": []
    },
    "en": {
      "term": "absolute value",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "数直線上で、ある数を表す点と原点との距離。たとえば 3 と −3 の絶対値はどちらも 3 である。",
    "definition_en": "How far a number is from zero on the number line, so it is never negative: |−3| = 3 and |3| = 3.",
    "latex": "|x|",
    "spoken_en": "the absolute value of x",
    "examples": [
      {
        "en": "The absolute value of negative 6 is 6, because negative 6 is 6 units away from zero.",
        "ja": "−6 の絶対値は 6。0 から 6 目盛り離れているからね。",
        "register": "spoken"
      },
      {
        "en": "Find the absolute value of each number: −12, 3/4, 0.",
        "ja": "次の数の絶対値を求めよ。−12、3/4、0",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "絶対値が 5 である数は 5 と −5 の 2 つ。絶対値だけでは符号は決まらない。",
      "|x| がいつも x になるわけではない。x が負の数なら |x| = −x（たとえば |−4| = 4 = −(−4)）。"
    ],
    "related": [
      "number-line",
      "distance",
      "opposite",
      "absolute-value-equation",
      "absolute-value-inequality"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "絶対値",
        "en": "Absolute value"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "absolute value": 1339
      },
      "written": {
        "absolute value": 411
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "acceleration",
    "ja": {
      "term": "加速度",
      "reading": "かそくど",
      "alt": []
    },
    "en": {
      "term": "acceleration",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "速度を時刻で微分したもの。",
    "definition_en": "How fast velocity is changing over time; it is the derivative of velocity, a(t) = v′(t).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Acceleration is the derivative of velocity, so it's the second derivative of position.",
        "ja": "加速度は速度の導関数なので、位置の第 2 次導関数。",
        "register": "spoken"
      },
      {
        "en": "Find the acceleration of the particle at t = 1 if v(t) = 3t² − 4.",
        "ja": "v(t) = 3t² − 4 のとき、t = 1 における加速度を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "velocity",
      "second-derivative",
      "velocity-vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 4.2（acceleration）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "acceleration": 222
      },
      "written": {
        "acceleration": 183
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "accumulation-function",
    "ja": {
      "term": "定積分で表された関数",
      "reading": "ていせきぶんであらわされたかんすう",
      "alt": [
        "累積関数",
        "積分で定義された関数"
      ]
    },
    "en": {
      "term": "accumulation function",
      "alt": [
        "function defined by an integral"
      ],
      "variants": [
        {
          "term": "area function",
          "register": "spoken",
          "note": "話し言葉ではこの言い方しか数えられず、3Blue1Brown が大半。accumulation function はコーパスに出てこないので、AP の CED（topic 6.4）の呼び方として見出しにした。書き言葉にはほとんど出てこない。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「定積分で表された関数」は学習指導要領解説に無い。上端が変数の ∫ₐˣ f(t)dt のほか、∫₀¹ f(t)dt のように値が定数になる積分を含む式もこの名前で扱うことがある。前者はAP の CED は topic 6.4・6.5・8.3 で accumulation functions（functions defined by integrals とも）と呼ぶ。後者の「定積分を定数 k とおく」型の問題が米国の教科書にあるかは教科書による。ja.alt の「累積関数」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "上端などに変数を含む定積分で定められた関数。F(x) = ∫ₐˣ f(t)dt は a から x までに積み重なった量を表し、微分すると f(x) になる。",
    "definition_en": "A function given by a definite integral with a variable limit, such as F(x) = ∫ₐˣ f(t) dt: the amount accumulated from a up to x. Its derivative is f(x).",
    "latex": "F(x) = \\int_a^x f(t)\\,dt",
    "spoken_en": "F of x equals the integral from a to x of f of t dt",
    "examples": [
      {
        "en": "Here g is an accumulation function, so g prime of x is just f of x.",
        "ja": "ここでの g は定積分で表された関数だから、g′(x) はただの f(x)。",
        "register": "spoken"
      },
      {
        "en": "Here g is defined by an integral, so to find g prime you don't integrate at all. You just use the fundamental theorem.",
        "ja": "g は定積分で表された関数だから、g′ を求めるのに積分はいらない。基本定理を使うだけ。",
        "register": "spoken"
      },
      {
        "en": "Let g(x) = ∫₀ˣ f(t) dt. Find g′(2).",
        "ja": "g(x) = ∫₀ˣ f(t)dt とする。g′(2) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "積分の中の変数を t にするのは、上端の x と区別するため。OpenStax Calculus と AP の CED も同じ書き方をする。"
    ],
    "related": [
      "net-change",
      "fundamental-theorem-of-calculus",
      "fundamental-theorem-of-calculus-part-1",
      "upper-limit",
      "definite-integral"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.4（accumulation functions ／ functions defined by integrals）、6.5、8.3"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「定積分で表された関数」は出てこない・「累積関数」は出てこない"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "5.3 The Fundamental Theorem of Calculus（積分の中の変数を t にする書き方）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "area function": 11
      },
      "written": {
        "area function": 3
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "acute-angle",
    "ja": {
      "term": "鋭角",
      "reading": "えいかく",
      "alt": []
    },
    "en": {
      "term": "acute angle",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1",
        "Pre-Algebra"
      ]
    },
    "definition_ja": "大きさが 0° より大きく 90° より小さい角。直角よりも開きが小さい。",
    "definition_en": "An angle that measures more than 0° and less than 90°, so it is smaller than a right angle.",
    "latex": "0^\\circ < \\theta < 90^\\circ",
    "spoken_en": "zero degrees is less than theta, which is less than ninety degrees",
    "examples": [
      {
        "en": "An acute angle is anything less than 90 degrees, so it's narrower than the corner of your paper.",
        "ja": "鋭角は 90° より小さい角のこと。だから紙の角（かど）よりもせまい。",
        "register": "spoken"
      },
      {
        "en": "Select all of the following that could be the measure of an acute angle: 15°, 64°, 90°, 91°.",
        "ja": "次の 15°、64°、90°、91° のうち、鋭角の大きさになりうるものをすべて選べ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "acute は日常語では「鋭い」「急性の」の意味で使う単語。数学では 90° より小さい角を表す。",
      "0° と 90° は含まない。ちょうど 90° の角は直角（エントリ right-angle）で、鋭角ではない。",
      "鋭角三角形（エントリ acute-triangle）は 3 つの角がすべて鋭角の三角形。直角三角形や鈍角三角形にも鋭角は 2 つある。"
    ],
    "related": [
      "right-angle",
      "obtuse-angle",
      "acute-triangle",
      "angle",
      "complementary-angle"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 鋭角の三角比（level.jp 数I）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 1.9・Grade 7 1.4・Grade 8 1.1（acute angle。level.us Pre-Algebra）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "acute angle": 4
      },
      "written": {
        "acute angle": 40
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "acute-triangle",
    "ja": {
      "term": "鋭角三角形",
      "reading": "えいかくさんかくけい",
      "alt": []
    },
    "en": {
      "term": "acute triangle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "3 つの角がすべて鋭角である三角形。いちばん大きい角も 90° より小さい。",
    "definition_en": "A triangle in which all three angles are acute, so even its largest angle is less than 90°.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "All three angles are less than 90, so this is an acute triangle.",
        "ja": "3 つの角が全部 90° より小さいから、これは鋭角三角形。",
        "register": "spoken"
      },
      {
        "en": "Two angles of a triangle measure 55° and 65°. Is the triangle an acute triangle? Explain your reasoning.",
        "ja": "三角形の 2 つの角が 55°、65° である。この三角形は鋭角三角形か。理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "角が 1 つ鋭角なだけでは鋭角三角形とはいえない。どの三角形にも鋭角が少なくとも 2 つあるので、いちばん大きい角が鋭角かどうかで見分ける。",
      "角による三角形の分け方は acute ／ right ／ obtuse triangle（鋭角・直角・鈍角三角形。エントリ right-triangle、obtuse-triangle）。CK-12 Geometry は 4.2 Classify Triangles by Angle Measurement でこの 3 つに分ける。",
      "3 辺の長さ a, b, c（c がいちばん長い）から見分けるときは、c² < a² + b² なら鋭角三角形。"
    ],
    "related": [
      "acute-angle",
      "right-triangle",
      "obtuse-triangle",
      "triangle",
      "scalene-triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "鋭角三角形",
        "en": "Acute triangle"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.2 Classify Triangles by Angle Measurement・4.29 Pythagorean Theorem to Classify Triangles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "acute triangle": 3
      },
      "written": {
        "acute triangle": 1
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CK-12 の呼び方 acute triangle（CK-12 Geometry 4.29 Pythagorean Theorem to Classify Triangles・CK-12 Geometry 4.2 Classify Triangles by Angle Measurement・CK-12 Geometry 4.1 Classify Triangles・計 23 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "add",
    "ja": {
      "term": "足す",
      "reading": "たす",
      "alt": []
    },
    "en": {
      "term": "add",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "2 つ以上の数や式を合わせて、その和を求める。",
    "definition_en": "To combine two or more numbers or expressions to find their sum.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If x − 5 = 7, add 5 to both sides, and you get x = 12.",
        "ja": "x − 5 = 7 なら、両辺に 5 を足して x = 12。",
        "register": "spoken"
      },
      {
        "en": "Add 4x − 7 and −x + 3.",
        "ja": "4x − 7 と −x + 3 を足せ。",
        "register": "written"
      },
      {
        "en": "When you add 2/3 and 1/6, you need a common denominator first.",
        "ja": "2/3 と 1/6 を足すときは、まず通分が必要だよ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "語順が日本語と逆。「A に B を足す」は add B to A（add 5 to x は x + 5）。",
      "add A and B は「A と B を足す」。その結果の和は sum（the sum of A and B）。",
      "名詞の「足し算・加法」は addition（エントリ addition）。式の + は plus と読む。"
    ],
    "related": [
      "addition",
      "subtract",
      "multiply",
      "the-four-operations",
      "move-term-to-other-side"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "add": 4463
      },
      "written": {
        "add": 2044
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "addition-principle",
    "ja": {
      "term": "和の法則",
      "reading": "わのほうそく",
      "alt": []
    },
    "en": {
      "term": "addition principle",
      "alt": [
        "rule of sum",
        "sum principle"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "同時には起こらない 2 つの事柄 A, B があり、A の起こり方が m 通り、B の起こり方が n 通りのとき、A または B が起こる場合の数は m + n 通りであるという法則。",
    "definition_en": "If one task can be done in m ways and another in n ways, and the two cannot happen at the same time, then there are m + n ways to do one or the other.",
    "latex": "|A \\cup B| = |A| + |B| \\quad (A \\cap B = \\varnothing)",
    "spoken_en": "the size of A union B equals the size of A plus the size of B, when A intersect B is empty",
    "examples": [
      {
        "en": "These two cases don't overlap, so by the addition principle we just add them: 6 plus 4 is 10 ways.",
        "ja": "この 2 つの場合は重ならないので、和の法則で足せばよい。6 + 4 で 10 通りだ。",
        "register": "spoken"
      },
      {
        "en": "A cafe offers 4 kinds of sandwiches and 3 kinds of salads. A customer chooses either one sandwich or one salad. Use the addition principle to find the number of possible choices.",
        "ja": "ある店にはサンドイッチが 4 種類、サラダが 3 種類ある。サンドイッチかサラダのどちらか 1 つを選ぶとき、選び方は何通りあるか。和の法則を用いて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 つの場合に重なりがあるときは、足すだけでは数えすぎる。重なりを引く包除原理（inclusion-exclusion principle）を使う。",
      "確率の加法定理（addition rule）とは別の語。和の法則は場合の数を数える法則。",
      "用例コーパスの MIT OCW 6.042（Mathematics for Computer Science）は和の法則を sum rule と呼ぶ。微分の sum rule（derivative-of-a-sum）と同じ名前なので、文脈で区別する。",
      "用例コーパスの addition principle は書き言葉ではすべて OpenStax Algebra and Trigonometry。話し言葉には出てこない。"
    ],
    "related": [
      "multiplication-principle",
      "counting",
      "general-addition-rule",
      "inclusion-exclusion-principle"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.2 Combining Outcomes（Sum Principle）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Addition_principle",
        "note": "記事「Addition principle」の冒頭（addition principle・rule of sum。数学カテゴリから 3 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "addition principle": 19
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "addition",
    "ja": {
      "term": "加法",
      "reading": "かほう",
      "alt": []
    },
    "en": {
      "term": "addition",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "2 つ以上の数を合わせた大きさを求める計算。足し算のことで、結果を和という。",
    "definition_en": "The operation of combining numbers to find a total. The answer to an addition problem is called the sum.",
    "latex": "a + b",
    "spoken_en": "a plus b",
    "examples": [
      {
        "en": "With addition, the order doesn't matter: 3 plus 5 is the same as 5 plus 3.",
        "ja": "加法では順番を入れかえてもいい。3 + 5 と 5 + 3 は同じだ。",
        "register": "spoken"
      },
      {
        "en": "Rewrite 7 − 12 as an addition problem, and then evaluate it.",
        "ja": "7 − 12 を加法の式に直して、計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "addition は演算の名前（名詞）。「足す」という動詞は add（エントリ add）、足し算の答えの「和」は sum。",
      "in addition は日常語で「そのうえ」の意味で、加法とは関係ない。"
    ],
    "related": [
      "add",
      "subtraction",
      "the-four-operations",
      "commutative-property",
      "associative-property"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "加法",
        "en": "Addition"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!in addition": 266
      },
      "written": {
        "!in addition": 514
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "adjacency-matrix",
    "ja": {
      "term": "隣接行列",
      "reading": "りんせつぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "adjacency matrix",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Discrete Math",
        "Linear Algebra"
      ]
    },
    "definition_ja": "グラフの頂点を番号づけし、頂点 i と j を結ぶ辺の本数を (i, j) 成分にした正方行列。",
    "definition_en": "For a graph with numbered vertices, the square matrix whose (i, j) entry is the number of edges joining vertex i to vertex j.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Square the adjacency matrix and you count the walks of length two.",
        "ja": "隣接行列を 2 乗すると、長さ 2 の道の数がわかる。",
        "register": "spoken"
      },
      {
        "en": "Write the adjacency matrix of the graph shown.",
        "ja": "図のグラフの隣接行列を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はすべて MIT の講義ノート（離散数学）。"
    ],
    "related": [
      "matrix",
      "vertex-graph"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "隣接行列",
        "en": "Adjacency matrix"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "adjacency matrix": 8
      },
      "written": {
        "adjacency matrix": 18
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "adjacent-angles",
    "ja": {
      "term": "隣接角",
      "reading": "りんせつかく",
      "alt": []
    },
    "en": {
      "term": "adjacent angles",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "CK-12 Geometry は adjacent angles と呼ぶ（1.13 Angle Properties and Theorems ほか、15 件）。IM も Grade 7 の glossary の見出しにしている（レッスンで 7 件）。用例コーパスでは話し言葉に少しあり（ほとんどが Khan Academy の中学の講義）、書き言葉には出てこない。見出しの「隣接角」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書はこの 2 つの角の関係に名前をつけない。英語の adjacent angles は、頂点と 1 辺を共有し、内部が重ならない 2 つの角を指す。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "頂点と 1 つの辺を共有し、内部が重ならない 2 つの角。",
    "definition_en": "Two angles that share a vertex and a side but have no interior points in common.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These two are adjacent angles: same vertex, they share this ray, and they don't overlap.",
        "ja": "この 2 つは隣接角。頂点が同じで、この半直線を共有していて、重なっていない。",
        "register": "spoken"
      },
      {
        "en": "∠AOB and ∠BOC are adjacent angles with a sum of 90°. If ∠AOB = 2∠BOC, find ∠BOC.",
        "ja": "∠AOB と ∠BOC は頂点 O と辺 OB を共有し、和が 90° である。∠AOB = 2∠BOC のとき、∠BOC を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「隣り合う角」として、平行四辺形などの多角形で 1 辺を共有する 2 つの内角を adjacent angles と言うこともある（平行四辺形の隣り合う角の和は 180°）。頂点を共有する上の意味とは別。",
      "adjacent side（隣辺）は直角三角形で角に隣り合う辺のことで、adjacent angles とは関係がない。"
    ],
    "related": [
      "angle",
      "angle-addition-postulate",
      "supplementary-angle",
      "complementary-angle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Grade 7 glossary・Grade 7 7.2 Adjacent Angles・Grade 7 7.1 Relationships of Angles"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.13 Angle Properties and Theorems・1.15 Supplementary Angles・3.1 Line Types"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "adjacent angles": 9
      },
      "sources": [
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 9 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 adjacent angles（CK-12 Geometry 1.13 Angle Properties and Theorems・CK-12 Geometry 1.15 Supplementary Angles・CK-12 Geometry 3.1 Line Types・計 15 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "adjacent-side",
    "ja": {
      "term": "隣辺",
      "reading": "りんぺん",
      "alt": []
    },
    "en": {
      "term": "adjacent side",
      "alt": [
        "adjacent leg"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直角三角形で、注目する鋭角 θ をはさむ 2 辺のうち、斜辺でない方の辺。",
    "definition_en": "In a right triangle, the side next to the acute angle θ that is not the hypotenuse.",
    "latex": "\\cos\\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}",
    "spoken_en": "cosine theta equals adjacent over hypotenuse",
    "examples": [
      {
        "en": "The adjacent side is the leg touching the angle, and remember, it's never the hypotenuse.",
        "ja": "隣辺は角に接している方の辺で、斜辺は隣辺にならないことに注意。",
        "register": "spoken"
      },
      {
        "en": "In a right triangle, the adjacent side to angle θ is 12 and the hypotenuse is 13. Find cos θ and tan θ.",
        "ja": "直角三角形で、角 θ の隣辺が 12、斜辺が 13 のとき、cos θ と tan θ の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "隣辺はどの角から見るかで決まる。もう一方の鋭角から見ると、隣辺と対辺が入れ替わる。",
      "adjacent sides は多角形の「隣り合う辺」の意味にも使う（平行四辺形の隣り合う 2 辺など）。IM Geometry は adjacent leg（4.6 ほか）、OpenStax Algebra and Trigonometry は the side adjacent to the angle と言う。"
    ],
    "related": [
      "cosine",
      "tangent",
      "sine",
      "trigonometric-ratio"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「直角三角形」",
        "url": "https://ja.wikipedia.org/wiki/%E7%9B%B4%E8%A7%92%E4%B8%89%E8%A7%92%E5%BD%A2",
        "note": "鋭角 A に対する斜辺 c・隣辺 a・対辺 b"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "4.6 Working with Trigonometric Ratios ほか: adjacent leg（42 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "adjacent side": 80
      },
      "written": {
        "adjacent leg": 1,
        "adjacent side": 37
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "adjugate",
    "ja": {
      "term": "余因子行列",
      "reading": "よいんしぎょうれつ",
      "alt": [
        "古典随伴行列"
      ]
    },
    "en": {
      "term": "adjugate",
      "alt": [
        "classical adjoint",
        "adjoint"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（adjugate ／ classical adjoint ／ adjoint は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列の各成分の余因子を並べた行列を転置したもの。A adj(A) = (det A)I が成り立つ。",
    "definition_en": "The transpose of the matrix of cofactors of a square matrix; it satisfies A adj(A) = (det A)I.",
    "latex": "A^{-1} = \\frac{1}{\\det A}\\operatorname{adj}(A)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Take the matrix of cofactors, transpose it, and you've got the adjugate.",
        "ja": "余因子を並べた行列を転置すると、余因子行列になる。",
        "register": "spoken"
      },
      {
        "en": "Use the adjugate to find the inverse of the matrix.",
        "ja": "余因子行列を用いて逆行列を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "台帳の見出し「随伴行列」は、日本語では多くの場合、複素共役をとって転置した行列（conjugate transpose、英語でも adjoint と呼ぶことがある）を指す。台帳の langlink（随伴行列 → Conjugate transpose）もそちらで、出典にしなかった。",
      "英語の adjoint は余因子行列（古い呼び方）と共役転置の両方に使われる。Nicholson は adjugate（12 件）。"
    ],
    "related": [
      "cofactor",
      "inverse-matrix",
      "determinant",
      "transpose"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.2 Determinants and Matrix Inverses"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "adjoint": 2
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは Nicholson の呼び方 adjugate（3.2 Determinants and Matrix Inverses・2.4 Matrix Inverses）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "algebraic-expression",
    "ja": {
      "term": "文字式",
      "reading": "もじしき",
      "alt": []
    },
    "en": {
      "term": "algebraic expression",
      "alt": [
        "variable expression"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説の「文字式」（文字を用いた式）は、文字を使って表した式のこと。英語では algebraic expression（variable expression とも）と言い、等号を含まない式（3x + 2 など）を指す。等号でつないだものは equation と呼んで区別する。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "x や a などの文字を使って、数量を表した式。3x + 2 や a² − 1 など。",
    "definition_en": "An expression that uses one or more variables together with numbers and operations, such as 3x + 2. It has no equals sign.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If each ticket costs 8 dollars, the cost of n tickets is the algebraic expression 8n.",
        "ja": "チケットが 1 枚 8 ドルなら、n 枚の代金は文字式で 8n と表せる。",
        "register": "spoken"
      },
      {
        "en": "Write an algebraic expression for the perimeter of a rectangle whose length is x cm and whose width is 4 cm.",
        "ja": "縦が x cm、横が 4 cm の長方形の周の長さを、文字式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "等号を含む 2x + 3 = 7 は algebraic expression ではなく equation。英語では expression と equation を分けて言う（エントリ expression）。",
      "「文字式」の「文字」は英語では variable（エントリ variable）。letter expression とは言わない。",
      "3x は three x と読む（three times x とも言う）。× を省いた書き方は英語でも同じ。"
    ],
    "related": [
      "expression",
      "variable",
      "term",
      "coefficient",
      "polynomial"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A 数と式「文字を用いた式」（文字式）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "algebraic expression": 15,
        "variable expression": 1
      },
      "written": {
        "algebraic expression": 193,
        "variable expression": 46
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "algebraic-function",
    "ja": {
      "term": "代数関数",
      "reading": "だいすうかんすう",
      "alt": []
    },
    "en": {
      "term": "algebraic function",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "多項式・分数式・根号を有限回使って表される関数。",
    "definition_en": "A function built from polynomials using a finite number of additions, multiplications, divisions, and roots.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Anything you can build with polynomials and roots is an algebraic function.",
        "ja": "多項式と根号でつくれるものは代数関数。",
        "register": "spoken"
      },
      {
        "en": "The function f(x) = √(x² + 1)/(x − 2) is an algebraic function.",
        "ja": "関数 f(x) = √(x² + 1)/(x − 2) は代数関数である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "transcendental-function",
      "rational-function",
      "radical-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "代数関数",
        "en": "Algebraic function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "algebraic function": 1
      },
      "written": {
        "algebraic function": 20
      },
      "sources": [
        "mit-18.01",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "algebraic-multiplicity",
    "ja": {
      "term": "代数的重複度",
      "reading": "だいすうてきちょうふくど",
      "alt": []
    },
    "en": {
      "term": "algebraic multiplicity",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（algebraic multiplicity は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "固有値 λ が特性多項式の根として何重に現れるかを表す数。",
    "definition_en": "The number of times an eigenvalue λ occurs as a root of the characteristic polynomial.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Lambda equals two is a double root, so its algebraic multiplicity is two.",
        "ja": "λ = 2 は重根なので、代数的重複度は 2 だ。",
        "register": "spoken"
      },
      {
        "en": "Find the algebraic and geometric multiplicity of each eigenvalue.",
        "ja": "各固有値の代数的重複度と幾何的重複度を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "幾何的重複度（固有空間の次元）はいつも代数的重複度以下。すべての固有値で等しいときに対角化可能。",
      "用例コーパスでは話し言葉（MIT OCW）に少しあるだけ。Nicholson は 5.5 で multiplicity と書く。"
    ],
    "related": [
      "geometric-multiplicity",
      "eigenvalue",
      "characteristic-polynomial",
      "diagonalizable"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.5 Similarity and Diagonalization"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "algebraic multiplicity": 4
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 algebraic multiplicity（5.5 Similarity and Diagonalization）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "all",
    "ja": {
      "term": "すべての",
      "reading": "すべての",
      "alt": [
        "任意の"
      ]
    },
    "en": {
      "term": "for all",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "for all",
          "register": "both",
          "note": "話し言葉・書き言葉とも首位。話し言葉は MIT OCW と Khan Academy が中心で、書き言葉は OpenStax Calculus が半分以上を占める。"
        },
        {
          "term": "for each",
          "register": "both",
          "note": "一つ一つについて、の感じが強い。話し言葉は Khan Academy が半分以上で、MIT OCW が次ぐ。書き言葉は OpenStax Calculus・OpenStax Algebra and Trigonometry・OpenStax Introductory Statistics の順に多い。"
        },
        {
          "term": "for any",
          "register": "both",
          "note": "「任意の」に近い。話し言葉では Khan Academy が半分を占め、MIT OCW が次ぐ。書き言葉は OpenStax Calculus・OpenStax Algebra and Trigonometry・MIT の講義ノートの順に多い。"
        },
        {
          "term": "for every",
          "register": "both",
          "note": "話し言葉は半分が Khan Academy の中学の講義で、MIT OCW・Professor Leonard が次ぐ。書き言葉では 4 つの言い方の中で最も少なく、MIT の講義ノートと OpenStax Calculus が中心。"
        }
      ]
    },
    "pos": "adjective",
    "mapping": "near",
    "mapping_note": "日本語の「すべての」「任意の」に当たる英語は 1 つに決まらず、用例コーパスでは for all ／ for any ／ for each ／ for every が話・書ともよく使われている。どれも「すべての〜について」の意味で使える。論理記号 ∀ は for all と読む。",
    "domains": [
      "algebra",
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "対象の範囲に入るものの一つ一つについて例外なく、の意味。「任意の」も同じ意味で、どれを選んでも成り立つことを表す。",
    "definition_en": "Without exception: for every object in the given range, so that the statement holds whichever one you pick.",
    "latex": "\\forall x \\in \\mathbb{R},\\ x^2 \\ge 0",
    "spoken_en": "for all real numbers x, x squared is greater than or equal to zero",
    "examples": [
      {
        "en": "This inequality holds for any x, so the graph never goes below the x-axis.",
        "ja": "この不等式は任意の x で成り立つから、グラフは x 軸より下に行かない。",
        "register": "spoken"
      },
      {
        "en": "Prove that x² − 2x + 2 > 0 for all real numbers x.",
        "ja": "すべての実数 x について x² − 2x + 2 > 0 であることを証明せよ。",
        "register": "written"
      },
      {
        "en": "For each positive integer n, let aₙ = 2n + 1.",
        "ja": "各正の整数 n について aₙ = 2n + 1 とする。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "true for all x",
        "ja": "すべての x で成り立つ"
      }
    ],
    "pitfalls": [
      "any は「どれか 1 つ」とも読めるので、「すべての」を強く言う答案では for all を使う（本プロジェクトの書き方の案内）。",
      "共通テスト（令和 5・6 年度 数学II 第3問）の「すべての実数 r に対して」は文頭や文中に置くが、用例コーパスの for all x は述語の後ろに置く（f(c) ≥ f(x) for all x in the domain）。"
    ],
    "related": [
      "exist",
      "at-least-one",
      "proof",
      "counterexample",
      "set"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "令和 5・6 年度 数学II 第3問: すべての実数 r に対して"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "for each": 305,
        "for every": 319,
        "for all": 436,
        "for any": 407
      },
      "written": {
        "for all": 672,
        "for any": 536,
        "for each": 670,
        "for every": 165
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "alternate-exterior-angles",
    "ja": {
      "term": "外錯角",
      "reading": "がいさっかく",
      "alt": []
    },
    "en": {
      "term": "alternate exterior angles",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「外錯角」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書は錯角（内側）だけを扱い、外側の組に名前を付けない。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 直線に 1 本の直線が交わってできる角のうち、2 直線の外側にあり、交わる直線をはさんで反対側にある 2 つの角。2 直線が平行なら等しい。",
    "definition_en": "A pair of angles formed when a transversal crosses two lines, lying outside the two lines and on opposite sides of the transversal. If the two lines are parallel, the angles are congruent.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Angles 1 and 8 are alternate exterior angles. They're both outside the parallel lines and on opposite sides of the transversal, so they're equal.",
        "ja": "∠1 と ∠8 は外錯角。どちらも平行線の外側にあって、横断線をはさんで反対側だから等しい。",
        "register": "spoken"
      },
      {
        "en": "Two parallel lines are cut by a transversal. A pair of alternate exterior angles measure (4x + 7)° and (6x − 25)°. Find x and the measure of each angle.",
        "ja": "平行な 2 直線に 1 本の直線が交わっている。外錯角の組の大きさが (4x + 7)° と (6x − 25)° のとき、x の値とそれぞれの角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "外錯角が等しいことは、対頂角が等しいことと、錯角（または同位角）が等しいことから導ける。",
      "逆の「外錯角が等しければ 2 直線は平行」は converse of the alternate exterior angles theorem と呼ぶ。",
      "用例コーパスで alternate exterior angles は話し言葉に少しあり（すべて The Organic Chemistry Tutor）、書き言葉には出てこない。IM では 0 件、CK-12 Geometry では 27 件（3.6 Alternate Exterior Angles ほか）。"
    ],
    "related": [
      "transversal",
      "alternate-interior-angles-theorem",
      "same-side-exterior-angles",
      "parallel-lines"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "3.6 Alternate Exterior Angles・2.15 Proofs- Angle Pairs and Segments・3.5 Alternate Interior Angles"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "alternate exterior angles": 9
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 9 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 alternate exterior angles（CK-12 Geometry 3.6 Alternate Exterior Angles・CK-12 Geometry 2.15 Proofs- Angle Pairs and Segments・CK-12 Geometry 3.5 Alternate Interior Angles・計 27 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "alternate-interior-angles-theorem",
    "ja": {
      "term": "錯角の定理",
      "reading": "さっかくのていり",
      "alt": [
        "平行線の錯角"
      ]
    },
    "en": {
      "term": "alternate interior angles theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「錯角の定理」は学習指導要領解説に無い、本プロジェクトの訳語。日本の中2 では「平行線の錯角は等しい」を平行線の性質として扱う。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "平行な 2 直線に 1 本の直線が交わるとき、錯角は等しい、という定理。",
    "definition_en": "The theorem that when a transversal crosses two parallel lines, each pair of alternate interior angles is congruent.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Since these two lines are parallel, angle 3 and angle 6 are congruent by the alternate interior angles theorem.",
        "ja": "この 2 直線は平行だから、錯角が等しくて ∠3 = ∠6 になる。",
        "register": "spoken"
      },
      {
        "en": "In the figure, AB ∥ CD, and line EF meets AB at G and CD at H. If m∠AGH = (3x + 10)° and m∠GHD = (5x − 30)°, find x.",
        "ja": "図で AB ∥ CD であり、直線 EF は AB と点 G で、CD と点 H で交わる。∠AGH = (3x + 10)°、∠GHD = (5x − 30)° のとき、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "逆の「錯角が等しければ 2 直線は平行」は converse of the alternate interior angles theorem と呼ぶ。2 直線が平行であることを示すときに使うのは逆のほう。",
      "英語では外側の組にも alternate exterior angles という名前があり、平行線なら同じく等しい。",
      "用例コーパスには alternate interior angles theorem は出てこない。IM は 1 件（Geometry 6.10）、CK-12 Geometry は 8 件（5.9 Parallelograms・3.5 Alternate Interior Angles ほか）。"
    ],
    "related": [
      "transversal",
      "corresponding-angles-postulate",
      "alternate-exterior-angles",
      "parallel-lines"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "6.10 Parallel Lines in the Plane（1 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.9 Parallelograms・3.5 Alternate Interior Angles・2.15 Proofs- Angle Pairs and Segments"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 alternate interior angles theorem（CK-12 Geometry 5.9 Parallelograms・CK-12 Geometry 3.5 Alternate Interior Angles・CK-12 Geometry 2.15 Proofs- Angle Pairs and Segments・計 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "alternate-interior-angles",
    "ja": {
      "term": "錯角",
      "reading": "さっかく",
      "alt": []
    },
    "en": {
      "term": "alternate interior angles",
      "alt": [],
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 本の直線に 1 本の直線が交わってできる角のうち、2 直線の内側にあり、交わる直線をはさんで反対側にある 2 つの角。2 直線が平行なら、錯角は等しい。",
    "definition_en": "Two angles made by a transversal with two lines, one at each line, lying between the lines on opposite sides of the transversal. When the lines are parallel, alternate interior angles are congruent.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "They're both between the lines but on opposite sides of the transversal, so they're alternate interior angles.",
        "ja": "どちらも 2 直線の間にあるけど、横断線をはさんで反対側にある。だから錯角だね。",
        "register": "spoken"
      },
      {
        "en": "A transversal crosses lines ℓ and m so that a pair of alternate interior angles both measure 63°. Are ℓ and m parallel? Explain.",
        "ja": "2 直線 ℓ、m に 1 本の直線が交わり、錯角の組がどちらも 63° になっている。ℓ と m は平行か。理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「錯角」は内側の組を指す。英語では外側の組 alternate exterior angles（エントリ alternate-exterior-angles）と区別するため interior を付ける。",
      "平行線の錯角が等しいことは alternate interior angles theorem（エントリ alternate-interior-angles-theorem）。",
      "alternate はここでは「互い違いの」の意味で、2 つの角が交わる直線の左右に互い違いにあることを表す。「代わりの」の意味の alternate（alternative）とは別。"
    ],
    "related": [
      "transversal",
      "alternate-interior-angles-theorem",
      "alternate-exterior-angles",
      "corresponding-angles",
      "parallel-lines"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "錯角",
        "en": "Alternate angles"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "3.5 Alternate Interior Angles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "alternate interior angles": 22
      },
      "written": {
        "alternate interior angles": 1
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "alternating-expression",
    "ja": {
      "term": "交代式",
      "reading": "こうたいしき",
      "alt": []
    },
    "en": {
      "term": "alternating polynomial",
      "alt": [
        "alternating expression"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（alternating polynomial ／ alternating expression は CED・OpenStax・IM・CK-12 とも 0 件）。用例コーパスでは alternating polynomial は出てこず、alternating expression も話し言葉にわずか（すべて 3Blue1Brown）で書き言葉には無い。見出しは英語版 Wikipedia の記事名「Alternating polynomial」（ja 記事「交代式」の langlink 先。冒頭で alternating polynomial を太字で定義する）。英語で説明するときは swapping two variables changes the sign と性質を言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": []
    },
    "definition_ja": "どの 2 つの文字を入れ替えても、符号だけが変わる多項式。3 文字の交代式は (a − b)(b − c)(c − a) で割り切れる。",
    "definition_en": "A polynomial that only changes sign when any two of its variables are interchanged, such as x − y or x²y − xy².",
    "latex": "f(y, x) = -f(x, y)",
    "spoken_en": "f of y comma x equals negative f of x comma y",
    "examples": [
      {
        "en": "Swap a and b and the whole thing just changes sign, so it's an alternating polynomial, and a − b has to be a factor.",
        "ja": "a と b を入れ替えると全体の符号だけが変わるので交代式だ。だから a − b を因数にもつ。",
        "register": "spoken"
      },
      {
        "en": "Factor the alternating polynomial a²(b − c) + b²(c − a) + c²(a − b).",
        "ja": "交代式 a²(b − c) + b²(c − a) + c²(a − b) を因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "因数分解では、交代式が (a − b)(b − c)(c − a) を因数にもつことを使い、残りは係数を比べて決める。",
      "alternating は交代級数（alternating series）の「符号が交互に変わる」の意味にもなるので、alternating expression だけでは交代式の意味に取られないことがある。"
    ],
    "related": [
      "symmetric-expression",
      "factor-theorem",
      "compare-coefficients",
      "polynomial"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "交代式",
        "en": "Alternating polynomial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「交代式」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BA%A4%E4%BB%A3%E5%BC%8F",
        "note": "交代多項式（英: alternating polynomial）: 任意の 2 変数を入れ替えると符号が反転する多項式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "alternating expression": 2
      },
      "sources": [
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 alternating polynomial（英語版「Alternating polynomial」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "alternating-series-error-bound",
    "ja": {
      "term": "交代級数の誤差限界",
      "reading": "こうたいきゅうすうのごさげんかい",
      "alt": []
    },
    "en": {
      "term": "alternating series error bound",
      "alt": [
        "alternating series remainder"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「交代級数の誤差限界」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "交代級数判定法の条件を満たす級数では、第 n 部分和で和を近似したときの誤差が、次の項の絶対値 bₙ₊₁ 以下になるという評価。",
    "definition_en": "For a series that meets the alternating series test, the error in using the nth partial sum is at most the absolute value of the next term, bₙ₊₁.",
    "latex": "|S - S_n| \\le b_{n+1}",
    "spoken_en": "the absolute value of S minus S sub n is at most b sub n plus one",
    "examples": [
      {
        "en": "The error is less than the first term you left out.",
        "ja": "誤差は、切り捨てた最初の項より小さい。",
        "register": "spoken"
      },
      {
        "en": "Use the alternating series error bound to find how many terms are needed to approximate Σ (−1)ⁿ/n³ to within 0.001.",
        "ja": "交代級数の誤差限界を用いて、Σ (−1)ⁿ/n³ を誤差 0.001 以内で近似するのに必要な項数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "alternating-series-test",
      "alternating-series",
      "partial-sum",
      "lagrange-error-bound"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.10（alternating series error bound）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "alternating series remainder": 1
      },
      "sources": [
        "khan-ap-calc"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CED の呼び方 alternating series error bound（topic 10.10・topic 10.12）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "alternating-series-test",
    "ja": {
      "term": "交代級数判定法",
      "reading": "こうたいきゅうすうはんていほう",
      "alt": [
        "ライプニッツの判定法"
      ]
    },
    "en": {
      "term": "alternating series test",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「交代級数判定法」は学習指導要領解説に無い（日本語版 Wikipedia に記事「交代級数判定法」がある）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "bₙ > 0 が減少して 0 に収束すれば、交代級数 Σ(−1)ⁿbₙ は収束するという判定法。",
    "definition_en": "If bₙ > 0 decreases to 0, then the alternating series Σ(−1)ⁿbₙ converges.",
    "latex": "b_{n+1} \\le b_n, \\quad \\lim_{n\\to\\infty} b_n = 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "Check two things for the alternating series test: decreasing, and the limit is zero.",
        "ja": "交代級数判定法では 2 つ確かめる。減少していることと、極限が 0 であること。",
        "register": "spoken"
      },
      {
        "en": "Use the alternating series test to show that Σ (−1)ⁿ⁺¹/n converges.",
        "ja": "交代級数判定法を用いて Σ (−1)ⁿ⁺¹/n が収束することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "この判定法で分かるのは収束することだけで、絶対収束か条件収束かは別に調べる。"
    ],
    "related": [
      "alternating-series",
      "alternating-series-error-bound",
      "absolute-convergence",
      "conditional-convergence"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.7（alternating series test）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「交代級数判定法」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BA%A4%E4%BB%A3%E7%B4%9A%E6%95%B0%E5%88%A4%E5%AE%9A%E6%B3%95",
        "note": "記事名（英: the alternating series test）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "alternating series test": 25
      },
      "written": {
        "alternating series test": 28
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "alternating-series",
    "ja": {
      "term": "交代級数",
      "reading": "こうたいきゅうすう",
      "alt": [
        "交項級数"
      ]
    },
    "en": {
      "term": "alternating series",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "項の符号が交互に変わる級数。",
    "definition_en": "A series whose terms alternate in sign.",
    "latex": "\\sum_{n=1}^{\\infty} (-1)^{n+1} b_n \\quad (b_n > 0)",
    "spoken_en": "the sum of negative one to the n plus one times b sub n",
    "examples": [
      {
        "en": "The (−1)ⁿ out front is what makes it an alternating series.",
        "ja": "前にある (−1)ⁿ で交代級数になっている。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the alternating series Σ (−1)ⁿ/√n converges.",
        "ja": "交代級数 Σ (−1)ⁿ/√n が収束するかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "alternating-series-test",
      "alternating-series-error-bound",
      "conditional-convergence",
      "harmonic-series"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "交項級数",
        "en": "Alternating series"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.7（alternating series）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "alternating series": 37
      },
      "written": {
        "alternating series": 69
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "alternative-hypothesis",
    "ja": {
      "term": "対立仮説",
      "reading": "たいりつかせつ",
      "alt": []
    },
    "en": {
      "term": "alternative hypothesis",
      "alt": [
        "alternate hypothesis"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "仮説検定で、帰無仮説が棄却されたときに採る仮説。Hₐ や H₁ と書く。",
    "definition_en": "In a hypothesis test, the hypothesis accepted when the null hypothesis is rejected, written Hₐ or H₁.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The alternative hypothesis is what we're actually trying to find evidence for.",
        "ja": "対立仮説は、実際に証拠を見つけたい主張のほう。",
        "register": "spoken"
      },
      {
        "en": "Write the alternative hypothesis for a right-tailed test.",
        "ja": "右側検定の対立仮説を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "書き方が分かれる。高等学校学習指導要領解説（数学B）は H₁、AP Statistics の CED と OpenStax Introductory Statistics は Hₐ と書く。読み上げでは the alternative hypothesis と言う（記号エントリ h-sub-a-alternative。記号を読む H sub one ／ H sub a は用例コーパスでは少ない）。"
    ],
    "related": [
      "null-hypothesis",
      "hypothesis-testing"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "仮説検定",
        "en": "Statistical hypothesis test"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B 統計的な推測: 対立仮説 H₁"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Unit 3（alternative hypothesis Hₐ）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "alternative hypothesis": 63,
        "alternate hypothesis": 1
      },
      "written": {
        "alternative hypothesis": 111,
        "alternate hypothesis": 23
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "always-positive",
    "ja": {
      "term": "つねに正",
      "reading": "つねにせい",
      "alt": [
        "常に正"
      ]
    },
    "en": {
      "term": "always positive",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "definition_ja": "変数がどんな値をとっても、式や関数の値が 0 より大きいこと。二次式 ax² + bx + c がつねに正となるのは a > 0 かつ判別式 D < 0 のとき。",
    "definition_en": "Greater than zero for every value of the variable. A quadratic ax² + bx + c is always positive exactly when a > 0 and its discriminant is negative.",
    "latex": "ax^2 + bx + c > 0 \\text{ for all } x \\iff a > 0,\\ b^2 - 4ac < 0",
    "spoken_en": "a x squared plus b x plus c is greater than zero for all x if and only if a is positive and b squared minus four a c is negative",
    "examples": [
      {
        "en": "The parabola opens up and never touches the x-axis, so this quadratic is always positive.",
        "ja": "放物線は上に開いていて x 軸と共有点をもたないから、この二次式はつねに正だ。",
        "register": "spoken"
      },
      {
        "en": "Find the range of values of k for which x² + 2kx + 4 is always positive.",
        "ja": "x² + 2kx + 4 がつねに正となるような定数 k の値の範囲を求めよ。",
        "register": "written"
      },
      {
        "en": "Since a = 1 > 0 and the discriminant is negative, f(x) is always positive.",
        "ja": "a = 1 > 0 かつ判別式が負だから、f(x) はつねに正である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「つねに 0 以上」（≧ 0）は always positive ではない。0 になる点がある二次式（D = 0）は正ではないので、条件は D < 0 で D ≦ 0 ではない。",
      "「すべての実数 x について」は for all x ／ for every real number x と添えて書くと答案で紛れない。"
    ],
    "related": [
      "discriminant",
      "quadratic-inequality",
      "opens-upward",
      "real-solution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "always positive": 68
      },
      "written": {
        "always positive": 25
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "am-gm-inequality",
    "ja": {
      "term": "相加平均と相乗平均の関係",
      "reading": "そうかへいきんとそうじょうへいきんのかんけい",
      "alt": [
        "相加相乗平均の不等式"
      ]
    },
    "en": {
      "term": "AM-GM inequality",
      "alt": [
        "inequality of arithmetic and geometric means"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（AM-GM inequality は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "algebra",
      "proof"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": []
    },
    "definition_ja": "正の数 a, b について (a+b)/2 ≧ √(ab) が成り立ち、等号は a=b のときに限るという関係。",
    "definition_en": "For positive a and b, the arithmetic mean (a+b)/2 is at least the geometric mean √(ab), with equality exactly when a = b.",
    "latex": "\\frac{a+b}{2} \\ge \\sqrt{ab}",
    "spoken_en": "a plus b over two is greater than or equal to the square root of a b",
    "examples": [
      {
        "en": "Use the AM-GM inequality here; equality holds when the two terms are equal.",
        "ja": "ここは相加相乗を使う。等号は 2 つが等しいときです。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "by AM-GM",
        "ja": "相加相乗平均の関係より"
      },
      {
        "en": "equality holds when a = b",
        "ja": "等号成立は a=b のとき"
      }
    ],
    "pitfalls": [
      "学習指導要領解説（数学II いろいろな式）は、相加・相乗平均の関係などを用いて不等式が成り立つことを証明する、と書く。センター試験・共通テストは、最小値を求める問題で相加平均と相乗平均の関係を使わせる（センター試験 平成27年度 本試験 数学II、共通テスト 令和3年度 第1日程 数学II ほか）。",
      "等号成立条件は equality holds when … と書く。"
    ],
    "related": [
      "geometric-middle-term",
      "arithmetic-middle-term",
      "arithmetic-mean",
      "geometric-mean",
      "cauchy-schwarz-inequality",
      "equality-holds"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式: 相加・相乗平均の関係などを用いて等式や不等式が成り立つことを証明する"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成27年度 大学入試センター試験 本試験 数学Ⅱ（問題）",
        "url": "https://web.archive.org/web/20160601/https://www.dnc.ac.jp/albums/abm.php?f=abm00004812.pdf&n=2701-0501+suugaku2.pdf",
        "note": "相加平均と相乗平均の関係を利用して最小値を求める"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和3年度 大学入学共通テスト 第1日程 数学Ⅱ（問題）",
        "url": "https://web.archive.org/web/20220601/https://www.dnc.ac.jp/albums/abm.php?d=88&f=abm00001261.pdf&n=R3_%E6%95%B0%E5%AD%A6%E2%85%A1%EF%BC%88%EF%BC%91%E6%9C%8816%E6%97%A5%E3%83%BB17%E6%97%A5%EF%BC%89%E3%81%AE%E5%95%8F%E9%A1%8C.pdf",
        "note": "相加平均と相乗平均の関係から最小値を求める"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/AM%E2%80%93GM_inequality",
        "note": "記事「AM–GM inequality」の冒頭の太字 inequality of arithmetic and geometric means（数学カテゴリから 3 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「二乗平均・算術平均・幾何平均・調和平均の大小関係」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BA%8C%E4%B9%97%E5%B9%B3%E5%9D%87%E3%83%BB%E7%AE%97%E8%A1%93%E5%B9%B3%E5%9D%87%E3%83%BB%E5%B9%BE%E4%BD%95%E5%B9%B3%E5%9D%87%E3%83%BB%E8%AA%BF%E5%92%8C%E5%B9%B3%E5%9D%87%E3%81%AE%E5%A4%A7%E5%B0%8F%E9%96%A2%E4%BF%82",
        "note": "相加相乗平均の不等式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-24。docs/DECISIONS.md「Phase 2 代数 2・Precalculus の単元の前の修正」の、残った ③ を人間が決めた行）。en.term は AM-GM inequality のまま。level.us は確かめられないので外した。register は主張しない。",
        "raised": "2026-09-24"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "ambiguous-case",
    "ja": {
      "term": "曖昧な場合",
      "reading": "あいまいなばあい",
      "alt": [
        "2 通りの三角形ができる場合"
      ]
    },
    "en": {
      "term": "ambiguous case",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「曖昧な場合」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書では、2 辺とその一方の対角が与えられたときに三角形が 2 つできる場合を、名前をつけずに扱う。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "2 辺とその一方の対角（SSA）が与えられたとき、条件を満たす三角形が 0 個・1 個・2 個のどれにもなりうる場合。",
    "definition_en": "The situation where two sides and an angle opposite one of them (SSA) are given, so there may be no triangle, one, or two.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is the ambiguous case, so check whether the supplement of B also works.",
        "ja": "これは曖昧な場合なので、B の補角でも三角形ができるか確かめよう。",
        "register": "spoken"
      },
      {
        "en": "In triangle ABC, a = 6, b = 8, and A = 30°. Find all possible values of B.",
        "ja": "三角形 ABC で a = 6, b = 8, A = 30° のとき、B として考えられる値をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは書き言葉（OpenStax Algebra and Trigonometry）に少しあり、話し言葉にはほとんど出てこない。"
    ],
    "related": [
      "solving-triangles"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "ambiguous case": 1
      },
      "written": {
        "ambiguous case": 7
      },
      "sources": [
        "openstax-algtrig",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 ambiguous case（本文 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "amplitude",
    "ja": {
      "term": "振幅",
      "reading": "しんぷく",
      "alt": []
    },
    "en": {
      "term": "amplitude",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "y = a sin(bθ + c) の |a|。グラフの中心線からの最大のずれ。",
    "definition_en": "The number |a| in y = a sin(bθ + c): how far the graph rises above its midline.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The 3 out front stretches it vertically, so the amplitude is 3.",
        "ja": "前の 3 で縦に伸びるので、振幅は 3。",
        "register": "spoken"
      },
      {
        "en": "State the amplitude and period of y = −2 cos(θ/2).",
        "ja": "y = −2 cos(θ/2) の振幅と周期を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は y = A sin(Bx − C) + D のグラフから amplitude・period・phase shift・midline（振動の中心線。エントリ midline）を読み取る問題を扱う。",
      "振幅は負にならない。a が負ならその絶対値。"
    ],
    "related": [
      "period",
      "phase-shift",
      "stretch-vertically",
      "trigonometric-function",
      "midline"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "振幅",
        "en": "Amplitude"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "amplitude": 179
      },
      "written": {
        "amplitude": 222
      },
      "sources": [
        "khan-algebra",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "analyze",
    "ja": {
      "term": "分析する",
      "reading": "ぶんせきする",
      "alt": []
    },
    "en": {
      "term": "analyze",
      "alt": [],
      "uk": "analyse",
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "データや問題を部分に分けて調べ、特徴や傾向を明らかにする。",
    "definition_en": "To examine data or a problem closely, part by part, to find its features and patterns.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Before we analyze the data, let's organize it in a table.",
        "ja": "データを分析する前に、表に整理しよう。",
        "register": "spoken"
      },
      {
        "en": "Analyze the data and state your conclusion in context.",
        "ja": "データを分析し、結論を問題の文脈で述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "data"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences#yse_yze",
        "note": "記事「American and British English spelling differences」の節 -yse, -yze（英国は analyse、米国は analyze）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "analyze": 96
      },
      "written": {
        "analyze": 110
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle-addition-formulas",
    "ja": {
      "term": "加法定理",
      "reading": "かほうていり",
      "alt": [
        "三角関数の加法定理"
      ]
    },
    "en": {
      "term": "sum and difference formulas",
      "alt": [
        "angle addition formula",
        "addition formula"
      ],
      "uk": null,
      "register": "written",
      "variants": [
        {
          "term": "sum and difference formulas",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry の本文の言い方。話し言葉では angle addition formula・addition formula も使い、どれも少ない。"
        },
        {
          "term": "sum and difference identities",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry の節の題（Sum and Difference Identities）と本文の言い方。話し言葉には出てこない。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "sin(α ± β)、cos(α ± β)、tan(α ± β) を α, β の三角関数で表す公式。",
    "definition_en": "The formulas expressing sin(α ± β), cos(α ± β), and tan(α ± β) in terms of trigonometric functions of α and β.",
    "latex": "\\sin(\\alpha + \\beta) = \\sin\\alpha\\cos\\beta + \\cos\\alpha\\sin\\beta",
    "spoken_en": "sine of alpha plus beta equals sine alpha cosine beta plus cosine alpha sine beta",
    "examples": [
      {
        "en": "Write 75 degrees as 45 plus 30 and use the addition formula for sine.",
        "ja": "75° を 45° + 30° として、sin の加法定理を使う。",
        "register": "spoken"
      },
      {
        "en": "Use the sum and difference formulas to find the exact value of cos 15°.",
        "ja": "加法定理を用いて cos 15° の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は「定理」だが、OpenStax Algebra and Trigonometry は sum and difference formulas と呼ぶ（節の題は Sum and Difference Identities）。英語版 Wikipedia「List of trigonometric identities」は angle addition and subtraction theorems (or formulae) とも書く。",
      "addition theorem と訳すと、英語版 Wikipedia「Addition theorem」の一般の加法定理（f(x + y) を f(x), f(y) で表す式）の名前になる。用例コーパスと参照には出てこない。",
      "「値を求めよ」は、OpenStax Algebra and Trigonometry では find the exact value（小数ではなく正確な値）と書く（41 件）。",
      "1 つ 1 つの式は、OpenStax Algebra and Trigonometry（9.2）では the sum formula for cosine（cos(α + β) の式）、the difference formula for sine（sin(α − β) の式）のように呼ぶ。",
      "数学A の確率の加法定理（互いに排反な事象の和の確率）は別のエントリ general-addition-rule（addition rule）。日本語版 Wikipedia「加法定理」はどちらも加法定理の例に挙げる。"
    ],
    "related": [
      "double-angle-formulas",
      "half-angle-formulas",
      "product-to-sum-formulas",
      "auxiliary-angle-form",
      "triple-angle-formulas",
      "sum-to-product-formulas"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Sum and Difference Identities（sum and difference formulas、find the exact value）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角関数の公式の一覧」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0%E3%81%AE%E5%85%AC%E5%BC%8F%E3%81%AE%E4%B8%80%E8%A6%A7",
        "note": "節「加法定理」"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/List_of_trigonometric_identities",
        "note": "記事「List of trigonometric identities」の節 Angle sum and difference identities（数学カテゴリから 2 段）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Addition_theorem",
        "note": "記事「Addition theorem」（一般の加法定理）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「加法定理」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8A%A0%E6%B3%95%E5%AE%9A%E7%90%86",
        "note": "三角関数の加法定理と確率の加法定理（和の法則）を例に挙げる一般の記事"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "angle addition formula": 4,
        "sum and difference formulas": 3,
        "addition formula": 2
      },
      "written": {
        "sum and difference formulas": 30,
        "sum and difference identities": 12,
        "angle addition formula": 3
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle-addition-postulate",
    "ja": {
      "term": "角の加法公理",
      "reading": "かくのかほうこうり",
      "alt": []
    },
    "en": {
      "term": "angle addition postulate",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "CK-12 Geometry は angle addition postulate と呼ぶ（1.9 Measuring Angles ほか、5 件）。用例コーパスには出てこない。見出しの「角の加法公理」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書は ∠AOB + ∠BOC = ∠AOC に名前をつけない。英語名は、点 B が ∠AOC の内部にあれば角の大きさがこのように足せるという性質を公理として立てたときの名前。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "点 B が ∠AOC の内部にあるとき、∠AOB + ∠BOC = ∠AOC が成り立つ、という公理。",
    "definition_en": "The postulate that if point B lies in the interior of angle AOC, then the measure of angle AOB plus the measure of angle BOC equals the measure of angle AOC.",
    "latex": "m\\angle AOB + m\\angle BOC = m\\angle AOC",
    "spoken_en": "the measure of angle AOB plus the measure of angle BOC equals the measure of angle AOC",
    "examples": [
      {
        "en": "By the angle addition postulate, these two little angles add up to the big one, so 3x + 20 + x = 100.",
        "ja": "角の加法公理より、小さい 2 つの角を足すと大きい角になるから、3x + 20 + x = 100。",
        "register": "spoken"
      },
      {
        "en": "Point B is in the interior of ∠AOC, m∠AOB = 38°, and m∠AOC = 115°. Use the angle addition postulate to find m∠BOC.",
        "ja": "点 B は ∠AOC の内部にあり、∠AOB = 38°、∠AOC = 115° である。∠BOC を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の幾何には、角の大きさを m∠AOB と書いて角そのもの ∠AOB と区別する書き方がある。日本の教科書は ∠AOB = 38° のように同じ記号で大きさも表す。",
      "三角関数の加法定理（sin(α + β) = …）は angle addition formulas で、この公理とは別物。"
    ],
    "related": [
      "angle",
      "axiom",
      "segment-addition-postulate",
      "adjacent-angles",
      "undefined-terms"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.9 Measuring Angles・4.17 Triangle Angle Sum Theorem"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 angle addition postulate（CK-12 Geometry 1.9 Measuring Angles・CK-12 Geometry 4.17 Triangle Angle Sum Theorem・計 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle-between-vectors",
    "ja": {
      "term": "ベクトルのなす角",
      "reading": "べくとるのなすかく",
      "alt": []
    },
    "en": {
      "term": "angle between … vectors",
      "alt": [
        "angle between two vectors"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "始点をそろえた 2 つのベクトルがつくる角 θ（0° ≤ θ ≤ 180°）。cos θ = a·b / (|a||b|) で求める。",
    "definition_en": "The angle θ (0° ≤ θ ≤ 180°) between two vectors placed tail to tail, found from cos θ = a·b / (|a||b|).",
    "latex": "\\cos\\theta = \\frac{\\vec{a}\\cdot\\vec{b}}{|\\vec{a}||\\vec{b}|}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Use the dot product to find the angle between the two vectors.",
        "ja": "内積を使って 2 つのベクトルのなす角を求めよう。",
        "register": "spoken"
      },
      {
        "en": "Find the angle between the vectors a = ⟨1, 1⟩ and b = ⟨−1, 3⟩.",
        "ja": "ベクトル a = ⟨1, 1⟩ と b = ⟨−1, 3⟩ のなす角を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「なす角」は日本語特有の言い方で、英語は angle between（〜の間の角）と言う。",
      "「なす角を求める」は find the angle between …。"
    ],
    "related": [
      "dot-product",
      "vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "angle between … vectors": 12,
        "angle between two vectors": 2
      },
      "written": {
        "angle between … vectors": 21,
        "angle between two vectors": 6
      },
      "sources": [
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle-bisector-theorem",
    "ja": {
      "term": "角の二等分線の定理",
      "reading": "かくのにとうぶんせんのていり",
      "alt": []
    },
    "en": {
      "term": "angle bisector theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3",
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "△ABC で ∠A の二等分線が辺 BC と交わる点を D とすると、BD : DC = AB : AC になるという定理。AB ≠ AC のとき、外角の二等分線と直線 BC の交点についても同じ比が成り立つ（こちらは外分）。",
    "definition_en": "In triangle ABC, if the bisector of angle A meets side BC at D, then BD : DC = AB : AC. In other words, the bisector cuts the opposite side in the ratio of the two sides that form the angle.",
    "latex": "\\frac{BD}{DC} = \\frac{AB}{AC}",
    "spoken_en": "BD over DC equals AB over AC",
    "examples": [
      {
        "en": "By the angle bisector theorem, BD over DC is 6 over 4, so BD is three fifths of BC.",
        "ja": "角の二等分線の定理より BD : DC = 6 : 4 なので、BD は BC の 5 分の 3。",
        "register": "spoken"
      },
      {
        "en": "In triangle ABC, AB = 6, AC = 4, and BC = 5. The bisector of angle A meets BC at D. Use the angle bisector theorem to find BD.",
        "ja": "△ABC において AB = 6, AC = 4, BC = 5 とする。∠A の二等分線が辺 BC と交わる点を D とおく。角の二等分線の定理を用いて、線分 BD の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "by the angle bisector theorem",
        "ja": "角の二等分線の定理より"
      }
    ],
    "pitfalls": [
      "比は角をはさむ 2 辺の比。BD : DC = AB : AC で、AB : BC などと取り違えない。",
      "高等学校学習指導要領解説（数学A）は角の二等分線と辺の比の関係を外角の場合も含めて扱い、中学校学習指導要領解説は相似の活用の例としてこの性質の証明を挙げる。",
      "米国の高校課程（CED・OpenStax・IM・CK-12）には比の定理としての angle bisector theorem が出てこない。CK-12 Geometry 4.21 の Angle Bisector Theorem は、角の二等分線上の点が角の 2 辺から等距離にあるという別の定理。名前は英語版 Wikipedia「Angle bisector theorem」（外角の節もある）。"
    ],
    "related": [
      "angle-bisector",
      "point-of-internal-division",
      "point-of-external-division",
      "similar-triangles"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "角の二等分線の定理",
        "en": "Angle bisector theorem"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.21 Angle Bisectors in Triangles: 同名の Angle Bisector Theorem は角の二等分線上の点が角の 2 辺から等距離という別の定理（比の定理は 0 件）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Angle_bisector_theorem",
        "note": "記事「Angle bisector theorem」（数学カテゴリから 4 段）: 冒頭で太字。外角の節もある"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 図形の性質: 角の二等分線と辺の比（外角の場合も）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形: 相似の活用の例（角の二等分線と辺の比の証明）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「角の二等分線の定理」",
        "url": "https://ja.wikipedia.org/wiki/%E8%A7%92%E3%81%AE%E4%BA%8C%E7%AD%89%E5%88%86%E7%B7%9A%E3%81%AE%E5%AE%9A%E7%90%86",
        "note": "外角の場合は AB ≠ AC"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（監査 4、2026-09-26）: 英語版 Wikipedia の記事名「Angle bisector theorem」（ja 記事「角の二等分線の定理」の langlink 先、数学カテゴリから 4 段、冒頭で太字）。corpus:decide が根拠にした CK-12 Geometry 4.21 の Angle Bisector Theorem（10 件）は「角の二等分線上の点は角の 2 辺から等距離」という別の定理で、比の定理は CED・OpenStax・IM・CK-12 に出てこない。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle-bisector",
    "ja": {
      "term": "角の二等分線",
      "reading": "かくのにとうぶんせん",
      "alt": []
    },
    "en": {
      "term": "angle bisector",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "1 つの角を、大きさの等しい 2 つの角に分ける半直線。コンパスと定規で作図できる。",
    "definition_en": "A ray (or line) that splits an angle into two equal angles. You can construct it with a compass and a straightedge.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The angle bisector cuts this 80-degree angle into two 40-degree angles.",
        "ja": "角の二等分線で、この 80° の角が 40° の角 2 つに分かれる。",
        "register": "spoken"
      },
      {
        "en": "Construct the angle bisector of angle ABC using a compass and a straightedge.",
        "ja": "コンパスと定規を使って、∠ABC の二等分線を作図せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "bisector は bisect（二等分する。エントリ bisect）の名詞形。線分の垂直二等分線は perpendicular bisector（エントリ perpendicular-bisector）で、角の二等分線とは別の語。",
      "三角形の 3 つの角の二等分線は 1 点で交わり、その点（内心）は incenter（エントリ incenter）。",
      "三角形の角の二等分線が対辺を分ける比の定理は angle bisector theorem（エントリ angle-bisector-theorem）。"
    ],
    "related": [
      "bisect",
      "perpendicular-bisector",
      "incenter",
      "angle-bisector-theorem",
      "construction"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二等分線",
        "en": "Bisection"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.11 Congruent Angles and Angle Bisectors・4.21 Angle Bisectors in Triangles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "angle bisector": 11
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle-inscribed-in-a-semicircle",
    "ja": {
      "term": "直径に対する円周角",
      "reading": "ちょっけいにたいするえんしゅうかく",
      "alt": []
    },
    "en": {
      "term": "angle inscribed in a semicircle",
      "alt": [
        "inscribed in a semicircle"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。日本語の「直径に対する円周角」に当たる名前は出てこず、an angle inscribed in a semicircle is a right angle のように性質を文で言う（本プロジェクトの書き方の例）。英語版 Wikipedia「Thales's theorem」は、この性質（直径に対する円周角は直角）を Thales's theorem と呼ぶ（記事は定理の名前で、角の名前ではない）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直径 AB の両端と、円周上の A、B 以外の点 P を結んでできる円周角 ∠APB。その大きさはいつも 90° になる。",
    "definition_en": "An inscribed angle APB whose sides pass through the endpoints of a diameter AB, so the arc it intercepts is a semicircle. It is always a right angle.",
    "latex": "\\angle APB = 90^\\circ",
    "spoken_en": "angle APB equals 90 degrees",
    "examples": [
      {
        "en": "AB is a diameter, so this is an angle inscribed in a semicircle, and that means it's 90 degrees.",
        "ja": "AB は直径だから、これは直径に対する円周角で、90° になる。",
        "register": "spoken"
      },
      {
        "en": "∠APB is an angle inscribed in a semicircle, so ∠APB = 90° and ∠PBA = 180° − 90° − 28° = 62°.",
        "ja": "∠APB は直径に対する円周角だから ∠APB = 90° で、∠PBA = 180° − 90° − 28° = 62° である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は直径の側から「直径に対する円周角」と言うが、英語の an angle inscribed in a semicircle は半円（弧）の側から言う。",
      "Thales' theorem の Thales は、英語では「セイリーズ」に近く発音する（Merriam-Webster: ˈthā-lēz）。日本語の「タレス」と聞き比べて覚える。",
      "逆も成り立つ。∠APB = 90° なら、AB は 3 点 A、B、P を通る円の直径になる。"
    ],
    "related": [
      "inscribed-angle-theorem",
      "semicircle",
      "diameter",
      "right-angle",
      "inscribed-angle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Thales%27s_theorem",
        "note": "記事「Thales's theorem」（数学カテゴリから 3 段。冒頭で太字）: 直径に対する円周角は直角"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「Thales」",
        "url": "https://www.merriam-webster.com/dictionary/Thales",
        "note": "発音 ˈthā-(ˌ)lēz（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle-of-depression",
    "ja": {
      "term": "俯角",
      "reading": "ふかく",
      "alt": []
    },
    "en": {
      "term": "angle of depression",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "上から下の物を見下ろしたとき、視線と水平線がなす角。",
    "definition_en": "The angle that your line of sight makes with the horizontal when you look down at something below eye level.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The angle of depression is measured down from the horizontal, not from the side of the cliff.",
        "ja": "俯角は水平線から下に向けて測る。崖の壁から測るんじゃないよ。",
        "register": "spoken"
      },
      {
        "en": "From the top of a 40 m cliff, the angle of depression to a boat is 25°. Find the distance from the boat to the base of the cliff.",
        "ja": "高さ 40 m の崖の上から船を見下ろした俯角は 25° であった。船と崖の真下の地点との距離を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "俯角は水平線から下向きに測る。平行線の錯角により、下の地点から見上げた仰角と等しい。"
    ],
    "related": [
      "angle-of-elevation",
      "tangent",
      "trigonometric-ratio",
      "bearing"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「仰俯角」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BB%B0%E4%BF%AF%E8%A7%92",
        "note": "仰角・俯角の記事"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "angle of depression": 6
      },
      "written": {
        "angle of depression": 12
      },
      "sources": [
        "openstax-algtrig",
        "openstax-precalculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle-of-elevation",
    "ja": {
      "term": "仰角",
      "reading": "ぎょうかく",
      "alt": []
    },
    "en": {
      "term": "angle of elevation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "下から上の物を見上げたとき、視線と水平線がなす角。",
    "definition_en": "The angle that your line of sight makes with the horizontal when you look up at something above eye level.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "You're standing 20 meters from the tree, and the angle of elevation to the top is 35 degrees, so the height is 20 tan 35°.",
        "ja": "木から 20 m 離れた所に立って、てっぺんへの仰角が 35° だから、高さは 20 tan 35° だ。",
        "register": "spoken"
      },
      {
        "en": "From a point 50 m from the base of a building, the angle of elevation to the top is 32°. Find the height of the building.",
        "ja": "建物の真下から 50 m 離れた地点から建物の頂上を見上げた仰角は 32° であった。建物の高さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "仰角は水平線から上向きに測る。鉛直線から測る角ではない。"
    ],
    "related": [
      "angle-of-depression",
      "tangent",
      "trigonometric-ratio",
      "bearing"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「仰俯角」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BB%B0%E4%BF%AF%E8%A7%92",
        "note": "仰角・俯角の記事"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "angle of elevation": 10
      },
      "written": {
        "angle of elevation": 55
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle-sum-of-a-triangle",
    "ja": {
      "term": "三角形の内角の和",
      "reading": "さんかくけいのないかくのわ",
      "alt": []
    },
    "en": {
      "term": "triangle sum theorem",
      "alt": [
        "triangle angle sum",
        "triangle angle sum theorem",
        "sum of the … angles of a triangle",
        "sum of the angles of a triangle"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語は「三角形の内角の和」という量の名前で言うが、CK-12 Geometry（4.17 Triangle Angle Sum Theorem）は本文でこれを定理の名前 triangle sum theorem で呼び、IM Geometry（1.21 One Hundred and Eighty）は Triangle Angle Sum Theorem と呼んで証明する。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "三角形の 3 つの内角の大きさの和。どんな三角形でも 180° になる。",
    "definition_en": "The theorem that the interior angles of a triangle always total 180°. It is proved by drawing a line through one vertex parallel to the opposite side and using alternate interior angles.",
    "latex": "\\angle A + \\angle B + \\angle C = 180^\\circ",
    "spoken_en": "angle A plus angle B plus angle C equals one hundred eighty degrees",
    "examples": [
      {
        "en": "By the triangle sum theorem, all three angles add up to 180, so the missing angle is 180 minus 50 minus 60, which is 70 degrees.",
        "ja": "三角形の内角の和は 180° だから、3 つの角を足すと 180° になる。残りの角は 180 − 50 − 60 = 70 で、70° だ。",
        "register": "spoken"
      },
      {
        "en": "In △ABC, m∠A = 48° and m∠B = 67°. Use the triangle sum theorem to find m∠C.",
        "ja": "△ABC で ∠A = 48°、∠B = 67° である。三角形の内角の和を用いて、∠C の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "triangle sum theorem は「3 つの内角の和は 180° である」という定理の名前で、和の値そのものを指す語ではない。「内角の和」という量を言うときは、OpenStax Prealgebra（9.3）の the sum of the measures of the angles of a triangle is 180° や、IM Grade 8（1.16）の the sum of the angles in a triangle のように、sum of the angles の形で言う。",
      "多角形の内角の和（エントリ sum-of-the-interior-angles）とは別の見出し。n 角形の内角の和は 180° × (n − 2) で、三角形の 180° は n = 3 の場合にあたる。",
      "三角形の 1 つの外角が、隣り合わない 2 つの内角の和に等しいこと（エントリ exterior-angle-theorem）は、内角の和が 180° であることから導ける。"
    ],
    "related": [
      "sum-of-the-interior-angles",
      "alternate-interior-angles",
      "interior-angle",
      "exterior-angle-theorem",
      "auxiliary-line"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.17 Triangle Angle Sum Theorem・4.18 Exterior Angles and Theorems"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 1.21 One Hundred and Eighty（Triangle Angle Sum Theorem。mapping_note・en.alt）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem（the sum of the measures of the angles of a triangle is 180°。pitfalls[0]・en.alt）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "1.16 Parallel Lines and the Angles in a Triangle（the sum of the angles in a triangle is always 180°。pitfalls[0]）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "sum of the angles of a triangle": 3
      },
      "written": {
        "sum of the … angles of a triangle": 6
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-elemalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-10-09"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは CK-12 の呼び方 triangle sum theorem（CK-12 Geometry 4.17 Triangle Angle Sum Theorem・CK-12 Geometry 4.18 Exterior Angles and Theorems・CK-12 Geometry 4.4 Isosceles Triangles・計 25 件）。register は主張しない。",
        "raised": "2026-10-09"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "angle",
    "ja": {
      "term": "角",
      "reading": "かく",
      "alt": []
    },
    "en": {
      "term": "angle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中2",
        "数I",
        "数A"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1",
        "Precalculus"
      ]
    },
    "definition_ja": "1 点から出る 2 本の半直線がつくる図形。また、その開きの大きさ（角度）。",
    "definition_en": "The figure formed by two rays that start from the same point, or the size of the opening between them.",
    "latex": "\\angle ABC",
    "spoken_en": "angle A B C",
    "examples": [
      {
        "en": "This is a right angle, so it's 90 degrees. Anything smaller is acute, and anything between 90 and 180 is obtuse.",
        "ja": "これは直角だから 90°。それより小さいのが鋭角、90° と 180° の間が鈍角だよ。",
        "register": "spoken"
      },
      {
        "en": "Find the measure of angle ABC.",
        "ja": "∠ABC の大きさを求めよ。",
        "register": "written"
      },
      {
        "en": "Classify each angle as acute, right, obtuse, or straight.",
        "ja": "それぞれの角が鋭角・直角・鈍角・平角のどれかを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "measure of an angle",
        "ja": "角の大きさ"
      },
      {
        "en": "find the angle",
        "ja": "角度を求める"
      }
    ],
    "pitfalls": [
      "鋭角 acute angle、直角 right angle、鈍角 obtuse angle、平角 straight angle。right angle の right は「右」ではなく「直」の意味。",
      "∠ABC は英語でも頂点が真ん中の B（the vertex is B）。英語では角そのもの ∠ABC と、その大きさ m∠ABC を記号で書き分けることがある。共通テストの問題文は区別せず ∠ABC = 60° のように書く。"
    ],
    "related": [
      "vertex",
      "triangle",
      "inscribed-angle",
      "degree-measure",
      "radian"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和4年度 大学入学共通テスト 本試験 数学Ⅰ（問題）",
        "url": "https://web.archive.org/web/20230601/https://www.dnc.ac.jp/albums/abm.php?d=67&f=abm00000843.pdf&n=R4%E5%95%8F%E9%A1%8C_%E5%95%8F%E9%A1%8C_%E6%95%B0%E5%AD%A6%E2%85%A0.pdf",
        "note": "∠ABC = … の書き方"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "angle": 3237
      },
      "written": {
        "angle": 2110
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "anova",
    "ja": {
      "term": "分散分析",
      "reading": "ぶんさんぶんせき",
      "alt": []
    },
    "en": {
      "term": "ANOVA",
      "alt": [
        "analysis of variance"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "3 つ以上の群の平均が等しいかどうかを、群の間のばらつきと群の中のばらつきの比（F 値）で検定する方法。",
    "definition_en": "A test of whether three or more group means are equal, comparing the variation between groups with the variation within groups (the F ratio).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "With four groups, running six t-tests inflates the error rate, so we use ANOVA instead.",
        "ja": "4 群で t 検定を 6 回やると誤りの確率が大きくなるので、代わりに分散分析を使う。",
        "register": "spoken"
      },
      {
        "en": "Use a one-way ANOVA to test whether the mean yields of the three fertilizers differ.",
        "ja": "一元配置分散分析を用いて、3 種類の肥料の平均収量が異なるか検定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ANOVA は analysis of variance の略で、アノーバと読む。用例コーパスの書き言葉はすべて OpenStax Introductory Statistics。",
      "one-way ANOVA（一元配置分散分析）は見出しを中に含むので候補にしなかった。"
    ],
    "related": [
      "f-distribution",
      "variance",
      "hypothesis-testing"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "分散分析",
        "en": "Analysis of variance"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "ANOVA": 65,
        "analysis of variance": 6
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "antiderivative",
    "ja": {
      "term": "原始関数",
      "reading": "げんしかんすう",
      "alt": []
    },
    "en": {
      "term": "antiderivative",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "意味は同じだが、言い分けが違う。OpenStax Calculus Volume 1 は F′(x) = f(x) となる関数 1 つを an antiderivative と呼び、+C を付けた全体は indefinite integral（または the most general antiderivative）と呼ぶ。学習指導要領解説（数学II・III）には「不定積分」だけが出てきて「原始関数」は出てこない。英語では 1 つを指すのか全体を指すのかで言い分ける。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "微分すると f(x) になる関数 F(x)。F′(x) = f(x) を満たす。",
    "definition_en": "A function F whose derivative is f, so that F′(x) = f(x).",
    "latex": "F'(x) = f(x)",
    "spoken_en": "F prime of x equals f of x",
    "examples": [
      {
        "en": "What's an antiderivative of cosine? Sine works, since the derivative of sine is cosine.",
        "ja": "cos の原始関数は何？ sin を微分すると cos だから sin でいい。",
        "register": "spoken"
      },
      {
        "en": "Since F′(x) = f(x) on the interval, F is an antiderivative of f there.",
        "ja": "その区間で F′(x) = f(x) だから、F は f の原始関数である。",
        "register": "written"
      },
      {
        "en": "Antiderivatives of the same function differ only by a constant.",
        "ja": "同じ関数の原始関数どうしは、定数の差しかない。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "find an antiderivative",
        "ja": "原始関数を 1 つ見つける"
      },
      {
        "en": "take the antiderivative",
        "ja": "原始関数を求める"
      }
    ],
    "pitfalls": [
      "原始関数は定数の差を除いて無数にあるので、英語では an antiderivative と不定冠詞で言うのが正確。講義（Khan Academy の AP Calculus・MIT OCW）では take the antiderivative と the を付けても言う。",
      "anti-derivative とハイフンを入れて書くこともある。意味は同じ。"
    ],
    "related": [
      "indefinite-integral",
      "constant-of-integration",
      "integrate",
      "reverse-of-differentiation",
      "find-an-antiderivative"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.8（antiderivatives と indefinite integrals）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II・III: 「不定積分」だけが出てきて「原始関数」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "antiderivative": 815
      },
      "written": {
        "antiderivative": 259
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "apollonian-circle",
    "ja": {
      "term": "アポロニウスの円",
      "reading": "あぽろにうすのえん",
      "alt": []
    },
    "en": {
      "term": "Apollonian circle",
      "alt": [
        "circle of Apollonius"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Apollonian circle ／ circle of Apollonius の言い方とも CED・OpenStax・IM・CK-12 で 0 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": []
    },
    "definition_ja": "2 定点 A, B からの距離の比が m : n（m ≠ n）で一定である点の軌跡。線分 AB を m : n に内分する点と外分する点を直径の両端とする円になる。",
    "definition_en": "The locus of points whose distances from two fixed points A and B are in a fixed ratio m : n (m ≠ n); it is the circle whose diameter joins the points dividing AB internally and externally in that ratio.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If the ratio of the distances is 2 to 1, the locus turns out to be a circle, called an Apollonian circle.",
        "ja": "距離の比が 2 : 1 なら、軌跡は円（アポロニウスの円）になる。",
        "register": "spoken"
      },
      {
        "en": "Find the locus of the point P such that PA : PB = 2 : 1, where A(0, 0) and B(3, 0).",
        "ja": "2 点 A(0, 0), B(3, 0) からの距離の比が PA : PB = 2 : 1 である点 P の軌跡を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパス（OpenStax 9 冊を含む）には出てこない。英語版 Wikipedia「Circles of Apollonius」は、アポロニウスにちなむ何種類かの円をまとめた記事で、この円はその 1 つ目（節 Apollonius's definition of a circle。冒頭で Apollonian circle と太字）。複数形の Apollonian circles は、互いに直交する 2 つの円の族を指す別の記事。"
    ],
    "related": [
      "locus",
      "point-of-internal-division",
      "equation-of-a-circle",
      "point-of-external-division"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "アポロニウスの円",
        "en": "Circles of Apollonius"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Circle",
        "note": "節 Circle of Apollonius（数学カテゴリから 3 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "apothem",
    "ja": {
      "term": "辺心距離",
      "reading": "へんしんきょり",
      "alt": []
    },
    "en": {
      "term": "apothem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "正多角形の中心から 1 辺に下ろした垂線の長さ。正多角形の内接円の半径に等しい。",
    "definition_en": "The distance from the center of a regular polygon to any of its sides, measured along a perpendicular. It equals the radius of the polygon's inscribed circle.",
    "latex": "a = \\frac{s}{2\\tan(180^\\circ/n)}",
    "spoken_en": "a equals s over two tangent of one hundred eighty degrees over n",
    "examples": [
      {
        "en": "The apothem goes from the center straight to the middle of a side, so it makes a right angle with that side.",
        "ja": "辺心距離は中心から辺のまん中までまっすぐ引いた線だから、その辺と直角に交わる。",
        "register": "spoken"
      },
      {
        "en": "A regular hexagon has side length 8. Find the apothem.",
        "ja": "1 辺の長さが 8 の正六角形の辺心距離を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "辺心距離は正多角形の内接円の半径と同じ長さ。日本の問題で「内接円の半径」と書かれている長さが、英語の apothem にあたることがある。",
      "apothem（中心から辺まで）と radius（中心から頂点まで）を取り違えない。正六角形では radius は 1 辺の長さと等しく、apothem は 1 辺の (√3/2) 倍。",
      "CK-12 Geometry では 5.21 Area of Regular and Irregular Polygons に 19 件。用例コーパスにはほとんど出てこない。"
    ],
    "related": [
      "area-of-a-regular-polygon",
      "inscribed-circle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "辺心距離",
        "en": "Apothem"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.21 Area of Regular and Irregular Polygons"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 apothem（CK-12 Geometry 5.21 Area of Regular and Irregular Polygons・計 19 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "approaches",
    "ja": {
      "term": "〜に限りなく近づく",
      "reading": "にかぎりなくちかづく",
      "alt": [
        "近づく"
      ]
    },
    "en": {
      "term": "approaches",
      "alt": [],
      "variants": [
        {
          "term": "approaches",
          "register": "both",
          "note": "高校向けの Khan Academy の言い方で、話し言葉の大半が Khan Academy。書き言葉（OpenStax）でもこれを使う。as x approaches 3 の形で数えた。"
        },
        {
          "term": "goes to",
          "register": "spoken",
          "note": "大学の講義（MIT OCW）の言い方で、書き言葉ではまれ。as x goes to 3 の形で数えた。"
        },
        {
          "term": "tends to",
          "register": "spoken",
          "note": "少数の言い方で、ほぼ MIT OCW だけ。as x tends to 3 の形で数えた。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "変数や関数の値が、ある値にいくらでも近くなっていく。",
    "definition_en": "To get as close as you like to a value.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "As x goes to 3, this whole thing goes to 9.",
        "ja": "x が 3 に近づくと、この式全体は 9 に近づく。",
        "register": "spoken"
      },
      {
        "en": "As n increases, aₙ approaches 1.",
        "ja": "n が大きくなると、aₙ は 1 に限りなく近づく。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では as x approaches 3 と as x goes to 3 の両方を聞く（高校向けの Khan Academy は approaches、大学の講義は goes to が多い）。教科書と答案では approaches。",
      "「限りなく」はふつう訳さない。gets arbitrarily close to は定義を説明するときの言い方で、用例コーパスには少ない。"
    ],
    "related": [
      "as-x-approaches-a",
      "limit",
      "take-the-limit"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.2（approaches）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "as … goes to": 455,
        "as … tends to": 15,
        "as … approaches": 1288
      },
      "written": {
        "as … approaches": 204,
        "as … goes to": 14,
        "as … tends to": 4
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "approximate-value",
    "ja": {
      "term": "近似値",
      "reading": "きんじち",
      "alt": [
        "概数",
        "近似"
      ]
    },
    "en": {
      "term": "approximation",
      "alt": [
        "approximate value"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic",
      "statistics"
    ],
    "level": {
      "jp": [
        "中3",
        "中1",
        "数B"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2",
        "Pre-Algebra",
        "Algebra 2"
      ]
    },
    "definition_ja": "真の値に近い値。測定したり四捨五入したりして得られ、真の値との間に少し差がある。",
    "definition_en": "A value that is close to, but not exactly equal to, the true value. The symbol ≈ means \"is approximately equal to.\"",
    "latex": "\\sqrt{2} \\approx 1.414",
    "spoken_en": "the square root of two is approximately one point four one four",
    "examples": [
      {
        "en": "3.14 is just an approximation of pi, so your answer won't be exact.",
        "ja": "3.14 は π の近似値にすぎないから、答えはぴったりにはならないよ。",
        "register": "spoken"
      },
      {
        "en": "Use a calculator to find an approximate value of √7 to the nearest hundredth.",
        "ja": "電卓を使い、√7 の近似値を小数第 3 位を四捨五入して求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "≈ は approximately equal to と読み、文の中では is approximately と言う（π ≈ 3.14 は pi is approximately 3.14）。",
      "四捨五入などで丸めた数（概数）は rounded number とも言う（エントリ round）。",
      "approximate は形容詞（approximate value、近似の）と動詞（近似する）の両方で使い、語末の発音が違う（Merriam-Webster: 形容詞は ə-ˈpräk-sə-mət「〜メット」、動詞は ə-ˈpräk-sə-ˌmāt「〜メイト」）。"
    ],
    "related": [
      "round",
      "decimal",
      "estimation",
      "linear-approximation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "近似",
        "en": "Approximation"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「approximate」",
        "url": "https://www.merriam-webster.com/dictionary/approximate",
        "note": "発音 形容詞 ə-ˈpräk-sə-mət、動詞 ə-ˈpräk-sə-ˌmāt（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "approximation": 641,
        "approximate value": 8
      },
      "written": {
        "approximation": 445,
        "approximate value": 30
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "arc-length-of-a-curve",
    "ja": {
      "term": "曲線の長さ",
      "reading": "きょくせんのながさ",
      "alt": [
        "弧長"
      ]
    },
    "en": {
      "term": "arc length",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線のある区間の長さ。y = f(x) なら ∫ₐᵇ √(1 + {f′(x)}²) dx で求まる。",
    "definition_en": "The length of a curve over an interval; for y = f(x) it is ∫ₐᵇ √(1 + (f′(x))²) dx.",
    "latex": "L = \\int_a^b \\sqrt{1 + \\bigl(f'(x)\\bigr)^2}\\,dx",
    "spoken_en": "L equals the integral from a to b of the square root of one plus f prime of x squared, dx",
    "examples": [
      {
        "en": "Arc length integrals almost never come out nice unless the problem was built for it.",
        "ja": "曲線の長さの積分は、きれいに解けるように作られた問題でない限り、まずきれいな形にならない。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の arc length は円の弧の長さにも一般の曲線の長さにも使う。",
      "媒介変数表示の曲線の長さ ∫√((dx/dt)² + (dy/dt)²) dt は CED の topic 9.3（BC）と OpenStax Calculus Volume 2 で扱う。極座標の曲線の長さ ∫√(r² + (dr/dθ)²) dθ も同じ arc length。",
      "円の弧の長さ（扇形の弧の長さ、s = rθ）は別のエントリ（弧の長さ）にした。英語はどちらも arc length。"
    ],
    "related": [
      "find-the-arc-length",
      "surface-area-of-revolution",
      "arc-length"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "弧長",
        "en": "Arc length"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.13・9.3（arc length）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "arc length": 249
      },
      "written": {
        "arc length": 192
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "arc-length",
    "ja": {
      "term": "弧の長さ",
      "reading": "このながさ",
      "alt": [
        "扇形の弧長"
      ]
    },
    "en": {
      "term": "arc length",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円の弧の部分の長さ。半径 r、中心角 θ（ラジアン）なら rθ、中心角が a° なら 2πr × a/360。",
    "definition_en": "How long an arc of a circle is. With radius r and central angle θ in radians it is rθ; with a central angle of a degrees it is 2πr · a/360.",
    "latex": "s = r\\theta",
    "spoken_en": "s equals r theta",
    "examples": [
      {
        "en": "Find the arc length of a sector with radius 8 cm and central angle 45°.",
        "ja": "半径 8 cm、中心角 45° の扇形の弧の長さを求めよ。",
        "register": "written"
      },
      {
        "en": "The arc length formula s = rθ holds only when θ is measured in radians.",
        "ja": "弧の長さの公式 s = rθ は、θ がラジアンのときだけ成り立つ。",
        "register": "written"
      },
      {
        "en": "The length of the arc is just the fraction of the circle times the whole circumference.",
        "ja": "弧の長さは、円全体に対する割合に円周の長さを掛けるだけ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "arc length とだけ言うと、積分で求める曲線の長さ（arc-length-of-a-curve）の意味になることが多い（用例コーパス）。",
      "度数法なら 2πr × a/360、弧度法（数II）なら rθ。rθ に度の値を入れない。"
    ],
    "related": [
      "arc-length-of-a-curve",
      "area-of-a-sector",
      "radian-measure",
      "chord"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "arc length of the sector | arc length of a sector | arc length of a circle | arc length of the circle | arc length formula | length of the arc | length of an arc": 9
      },
      "written": {
        "arc length of the sector | arc length of a sector | arc length of a circle | arc length of the circle | arc length formula | length of the arc | length of an arc": 24
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "arc-measure",
    "ja": {
      "term": "弧の度数",
      "reading": "このどすう",
      "alt": []
    },
    "en": {
      "term": "measure of the arc",
      "alt": [
        "arc measure",
        "measure of an arc"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「弧の度数」は学習指導要領解説に無い、本プロジェクトの訳語。日本では弧の大きさを中心角で言い、弧そのものに度数を付けない。米国の Geometry では弧 AB の度数を、AB の上に弧の記号を付けた記号に m を添えて書き、中心角と同じ度数で表す（長さの arc length とは別）。CK-12 Geometry は measure of the arc と呼ぶ（6.9 Arcs in Circles ほか、5 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "弧の大きさを、その弧に対する中心角の大きさ（度）で表したもの。半円の弧は 180°、円全体は 360° になる。",
    "definition_en": "The size of an arc given in degrees, equal to the measure of its central angle. A semicircle measures 180°, and a full circle measures 360°.",
    "latex": "m\\overset{\\frown}{AB} = m\\angle AOB",
    "spoken_en": "the measure of arc AB equals the measure of angle AOB",
    "examples": [
      {
        "en": "The central angle is 70 degrees, so the measure of the arc is 70 degrees too. That's in degrees, not a length.",
        "ja": "中心角が 70° だから、弧の度数も 70°。これは角度で、長さではない。",
        "register": "spoken"
      },
      {
        "en": "In circle O, m∠AOB = 64°. Find the measure of minor arc AB and the measure of major arc ACB.",
        "ja": "円 O で ∠AOB = 64° である。劣弧 AB と優弧 ACB に対する中心角の大きさをそれぞれ求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "弧の度数（単位は度）と arc length（弧の長さ）は別物。半径が違えば、度数が同じ弧でも長さは違う。",
      "優弧の度数は 360° から劣弧の度数を引いたもの。",
      "用例コーパスでは arc measure は話し言葉・書き言葉ともほとんど出てこない。IM Geometry は measure of the arc の形で 4 件使う。"
    ],
    "related": [
      "arc-length",
      "intercepted-arc",
      "major-arc",
      "minor-arc",
      "inscribed-angle"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "7.2 Inscribed Angles・7.5・7.8 Arcs and Sectors（4 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.9 Arcs in Circles・6.16 Angles On and Inside a Circle・6.17 Angles Outside a Circle"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "arc measure": 1
      },
      "written": {
        "arc measure": 1
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CK-12 の呼び方 measure of the arc（CK-12 Geometry 6.9 Arcs in Circles・CK-12 Geometry 6.16 Angles On and Inside a Circle・CK-12 Geometry 6.17 Angles Outside a Circle・計 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "arc",
    "ja": {
      "term": "弧",
      "reading": "こ",
      "alt": []
    },
    "en": {
      "term": "arc",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円周の一部分。円周上の 2 点 A、B の間の部分を弧 AB といい、⌒AB と書く。",
    "definition_en": "A part of a circle between two points on the circle. The arc between A and B is called arc AB and is written with a small curved mark over AB.",
    "latex": "\\overset{\\frown}{AB}",
    "spoken_en": "arc AB",
    "examples": [
      {
        "en": "The arc is the curved part, and the chord is the straight segment connecting its endpoints.",
        "ja": "弧は曲がった部分で、弦はその両端を結ぶまっすぐな線分だよ。",
        "register": "spoken"
      },
      {
        "en": "Points A and B are on circle O, the radius is 6 cm, and ∠AOB = 60°. Find the length of arc AB.",
        "ja": "点 A、B は円 O の周上にあり、半径は 6 cm、∠AOB = 60° である。弧 AB の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "記号 ⌒AB は arc AB と読む。2 点の間の弧は 2 つあり、大きいほうを major arc（エントリ major-arc）、小さいほうを minor arc（エントリ minor-arc）と区別する。",
      "弧 arc は曲線で、弦 chord（エントリ chord）は弧の両端を結ぶ線分。",
      "弧の長さは arc length（エントリ arc-length）。弧の度数（エントリ arc-measure）は度で表す別の量。"
    ],
    "related": [
      "arc-length",
      "chord",
      "major-arc",
      "minor-arc",
      "central-angle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "弧 (幾何学)",
        "en": "Arc (geometry)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "arc !length": 165
      },
      "written": {
        "arc !length": 71
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "arccosine",
    "ja": {
      "term": "アークコサイン",
      "reading": "あーくこさいん",
      "alt": [
        "逆余弦",
        "cos⁻¹"
      ]
    },
    "en": {
      "term": "inverse cosine",
      "alt": [
        "arccosine"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "cos x を 0 ≦ x ≦ π に制限した関数の逆関数。arccos x または cos⁻¹ x と書く。",
    "definition_en": "The inverse of cos x restricted to 0 ≤ x ≤ π, written arccos x or cos⁻¹ x.",
    "latex": "y = \\cos^{-1} x \\iff \\cos y = x,\\ 0 \\le y \\le \\pi",
    "spoken_en": null,
    "examples": [
      {
        "en": "Inverse cosine always gives an angle between 0 and π.",
        "ja": "arccos はいつも 0 から π の間の角を返す。",
        "register": "spoken"
      },
      {
        "en": "Find the angle between the vectors u and v by computing cos⁻¹((u · v)/(|u||v|)).",
        "ja": "cos⁻¹((u·v)/(|u||v|)) を計算して、ベクトル u, v のなす角を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数式の中の arccos は書き言葉によく出てくる（OpenStax Calculus Volume 3 のベクトルのなす角など）。読むときは inverse cosine か arc cosine。"
    ],
    "related": [
      "arcsine",
      "arctangent",
      "inverse-trigonometric-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "逆三角関数",
        "en": "Inverse trigonometric functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "用語集（arccosine）。level.us の Geometry"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.3 The Dot Product（ベクトルのなす角を arccos で求める）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "inverse cosine": 17,
        "arccosine": 1
      },
      "written": {
        "inverse cosine": 15,
        "arccosine": 3
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "arcsine",
    "ja": {
      "term": "アークサイン",
      "reading": "あーくさいん",
      "alt": [
        "逆正弦",
        "sin⁻¹"
      ]
    },
    "en": {
      "term": "inverse sine",
      "alt": [],
      "variants": [
        {
          "term": "inverse sine",
          "register": "both",
          "note": "話し言葉では arcsine と同じくらいで、Khan Academy に多い。書き言葉ではこちらがずっと多い（OpenStax Algebra and Trigonometry・OpenStax Calculus）。"
        },
        {
          "term": "arcsine",
          "register": "spoken",
          "note": "話し言葉はすべて MIT OCW。書き言葉は少ない（OpenStax Algebra and Trigonometry だけ）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "sin x を −π/2 ≦ x ≦ π/2 に制限した関数の逆関数。arcsin x または sin⁻¹ x と書く。",
    "definition_en": "The inverse of sin x restricted to −π/2 ≤ x ≤ π/2, written arcsin x or sin⁻¹ x.",
    "latex": "y = \\sin^{-1} x \\iff \\sin y = x,\\ -\\frac{\\pi}{2} \\le y \\le \\frac{\\pi}{2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Inverse sine of negative one half is −π/6, not 7π/6 — it has to be between −π/2 and π/2.",
        "ja": "arcsin(−1/2) は 7π/6 ではなく −π/6。−π/2 と π/2 の間でないといけない。",
        "register": "spoken"
      },
      {
        "en": "Evaluate arcsin(√3/2).",
        "ja": "arcsin(√3/2) の値を求めよ。",
        "register": "written"
      },
      {
        "en": "In right triangle ABC, sin A = 5/13. Use inverse sine to find the measure of angle A to the nearest degree.",
        "ja": "直角三角形 ABC で sin A = 5/13 のとき、∠A の大きさを 1° 単位で求めよ（逆正弦を使う）。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では inverse sine と arcsine が並ぶ（arcsine はすべて MIT OCW）。書き言葉は inverse sine がずっと多い（記号 arcsin は語として数えない）。",
      "IM Geometry は、直角三角形の辺の比から角を求めるときに arcsine（arccos・arctan）を使う。日本の共通テスト（数学I）は三角比の表を添えて角を読ませる。"
    ],
    "related": [
      "arccosine",
      "arctangent",
      "inverse-trigonometric-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "逆三角関数",
        "en": "Inverse trigonometric functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和4年度 大学入学共通テスト 本試験 数学Ⅰ（問題）",
        "url": "https://web.archive.org/web/20230601/https://www.dnc.ac.jp/albums/abm.php?d=67&f=abm00000843.pdf&n=R4%E5%95%8F%E9%A1%8C_%E5%95%8F%E9%A1%8C_%E6%95%B0%E5%AD%A6%E2%85%A0.pdf",
        "note": "三角比の表を用いてもよい"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 4.9 Using Trigonometric Ratios to Find Angles（arcsine・arccos・arctan）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "arcsine": 12,
        "inverse sine": 18
      },
      "written": {
        "inverse sine": 32,
        "arcsine": 3
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "arctangent",
    "ja": {
      "term": "アークタンジェント",
      "reading": "あーくたんじぇんと",
      "alt": [
        "逆正接",
        "tan⁻¹"
      ]
    },
    "en": {
      "term": "inverse tangent",
      "alt": [],
      "variants": [
        {
          "term": "arctangent",
          "register": "spoken",
          "note": "話し言葉は Khan Academy と MIT OCW。書き言葉には少ない。"
        },
        {
          "term": "inverse tangent",
          "register": "both",
          "note": "話し言葉は blackpenredpen と MIT OCW に多い。書き言葉では arctangent よりずっと多い（OpenStax Algebra and Trigonometry・OpenStax Calculus）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "tan x を −π/2 < x < π/2 に制限した関数の逆関数。arctan x または tan⁻¹ x と書く。",
    "definition_en": "The inverse of tan x restricted to −π/2 < x < π/2, written arctan x or tan⁻¹ x.",
    "latex": "y = \\tan^{-1} x \\iff \\tan y = x,\\ -\\frac{\\pi}{2} < y < \\frac{\\pi}{2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The antiderivative of one over one plus x squared is arctangent of x.",
        "ja": "1/(1 + x²) の原始関数は arctan x。",
        "register": "spoken"
      },
      {
        "en": "Evaluate tan⁻¹(−1) and sketch the graph of y = tan⁻¹ x.",
        "ja": "tan⁻¹(−1) の値を求め、y = tan⁻¹ x のグラフをかけ。",
        "register": "written"
      },
      {
        "en": "Inverse tangent of 1 is pi over 4, because the tangent of pi over 4 is 1.",
        "ja": "アークタンジェント 1 は π/4。tan(π/4) = 1 だから。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説には逆三角関数が出てこず、数学III の置換積分法で主に扱う置き換えは ax + b = t と x = a sin θ（〔内容の取扱い〕）。逆三角関数を使わずに求めるなら、∫₀¹ dx/(1 + x²) のような定積分は x = tan θ と置換して値を出す。OpenStax Calculus は答えを tan⁻¹ x + C の形で書くことが多く、arctan x とも書く。話し言葉の arctangent は半分以上が Khan Academy。"
    ],
    "related": [
      "arcsine",
      "arccosine",
      "inverse-trigonometric-function",
      "integrals-giving-inverse-trig-functions"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "逆三角関数",
        "en": "Inverse trigonometric functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "逆三角関数が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "inverse tangent": 37,
        "arctangent": 49
      },
      "written": {
        "inverse tangent": 15,
        "arctangent": 4
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-between-two-curves",
    "ja": {
      "term": "2 曲線間の面積",
      "reading": "にきょくせんかんのめんせき",
      "alt": [
        "曲線間の面積",
        "曲線で囲まれた面積",
        "放物線と直線で囲まれた面積"
      ]
    },
    "en": {
      "term": "area between two curves",
      "alt": [
        "area between curves",
        "area bounded by curves"
      ],
      "variants": [
        {
          "term": "area between the curves",
          "register": "written",
          "note": "OpenStax Calculus の書き方。話し言葉は少なく（ほとんどが Khan Academy）、書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "area between two curves",
          "register": "written",
          "note": "OpenStax Calculus と MIT の講義ノート。話し言葉では Professor Leonard がこの言い方。話し言葉は少ない。書き言葉では area between the curves と同じくらい使う。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "2 つ以上の曲線（や直線）にはさまれた、または囲まれた部分の面積。交点で区間を分け、上の曲線から下の曲線を引いて積分する。左右の曲線で決まる部分は y で積分する（右 − 左）。",
    "definition_en": "The area of a region between or enclosed by two or more curves: split at the intersection points and integrate top minus bottom, or, for a region bounded on the left and right, right minus left with respect to y.",
    "latex": "A = \\int_a^b \\bigl(f(x) - g(x)\\bigr)\\,dx",
    "spoken_en": "A equals the integral from a to b of f of x minus g of x, dx",
    "examples": [
      {
        "en": "For the area between two curves, it's always the top one minus the bottom one.",
        "ja": "2 曲線間の面積は、いつも上の曲線から下の曲線を引く。",
        "register": "spoken"
      },
      {
        "en": "If curves bound the region on the left and on the right, slice it horizontally and integrate with respect to y.",
        "ja": "部分が左右の曲線で決まるなら、横に切って y で積分する。",
        "register": "spoken"
      },
      {
        "en": "Find the area between the two curves y = x and y = x² on [0, 1].",
        "ja": "区間 [0, 1] で 2 曲線 y = x、y = x² にはさまれた部分の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "区間の途中で曲線の上下が入れ替わるときは、交点で積分を分けて、それぞれ上 − 下にする。",
      "OpenStax Calculus は同じ節で、y で積分するやり方（右の曲線 − 左の曲線）も扱う。AP の CED は x の関数・y の関数・3 点以上で交わる場合を別の topic（8.4・8.5・8.6）にしている。",
      "OpenStax Calculus と AP の CED の問題文は、Find the area of the region bounded by the parabola y = x² and the line y = x + 2. のように the region bounded by と書く。",
      "媒介変数表示の曲線の下の面積は、OpenStax では Calculus Volume 2 の parametric curves の章で扱う。"
    ],
    "related": [
      "area-under-the-curve",
      "one-sixth-formula"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.4–8.6（area between curves。x の関数・y の関数・3 点以上で交わる場合）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "7.2 Calculus of Parametric Curves（媒介変数表示の曲線の下の面積）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "area between curves": 5,
        "area between the curves": 6,
        "area between two curves": 8
      },
      "written": {
        "area bounded by curves": 1,
        "area between the curves": 15,
        "area between two curves": 13,
        "area between curves": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-formula-with-sine",
    "ja": {
      "term": "三角形の面積 ½ab sin C",
      "reading": "さんかくけいのめんせき",
      "alt": []
    },
    "en": {
      "term": "area of an oblique triangle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "2 辺 a, b とその間の角 C から、三角形の面積を S = (1/2)ab sin C で求める公式。",
    "definition_en": "The formula S = (1/2)ab sin C for the area of a triangle from two sides a, b and the included angle C.",
    "latex": "S = \\frac{1}{2}ab\\sin C",
    "spoken_en": "S equals one half a b sine C",
    "examples": [
      {
        "en": "You don't need the height — one half times two sides times the sine of the angle between them.",
        "ja": "高さはいらない。2 辺と、その間の角の sin の積の半分。",
        "register": "spoken"
      },
      {
        "en": "Find the area of an oblique triangle with sides 8 and 11 and included angle 40°.",
        "ja": "2 辺が 8, 11 でその間の角が 40° の三角形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "oblique triangle は直角でない三角形のこと。OpenStax Algebra and Trigonometry は節の中で area of an oblique triangle と呼ぶ。授業では単に the area formula with sine と説明する。"
    ],
    "related": [
      "solving-triangles",
      "area-of-a-sector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "area of an oblique triangle": 7
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 area of an oblique triangle（本文 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-in-polar-coordinates",
    "ja": {
      "term": "極座標での面積",
      "reading": "きょくざひょうでのめんせき",
      "alt": [
        "極座標の面積"
      ]
    },
    "en": {
      "term": "area of a polar region",
      "alt": [
        "area in polar coordinates",
        "area of a region bounded by a polar curve"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「極座標での面積」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "極方程式 r = f(θ) で表された曲線と 2 本の半直線で囲まれた部分の面積。½∫r² dθ で求める。",
    "definition_en": "The area of a region bounded by a polar curve r = f(θ) and two rays, found as ½∫r² dθ.",
    "latex": "A = \\frac{1}{2}\\int_{\\alpha}^{\\beta} r^2\\,d\\theta",
    "spoken_en": "A equals one half the integral from alpha to beta of r squared d theta",
    "examples": [
      {
        "en": "For area in polar, think thin pie slices, not rectangles.",
        "ja": "極座標の面積は、長方形ではなく細い扇形で考える。",
        "register": "spoken"
      },
      {
        "en": "Find the area of the region enclosed by one petal of r = sin 2θ.",
        "ja": "r = sin 2θ の花びら 1 枚で囲まれた部分の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 曲線の間の面積は ½∫(r₁² − r₂²) dθ（CED topic 9.9）。"
    ],
    "related": [
      "polar-coordinates",
      "polar-equation",
      "area-between-two-curves"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 9.8・9.9（area of a polar region）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "area in polar coordinates": 3
      },
      "written": {
        "area of a polar region": 1,
        "area of a region bounded by a polar curve": 3,
        "area in polar coordinates": 4
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは CED の呼び方 area of a polar region（topic 9.8）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-model",
    "ja": {
      "term": "面積図",
      "reading": "めんせきず",
      "alt": [
        "面積モデル"
      ]
    },
    "en": {
      "term": "area diagram",
      "alt": [
        "area model"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の中 3 の教科書も、長方形の面積で式の展開を説明するが、図に決まった名前はない。英語でも IM は area diagram、ほかの教材では area model と呼び、呼び方は教材による。IM Grade 6 は分配法則や小数の掛け算の説明にこの図を使う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "積を長方形の面積として表す図。辺をいくつかに分け、全体の面積を小さな長方形の面積の和として見ることで、分配法則や式の展開を説明する。",
    "definition_en": "A diagram that shows a product as the area of a rectangle. The sides are split into parts, and the whole area is the sum of the areas of the smaller rectangles.",
    "latex": "(x + 3)(x + 2) = x^2 + 2x + 3x + 6",
    "spoken_en": "the quantity x plus three times the quantity x plus two equals x squared plus two x plus three x plus six",
    "examples": [
      {
        "en": "Let's draw an area diagram: split one side into x and 3, and the other side into x and 2.",
        "ja": "面積図をかこう。1 辺を x と 3 に、もう 1 辺を x と 2 に分ける。",
        "register": "spoken"
      },
      {
        "en": "Draw an area diagram to show that 6(40 + 7) = 6 · 40 + 6 · 7.",
        "ja": "6(40 + 7) = 6 × 40 + 6 × 7 であることを面積図で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "同じ図を area model と呼ぶ教材もある。IM は area diagram と呼ぶ。",
      "因数分解の box method は、同じ長方形の図を逆向きに使い、面積の和から辺の長さ（因数）を求める方法。"
    ],
    "related": [
      "distributive-property",
      "expand",
      "cross-method",
      "foil",
      "rectangle"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 5.7 Using Diagrams to Represent Multiplication・6.10 The Distributive Property, Part 2（area diagram）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "area model": 1
      },
      "written": {
        "area model": 1
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 area diagram（Grade 6 5.7 Using Diagrams to Represent Multiplication・Grade 6 6.10 The Distributive Property, Part 2・Grade 6 5.8 Calculating Products of Decimals・計 23 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-of-a-regular-polygon",
    "ja": {
      "term": "正多角形の面積",
      "reading": "せいたかくけいのめんせき",
      "alt": []
    },
    "en": {
      "term": "area of a regular polygon",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "正多角形の面積。辺心距離を a、周の長さを p とすると A = (1/2)ap で求められる。",
    "definition_en": "The area enclosed by a regular polygon. It equals one half of the apothem times the perimeter.",
    "latex": "A = \\frac{1}{2}ap",
    "spoken_en": "A equals one half a p",
    "examples": [
      {
        "en": "For the area of a regular polygon, it's one half times the apothem times the perimeter, because you're really adding up a bunch of congruent triangles.",
        "ja": "正多角形の面積は ½ × 辺心距離 × 周の長さ。実際には合同な三角形をたくさん足しているのと同じだから。",
        "register": "spoken"
      },
      {
        "en": "A regular octagon has sides of length 10 cm and an apothem of 12.1 cm. Use the formula for the area of a regular polygon to find its area.",
        "ja": "1 辺の長さが 10 cm、辺心距離が 12.1 cm の正八角形がある。正多角形の面積の公式を使って、面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数I では、中心と各頂点を結んで n 個の合同な二等辺三角形に分け、外接円の半径 r を使って (n/2)r² sin(360°/n) と求める。CK-12 Geometry（5.21 Area of Regular and Irregular Polygons）は辺心距離（apothem）を使う A = (1/2)ap で求める。",
      "式の a は apothem（辺心距離）、p は perimeter（周の長さ）。a を 1 辺の長さと取り違えない。",
      "CK-12 Geometry で area of a regular polygon は 3 件（5.21）。用例コーパスにはほとんど出てこない。"
    ],
    "related": [
      "apothem",
      "area-formula-with-sine"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.21 Area of Regular and Irregular Polygons"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 area of a regular polygon（CK-12 Geometry 5.21 Area of Regular and Irregular Polygons・計 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-of-a-sector",
    "ja": {
      "term": "扇形の面積",
      "reading": "おうぎがたのめんせき",
      "alt": []
    },
    "en": {
      "term": "area of a sector",
      "alt": [
        "sector area"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry",
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "半径 r、中心角 θ（ラジアン）の扇形の面積 (1/2)r²θ。弧の長さ l を使うと (1/2)rl。",
    "definition_en": "The area (1/2)r²θ of a sector with radius r and central angle θ in radians, or (1/2)rl with arc length l.",
    "latex": "S = \\frac{1}{2}r^2\\theta = \\frac{1}{2}rl",
    "spoken_en": "S equals one half r squared theta, which is one half r l",
    "examples": [
      {
        "en": "With θ in radians, the area of a sector is just one half r squared θ.",
        "ja": "θ がラジアンなら、扇形の面積は (1/2)r²θ。",
        "register": "spoken"
      },
      {
        "en": "Find the area of a sector with radius 6 cm and central angle 2π/3.",
        "ja": "半径 6 cm、中心角 2π/3 の扇形の面積を求めよ。",
        "register": "written"
      },
      {
        "en": "Find the area of a sector of a circle with radius 4 whose central angle measures 45°.",
        "ja": "半径 4、中心角 45° の扇形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "度数法で表すと (θ/360°)πr²。弧度法の (1/2)r²θ は OpenStax Algebra and Trigonometry にある。"
    ],
    "related": [
      "radian-measure",
      "radian",
      "sector",
      "arc-length",
      "central-angle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles（area of a sector）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第1学年 B(2) 空間図形: 扇形の弧の長さと面積を求めること"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II (4) 三角関数: 扇形の弧の長さや面積を求めるなどの活動を通して，弧度法に関する理解を深める"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.8 Arcs and Sectors・7.13 Using Radians・7.14 Putting It All Together（area of a sector、sector area）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.10 Area of Sectors and Segments（area of a sector、sector area）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "area of a sector": 13,
        "sector area": 1
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-of-a-triangle-using-vectors",
    "ja": {
      "term": "三角形の面積とベクトル",
      "reading": "さんかくけいのめんせきとべくとる",
      "alt": []
    },
    "en": {
      "term": "area of a triangle using vectors",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（公式に名前がない）。外積（cross product）を使うと、三角形の面積は |a × b| の半分（2 辺が作る平行四辺形の面積の半分）になる。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "2 辺を表すベクトル a, b から三角形の面積を求める公式 S = (1/2)√(|a|²|b|² − (a·b)²)。成分では (1/2)|a₁b₂ − a₂b₁|。",
    "definition_en": "The formula S = (1/2)√(|a|²|b|² − (a·b)²) for the area of a triangle with sides given by vectors a and b; in components, (1/2)|a₁b₂ − a₂b₁|.",
    "latex": "S = \\frac{1}{2}\\sqrt{|\\vec{a}|^2|\\vec{b}|^2 - (\\vec{a}\\cdot\\vec{b})^2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "You can get the area straight from the dot product, without finding the angle.",
        "ja": "角を求めなくても、内積から直接面積が出る。",
        "register": "spoken"
      },
      {
        "en": "Find the area of the triangle with vertices O(0, 0), A(3, 1), and B(1, 4).",
        "ja": "3 点 O(0, 0), A(3, 1), B(1, 4) を頂点とする三角形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "dot-product",
      "cross-product",
      "determinant"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-of-a-triangle",
    "ja": {
      "term": "三角形の面積",
      "reading": "さんかくけいのめんせき",
      "alt": []
    },
    "en": {
      "term": "area of a triangle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "三角形が囲む部分の広さ。底辺と高さから (1/2) × 底辺 × 高さ、2 辺とその間の角から (1/2)bc sin A、3 辺からヘロンの公式で求める。",
    "definition_en": "The amount of space inside a triangle. It can be found as one half base times height, as (1/2)bc sin A from two sides and the included angle, or from the three sides by Heron's formula.",
    "latex": "S = \\frac{1}{2}bh",
    "spoken_en": "one half base times height",
    "examples": [
      {
        "en": "The area of a triangle is one half base times height, but here we don't know the height, so let's use the sine version instead.",
        "ja": "三角形の面積は底辺 × 高さ ÷ 2 だけど、ここでは高さがわからないから sin を使う公式にしよう。",
        "register": "spoken"
      },
      {
        "en": "Find the area of a triangle with b = 6, c = 10, and A = 150°.",
        "ja": "b = 6, c = 10, A = 150° である三角形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 辺と間の角を使う公式 S = (1/2)bc sin A そのものは別エントリ area-formula-with-sine（OpenStax Algebra and Trigonometry の area of an oblique triangle）。このエントリは三角形の面積一般。",
      "「底辺」は base、「高さ」は height。底辺と高さは垂直で、高さは対辺の頂点から底辺（またはその延長）に下ろした垂線の長さ。"
    ],
    "related": [
      "area-formula-with-sine",
      "herons-formula",
      "area",
      "triangle",
      "inscribed-circle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "area of a triangle": 31
      },
      "written": {
        "area of a triangle": 48
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-of-the-base",
    "ja": {
      "term": "底面積",
      "reading": "ていめんせき",
      "alt": []
    },
    "en": {
      "term": "area of the base",
      "alt": [
        "base area"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "角柱・角錐・円柱・円錐などの、底面 1 つの面積。角柱や円柱の体積は、底面積 × 高さで求められる。",
    "definition_en": "The area of one base of a solid such as a prism, pyramid, cylinder, or cone. In formulas it is written as a capital B, as in V = Bh.",
    "latex": "V = Bh",
    "spoken_en": "V equals B times h",
    "examples": [
      {
        "en": "Big B is the area of the base, so for a cylinder that's pi r squared.",
        "ja": "大文字の B は底面積だから、円柱なら πr² だね。",
        "register": "spoken"
      },
      {
        "en": "The base of a prism is a right triangle with legs of 3 cm and 4 cm, and the prism is 10 cm tall. Find the area of the base and the volume of the prism.",
        "ja": "底面が、直角をはさむ 2 辺の長さが 3 cm と 4 cm の直角三角形で、高さが 10 cm の三角柱がある。この三角柱の底面積と体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語版 Wikipedia「角錐」は底面積を S として V = Sh/3 と書くが、英語の公式では大文字の B を使う（V = Bh）。小文字の b は三角形などの底辺の長さ（A = ½bh）なので区別する。",
      "角柱・円柱は底面が 2 つあるが、V = Bh の B は 1 つ分の面積。表面積を求めるときは底面積を 2 倍する。",
      "底面は base（2 つなら bases）。英語の base は三角形の底辺や、累乗の底（エントリ base-of-a-power）にも使う。"
    ],
    "related": [
      "volume",
      "surface-area",
      "lateral-area",
      "prism",
      "cylinder"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「角錐」",
        "url": "https://ja.wikipedia.org/wiki/%E8%A7%92%E9%8C%90",
        "note": "V = Sh/3（底面積 S）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "area of the base": 51,
        "base area": 1
      },
      "written": {
        "area of the base": 11,
        "base area": 2
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-preserving-transformation",
    "ja": {
      "term": "等積変形",
      "reading": "とうせきへんけい",
      "alt": []
    },
    "en": {
      "term": "area-preserving transformation",
      "alt": [
        "same base and height"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。等積変形に当たる名前は出てこず、IM Grade 6（1.6 Area of Parallelograms）は、底辺と高さが等しい平行四辺形は形によらず面積が等しいと文で説明し、三角形の面積は同じ底辺と高さの平行四辺形の半分とする（1.9 Formula for the Area of a Triangle）。見出しの area-preserving transformation は本プロジェクトの説明的な訳。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "図形の面積を変えずに、形を変えること。底辺が共通で高さが等しい三角形は面積が等しいことを使い、底辺に平行な直線にそって頂点を動かす。",
    "definition_en": "Changing the shape of a figure without changing its area. It uses the fact that triangles with the same base and the same height have equal areas, so a vertex can slide along a line parallel to the base.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Slide point C along this parallel line, and the triangle keeps the same base and height, so the area doesn't change.",
        "ja": "点 C をこの平行線にそって動かしても、底辺と高さは同じままだから、面積は変わらないよ。",
        "register": "spoken"
      },
      {
        "en": "Since AD ∥ BC, triangles ABC and DBC have the same base and height, so their areas are equal.",
        "ja": "AD ∥ BC より、△ABC と △DBC は底辺が共通で高さが等しいから、面積は等しい。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高さが等しいと言えるのは、動かす頂点が底辺に平行な直線の上にあるとき。平行でない直線にそって動かすと高さが変わり、面積も変わる。",
      "等積変形では形が変わる。平行移動・回転・対称移動（エントリ rigid-motion）のように形も大きさも変わらない移動とは別のもの。",
      "見出しの area-preserving transformation は本プロジェクトの説明的な訳で、用例コーパスと参照には出てこない。"
    ],
    "related": [
      "area",
      "area-of-a-triangle",
      "parallel-lines",
      "height",
      "base-of-a-triangle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 1.6 Area of Parallelograms（parallelograms that have the same base and the same height will have the same area）・1.9 Formula for the Area of a Triangle"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "same base and height": 2
      },
      "sources": [
        "openstax-prealgebra"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 2 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "area-problem",
    "ja": {
      "term": "面積問題",
      "reading": "めんせきもんだい",
      "alt": []
    },
    "en": {
      "term": "area problem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「面積問題」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "曲線の下の部分の面積を求めるという問題。米国の教科書（OpenStax Calculus）では積分の導入に使う。",
    "definition_en": "The problem of finding the area under a curve, used to introduce the integral.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The area problem is where integrals come from.",
        "ja": "積分は面積問題から生まれた。",
        "register": "spoken"
      },
      {
        "en": "The area problem asks for the area of the region under y = x² from x = 0 to x = 1.",
        "ja": "面積問題とは、たとえば x = 0 から x = 1 までの y = x² の下の部分の面積を求める問題である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "tangent-problem",
      "riemann-sum",
      "area-under-the-curve"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「面積問題」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "area problem": 5
      },
      "written": {
        "area problem": 6
      },
      "sources": [
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 area problem（本文 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "area-under-the-curve",
    "ja": {
      "term": "曲線と x 軸の間の面積",
      "reading": "きょくせんとえっくすじくのあいだのめんせき",
      "alt": []
    },
    "en": {
      "term": "area under the curve",
      "alt": [],
      "variants": [
        {
          "term": "area under the curve",
          "register": "both",
          "note": "最も多い言い方。話し言葉は Khan Academy が中心。話し言葉では area under a curve よりずっと多い。書き言葉は OpenStax Calculus が中心で、OpenStax Introductory Statistics にも出てくる。"
        },
        {
          "term": "area under a curve",
          "register": "both",
          "note": "書き言葉では OpenStax Calculus がこちらも使う。話し言葉では area under the curve よりずっと少ない。書き言葉はほとんどが OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線 y = f(x) と x 軸、2 直線 x = a、x = b で囲まれた部分の面積。f(x) ≧ 0 なら定積分 ∫ₐᵇ f(x)dx に等しい。",
    "definition_en": "For y = f(x) on [a, b], how much area lies between the graph and the x-axis; if f(x) ≥ 0 there, this area is ∫ₐᵇ f(x) dx.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "As long as the curve stays above the axis, the integral from zero to three is just the area under the curve.",
        "ja": "曲線が x 軸より上にある限り、0 から 3 までの積分はそのまま曲線と x 軸の間の面積。",
        "register": "spoken"
      },
      {
        "en": "Find the area under the curve y = x² from x = 0 to x = 1.",
        "ja": "曲線 y = x² と x 軸、直線 x = 1 で囲まれた部分の面積を求めよ。",
        "register": "written"
      },
      {
        "en": "The net signed area is 0, but the total area is 4.",
        "ja": "定積分の値は 0 だが、面積は 4 である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "approximate the area under the curve",
        "ja": "曲線の下の面積を近似する"
      }
    ],
    "pitfalls": [
      "OpenStax Calculus Volume 1（5.2 The Definite Integral）は、f(x) が負のところを含む area under the curve を net signed area として扱い、6.7 では ln x を x < 1 のとき the negative of the area under the curve と書く。正の面積の合計がほしいときは total area と言って区別する。",
      "定積分の値は、x 軸より上を正、下を負として足した面積（signed area、net signed area とも）。面積を求めるときは x 軸より下の部分の定積分に − を付けるか、区間を分けて絶対値にする。"
    ],
    "related": [
      "definite-integral",
      "below-the-x-axis",
      "area-between-two-curves",
      "riemann-sum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "5.2 The Definite Integral（net signed area）・6.7 Integrals, Exponential Functions, and Logarithms（the negative of the area under the curve）"
      },
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "area under the curve": 140,
        "area under a curve": 19
      },
      "written": {
        "area under a curve": 26,
        "area under the curve": 61
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "area",
    "ja": {
      "term": "面積",
      "reading": "めんせき",
      "alt": []
    },
    "en": {
      "term": "area",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Precalculus"
      ]
    },
    "definition_ja": "平面図形や曲面が占める広さを、単位の正方形がいくつ分かで表した量。",
    "definition_en": "The amount of surface a region or figure covers, measured by how many unit squares it takes to cover it.",
    "latex": "A = \\frac{1}{2}bh",
    "spoken_en": "A equals one half base times height",
    "examples": [
      {
        "en": "The area of a triangle is one half base times height, so here it's one half times 6 times 4, which is 12 square centimeters.",
        "ja": "三角形の面積は底辺 × 高さ ÷ 2 だから、ここでは 6 × 4 ÷ 2 で 12 cm² だね。",
        "register": "spoken"
      },
      {
        "en": "A triangle has two sides of length 5 and 8 with an included angle of 30°. Find its area.",
        "ja": "2 辺の長さが 5 と 8 で、その間の角が 30° である三角形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "area of a triangle",
        "ja": "三角形の面積"
      },
      {
        "en": "find the area",
        "ja": "面積を求める"
      }
    ],
    "pitfalls": [
      "面積の単位 cm² は square centimeters と読む（centimeters squared とも言う）。答えに単位を付けるときは square units。",
      "立体の表面の広さ（表面積）は surface area で、area だけとは言い分ける。",
      "統計では、正規分布などの曲線の下の面積が確率を表す（the area under the curve）。"
    ],
    "related": [
      "triangle",
      "area-formula-with-sine",
      "area-of-a-sector",
      "area-under-the-curve",
      "ratio-of-areas-of-similar-figures"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "面積",
        "en": "Area"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "area": 3540
      },
      "written": {
        "area": 2349
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "argument-of-a-complex-number",
    "ja": {
      "term": "複素数の偏角",
      "reading": "ふくそすうのへんかく",
      "alt": []
    },
    "en": {
      "term": "argument",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "複素数 z を複素数平面の点とみたとき、実軸の正の向きから測った角 θ。arg z と書く。",
    "definition_en": "The angle θ, measured from the positive real axis, of the point z in the complex plane; written arg z.",
    "latex": "\\arg z = \\theta",
    "spoken_en": "the argument of z is theta",
    "examples": [
      {
        "en": "When you multiply complex numbers, the arguments add.",
        "ja": "複素数を掛けると、偏角は足し算になる。",
        "register": "spoken"
      },
      {
        "en": "Find the modulus and argument of z = −1 + i.",
        "ja": "z = −1 + i の絶対値と偏角を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "関数の引数（f(x) の x、log の真数）も英語では argument。偏角の意味で言うときは the argument of z ／ the argument of the complex number と、何の argument かを添える（真数は argument を参照）。",
      "argument は関数の引数（the argument of the logarithm）の意味でも使う語。偏角の意味では MIT OCW（18.03）の講義と OpenStax Algebra and Trigonometry が使う。",
      "極座標の θ（偏角）には名前をつけないことが多い（polar-angle を参照）。"
    ],
    "related": [
      "modulus-of-a-complex-number",
      "polar-form",
      "polar-angle",
      "complex-plane",
      "de-moivres-theorem",
      "argument"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "複素数の偏角",
        "en": "Argument (complex analysis)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "argument of … complex number": 1
      },
      "written": {
        "argument of … complex number": 1
      },
      "sources": [
        "mit-18.03",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-24。docs/DECISIONS.md「Phase 2 代数 2・Precalculus の単元の前の修正」の、残った ③ を人間が決めた行）。en.term は argument のまま。register は主張しない。",
        "raised": "2026-09-24"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "argument",
    "ja": {
      "term": "真数",
      "reading": "しんすう",
      "alt": []
    },
    "en": {
      "term": "argument",
      "alt": [
        "argument of the log"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "log_a M の M。正の数でなければならない（真数条件 M > 0）。",
    "definition_en": "The number M in log_a M; it must be positive.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The argument of the log has to be positive, so x has to be bigger than 3.",
        "ja": "真数は正でないといけないので、x は 3 より大きい。",
        "register": "spoken"
      },
      {
        "en": "Solve log₂(x − 1) + log₂(x + 1) = 3. Check that each argument is positive.",
        "ja": "方程式 log₂(x − 1) + log₂(x + 1) = 3 を解け（真数条件を確かめよ）。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the argument of a logarithm must be positive",
        "ja": "真数は正"
      }
    ],
    "pitfalls": [
      "「真数条件」に当たる名前は用例コーパスと参照（CED・OpenStax・IM・CK-12）に出てこない。OpenStax Algebra and Trigonometry は the argument of a logarithm must be positive と文で言い（6.5）、定義域を求めるときは真数が 0 より大きいという不等式を立てる（6.4）。",
      "argument は関数の引数（入力）一般を指す語で、複素数の偏角（argument-of-a-complex-number）も argument。"
    ],
    "related": [
      "logarithm",
      "argument-of-a-complex-number",
      "logarithmic-equation",
      "logarithmic-inequality"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.4〜6.6（argument of the logarithm、must be positive）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "argument of the logarithm": 1,
        "argument of the log": 1
      },
      "written": {
        "argument of the logarithm": 4
      },
      "sources": [
        "openstax-algtrig",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは OpenStax の呼び方 argument of the logarithm（本文 4 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "arithmetic-mean",
    "ja": {
      "term": "相加平均",
      "reading": "そうかへいきん",
      "alt": [
        "算術平均",
        "平均値",
        "平均"
      ]
    },
    "en": {
      "term": "mean",
      "alt": [
        "arithmetic mean"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics",
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "数I",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 1",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "n 個の数（データの値）の和を n で割った値。2 数 a, b では (a + b)/2。データの代表値としては平均値という。",
    "definition_en": "The sum of n numbers or data values divided by n; for two numbers a and b, (a + b)/2. As a measure of center it is called the mean.",
    "latex": "\\frac{a + b}{2}",
    "spoken_en": "a plus b over two",
    "examples": [
      {
        "en": "The mean is pulled toward the outlier, but the median isn't.",
        "ja": "平均値は外れ値のほうへ引っぱられるが、中央値はそうならない。",
        "register": "spoken"
      },
      {
        "en": "Find the mean of the data: 12, 15, 15, 18, 20.",
        "ja": "データ 12, 15, 15, 18, 20 の平均値を求めよ。",
        "register": "written"
      },
      {
        "en": "The arithmetic mean of 4 and 16 is 10, and the geometric mean is 8.",
        "ja": "4 と 16 の相加平均は 10、相乗平均は 8。",
        "register": "spoken"
      },
      {
        "en": "For positive numbers a and b, compare the arithmetic mean (a + b)/2 with the geometric mean √(ab).",
        "ja": "正の数 a, b について、相加平均 (a + b)/2 と相乗平均 √(ab) の大小を比較せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "データの平均値は mean、相加・相乗平均の不等式の文脈では arithmetic mean（相乗平均 geometric mean と対にする）と言う。用例コーパスでは the mean（the mean value theorem を除く）が話し言葉・書き言葉とも arithmetic mean よりずっと多い。",
      "日常語では average とも言うが、average rate of change（平均変化率）や average value（関数の平均値）は別の概念。",
      "話し言葉の arithmetic mean は大半が The Organic Chemistry Tutor。"
    ],
    "related": [
      "geometric-mean",
      "am-gm-inequality",
      "arithmetic-middle-term",
      "median",
      "mode",
      "measure-of-center",
      "take-the-average"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "算術平均",
        "en": "Arithmetic mean"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the mean !value": 878,
        "arithmetic mean": 69
      },
      "written": {
        "the mean !value": 1005,
        "arithmetic mean": 8
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "arithmetic-middle-term",
    "ja": {
      "term": "等差中項",
      "reading": "とうさちゅうこう",
      "alt": []
    },
    "en": {
      "term": "arithmetic mean",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスでは arithmetic mean of … and … の形も少なく、書き言葉には出てこない）。3 数 a, b, c が等差数列をなすときの b に名前はつけず、b is the arithmetic mean of a and c（b = (a + c)/2）と相加平均で言う。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2"
      ]
    },
    "definition_ja": "3 つの数 a, b, c がこの順に等差数列をなすときの b。2b = a + c を満たす。",
    "definition_en": "The number b when a, b, c form an arithmetic sequence in that order, so that 2b = a + c.",
    "latex": "2b = a + c",
    "spoken_en": null,
    "examples": [
      {
        "en": "The middle term is just the arithmetic mean of its neighbors: b is a plus c over two.",
        "ja": "等差中項は両隣の相加平均、b = (a + c)/2。",
        "register": "spoken"
      },
      {
        "en": "If x + 1, 2x, and 4x − 3 form an arithmetic sequence, find x.",
        "ja": "x + 1、2x、4x − 3 がこの順に等差数列をなすとき、x を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "等差中項の意味では the arithmetic mean of a and c と 2 数を添えて言う。arithmetic mean だけだと統計の平均（arithmetic-mean）と同じ語。"
    ],
    "related": [
      "arithmetic-sequence",
      "geometric-middle-term",
      "arithmetic-mean",
      "am-gm-inequality"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "arithmetic mean of … and": 7
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 7 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "arithmetic-sequence",
    "ja": {
      "term": "等差数列",
      "reading": "とうさすうれつ",
      "alt": [
        "算術数列"
      ]
    },
    "en": {
      "term": "arithmetic sequence",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "隣り合う項の差が一定の数列。その一定の差を公差という。",
    "definition_en": "A sequence in which the difference between each term and the one before it is always the same number, the common difference.",
    "latex": "a_n = a_1 + (n-1)d",
    "spoken_en": "a sub n equals a sub one plus n minus one times d",
    "examples": [
      {
        "en": "It's arithmetic, because you keep adding 4.",
        "ja": "4 ずつ足しているから等差数列。",
        "register": "spoken"
      },
      {
        "en": "Show that a, b, c form an arithmetic sequence if and only if 2b = a + c.",
        "ja": "a, b, c がこの順に等差数列をなすことと 2b = a + c が同値であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "form an arithmetic sequence",
        "ja": "等差数列をなす"
      }
    ],
    "pitfalls": [
      "英語版 Wikipedia の記事名は Arithmetic progression だが、参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）はどれも arithmetic sequence と書き、arithmetic progression は出てこない。"
    ],
    "related": [
      "common-difference",
      "geometric-sequence",
      "sum-of-an-arithmetic-sequence",
      "arithmetic-middle-term",
      "sequence"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "等差数列",
        "en": "Arithmetic progression"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "arithmetic sequence": 28
      },
      "written": {
        "arithmetic sequence": 202
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "arrange-by-the-variable-of-lowest-degree",
    "ja": {
      "term": "最低次の文字について整理する",
      "reading": "さいていじのもじについてせいりする",
      "alt": []
    },
    "en": {
      "term": "arrange by the variable of lowest degree",
      "alt": [],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスに arrange by the variable of lowest degree は出てこない）。英語の見出しは本プロジェクトの訳語。因数分解の手順の一つで、英語で説明するなら「次数の最も低い文字について式を整理する」と手順をそのまま言う。参照でいちばん近い手法は OpenStax Elementary Algebra・Intermediate Algebra の Factor by Grouping（項を組にして共通因数をくくる。17・19 件）だが、同じ手法ではない。見出しの日本語は学習指導要領解説に出てこない。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "definition_ja": "複数の文字を含む式を因数分解するとき、次数が最も低い文字に注目し、その文字についての多項式として降べきの順に整理する。",
    "definition_en": "In factoring an expression with several variables, to rewrite it as a polynomial in the variable of lowest degree, collecting the terms by powers of that variable.",
    "latex": "a^2b - a^2c - b^3 + b^2c = -(a^2 - b^2)c + b(a^2 - b^2)",
    "spoken_en": null,
    "examples": [
      {
        "en": "In x² + xy + 2x + y + 1, y only shows up to the first power, so arrange by the variable of lowest degree: y(x + 1) + (x + 1)² = (x + 1)(x + y + 1).",
        "ja": "x² + xy + 2x + y + 1 では y が 1 次までしか出てこないので、最低次の文字 y について整理すると y(x + 1) + (x + 1)² = (x + 1)(x + y + 1) になる。",
        "register": "spoken"
      },
      {
        "en": "Factor a²b − a²c − b³ + b²c. Arrange by the variable of lowest degree, c: −(a² − b²)c + b(a² − b²) = (a + b)(a − b)(b − c).",
        "ja": "a²b − a²c − b³ + b²c を因数分解せよ。最低次の文字 c について整理すると −(a² − b²)c + b(a² − b²) = (a + b)(a − b)(b − c)。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の見出しは訳語で、決まった言い方ではない。英語で説明するときは手順を言葉にする: 「c について整理する」は write it as a polynomial in c、「c でくくる」は factor out c。",
      "factoring by grouping は項を 2 つずつ組にしてくくる手法で、1 つの文字について整理する手順とは別物。結果として同じ組分けになることはある。"
    ],
    "related": [
      "polynomial",
      "common-factor",
      "write-in-descending-order",
      "symmetric-expression",
      "alternating-expression"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Greatest Common Factor and Factor by Grouping: factor by grouping（17 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Factor by Grouping（19 件）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 数と式: 「最低次の文字について整理する」は 0 件（見出しの日本語は解説に出てこない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "arrange",
    "ja": {
      "term": "並べる",
      "reading": "ならべる",
      "alt": []
    },
    "en": {
      "term": "arrange",
      "alt": [],
      "uk": null
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "いくつかのものを、順序を付けて 1 列や円形などに置く。場合の数では、その置き方が何通りあるかを数える。",
    "definition_en": "To put a set of objects in some order, such as in a row or around a circle; in counting problems, the question is how many different orders are possible.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "How many ways can we arrange these 5 books on a shelf? Five choices for the first spot, four for the next, so 5! = 120.",
        "ja": "この 5 冊の本を棚に並べる方法は何通り？ 最初の場所に 5 通り、次に 4 通り…で 5! = 120 通り。",
        "register": "spoken"
      },
      {
        "en": "In how many ways can 6 people be arranged in a row if two of them must sit next to each other?",
        "ja": "6 人を 1 列に並べるとき、特定の 2 人が隣り合う並べ方は何通りあるか。",
        "register": "written"
      },
      {
        "en": "Find the number of arrangements of the letters in the word LEVEL.",
        "ja": "LEVEL の 5 文字を 1 列に並べる方法は何通りあるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "arrange は「式を整理する」「段取りをする」の意味でも使う。",
      "「並べ方」は arrangement（数えられる名詞）。「並べ方は何通りか」は How many arrangements are there? ／ In how many ways can … be arranged? と言う。"
    ],
    "related": [
      "permutation",
      "circular-permutation",
      "permutation-of-a-multiset",
      "order-matters",
      "counting",
      "choose"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles・13.7 Probability（練習問題）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.4 Combinations and Permutations（ways to arrange）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "ways to arrange | ways can … be arranged | arrange … in a row | ways of arranging | number of arrangements": 4
      },
      "written": {
        "ways to arrange | ways can … be arranged | arrange … in a row | ways of arranging | number of arrangements": 3
      },
      "sources": [
        "mit-18.02",
        "openstax-algtrig",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは OpenStax の呼び方 ways to arrange | ways can … be arranged | arrange … in a row | ways of arranging | number of arrangements（本文 3 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "as-n-approaches-infinity",
    "ja": {
      "term": "n → ∞ のとき",
      "reading": "えぬがむげんだいのとき",
      "alt": []
    },
    "en": {
      "term": "as n approaches infinity",
      "alt": [],
      "variants": [
        {
          "term": "as n approaches infinity",
          "register": "spoken",
          "note": "Khan Academy の言い方で、ほかでは Professor Leonard だけが使う。書き言葉にはほとんど出てこない。"
        },
        {
          "term": "as n goes to infinity",
          "register": "spoken",
          "note": "大学の講義（MIT OCW）と YouTube の解説の多くはこちら。書き言葉にはほとんど出てこない。"
        }
      ],
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "sequences"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "n を限りなく大きくするとき。極限を考える条件を表す。",
    "definition_en": "As n gets larger and larger without bound; the condition under which a limit is taken.",
    "latex": "\\lim_{n \\to \\infty} \\frac{1}{n} = 0",
    "spoken_en": "the limit as n approaches infinity of one over n is zero",
    "examples": [
      {
        "en": "As n approaches infinity, the rectangles get thinner and the sum gets closer to the area.",
        "ja": "n → ∞ のとき、長方形は細くなって、和は面積に近づく。",
        "register": "spoken"
      },
      {
        "en": "As n → ∞, the sum converges to ∫₀¹ x² dx = 1/3.",
        "ja": "n → ∞ のとき、この和は ∫₀¹ x² dx = 1/3 に収束する。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "答案では as n → ∞ と記号で書いてもよい。"
    ],
    "related": [
      "limit-of-a-riemann-sum",
      "riemann-sum"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "as n goes to infinity": 78,
        "as n approaches infinity": 112
      },
      "written": {
        "as n goes to infinity": 2,
        "as n approaches infinity": 4
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-6.042",
        "openstax-calculus",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "as-x-approaches-a",
    "ja": {
      "term": "x → a のとき",
      "reading": "えっくすがえーにちかづくとき",
      "alt": []
    },
    "en": {
      "term": "as x approaches a",
      "alt": [
        "as x approaches …"
      ],
      "variants": [
        {
          "term": "as x goes to …",
          "register": "spoken",
          "note": "MIT OCW の言い方。話し言葉はほとんどが MIT OCW。書き言葉は少ない（OpenStax Calculus だけ）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x を a に限りなく近づけていくとき（x は a に等しくならない）。",
    "definition_en": "As x gets closer and closer to a (without being equal to a).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "As x approaches 2, x squared approaches 4.",
        "ja": "x → 2 のとき、x² は 4 に近づく。",
        "register": "spoken"
      },
      {
        "en": "Find the limit of (x² − 4)/(x − 2) as x approaches 2.",
        "ja": "x → 2 のときの (x² − 4)/(x − 2) の極限を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "矢印 → は approaches（口頭では goes to）と読む。tends to は用例コーパスでは MIT OCW にわずかにあるだけ。"
    ],
    "related": [
      "limit",
      "as-x-approaches-infinity",
      "take-the-limit"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.2（as x approaches）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "as x goes to …": 164,
        "as x approaches a": 639
      },
      "written": {
        "as x approaches a": 131,
        "as x goes to …": 6
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "as-x-approaches-infinity",
    "ja": {
      "term": "x → ∞ のとき",
      "reading": "えっくすがむげんだいのとき",
      "alt": []
    },
    "en": {
      "term": "as x approaches infinity",
      "alt": [],
      "variants": [
        {
          "term": "as x goes to infinity",
          "register": "spoken",
          "note": "話し言葉では approaches とほぼ同じ数で、MIT OCW に多い。書き言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x を限りなく大きくしていくとき。",
    "definition_en": "As x grows larger without bound.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "As x goes to infinity, e to the negative x goes to zero.",
        "ja": "x → ∞ のとき、e⁻ˣ は 0 に近づく。",
        "register": "spoken"
      },
      {
        "en": "Find the limit of (3x² + 1)/(x² − 5) as x approaches infinity.",
        "ja": "x → ∞ のときの (3x² + 1)/(x² − 5) の極限を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "as-n-approaches-infinity",
      "as-x-approaches-a",
      "infinity",
      "limit"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "as x goes to infinity": 67,
        "as x approaches infinity": 77
      },
      "written": {
        "as x approaches infinity": 9,
        "as x goes to infinity": 3
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "asa-congruence",
    "ja": {
      "term": "1 組の辺とその両端の角がそれぞれ等しい",
      "reading": "ひとくみのへんとそのりょうたんのかくがそれぞれひとしい",
      "alt": [
        "ASA"
      ]
    },
    "en": {
      "term": "angle-side-angle",
      "alt": [
        "ASA",
        "angle-side-angle triangle congruence theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの三角形で、1 組の辺とその両端の角がそれぞれ等しければ合同であるという合同条件。",
    "definition_en": "A congruence criterion: two triangles are congruent when two angles and the side between them in one triangle match two angles and the side between them in the other.",
    "latex": "\\angle B = \\angle E,\\ BC = EF,\\ \\angle C = \\angle F \\implies \\triangle ABC \\cong \\triangle DEF",
    "spoken_en": "if angle B equals angle E, BC equals EF, and angle C equals angle F, then triangle ABC is congruent to triangle DEF",
    "examples": [
      {
        "en": "The side we know is sandwiched between the two angles we know, so this is ASA.",
        "ja": "分かっている辺が、分かっている 2 つの角にはさまれているから、1 組の辺とその両端の角がそれぞれ等しい場合だ。",
        "register": "spoken"
      },
      {
        "en": "In the figure, ∠ABD = ∠CBD and ∠ADB = ∠CDB. Prove that △ABD ≅ △CBD.",
        "ja": "図で ∠ABD = ∠CBD, ∠ADB = ∠CDB のとき、△ABD ≡ △CBD であることを証明せよ。",
        "register": "written"
      },
      {
        "en": "Angle-side-angle means the side you know is the one between the two angles you know.",
        "ja": "1 組の辺とその両端の角がそれぞれ等しい、というのは、分かっている辺が分かっている 2 つの角の間にあるということだ。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "by ASA",
        "ja": "1 組の辺とその両端の角がそれぞれ等しいから"
      }
    ],
    "pitfalls": [
      "日本語は辺から見て「両端の角」と言い、英語は角から見て included side（2 角にはさまれた辺）と言う。A-S-A の並びで、辺が 2 角の間にあることを表す。",
      "2 角と 1 辺が等しくても、その辺が 2 角の間にないときは AAS（angle-angle-side）と呼んで区別する。",
      "統計の文脈（AP Statistics の CED など）で出てくる ASA は American Statistical Association で、合同条件とは関係ない。",
      "IM Geometry はこの合同条件を Angle-Side-Angle Triangle Congruence Theorem と呼び（Geometry 2.7 ほか、13 件）、CK-12 Geometry は略号 ASA の postulate として扱う（4.15 ASA and AAS）。"
    ],
    "related": [
      "sss-congruence",
      "sas-congruence",
      "aas-congruence",
      "congruent"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "2.7 Angle-Side-Angle Triangle Congruence ほか（21 件）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "ASA は American Statistical Association（3 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.15 ASA and AAS（ASA を postulate として扱う）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "angle-side-angle": 1
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 angle-side-angle（Geometry 2.7 Angle-Side-Angle Triangle Congruence・Geometry 2.10 Practicing Proofs・Geometry 2.9 Side-Side-Side Triangle Congruence・計 21 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "ascending-order",
    "ja": {
      "term": "昇べきの順",
      "reading": "しょうべきのじゅん",
      "alt": [
        "昇冪の順"
      ]
    },
    "en": {
      "term": "increasing powers",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（多項式の項を次数の低い順に並べる言い方。用例コーパスの increasing powers は部分分数分解の分母の累乗や i の累乗の並び、ascending order は素数や数の並び）。OpenStax Elementary Algebra は次数の高い順の並びだけを standard form と呼ぶ。見出しの increasing powers は本プロジェクトの説明の訳。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "多項式の項を、次数の低いものから高いものへ順に並べること。1 − 5x + 3x² のような並べ方。",
    "definition_en": "An arrangement of the terms of a polynomial from the lowest power of the variable to the highest, as in 1 − 5x + 3x².",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's put the terms in order from the lowest power to the highest, so the constant goes in front.",
        "ja": "項を昇べきの順に並べよう。次数の低いほうからだから、定数項が先頭に来る。",
        "register": "spoken"
      },
      {
        "en": "Write the polynomial 3x² + 1 − 5x in increasing powers of x.",
        "ja": "多項式 3x² + 1 − 5x を x について昇べきの順に整理せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ascending order だけでは、数を小さい順に並べる意味になる（OpenStax・IM では素数や定義域の値の並び）。多項式の項の並べ方だとはっきりさせるなら、in increasing powers of x のように何の順かを添える（本プロジェクトの書き方の例）。",
      "文字が 2 つ以上ある式では、どの文字についてかを of x のように付けて言う（本プロジェクトの書き方の例）。",
      "次数の高い順に並べるのは降べきの順（エントリ write-in-descending-order）。"
    ],
    "related": [
      "write-in-descending-order",
      "polynomial",
      "degree",
      "term",
      "constant-term"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "ascending order は数の並び（多項式の項の順ではない）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "次数の高い順の並びだけを standard form と呼ぶ"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「単項式順序」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8D%98%E9%A0%85%E5%BC%8F%E9%A0%86%E5%BA%8F",
        "note": "1 変数多項式は昇冪の順または降冪の順に並べる（ja.alt 昇冪の順）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "increasing powers of x | increasing powers of the variable": 1
      },
      "sources": [
        "mit-18.01"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 1 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "associative-property",
    "ja": {
      "term": "結合法則",
      "reading": "けつごうほうそく",
      "alt": []
    },
    "en": {
      "term": "associative property",
      "alt": [
        "associative law"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "3 つの数の和や積で、どの 2 つを先に計算しても結果が変わらないという法則。数の順番は変えずに、かっこの位置だけを変える。",
    "definition_en": "The rule that changing how three numbers are grouped does not change their sum or their product: (a + b) + c = a + (b + c) and (ab)c = a(bc). The numbers stay in the same order; only the parentheses move.",
    "latex": "(a + b) + c = a + (b + c)",
    "spoken_en": "the quantity a plus b, plus c, equals a plus the quantity b plus c",
    "examples": [
      {
        "en": "With the associative property, I can regroup (17 + 6) + 4 as 17 + (6 + 4), which is 17 + 10, or 27.",
        "ja": "結合法則を使うと、(17 + 6) + 4 を 17 + (6 + 4) とまとめ直せる。17 + 10 で 27 だね。",
        "register": "spoken"
      },
      {
        "en": "Use the associative property of multiplication to find (3 · 25) · 4.",
        "ja": "乗法の結合法則を使って、(3 × 25) × 4 を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "交換法則（エントリ commutative-property）は数の順番を入れかえる法則、結合法則は順番は変えずにかっこの位置（どこを先に計算するか）を変える法則。",
      "成り立つのは加法と乗法だけ。減法では (10 − 4) − 2 = 4 だが 10 − (4 − 2) = 8 で、結果が変わる。",
      "OpenStax・CK-12 は演算を付けて associative property of addition ／ of multiplication とも言う。"
    ],
    "related": [
      "commutative-property",
      "distributive-property",
      "parentheses",
      "addition",
      "multiplication"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "結合法則",
        "en": "Associative property"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "1.6.8 Associative and Commutative Property with Fractions・1.6.9 Addition and Multiplication Properties with Real Numbers（associative property of addition ／ of multiplication）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "associative law": 8,
        "associative property": 20
      },
      "written": {
        "associative property": 85,
        "associative law": 1
      },
      "sources": [
        "khan-middle",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "assume",
    "ja": {
      "term": "仮定する",
      "reading": "かていする",
      "alt": []
    },
    "en": {
      "term": "suppose",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "suppose",
          "register": "both",
          "note": "MIT OCW の講義に多い。話し言葉は大半が MIT OCW で、Professor Leonard・patrickJMT が次ぐ。書き言葉は OpenStax Introductory Statistics・OpenStax Calculus・MIT の講義ノートに多い。"
        },
        {
          "term": "assume",
          "register": "both",
          "note": "話し言葉の半分は Khan Academy。話し言葉では MIT OCW が次ぐ。書き言葉は OpenStax Calculus・MIT の講義ノート・OpenStax Introductory Statistics の順に多い。"
        }
      ]
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "議論の出発点として、ある条件が成り立つものとして話を進める。背理法では、示したい結論の否定を仮定する。",
    "definition_en": "To take a condition as true for the sake of argument; in a proof by contradiction, you suppose the opposite of what you want to prove.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "OK, so suppose x is some number bigger than 2. What happens to f(x)?",
        "ja": "じゃあ x を 2 より大きいある数だと仮定しよう。f(x) はどうなる？",
        "register": "spoken"
      },
      {
        "en": "Let's assume the data come from a normal distribution.",
        "ja": "データは正規分布に従うと仮定しよう。",
        "register": "spoken"
      },
      {
        "en": "Suppose that √2 is rational, so √2 = p/q for some integers p and q with no common factor.",
        "ja": "√2 が有理数であると仮定すると、互いに素な整数 p, q を用いて √2 = p/q と表せる。",
        "register": "written"
      },
      {
        "en": "Assume that n is an even integer. Show that n² is even.",
        "ja": "n を偶数と仮定する。n² が偶数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Suppose … と Assume … はどちらも「〜と仮定する」。",
      "名詞の「仮定」は assumption。命題「p ならば q」の p（仮定）は hypothesis と呼ぶ。",
      "that は省略できる（Suppose x > 0. ／ Assume n is even.）。"
    ],
    "related": [
      "proof-by-contradiction",
      "contradiction",
      "lead-to-a-contradiction",
      "condition"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 図形: 命題は「仮定」と「結論」からなる、仮定から結論を導く（仮定 15 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "assume": 1067,
        "suppose": 962
      },
      "written": {
        "suppose": 1082,
        "assume": 768
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "asymptote",
    "ja": {
      "term": "漸近線",
      "reading": "ぜんきんせん",
      "alt": []
    },
    "en": {
      "term": "asymptote",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "geometry"
    ],
    "level": {
      "jp": [
        "数II",
        "数III",
        "数C"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 1",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "曲線が限りなく近づいていく直線。",
    "definition_en": "A line that a curve gets closer and closer to.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "y = 2^x never touches the x-axis — the x-axis is a horizontal asymptote.",
        "ja": "y = 2ˣ は x 軸に触れない。x 軸が水平漸近線。",
        "register": "spoken"
      },
      {
        "en": "Find the asymptotes of the hyperbola x²/9 − y²/4 = 1.",
        "ja": "双曲線 x²/9 − y²/4 = 1 の漸近線を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax は垂直（vertical）・水平（horizontal）・斜め（Precalculus と Algebra and Trigonometry は slant、Calculus は oblique）と漸近線の向きで呼び分ける。find-the-asymptotes を参照。"
    ],
    "related": [
      "find-the-asymptotes",
      "exponential-function",
      "logarithmic-function",
      "limit-at-infinity"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "漸近線",
        "en": "Asymptote"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "asymptote": 386
      },
      "written": {
        "asymptote": 498
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "at-least-one",
    "ja": {
      "term": "少なくとも 1 つ",
      "reading": "すくなくともひとつ",
      "alt": [
        "少なくとも一方"
      ]
    },
    "en": {
      "term": "at least one",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "logic",
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Geometry",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "1 つ以上あること。「少なくとも一方」は 2 つのうち 1 つ以上（両方も含む）を表す。",
    "definition_en": "One or more; for two things, it means one of them or both.",
    "latex": "P(\\text{at least one}) = 1 - P(\\text{none})",
    "spoken_en": "the probability of at least one equals one minus the probability of none",
    "examples": [
      {
        "en": "If ab = 0, then at least one of a and b has to be zero.",
        "ja": "ab = 0 ならば、a と b の少なくとも一方は 0 だ。",
        "register": "spoken"
      },
      {
        "en": "A die is rolled three times. Find the probability of getting at least one 6.",
        "ja": "さいころを 3 回投げる。少なくとも 1 回 6 の目が出る確率を求めよ。",
        "register": "written"
      },
      {
        "en": "Show that the equation x³ + x − 1 = 0 has at least one real solution between 0 and 1.",
        "ja": "方程式 x³ + x − 1 = 0 は 0 と 1 の間に少なくとも 1 つの実数解をもつことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "at least one of",
        "ja": "〜のうち少なくとも 1 つ（一方）"
      },
      {
        "en": "at least one real solution",
        "ja": "少なくとも 1 つの実数解"
      }
    ],
    "pitfalls": [
      "「少なくとも 1 回」の確率は余事象で求めるのが定番：P(at least one 6) = 1 − P(no 6s)。英語でも 1 − P(none) の形で書ける。",
      "「a, b の少なくとも一方が 0」は英語では a = 0 or b = 0 とも言う。数学の or は「両方」も含む。"
    ],
    "related": [
      "probability",
      "exist",
      "all",
      "proof"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "at least one": 153
      },
      "written": {
        "at least one": 168
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "augmented-matrix",
    "ja": {
      "term": "拡大係数行列",
      "reading": "かくだいけいすうぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "augmented matrix",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "連立 1 次方程式の係数と右辺の定数を並べてつくった行列。",
    "definition_en": "The matrix made of the coefficients of a system of linear equations together with the constants on the right side.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Write the augmented matrix, then row-reduce it.",
        "ja": "拡大係数行列を書いて、行基本変形で簡単にしていく。",
        "register": "spoken"
      },
      {
        "en": "Write the augmented matrix for the system x + 2y − z = 3, 2x − y + z = 1, x + y + z = 6.",
        "ja": "連立方程式 x + 2y − z = 3, 2x − y + z = 1, x + y + z = 6 の拡大係数行列を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校の学習指導要領には含まれないが、米国では Precalculus（OpenStax Algebra and Trigonometry）で扱う。用例コーパスでは書き言葉（OpenStax Algebra and Trigonometry・OpenStax Intermediate Algebra）に多く、話し言葉にはほとんど出てこない。"
    ],
    "related": [
      "row-operations",
      "gaussian-elimination",
      "row-echelon-form",
      "system-of-three-equations"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "拡大行列",
        "en": "Augmented matrix"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "augmented matrix": 3
      },
      "written": {
        "augmented matrix": 95
      },
      "sources": [
        "mit-18.06",
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "auxiliary-angle-form",
    "ja": {
      "term": "三角関数の合成",
      "reading": "さんかくかんすうのごうせい",
      "alt": []
    },
    "en": {
      "term": "rewriting a sin θ + b cos θ as a single sine function",
      "alt": [],
      "variants": [
        {
          "term": "amplitude-phase form",
          "register": "spoken",
          "note": "大学の微分方程式の講義（MIT OCW 18.03）の言い方で、合成してできた形（A cos(ωt − φ) の形）を呼ぶ。同じ講義は複素数を極形式に直すことにも使う。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。米国の高校の資料（CED・OpenStax・IM・CK-12）には名前が出てこない。大学の微分方程式の講義（MIT OCW 18.03）では、できた形を amplitude-phase form と呼ぶ。OpenStax Calculus Volume 3（7.3 Applications）は、単振動の x(t) = c₁ cos(ωt) + c₂ sin(ωt) を A sin(ωt + ϕ)（A = √(c₁² + c₂²)）の形に書き、例題で Express … in the form A sin(ωt + ϕ) と問う。英語版 Wikipedia「List of trigonometric identities」は節 Linear combinations（Sine and cosine）でこの変形を扱い、本文で harmonic addition とも呼ぶ。見出しの rewriting a sin θ + b cos θ as a single sine function は本プロジェクトの説明的な訳で、英語の名前ではない。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "a sin θ + b cos θ を r sin(θ + α)（r = √(a² + b²)）の形に直すこと。最大・最小や方程式に使う。",
    "definition_en": "Rewriting a sin θ + b cos θ as r sin(θ + α) with r = √(a² + b²), used to find maximum and minimum values and to solve equations.",
    "latex": "a\\sin\\theta + b\\cos\\theta = \\sqrt{a^2 + b^2}\\,\\sin(\\theta + \\alpha)",
    "spoken_en": "a sine theta plus b cosine theta equals the square root of a squared plus b squared, times sine of theta plus alpha",
    "examples": [
      {
        "en": "Combine sine and cosine into a single sine: the amplitude is the square root of a squared plus b squared.",
        "ja": "sin と cos を 1 つの sin にまとめる。振幅は √(a² + b²)。",
        "register": "spoken"
      },
      {
        "en": "Rewrite √3 sin θ + cos θ as a single sine function r sin(θ + α), and find its maximum value.",
        "ja": "√3 sin θ + cos θ を 1 つの sin の式 r sin(θ + α) に直し、最大値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「合成」を直訳した synthesis of trigonometric functions は、用例コーパスにも参照（CED・OpenStax・IM・CK-12）にも出てこない。"
    ],
    "related": [
      "angle-addition-formulas",
      "amplitude",
      "trigonometric-function",
      "phase",
      "phase-shift"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "7.3 Applications（in the form A sin(ωt + ϕ)）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II (4) 三角関数 ア(エ): 三角関数の加法定理や2倍角の公式，三角関数の合成について理解すること"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和3年度 大学入学共通テスト 第1日程 数学Ⅱ（問題）",
        "url": "https://web.archive.org/web/20220601/https://www.dnc.ac.jp/albums/abm.php?d=88&f=abm00001261.pdf&n=R3_%E6%95%B0%E5%AD%A6%E2%85%A1%EF%BC%88%EF%BC%91%E6%9C%8816%E6%97%A5%E3%83%BB17%E6%97%A5%EF%BC%89%E3%81%AE%E5%95%8F%E9%A1%8C.pdf",
        "note": "第1問: 三角関数の合成"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/List_of_trigonometric_identities",
        "note": "記事「List of trigonometric identities」の節 Linear combinations ＞ Sine and cosine（a cos x + b sin x を 1 つの cos で書く。本文で harmonic addition とも呼ぶ）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "amplitude-phase form": 5
      },
      "sources": [
        "mit-18.03"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 5 件 ／ 書 0 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "auxiliary-line",
    "ja": {
      "term": "補助線",
      "reading": "ほじょせん",
      "alt": []
    },
    "en": {
      "term": "auxiliary line",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "証明や問題を解くために、もとの図にかき加える直線や線分。",
    "definition_en": "An extra line, segment, or ray that you add to a diagram to help you solve a problem or write a proof.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's draw an auxiliary line here. If we connect B and D, the quadrilateral splits into two triangles.",
        "ja": "ここに補助線を引こう。B と D を結ぶと、四角形が 2 つの三角形に分かれる。",
        "register": "spoken"
      },
      {
        "en": "Draw an auxiliary line through vertex C parallel to side AB, and use it to prove that the sum of the interior angles of △ABC is 180°.",
        "ja": "頂点 C を通り辺 AB に平行な補助線を引き、それを用いて △ABC の内角の和が 180° であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「補助線を引く」は draw an auxiliary line。この draw は「（線を）かく」の意味で、袋から玉を取り出す draw（エントリ draw）と同じ語。引き算の「引く」（subtract、エントリ subtract）とは別の語。",
      "auxiliary は 2 つ目の音節（zil）を強く読む（Merriam-Webster: ȯg-ˈzil-yə-rē）。",
      "IM Geometry は auxiliary line を用語集（glossary）の見出しにしている。"
    ],
    "related": [
      "proof",
      "parallel-lines",
      "alternate-interior-angles",
      "angle-sum-of-a-triangle",
      "draw"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary・2.6 Side-Angle-Side Triangle Congruence・2.10 Practicing Proofs"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「auxiliary」",
        "url": "https://www.merriam-webster.com/dictionary/auxiliary",
        "note": "発音 ȯg-ˈzil-yə-rē（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 auxiliary line（Geometry glossary・Geometry 2.6 Side-Angle-Side Triangle Congruence・Geometry 2.10 Practicing Proofs・計 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "average-rate-of-change",
    "ja": {
      "term": "平均変化率",
      "reading": "へいきんへんかりつ",
      "alt": []
    },
    "en": {
      "term": "average rate of change",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x が a から b まで変わるときの (f(b) − f(a))/(b − a)。2 点を結ぶ直線（割線）の傾きに等しい。",
    "definition_en": "The change in f(x) divided by the change in x between two points; the slope of the secant line through them.",
    "latex": "\\frac{f(b) - f(a)}{b - a}",
    "spoken_en": "f of b minus f of a over b minus a",
    "examples": [
      {
        "en": "From x = 1 to x = 3, the average rate of change is just the slope of the secant line.",
        "ja": "x = 1 から x = 3 までの平均変化率は、割線の傾きにほかならない。",
        "register": "spoken"
      },
      {
        "en": "Find the average rate of change of f(x) = x² on the interval [1, 3].",
        "ja": "区間 [1, 3] における f(x) = x² の平均変化率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 点を結ぶ直線は英語で secant line（割線）。三角関数の secant と同じ語。"
    ],
    "related": [
      "rate-of-change",
      "slope-of-the-tangent-line",
      "derivative"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.1・2.1・5.1（average rate of change）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "average rate of change": 92
      },
      "written": {
        "average rate of change": 101
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "average-value-of-a-function",
    "ja": {
      "term": "関数の平均値",
      "reading": "かんすうのへいきんち",
      "alt": []
    },
    "en": {
      "term": "average value of a function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "区間 a ≦ x ≦ b での関数の値の平均。定積分を区間の幅で割った (1/(b − a))∫ₐᵇ f(x)dx。",
    "definition_en": "The average of f over [a, b]: the integral divided by the length of the interval, (1/(b − a)) ∫ₐᵇ f(x) dx.",
    "latex": "f_{\\text{avg}} = \\frac{1}{b-a}\\int_a^b f(x)\\,dx",
    "spoken_en": "f average equals one over b minus a times the integral from a to b of f of x dx",
    "examples": [
      {
        "en": "The average value of a function is just the integral divided by the width of the interval.",
        "ja": "関数の平均値は、積分を区間の幅で割るだけ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説には出てこない。積分の平均値の定理は mean value theorem for integrals と言う。",
      "平均変化率（average rate of change）とは別物。"
    ],
    "related": [
      "definite-integral",
      "net-change"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「関数の平均値」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "average value of a function": 17
      },
      "written": {
        "average value of a function": 28
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "axiom",
    "ja": {
      "term": "公理",
      "reading": "こうり",
      "alt": []
    },
    "en": {
      "term": "axiom",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Intro Statistics",
        "Linear Algebra"
      ]
    },
    "definition_ja": "証明なしに正しいと認め、理論の出発点とする命題。",
    "definition_en": "A statement accepted as true without proof and used as a starting point from which other results are derived.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This one's an axiom, so we don't prove it. We just take it as given.",
        "ja": "これは公理なので証明はしない。そういうものとして受け入れる。",
        "register": "spoken"
      },
      {
        "en": "Using only the axioms of probability, show that P(∅) = 0.",
        "ja": "確率の公理だけを用いて P(∅) = 0 を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "CK-12 Geometry（1.1 Geometry Terms ほか）はユークリッド幾何の公理を postulate と呼ぶ（81 件。axiom は 0 件。エントリ postulate）。",
      "用例コーパスでは axiom は話し言葉のほとんどが MIT OCW、書き言葉のほとんどが MIT の講義ノートで、大学の講義寄りの語。"
    ],
    "related": [
      "proof",
      "lemma",
      "corollary",
      "proposition",
      "postulate",
      "theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "公理",
        "en": "Axiom"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "postulate（81 件。axiom は 0 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "1.6.1 Real Number Properties and Axioms（3 件）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "axiom（98 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "axiom": 58
      },
      "written": {
        "axiom": 19
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "axis-of-revolution",
    "ja": {
      "term": "回転軸",
      "reading": "かいてんじく",
      "alt": []
    },
    "en": {
      "term": "axis of revolution",
      "alt": [],
      "variants": [
        {
          "term": "axis of rotation",
          "register": "written",
          "note": "少数の言い方（書き言葉の次点）。axes of rotation（複数形）を含む。話し言葉にはほとんど出てこない。書き言葉は OpenStax Calculus が中心。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "回転体をつくるときに、図形を回転させる軸となる直線。",
    "definition_en": "The line a region is revolved around to form a solid of revolution.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When the axis of revolution is y equals negative one, the radius is f of x plus one.",
        "ja": "回転軸が y = −1 なら、半径は f(x) + 1 になる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は、座標軸のまわりの回転（topic 8.9・8.11）と、座標軸以外の直線（y = −1 や x = 3）のまわりの回転（topic 8.10・8.12）を別の topic にしている。"
    ],
    "related": [
      "volume-of-a-solid-of-revolution",
      "revolve-around-the-x-axis",
      "washer-method"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.9–8.12（revolving around the x- or y-axis ／ other axes）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "axis of rotation": 4
      },
      "written": {
        "axis of revolution": 13,
        "axis of rotation": 9
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "axis-of-symmetry",
    "ja": {
      "term": "軸",
      "reading": "じく",
      "alt": [
        "放物線の軸",
        "対称軸"
      ]
    },
    "en": {
      "term": "axis of symmetry",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中3",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2",
        "Algebra 2"
      ]
    },
    "definition_ja": "放物線がそれに関して線対称になる直線。y = a(x − p)² + q のグラフでは直線 x = p で、頂点を通る。",
    "definition_en": "The line a parabola is symmetric about. For y = a(x − h)² + k it is the vertical line x = h, which passes through the vertex.",
    "latex": "x = -\\frac{b}{2a}",
    "spoken_en": "x equals negative b over two a",
    "examples": [
      {
        "en": "The axis of symmetry always passes through the vertex, so for this parabola it's x equals 3.",
        "ja": "軸は頂点を通るので、ここでは x = 3。",
        "register": "spoken"
      },
      {
        "en": "For the parabola y = 2x² − 8x + 5, find the vertex and write the equation of its axis of symmetry.",
        "ja": "放物線 y = 2x² − 8x + 5 の頂点と軸の方程式を求めよ。",
        "register": "written"
      },
      {
        "en": "The axis of symmetry of the parabola y = −(x + 1)² + 4 is the line x = −1.",
        "ja": "放物線 y = −(x + 1)² + 4 の軸は直線 x = −1 である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "equation of the axis of symmetry",
        "ja": "軸の方程式"
      }
    ],
    "pitfalls": [
      "日本語は「軸」だけで放物線の軸を指すが、英語で axis とだけ言うと x 軸・y 軸（x-axis, y-axis）と紛れる。放物線の軸は axis of symmetry と言う。",
      "line of symmetry は図形一般の対称の軸（エントリ line-of-reflection）にも使う。",
      "軸の方程式は x = p の形の直線の式で答える。y = p と書き間違えない。"
    ],
    "related": [
      "vertex",
      "completing-the-square",
      "x-intercept",
      "quadratic-formula"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Geometry 6.8: 放物線の line of symmetry"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 二次関数: グラフの対称軸（直線 x = p）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "axis of symmetry": 19
      },
      "written": {
        "axis of symmetry": 196
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "bar-chart",
    "ja": {
      "term": "棒グラフ",
      "reading": "ぼうぐらふ",
      "alt": []
    },
    "en": {
      "term": "bar graph",
      "alt": [
        "bar chart"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "小学校"
      ],
      "us": [
        "Pre-Algebra",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "項目ごとの量を、離れた長方形の棒の長さで表したグラフ。",
    "definition_en": "A graph that shows the amount for each category by the length of a separate bar.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a bar chart the bars don't touch, because the categories are separate.",
        "ja": "棒グラフでは、カテゴリが別々なので棒がくっつかない。",
        "register": "spoken"
      },
      {
        "en": "Make a bar graph of the number of students in each club.",
        "ja": "部ごとの部員数の棒グラフをかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は bar graph と書く。Khan Academy はどちらも使う（中学の講義では bar graph、AP Statistics では bar chart が多い）。"
    ],
    "related": [
      "histogram",
      "pie-chart",
      "statistical-graph",
      "categorical-variable"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "棒グラフ",
        "en": "Bar chart"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "bar graph": 10,
        "bar chart": 6
      },
      "written": {
        "bar graph": 34
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "base-angle",
    "ja": {
      "term": "底角",
      "reading": "ていかく",
      "alt": []
    },
    "en": {
      "term": "base angle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "二等辺三角形の底辺の両端にある 2 つの角。2 つの底角の大きさは等しい。",
    "definition_en": "One of the two angles at the ends of the base of an isosceles triangle. The two base angles of an isosceles triangle are equal.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If the vertex angle is 36 degrees, then each base angle is 72, because 180 minus 36 is 144, and half of that is 72.",
        "ja": "頂角が 36° なら、底角はそれぞれ 72°。180 − 36 = 144 で、その半分が 72 だからね。",
        "register": "spoken"
      },
      {
        "en": "In isosceles triangle ABC with AB = AC, each base angle measures 65°. Find the measure of the vertex angle.",
        "ja": "AB = AC の二等辺三角形 ABC で、底角がそれぞれ 65° のとき、頂角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「二等辺三角形の底角は等しい」という定理は base angles theorem と呼ぶ（エントリ isosceles-triangle-theorem）。",
      "底角の「底」は底辺（base、エントリ base-of-a-triangle）の「底」。等しい 2 辺にはさまれた角は頂角（vertex angle、エントリ vertex-angle）。",
      "英語の base angle は台形の底辺の両端の角にも使う（CK-12 Geometry 5.13 Trapezoids）。"
    ],
    "related": [
      "vertex-angle",
      "isosceles-triangle",
      "isosceles-triangle-theorem",
      "base-of-a-triangle",
      "trapezoid"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.4 Isosceles Triangles・4.5 Equilateral Triangles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "base angle": 4
      },
      "sources": [
        "khan-algebra",
        "mit-18.01"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 base angle（CK-12 Geometry 4.4 Isosceles Triangles・CK-12 Geometry 4.5 Equilateral Triangles・CK-12 Geometry 5.13 Trapezoids・計 37 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "base-case",
    "ja": {
      "term": "n=1 のとき",
      "reading": "えぬいこーるいちのとき",
      "alt": []
    },
    "en": {
      "term": "base case",
      "alt": [
        "basis step"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「数学的帰納法」は手順を番号で書くだけでこの段に名前をつけず、「多項定理」などの証明は「n=1 のとき」と書き出す。英語では base case と名前で呼ぶ（英語版 Wikipedia「Mathematical induction」）。見出しの「n=1 のとき」は答案の書き方。",
    "domains": [
      "proof"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "数学的帰納法の最初の段。最初の値（ふつう n = 1）で成り立つことを確かめる。",
    "definition_en": "The first step of a proof by induction: checking the statement for the first value, usually n = 1.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The base case is easy: when n equals 1, both sides are 1.",
        "ja": "n = 1 のときは簡単。両辺とも 1 になる。",
        "register": "spoken"
      },
      {
        "en": "Base case: when n = 1, the left side is 1 and the right side is 1(1 + 1)/2 = 1.",
        "ja": "[1] n = 1 のとき、左辺は 1、右辺は 1(1 + 1)/2 = 1。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉・書き言葉とも、ほぼすべて MIT 6.042（講義とそのテキスト）。"
    ],
    "related": [
      "proof-by-induction",
      "inductive-hypothesis",
      "inductive-step",
      "prove-by-induction",
      "strong-induction"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Mathematical_induction",
        "note": "記事「Mathematical induction」の冒頭と節 Description で base case を太字で定義（数学カテゴリから 3 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学的帰納法」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%E7%9A%84%E5%B8%B0%E7%B4%8D%E6%B3%95",
        "note": "手順を番号で書き、最初の段に名前をつけない"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「多項定理」",
        "url": "https://ja.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%AE%9A%E7%90%86",
        "note": "証明を「n=1 のとき」と書き出す"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.5 Proof by Induction・4.6 Strong Induction（base case）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "base case": 61,
        "basis step": 1
      },
      "written": {
        "base case": 56,
        "basis step": 1
      },
      "sources": [
        "khan-algebra",
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "base-n",
    "ja": {
      "term": "n 進法",
      "reading": "えぬしんほう",
      "alt": []
    },
    "en": {
      "term": "base n",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語は表し方（n 進法）と表された数（n 進数）を呼び分けず、どちらも base n を使って言う（in base 5 ／ a base-5 number）。英語版 Wikipedia「Positional notation」も底を base（radix）と呼び、1111011₂ のように書く。「n 進法で表す」は write … in base n。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "位が 1 つ上がるごとに n 倍になるように、0 から n − 1 までの n 種類の数字で数を表す方法。",
    "definition_en": "A way of writing numbers with the n digits 0 through n − 1, where each place is worth n times the place to its right.",
    "latex": "142_{(5)} = 1 \\cdot 5^2 + 4 \\cdot 5 + 2 = 47",
    "spoken_en": "one four two in base five equals forty-seven",
    "examples": [
      {
        "en": "Let's write 13 in base 2: that's 1101.",
        "ja": "13 を 2 進法で書くと 1101 になる。",
        "register": "spoken"
      },
      {
        "en": "Write 47 in base 5.",
        "ja": "47 を 5 進法で表せ。",
        "register": "written"
      },
      {
        "en": "Convert the base-3 number 212 to base 10.",
        "ja": "3 進法で表された数 212₍₃₎ を 10 進法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "共通テスト（令和 6 年度 本試験 数学I・数学A 第 4 問）は「n 進数とは n 進法で表された数のこと」と書き、表し方と数を呼び分ける。英語はどちらも base n を使い、in base 5（5 進法で）と a base-5 number（5 進数）のように言い分ける。",
      "センター試験（平成 28 年度 本試験 数学I・数学A 第 4 問）は 11011₍₂₎ のように括弧付きの添え字で書く。英語版 Wikipedia「Positional notation」は 1111011₂ と括弧なしで書く。",
      "10 進法の decimal は「小数」の意味でも使う。「10 進法で」は in base 10 と言うと紛れない。",
      "base は対数・累乗の底（log base 2、エントリ base-of-a-power）にも使う。「n 進法で」は in base n と in を付けて言う。"
    ],
    "related": [
      "binary",
      "hexadecimal",
      "numeral-system",
      "digit",
      "place",
      "decimal-system"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Positional_notation",
        "note": "記事「Positional notation」（base、radix、1111011₂。数学カテゴリから 2 段）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和6年度 大学入学共通テスト 本試験 数学Ⅰ・数学Ａ（問題）",
        "note": "第4問: n 進数とは n 進法で表された数のこと"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成28年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "note": "第4問: 11011(2)、4 進法・6 進法"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 数学と人間の活動: 二進法・六十進法"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "in base !e !on | written in base | base-n representation": 17
      },
      "sources": [
        "khan-middle"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "base-of-a-power",
    "ja": {
      "term": "底",
      "reading": "てい",
      "alt": []
    },
    "en": {
      "term": "base",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "累乗 aⁿ で、繰り返し掛け合わせる数 a のこと。指数関数 y = aˣ や対数（a を底とする log）の a も底という。",
    "definition_en": "In a power aⁿ, the base a is the number being multiplied by itself, and the exponent n tells how many times it is used as a factor. The same word is used for the base of a logarithm, as in log base 2.",
    "latex": "a^n",
    "spoken_en": "a to the n",
    "examples": [
      {
        "en": "In 2 to the fifth, 2 is the base and 5 is the exponent, so you multiply five 2s together.",
        "ja": "2 の 5 乗では、2 が底で 5 が指数。だから 2 を 5 個掛け合わせる。",
        "register": "spoken"
      },
      {
        "en": "Write 3 · 3 · 3 · 3 as a power, and identify the base and the exponent.",
        "ja": "3 × 3 × 3 × 3 を累乗の形で表し、底と指数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の base は三角形の底辺（エントリ base-of-a-triangle）、立体の底面（エントリ base-of-a-solid）、10 進法（base ten、エントリ base-n）にも使う同じ単語。文脈で見分ける。",
      "対数の底も base で、log₂ 8 は log base 2 of 8 と読む。",
      "−3² の底は 3 で、値は −9。(−3)² の底は −3 で、値は 9。かっこがあるかどうかで底が変わる。"
    ],
    "related": [
      "power",
      "exponent",
      "logarithm",
      "laws-of-exponents",
      "base-n",
      "base-of-a-triangle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the base is | same base | base and exponent | base and the exponent | base of the power | base of the exponent | bases are the same": 171
      },
      "written": {
        "the base is | same base | base and exponent | base and the exponent | base of the power | base of the exponent | bases are the same": 131
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "base-of-a-solid",
    "ja": {
      "term": "底面",
      "reading": "ていめん",
      "alt": []
    },
    "en": {
      "term": "base",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "角柱・円柱で、向かい合って平行な 2 つの面。角錐・円錐では、頂点と向かい合う 1 つの面をいう。",
    "definition_en": "One of the two parallel, congruent faces of a prism or cylinder, or the face of a pyramid or cone that is opposite its vertex. The height of the solid is measured perpendicular to the base.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A cone has just one base, and it's a circle, but a cylinder has two.",
        "ja": "円錐の底面は 1 つだけで、円。でも円柱には底面が 2 つある。",
        "register": "spoken"
      },
      {
        "en": "The base of a pyramid is a regular hexagon. Find the number of faces, edges, and vertices of the pyramid.",
        "ja": "底面が正六角形の角錐がある。この角錐の面の数、辺の数、頂点の数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の base は、三角形の底辺（エントリ base-of-a-triangle）や累乗の底（エントリ base-of-a-power）にも使う同じ単語。立体の底面かどうかは、the base of the prism のような前後の語で見分ける。",
      "角柱・円柱には底面が 2 つあり、複数形は bases。底面積は area of the base（エントリ area-of-the-base）で、公式 V = Bh では大文字の B で書く。",
      "底面は、下に置いた面とは限らない。横に倒して置いた三角柱でも、底面は三角形の 2 つの面。"
    ],
    "related": [
      "area-of-the-base",
      "prism",
      "pyramid",
      "lateral-face",
      "base-of-a-power",
      "base-of-a-triangle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7・Grade 8) / IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "Grade 8 5.13 The Volume of a Cylinder・Geometry 5.13 Building a Volume Formula for a Pyramid・Grade 7 7.12 Volume of Right Prisms"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "base of the prism | base of a prism | base of the cylinder | base of a cylinder | base of the pyramid | base of a pyramid | base of the cone | base of a cone": 5
      },
      "written": {
        "base of the prism | base of a prism | base of the cylinder | base of a cylinder | base of the pyramid | base of a pyramid | base of the cone | base of a cone": 4
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは IM の呼び方 base of the prism | base of a prism | base of the cylinder | base of a cylinder | base of the pyramid | base of a pyramid | base of the cone | base of a cone（Grade 8 5.13 The Volume of a Cylinder・Geometry 5.13 Building a Volume Formula for a Pyramid・Grade 7 7.12 Volume of Right Prisms・計 10 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "base-of-a-triangle",
    "ja": {
      "term": "底辺",
      "reading": "ていへん",
      "alt": []
    },
    "en": {
      "term": "base",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "三角形の面積や高さを考えるときに基準にする辺。どの辺を底辺にしてもよく、高さはその辺に垂直に測る。",
    "definition_en": "The side of a triangle that the height is measured from; any side can be chosen as the base. The height is the perpendicular distance from the opposite vertex to the line containing the base.",
    "latex": "A = \\frac{1}{2}bh",
    "spoken_en": "A equals one half b h",
    "examples": [
      {
        "en": "You can use any side as the base, as long as the height is perpendicular to it.",
        "ja": "どの辺を底辺にしてもいいよ。高さがその辺に垂直になっていればね。",
        "register": "spoken"
      },
      {
        "en": "A triangle has a base of 12 cm and a height of 5 cm. Find its area.",
        "ja": "底辺が 12 cm、高さが 5 cm の三角形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の base は、累乗の底（エントリ base-of-a-power）や立体の底面（エントリ base-of-a-solid）と同じ語。三角形の底辺かどうかは、the base of the triangle のような前後の語で見分ける。",
      "底辺は下にある辺とは限らない。鈍角三角形で鈍角をはさむ辺を底辺にすると、高さは三角形の外にとり、底辺を延ばした直線に垂直に測る。",
      "平行四辺形・台形の底辺も base で、台形の平行な 2 辺はまとめて bases と言う。二等辺三角形の底辺の両端の角は底角（base angles、エントリ base-angle）。"
    ],
    "related": [
      "height",
      "area-of-a-triangle",
      "base-angle",
      "base-of-a-power",
      "base-of-a-solid"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "base of the triangle | base of a triangle | base times height | base and height | base and the height": 80
      },
      "written": {
        "base of the triangle | base of a triangle | base times height | base and height | base and the height": 28
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "base-of-the-natural-logarithm",
    "ja": {
      "term": "自然対数の底",
      "reading": "しぜんたいすうのてい",
      "alt": [
        "e",
        "ネイピア数"
      ]
    },
    "en": {
      "term": "the number e",
      "alt": [
        "base of the natural logarithm"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "lim_{h→0} (1 + h)^(1/h) で定まる無理数 e ≈ 2.718。自然対数の底。",
    "definition_en": "The irrational number e ≈ 2.718, the limit of (1 + 1/n)ⁿ as n → ∞; the base of the natural logarithm.",
    "latex": "e = \\lim_{n \\to \\infty} \\left(1 + \\frac{1}{n}\\right)^n",
    "spoken_en": "e equals the limit as n approaches infinity of one plus one over n, to the n",
    "examples": [
      {
        "en": "The number e is about 2.718, and e to the x is its own derivative.",
        "ja": "e はおよそ 2.718 で、eˣ は微分しても eˣ のまま。",
        "register": "spoken"
      },
      {
        "en": "Show that lim_{h→0} (1 + h)^(1/h) = e.",
        "ja": "lim_{h→0} (1 + h)^(1/h) = e を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Euler's number とも言う（用例コーパスにはほとんど出てこない）。",
      "OpenStax Algebra and Trigonometry は自然対数を ln x、底を省いた log x を常用対数（底 10）と書く（学習指導要領解説には ln が出てこない）。"
    ],
    "related": [
      "derivative-of-the-exponential-function",
      "derivative-of-the-logarithm",
      "properties-of-logarithms"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ネイピア数",
        "en": "E (mathematical constant)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "ln が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the number e": 39
      },
      "written": {
        "the number e": 27
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "basic-properties-of-probability",
    "ja": {
      "term": "確率の基本性質",
      "reading": "かくりつのきほんせいしつ",
      "alt": []
    },
    "en": {
      "term": "rules of probability",
      "alt": [
        "probability rules"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "AP Statistics の CED（Unit 2 の概要・topic 2.8・2.10）は、確率を求める規則全体を the rules of probability と呼ぶ。OpenStax Introductory Statistics の節 Two Basic Rules of Probability は乗法定理（multiplication rule）と加法定理（addition rule）の 2 つを指す。学習指導要領解説（数学A）が確率の性質や法則として挙げるのは 0 ≦ P(A) ≦ 1、P(∅) = 0・P(U) = 1、互いに排反な事象の P(A ∪ B) = P(A) + P(B)、余事象の確率で、乗法定理は入らない。",
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "AP Statistics",
        "Integrated Math 2",
        "Intro Statistics"
      ]
    },
    "definition_ja": "確率がもつ基本的な性質。0 ≤ P(A) ≤ 1、全事象の確率は 1、互いに排反な事象の和事象の確率は各確率の和、など。",
    "definition_en": "The basic properties every probability has: 0 ≤ P(A) ≤ 1, P(S) = 1 for the whole sample space S, and the probability of a union of mutually exclusive events is the sum of their probabilities.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Any probability has to be between zero and one; that's one of the basic rules.",
        "ja": "確率は必ず 0 と 1 の間にある。それが基本の性質の 1 つ。",
        "register": "spoken"
      },
      {
        "en": "If A and B are mutually exclusive, use the rules of probability to find P(A or B).",
        "ja": "A と B が互いに排反のとき、確率の基本性質を使って P(A または B) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "確率の公理（日本語版 Wikipedia「確率の公理」、英語版 Wikipedia「Probability axioms」）は、P(A) ≥ 0、全事象の確率が 1、互いに排反な事象の和の確率は確率の和、の 3 つ。P(A) ≤ 1 や P(∅) = 0 はそこから導かれる性質で、公理そのものではない。"
    ],
    "related": [
      "sample-space",
      "general-addition-rule",
      "general-multiplication-rule",
      "probability"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.3 Two Basic Rules of Probability"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Unit 2 の概要・Topic 2.8・2.10（rules of probability）、Topic 2.4（the sample space、between 0 and 1）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A (2) 場合の数と確率: 確率の性質や法則（0≦P(A)≦1、P(φ)=0・P(U)=1、排反な事象の和、余事象の確率）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「確率の公理」",
        "url": "https://ja.wikipedia.org/wiki/%E7%A2%BA%E7%8E%87%E3%81%AE%E5%85%AC%E7%90%86",
        "note": "公理は P(A) ≥ 0・全事象の確率 1・排反な事象の和"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Probability_axioms",
        "note": "記事「Probability axioms」（First axiom: non-negativity、節 Elementary consequences）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学A: 確率の基本性質"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "probability rules": 1
      },
      "written": {
        "rules of probability": 3,
        "probability rules": 4
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは AP Statistics の CED の呼び方 rules of probability（topic 2.8・topic 2.10）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "basic-variable",
    "ja": {
      "term": "基本変数",
      "reading": "きほんへんすう",
      "alt": [
        "主変数"
      ]
    },
    "en": {
      "term": "pivot variable",
      "alt": [
        "basic variable",
        "leading variable"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "連立一次方程式を行階段形にしたとき、ピボットのある列に対応する変数。自由変数を使って表される。",
    "definition_en": "In a row-reduced linear system, a variable whose column contains a pivot; it is expressed in terms of the free variables.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "x1 and x2 are pivot variables; x3 is free.",
        "ja": "x1 と x2 が基本変数で、x3 は自由変数。",
        "register": "spoken"
      },
      {
        "en": "Express the leading variables in terms of the parameters.",
        "ja": "基本変数をパラメータで表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "教科書により呼び方が違う: MIT 18.06 は pivot variable、Nicholson は leading variable。basic variable は用例コーパスに出てこない（線形計画法の基底変数も basic variable と言う）。"
    ],
    "related": [
      "free-variable",
      "pivot",
      "reduced-row-echelon-form"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "1.2 Gaussian Elimination（leading variable）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "pivot variable": 29
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "basis",
    "ja": {
      "term": "基底",
      "reading": "きてい",
      "alt": []
    },
    "en": {
      "term": "basis",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "ベクトル空間を張る、1 次独立なベクトルの組。空間のどのベクトルも、それらの 1 次結合でただ 1 通りに表せる。",
    "definition_en": "A linearly independent set of vectors that spans a vector space; every vector in the space is a unique linear combination of them.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A basis has to do two jobs: span the space and be independent.",
        "ja": "基底には 2 つの役目がある。空間を張ることと、1 次独立であることだ。",
        "register": "spoken"
      },
      {
        "en": "Find a basis for the column space of A.",
        "ja": "A の列空間の基底を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形は bases（ベイシーズ）。base（底）の複数形 bases と同じつづり。",
      "台帳の Wikipedia（基底 (位相空間論)）は位相の開基で別の概念。"
    ],
    "related": [
      "span",
      "linearly-independent",
      "dimension",
      "vector-space",
      "coordinate-vector"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.2 Independence and Dimension"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "basis for … space | basis vector | orthonormal basis | standard basis | basis for the column | basis for the null | basis for r": 273
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "bayes-theorem",
    "ja": {
      "term": "ベイズの定理",
      "reading": "べいずのていり",
      "alt": []
    },
    "en": {
      "term": "Bayes' theorem",
      "alt": [
        "Bayes' rule"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "結果 B が起きたときに原因 A が起きていた確率を、P(A | B) = P(B | A) P(A) / P(B) で求める定理。",
    "definition_en": "The theorem P(A | B) = P(B | A) P(A) / P(B), giving the probability of a cause A once the result B is known.",
    "latex": "P(A \\mid B) = \\frac{P(B \\mid A)\\,P(A)}{P(B)}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Bayes' theorem lets you flip the condition: from B given A to A given B.",
        "ja": "ベイズの定理を使うと、条件をひっくり返せる。A のもとでの B から、B のもとでの A へ。",
        "register": "spoken"
      },
      {
        "en": "A test is 95% accurate and 1% of people have the disease. Use Bayes' theorem to find the probability that a person who tests positive has the disease.",
        "ja": "ある検査の精度は 95%、病気の人は全体の 1% である。ベイズの定理を用いて、陽性と判定された人が実際に病気である確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは Bayes' rule がわずかにあるだけ（3Blue1Brown・MIT の講義ノート）。Bayes' theorem ／ Bayes theorem は出てこない。AP Statistics の CED（2026 年版）にも名前は出てこない。"
    ],
    "related": [
      "general-multiplication-rule",
      "tree-diagram"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ベイズの定理",
        "en": "Bayes' theorem"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "Bayes' rule": 1
      },
      "written": {
        "Bayes' rule": 2
      },
      "sources": [
        "mit-notes",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "be-circumscribed-about",
    "ja": {
      "term": "外接する",
      "reading": "がいせつする",
      "alt": []
    },
    "en": {
      "term": "circumscribed about",
      "alt": [],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。IM Geometry は 7.4 で circumscribed about を 1 回使い、用語集は a polygon is circumscribed by a circle と内側の図形を主語にする。日本語の「外接する」は、2 円が互いの外側にあって 1 点を共有する場合にも使う（日本語版 Wikipedia「円 (数学)」の節「位置関係」）が、IM Geometry の用語集は circumscribed を多角形と円の関係で定義し、2 円が外側で接する場合は CK-12 Geometry（6.2 Identify Circle Components）が externally tangent と書く。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "図形が別の図形の外側にあって、内側の図形の各頂点を周上に通る（円が多角形に外接する）か、内側の円に各辺が接する（多角形が円に外接する）こと。",
    "definition_en": "To lie outside another figure and pass through all of its vertices (a circle circumscribed about a polygon), or to have every side tangent to it (a polygon circumscribed about a circle).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The circle circumscribed about the triangle goes through all three vertices, and its center is where the perpendicular bisectors meet.",
        "ja": "三角形に外接する円は 3 つの頂点すべてを通り、その中心は垂直二等分線の交点になる。",
        "register": "spoken"
      },
      {
        "en": "A square is circumscribed about a circle, so each side of the square touches the circle once.",
        "ja": "正方形が円に外接しているので、正方形の各辺は円に 1 点で接する。",
        "register": "spoken"
      },
      {
        "en": "A circle is circumscribed about a square with side length 4. Find the radius of the circle.",
        "ja": "1 辺の長さが 4 の正方形に円が外接している。円の半径を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "向きは日本語と同じ。A is circumscribed about B は「A が B に外接する」で、A が外側。内側の図形を主語にして B is inscribed in A と言い換えられる（英語版 Wikipedia「Inscribed figure」は 2 つの言い方を同じ意味とする）。",
      "IM Geometry の用語集は、a polygon is circumscribed by a circle（多角形が円に内接する）と by を使って内側の図形を主語にする。about と by で主語が入れ替わるので注意。",
      "CK-12 Geometry（4.22 Concurrence and Constructions）は三角形の外接円を、3 つの頂点すべてを通る円として説明する。",
      "「2 円が外接する」は externally tangent（CK-12 Geometry 6.2 Identify Circle Components）。多角形と円の関係の circumscribed と取り違えない。"
    ],
    "related": [
      "be-inscribed-in",
      "circumscribed-circle",
      "circumcenter",
      "relative-positions-of-two-circles"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.4 Quadrilaterals in Circles（circumscribed about）・glossary（circumscribed by）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.22 Concurrence and Constructions・6.2 Identify Circle Components（externally tangent）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 正弦定理: 三角形に外接する円・数学A 図形の性質: 与えられた四角形に外接する正方形の作図（level.jp 数A）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「円 (数学)」",
        "url": "https://ja.wikipedia.org/wiki/%E5%86%86_(%E6%95%B0%E5%AD%A6)",
        "note": "節「位置関係」: 2 円が 1 点だけを共有し、互いの外側にある場合も「外接する」と呼ぶ（mapping_note）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Inscribed_figure",
        "note": "記事「Inscribed figure」の冒頭（figure F is inscribed in figure G と figure G is circumscribed about figure F を同じ意味とする。pitfalls[0]・definition_en の 2 つ目の場合。数学カテゴリから 4 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-10-09"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-10-09"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "be-inscribed-in",
    "ja": {
      "term": "内接する",
      "reading": "ないせつする",
      "alt": []
    },
    "en": {
      "term": "inscribed in",
      "alt": [],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「内接する」は、2 円の一方が他方の内側にあって 1 点を共有する場合にも使う（日本語版 Wikipedia「円 (数学)」、共通テスト 令和3年度 第1日程 数学Ⅰ・数学Ａ 第5問の「外接円 O に内接する円」）が、IM Geometry の用語集は inscribed を多角形が円に・円が多角形に内接する場合で定義し、2 円が内側で接する場合は英語版 Wikipedia「Tangent」（節 Tangent circles）が internally tangent と呼ぶ。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "図形が別の図形の内側にあって、各頂点が外側の図形の周上にある（多角形が円に内接する）か、各辺に接する（円が多角形に内接する）こと。",
    "definition_en": "To lie inside another figure with every vertex on its boundary (a polygon inscribed in a circle), or with every side of it tangent to the figure inside (a circle inscribed in a polygon).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A circle inscribed in a triangle touches all three sides, and its center is where the angle bisectors meet.",
        "ja": "三角形に内接する円は 3 辺すべてに接し、その中心は角の二等分線の交点になる。",
        "register": "spoken"
      },
      {
        "en": "If a triangle is inscribed in a circle and one side is a diameter, the angle opposite that side is a right angle.",
        "ja": "三角形が円に内接し、1 辺が直径なら、その辺に向かい合う角は直角になる。",
        "register": "spoken"
      },
      {
        "en": "A square is inscribed in a circle of radius 5. Find the area of the square.",
        "ja": "半径 5 の円に正方形が内接している。正方形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "向きは日本語と同じ。A is inscribed in B は「A が B に内接する」で、A が内側。",
      "「2 円が内接する」は、英語版 Wikipedia「Tangent」（節 Tangent circles）の internally tangent。用例コーパスと参照の inscribed in は、多角形と円・楕円の関係（と円周角）にだけ出てくる。",
      "用例コーパスでは inscribed in は話し言葉・書き言葉とも少ない（書き言葉は OpenStax Calculus・OpenStax Algebra and Trigonometry）。IM Geometry は glossary の見出し（inscribed）にしてレッスンで 26 件、CK-12 Geometry は 31 件（6.15 Inscribed Quadrilaterals in Circles ほか）使う。"
    ],
    "related": [
      "inscribed-circle",
      "incenter",
      "be-circumscribed-about",
      "cyclic-quadrilateral",
      "circumscribed-circle",
      "inscribed-polygon"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "4.7 Applied Optimization Problems（A rectangle is to be inscribed in the ellipse。pitfalls[1]）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary・Geometry 1.8 Using Technology for Constructions・Geometry 4.10 Solving Problems with Trigonometry"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.15 Inscribed Quadrilaterals in Circles・5.23 Construct Regular Polygons・6.18 Tangent Lines"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Tangent",
        "note": "記事「Tangent」の節 Tangent circles（internally tangent。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「円 (数学)」",
        "url": "https://ja.wikipedia.org/wiki/%E5%86%86_(%E6%95%B0%E5%AD%A6)",
        "note": "節「位置関係」: 2 円が 1 点だけを共有し、一方が他方の内側にある場合も「内接する」と呼ぶ（mapping_note）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和3年度 大学入学共通テスト 第1日程 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20220601/https://www.dnc.ac.jp/albums/abm.php?d=88&f=abm00001267.pdf&n=R3_%E6%95%B0%E5%AD%A6%E2%85%A0%E3%83%BB%E6%95%B0%E5%AD%A6%EF%BC%A1%EF%BC%88%EF%BC%91%E6%9C%8816%E6%97%A5%E3%83%BB17%E6%97%A5%EF%BC%89%E3%81%AE%E5%95%8F%E9%A1%8C.pdf",
        "note": "第5問: 外接円 O に内接する円の中心を P（mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "inscribed in": 4
      },
      "written": {
        "inscribed in": 8
      },
      "sources": [
        "khan-middle",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-10-09"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは CK-12 の呼び方 inscribed in（CK-12 Geometry 6.15 Inscribed Quadrilaterals in Circles・CK-12 Geometry 5.23 Construct Regular Polygons・CK-12 Geometry 6.18 Tangent Lines・計 31 件）。register は主張しない。",
        "raised": "2026-10-09"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "be-skewed",
    "ja": {
      "term": "偏る",
      "reading": "かたよる",
      "alt": [
        "右に歪んだ",
        "分布が偏る"
      ]
    },
    "en": {
      "term": "skewed",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「偏る」は分布の形にも標本の選び方にも使う。英語は分布の形なら skewed（skewed right ／ skewed left）、標本の選び方なら biased と言い分ける。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "AP Statistics",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "分布の形が左右対称でなく、一方に長い裾を引いていること。裾が右に長いものを右に歪んだ（skewed right）分布という。",
    "definition_en": "Of a distribution: not symmetric, with a long tail on one side; skewed right when the long tail is on the right.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The income data are skewed right, so the mean is pulled up above the median.",
        "ja": "収入のデータは右に歪んでいるので、平均値が中央値より大きくなる。",
        "register": "spoken"
      },
      {
        "en": "Is the distribution symmetric, skewed left, or skewed right?",
        "ja": "分布は左右対称か、左に歪んでいるか、右に歪んでいるか。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "skewed right",
        "ja": "右に歪んだ"
      },
      {
        "en": "skewed to the left",
        "ja": "左に歪んだ"
      }
    ],
    "pitfalls": [
      "skewed right は裾（tail）が右に長い分布で、山は左にある。山の位置で左右を決めない。",
      "空間の skew lines（ねじれの位置にある 2 直線）と同じ語。用例コーパスの書き言葉でも、文脈を見ると一部がこの意味だった。"
    ],
    "related": [
      "distribution",
      "median",
      "arithmetic-mean"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "skewed": 111
      },
      "written": {
        "skewed": 58
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "be-tangent-to",
    "ja": {
      "term": "接する",
      "reading": "せっする",
      "alt": []
    },
    "en": {
      "term": "be tangent to",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直線が円とただ 1 つの点だけを共有するとき、直線は円に接するという。2 つの円が 1 点だけを共有するときにも使う。",
    "definition_en": "Describes a line that touches a circle at exactly one point. Two circles that touch at exactly one point are also tangent to each other.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "See how the line just touches the circle here and doesn't cut through it? It's tangent to the circle at P.",
        "ja": "直線がここで円にちょうどふれていて、円の中を通っていないでしょ？ この直線は点 P で円に接しているんだ。",
        "register": "spoken"
      },
      {
        "en": "Construct a line that is tangent to circle O at point A.",
        "ja": "点 A で円 O に接する直線を作図せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "tangent は形容詞として使う。「接する」は is tangent to で、接する相手は to、接点は at で示す（The line is tangent to the circle at P.）。",
      "名詞の tangent は接線（tangent line、エントリ tangent-line）のことも、三角比の tan（エントリ tangent）のこともある。接点は point of tangency（エントリ point-of-tangency）。",
      "円と 2 点で交わる直線は tangent ではない（接線は円と 1 点だけを共有する）。「交わる」は intersect（エントリ intersect）。"
    ],
    "related": [
      "tangent-line",
      "point-of-tangency",
      "tangent",
      "circle",
      "position-of-a-line-relative-to-a-circle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "tangent to": 73
      },
      "written": {
        "tangent to": 107
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "bearing",
    "ja": {
      "term": "方位角",
      "reading": "ほういかく",
      "alt": [
        "方位"
      ]
    },
    "en": {
      "term": "bearing",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "ある地点から見た方向を、北を基準に測った角度で表したもの。北から時計回りに測る表し方（120° など）と、N 15° E のように南北から東西へ測る表し方がある。",
    "definition_en": "A direction given as an angle measured from north, either clockwise from north (a bearing of 120°) or from north or south toward east or west (N 15° E).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A bearing of 120 degrees means you face north and turn 120 degrees clockwise.",
        "ja": "方位 120° とは、北を向いてから時計回りに 120° 回った方向のことだ。",
        "register": "spoken"
      },
      {
        "en": "A ship leaves port on a bearing of 60° and travels 12 km. How far east of the port is the ship?",
        "ja": "船が港から北から東へ 60° の方向に 12 km 進んだ。船は港から東へ何 km のところにいるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "方位のうち東西南北のような向きの名前（compass directions）は bearing ではない。bearing は北から測った角度で方向を表したもの。",
      "N 15° E は北から東へ 15° の方向。OpenStax Calculus Volume 3 は a bearing of N 15 E とも書き、OpenStax は 52° east of north のように向きを書くこともある。"
    ],
    "related": [
      "angle-of-elevation",
      "angle-of-depression",
      "clockwise",
      "law-of-cosines",
      "law-of-sines"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「方位角」",
        "url": "https://ja.wikipedia.org/wiki/%E6%96%B9%E4%BD%8D%E8%A7%92",
        "note": "北から測った角で方向を表す"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "bearing of": 8
      },
      "written": {
        "bearing of": 10
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "below-the-x-axis",
    "ja": {
      "term": "x 軸より下",
      "reading": "えっくすじくよりした",
      "alt": []
    },
    "en": {
      "term": "below the x-axis",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "グラフが x 軸の下側にあること。その区間では関数の値が負になる。",
    "definition_en": "Lying under the x-axis, where the function's values are negative.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This piece is below the x-axis, so its integral comes out negative.",
        "ja": "この部分は x 軸より下にあるから、積分は負になる。",
        "register": "spoken"
      },
      {
        "en": "On (2, 3), the graph lies below the x-axis.",
        "ja": "区間 (2, 3) では、グラフは x 軸より下にある。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "反対は above the x-axis。「x 軸より下の部分」は the part below the x-axis。"
    ],
    "related": [
      "area-under-the-curve"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "below the x-axis": 47
      },
      "written": {
        "below the x-axis": 25
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "bezouts-identity",
    "ja": {
      "term": "ベズーの等式",
      "reading": "べずーのとうしき",
      "alt": []
    },
    "en": {
      "term": "Bézout's identity",
      "alt": [
        "Bezout's identity"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Bézout's identity ／ Bezout's identity は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "definition_ja": "少なくとも一方が 0 でない整数 a, b に対し、ax + by = gcd(a, b) を満たす整数 x, y が存在するという定理。特に a, b が互いに素なら、ax + by = 1 は整数解をもつ。",
    "definition_en": "For integers a and b, not both zero, there exist integers x and y with ax + by = gcd(a, b). In particular, if a and b are relatively prime, then ax + by = 1 has an integer solution.",
    "latex": "ax + by = \\gcd(a, b)",
    "spoken_en": "a x plus b y equals the G C D of a and b",
    "examples": [
      {
        "en": "Bézout's identity says we can write the gcd as a combination of a and b, and the extended Euclidean algorithm actually finds the x and y.",
        "ja": "ベズーの等式によれば、最大公約数は a と b を使った ax + by の形で書ける。実際の x, y は拡張ユークリッド互除法で求まる。",
        "register": "spoken"
      },
      {
        "en": "Use Bézout's identity to show that 7x + 12y = 1 has an integer solution, and find one.",
        "ja": "7x + 12y = 1 が整数解をもつことをベズーの等式を用いて示し、解を 1 組求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数A では、一次不定方程式の単元で「a, b が互いに素なら ax + by = 1 は整数解をもつ」という形で出てくる。",
      "Bézout's theorem は代数曲線の交点の個数についての別の定理。identity と theorem を取り違えない。"
    ],
    "related": [
      "linear-diophantine-equation",
      "euclidean-algorithm",
      "extended-euclidean-algorithm",
      "greatest-common-divisor",
      "relatively-prime"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ベズーの等式",
        "en": "Bézout's identity"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 Bézout's identity（英語版「Bézout's identity」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "biconditional",
    "ja": {
      "term": "双条件文",
      "reading": "そうじょうけんぶん",
      "alt": [
        "同値な命題"
      ]
    },
    "en": {
      "term": "biconditional statement",
      "alt": [
        "biconditional"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof",
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "「p ならば q」と「q ならば p」を合わせた「p ⇔ q」の形の命題。p と q の真偽が一致するときに真になる。",
    "definition_en": "A statement of the form \"p if and only if q,\" which combines \"if p, then q\" and \"if q, then p.\" It is true exactly when p and q have the same truth value.",
    "latex": "p \\Leftrightarrow q",
    "spoken_en": "p if and only if q",
    "examples": [
      {
        "en": "The statement and its converse are both true, so we can combine them into a biconditional with \"if and only if.\"",
        "ja": "命題もその逆も真なので、「p ⇔ q」の形の 1 つの命題にまとめられる。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the biconditional statement \"x² = 4 if and only if x = 2\" is true. If it is false, give a counterexample.",
        "ja": "命題「x² = 4 ⇔ x = 2」の真偽を調べよ。偽ならば反例を挙げよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数I では「双条件文」という名前を使わず、「p ⇔ q」「p は q であるための必要十分条件」「p と q は同値」と言う。",
      "書き言葉では if and only if を iff と略すことがある。",
      "用例コーパスで biconditional は話し言葉にわずか（Professor Leonard）で、書き言葉には出てこない。Levin の本では 5 件、CK-12 Geometry では 8 件（2.12 Converse Inverse and Contrapositive Statements ほか）。"
    ],
    "related": [
      "conditional-statement",
      "necessary-and-sufficient-condition",
      "equivalent",
      "contrapositive"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1 Mathematical Statements, 1.2 Implications"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.12 Converse Inverse and Contrapositive Statements・6.18 Tangent Lines"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "biconditional statement": 1
      },
      "sources": [
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 biconditional statement（CK-12 Geometry 2.12 Converse Inverse and Contrapositive Statements・CK-12 Geometry 6.18 Tangent Lines・計 6 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "bijection",
    "ja": {
      "term": "全単射",
      "reading": "ぜんたんしゃ",
      "alt": []
    },
    "en": {
      "term": "bijection",
      "alt": [
        "one-to-one correspondence",
        "bijective function"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "set-theory",
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "単射（1 対 1）かつ全射（上への写像）である写像のこと。行き先の集合のどの要素にも、元の集合のちょうど 1 つの要素が対応する。",
    "definition_en": "A function that is both one-to-one (injective) and onto (surjective). Every element of the target set is matched with exactly one element of the domain.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To show it's a bijection, you need two things: it's one-to-one, and it's onto.",
        "ja": "全単射であることを示すには 2 つ必要だ。単射であることと、全射であることだ。",
        "register": "spoken"
      },
      {
        "en": "Show that f: ℝ → ℝ, f(x) = 2x + 1, is a bijection, and find its inverse.",
        "ja": "f: ℝ → ℝ, f(x) = 2x + 1 が全単射であることを示し、逆関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "one-to-one だけなら単射（injective）の意味。one-to-one correspondence（1 対 1 の対応）は全単射の意味で、別物である。",
      "全単射であることと、逆関数（inverse function）が存在することは同値。",
      "用例コーパスでは bijection は話し言葉がすべて MIT OCW、書き言葉がすべて MIT の講義ノート。one-to-one correspondence は話・書ともほとんど出てこない（3Blue1Brown・OpenStax Algebra and Trigonometry）。"
    ],
    "related": [
      "one-to-one",
      "inverse-function",
      "onto",
      "countable"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "全単射",
        "en": "Bijection"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "bijection": 39,
        "bijective function": 2,
        "one-to-one correspondence": 1
      },
      "written": {
        "one-to-one correspondence": 1,
        "bijection": 54
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "binary",
    "ja": {
      "term": "2 進法",
      "reading": "にしんほう",
      "alt": [
        "二進法"
      ]
    },
    "en": {
      "term": "binary",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "0 と 1 の 2 種類の数字だけを使い、位が 1 つ上がるごとに 2 倍になるように数を表す方法。",
    "definition_en": "The base-2 way of writing numbers, using only the digits 0 and 1, where each place is worth twice the place to its right.",
    "latex": "1101_{(2)} = 2^3 + 2^2 + 1 = 13",
    "spoken_en": "one one zero one in binary equals thirteen",
    "examples": [
      {
        "en": "In binary, 13 is 1101, and you read it one, one, zero, one.",
        "ja": "2 進法では 13 は 1101 で、「いち いち ぜろ いち」と読む。",
        "register": "spoken"
      },
      {
        "en": "Write 45 as a binary number.",
        "ja": "45 を 2 進法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "1101 は「千百一」ではなく、桁ごとに one, one, zero, one と読む。",
      "英語の base 2 は log base 2（2 を底とする対数）の意味でも使う。「2 進法で」は in binary か in base 2 と言うと紛れない。"
    ],
    "related": [
      "base-n",
      "hexadecimal",
      "numeral-system",
      "digit",
      "place",
      "decimal-system"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences（練習問題: binary representation、binary expansion）"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "二進法",
        "en": "Binary number"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 数学と人間の活動: ユークリッドの互除法や二進法"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.1 Pascal’s Arithmetical Triangle（bit strings、binary digit）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "in binary | binary number | binary representation | in base 2 | in base two": 35
      },
      "written": {
        "in binary | binary number | binary representation | in base 2 | in base two": 6
      },
      "sources": [
        "khan-middle",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "binomial-coefficient",
    "ja": {
      "term": "二項係数",
      "reading": "にこうけいすう",
      "alt": []
    },
    "en": {
      "term": "binomial coefficient",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A",
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "(a + b)ⁿ の展開に現れる係数 ₙCₖ。n 個から k 個を選ぶ組合せの数に等しい。",
    "definition_en": "A coefficient C(n, k) in the expansion of (a + b)ⁿ; it equals the number of ways to choose k objects from n.",
    "latex": "\\binom{n}{k} = \\frac{n!}{k!\\,(n-k)!}",
    "spoken_en": "n choose k",
    "examples": [
      {
        "en": "That binomial coefficient, 6 choose 2, is 15.",
        "ja": "この二項係数 ₆C₂ は 15 だ。",
        "register": "spoken"
      },
      {
        "en": "The binomial coefficients in the expansion of (a + b)⁴ are 1, 4, 6, 4, and 1.",
        "ja": "(a + b)⁴ の展開式の二項係数は 1, 4, 6, 4, 1 である。",
        "register": "written"
      },
      {
        "en": "Evaluate the binomial coefficient (9 choose 3).",
        "ja": "二項係数 ₉C₃ の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry（13.6 Binomial Theorem）は (n r) を binomial coefficient と呼び、C(n, r) と同じ数だと書く。",
      "記号 (n k) は縦に並べて書き、分数の線は引かない。n/k と書くと別物になる。読みは n choose k。"
    ],
    "related": [
      "combination",
      "binomial-theorem",
      "pascals-triangle",
      "choose"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二項係数",
        "en": "Binomial coefficient"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.6 Binomial Theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "binomial coefficient": 10
      },
      "written": {
        "binomial coefficient": 49
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "binomial-distribution",
    "ja": {
      "term": "二項分布",
      "reading": "にこうぶんぷ",
      "alt": []
    },
    "en": {
      "term": "binomial distribution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "成功の確率が p の試行を独立に n 回繰り返したときの成功の回数 X の確率分布。B(n, p) と書く。",
    "definition_en": "The probability distribution of the number of successes X in n independent trials, each with success probability p; written B(n, p).",
    "latex": "P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k}",
    "spoken_en": "the probability that X equals k is n choose k times p to the k times one minus p to the n minus k",
    "examples": [
      {
        "en": "A fixed number of independent trials, each a success or failure with the same p, so this is binomial.",
        "ja": "回数が決まった独立な試行で、各回が成功か失敗、p も同じなので、これは二項分布。",
        "register": "spoken"
      },
      {
        "en": "X has a binomial distribution with n = 20 and p = 0.3. Find P(X = 5).",
        "ja": "X は n = 20, p = 0.3 の二項分布に従う。P(X = 5) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "平均は np、分散は np(1 − p)。",
      "OpenStax Introductory Statistics（4.3）は X ~ B(n, p) と書く（学習指導要領解説（数学B）の B(n, p) と同じ文字）。"
    ],
    "related": [
      "random-variable",
      "normal-approximation-to-the-binomial",
      "expected-value",
      "probability-distribution",
      "binomial-probability"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二項分布",
        "en": "Binomial distribution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "4.3 Binomial Distribution（X ~ B(n, p)）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B (2) 統計的な推測（二項分布 B(n, p)）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.10（binomial distribution）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "binomial distribution": 25
      },
      "written": {
        "binomial distribution": 66
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "binomial-identities",
    "ja": {
      "term": "二項係数の恒等式",
      "reading": "にこうけいすうのこうとうしき",
      "alt": []
    },
    "en": {
      "term": "binomial identity",
      "alt": [
        "binomial coefficient identity"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（binomial identity ／ binomial coefficient identity は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "二項係数 nCk の間に成り立つ等式のこと。nCk = nCn−k や nCk = n−1Ck−1 + n−1Ck などがあり、式の変形や数え上げの議論で示す。",
    "definition_en": "An equation that holds between binomial coefficients, such as C(n, k) = C(n, n − k) or C(n, k) = C(n − 1, k − 1) + C(n − 1, k). It can be proved with algebra or by counting the same set in two ways.",
    "latex": "\\binom{n}{k} = \\binom{n-1}{k-1} + \\binom{n-1}{k}",
    "spoken_en": "n choose k equals n minus one choose k minus one plus n minus one choose k",
    "examples": [
      {
        "en": "To prove this binomial identity, count the same committees in two different ways.",
        "ja": "この二項係数の恒等式を示すには、同じ委員会の選び方を 2 通りに数えればよい。",
        "register": "spoken"
      },
      {
        "en": "Prove the binomial identity C(n, 0) + C(n, 1) + ⋯ + C(n, n) = 2ⁿ by counting the subsets of an n-element set.",
        "ja": "n 個の要素からなる集合の部分集合の個数を数えることで、等式 nC0 + nC1 + ⋯ + nCn = 2ⁿ を証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の教科書は nCk と書くが、英語の本では縦に並べた (n k) や C(n, k) と書き、n choose k と読む。",
      "個々の等式には名前が付いたものがある。nCk = n−1Ck−1 + n−1Ck は Pascal's identity（パスカルの三角形の作り方にあたる）。",
      "binomial identity は Levin の本の 3.6 Combinatorial Proofs で 18 件。CED・OpenStax・IM・CK-12 には出てこない。"
    ],
    "related": [
      "binomial-coefficient",
      "binomial-theorem",
      "pascals-triangle",
      "combination",
      "subset"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.6 Combinatorial Proofs"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Levin の呼び方 binomial identity（3.6 Combinatorial Proofs）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "binomial-probability",
    "ja": {
      "term": "反復試行の確率",
      "reading": "はんぷくしこうのかくりつ",
      "alt": []
    },
    "en": {
      "term": "binomial probability",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics",
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "1 回の確率が p の試行を独立に n 回繰り返すとき、その事象がちょうど k 回起こる確率。ₙCₖ pᵏ(1 − p)ⁿ⁻ᵏ で求める。",
    "definition_en": "The probability that an event with probability p on each trial happens exactly k times in n independent trials, given by C(n, k) pᵏ(1 − p)ⁿ⁻ᵏ.",
    "latex": "P(X = k) = \\binom{n}{k} p^k (1-p)^{n-k}",
    "spoken_en": "the probability that X equals k is n choose k times p to the k times one minus p to the n minus k",
    "examples": [
      {
        "en": "To get exactly three heads in five flips, the binomial probability is five choose three times one half to the fifth.",
        "ja": "5 回投げてちょうど 3 回表が出る確率（反復試行の確率）は ₅C₃ × (1/2)⁵ だ。",
        "register": "spoken"
      },
      {
        "en": "A multiple-choice quiz has 10 questions with 4 choices each. A student guesses on every question. Find the binomial probability that the student gets exactly 4 correct.",
        "ja": "4 択の問題が 10 問ある小テストで、すべて当てずっぽうで答える。ちょうど 4 問正解する確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高等学校学習指導要領〔用語・記号〕（数学A）は nCr と書く。AP Statistics の CED（topic 2.10）は二項係数を縦に並べた (n k) で書き、OpenStax Intermediate Algebra（12.4 Binomial Theorem）はそれを n choose k と読む。",
      "「ちょうど k 回」は exactly k times。「少なくとも 1 回」（at least once）の確率は、余事象（1 回も起こらない）から 1 − (1 − p)ⁿ と求められる。",
      "用例コーパスでは書き言葉がすべて OpenStax Introductory Statistics で、話し言葉は少ない（Professor Leonard・Khan Academy）。"
    ],
    "related": [
      "binomial-distribution",
      "repeated-trials",
      "binomial-coefficient",
      "combination",
      "sampling-with-replacement"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）〔用語・記号〕",
        "note": "数学A 場合の数と確率: nCr"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.10（the binomial probability function、二項係数の縦書き）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.4 Binomial Theorem（n choose r）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "binomial probability !distribution !distributions": 4
      },
      "written": {
        "binomial probability !distribution !distributions": 7
      },
      "sources": [
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは AP Statistics の CED の呼び方 binomial probability !distribution !distributions（topic 2.10）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "binomial-series",
    "ja": {
      "term": "二項級数",
      "reading": "にこうきゅうすう",
      "alt": []
    },
    "en": {
      "term": "binomial series",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "任意の実数 α について (1 + x)^α を表すべき級数。|x| < 1 で収束する。",
    "definition_en": "The power series for (1 + x)^α with any real exponent α, which converges for |x| < 1.",
    "latex": "(1+x)^{\\alpha} = \\sum_{n=0}^{\\infty} \\binom{\\alpha}{n} x^n",
    "spoken_en": "one plus x to the alpha equals the sum from n equals zero to infinity of alpha choose n times x to the n",
    "examples": [
      {
        "en": "When the exponent isn't a whole number, the binomial series never stops.",
        "ja": "指数が自然数でなければ、二項級数は終わらない。",
        "register": "spoken"
      },
      {
        "en": "Use the binomial series to find the Maclaurin series for √(1 + x).",
        "ja": "二項級数を用いて √(1 + x) のマクローリン級数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉には出てこず、書き言葉はすべて OpenStax Calculus。",
      "α が自然数なら有限個の項で終わり、二項定理（binomial theorem）になる。"
    ],
    "related": [
      "taylor-series",
      "maclaurin-series",
      "power-series"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二項級数",
        "en": "Binomial series"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "binomial series": 30
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "binomial-theorem",
    "ja": {
      "term": "二項定理",
      "reading": "にこうていり",
      "alt": []
    },
    "en": {
      "term": "binomial theorem",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "(a + b)ⁿ を展開したときの各項の係数が二項係数 ₙCₖ になることを示す定理。",
    "definition_en": "The theorem that expanding (a + b)ⁿ gives terms whose coefficients are the binomial coefficients C(n, k).",
    "latex": "(a+b)^n = \\sum_{k=0}^{n} \\binom{n}{k} a^{n-k} b^{k}",
    "spoken_en": "a plus b to the n equals the sum from k equals zero to n of n choose k times a to the n minus k times b to the k",
    "examples": [
      {
        "en": "Instead of multiplying out (x + 2) to the fifth by hand, just use the binomial theorem.",
        "ja": "(x + 2)⁵ を手で掛け算していくのではなく、二項定理を使えばいい。",
        "register": "spoken"
      },
      {
        "en": "Use the binomial theorem to find the coefficient of x³ in the expansion of (2x − 1)⁷.",
        "ja": "二項定理を用いて、(2x − 1)⁷ の展開式における x³ の係数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "(2x − 1)ⁿ の係数を求めるときは、2 や −1 の累乗も係数に入る。二項係数 ₙCₖ だけを答えにしない。",
      "一般項 ₙCₖ aⁿ⁻ᵏ bᵏ は k = 0 から数えるので第 k + 1 項。第 10 項なら k = 9。OpenStax Algebra and Trigonometry（13.6）も (r + 1)th term の形で書く。"
    ],
    "related": [
      "binomial-coefficient",
      "pascals-triangle",
      "expansion",
      "general-term",
      "binomial-series"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二項定理",
        "en": "Binomial theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.6 Binomial Theorem（the (r+1)th term）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II (1) いろいろな式: 二項定理（記号 nCr は数学A）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）〔用語・記号〕",
        "note": "数学II いろいろな式: 二項定理"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "binomial theorem": 31
      },
      "written": {
        "binomial theorem": 60
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "binomial",
    "ja": {
      "term": "二項式",
      "reading": "にこうしき",
      "alt": []
    },
    "en": {
      "term": "binomial",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "項が 2 つの多項式。x + 3 や 2a − b のような式。",
    "definition_en": "A polynomial with exactly two terms, such as x + 3 or 2a − b.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "See how x + 3 has two terms? That makes it a binomial.",
        "ja": "x + 3 には項が 2 つあるよね。だからこれは二項式。",
        "register": "spoken"
      },
      {
        "en": "Classify each polynomial as a monomial, binomial, or trinomial: 4x² − 9, 7ab, x² + 2x − 5.",
        "ja": "次の多項式を、単項式・二項式・三項式に分類せよ。4x² − 9、7ab、x² + 2x − 5",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "bi- は「2 つの」の意味。項が 1 つなら monomial（エントリ monomial）、3 つなら trinomial（エントリ trinomial）と、項の数で呼び名が変わる。",
      "二項定理（binomial theorem）や二項分布（binomial distribution）の binomial は「二項の」という形容詞で、それぞれ別の用語（エントリ binomial-theorem、binomial-distribution）。",
      "項の数は同類項をまとめてから数える。3x + 2x は 5x になるので、二項式ではなく単項式。"
    ],
    "related": [
      "monomial",
      "trinomial",
      "polynomial",
      "term",
      "square-of-a-binomial"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二項式",
        "en": "Binomial (polynomial)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "binomial !theorem !distribution !distributions !coefficient !coefficients !expansion !probability !random !experiment !setting !formula !model !squares !square !variable !series": 114
      },
      "written": {
        "binomial !theorem !distribution !distributions !coefficient !coefficients !expansion !probability !random !experiment !setting !formula !model !squares !square !variable !series": 669
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "bipartite-graph",
    "ja": {
      "term": "二部グラフ",
      "reading": "にぶぐらふ",
      "alt": [
        "2部グラフ"
      ]
    },
    "en": {
      "term": "bipartite graph",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "頂点全体を 2 つの組に分けて、どの辺も異なる組の頂点どうしを結ぶようにできるグラフのこと。",
    "definition_en": "A graph whose vertices can be split into two sets so that every edge joins a vertex in one set to a vertex in the other.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A bipartite graph is one you can color with just two colors, so no edge ever joins two vertices of the same color.",
        "ja": "二部グラフは 2 色だけで塗り分けられるグラフで、同じ色の頂点どうしを結ぶ辺が 1 本もない。",
        "register": "spoken"
      },
      {
        "en": "Show that a bipartite graph cannot contain a cycle of odd length.",
        "ja": "二部グラフは長さが奇数の閉路を含まないことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「頂点を 2 色で塗り分けられる」「長さが奇数の閉路をもたない」は、どちらも二部グラフであることと同値。",
      "一方の組の m 個の頂点と他方の組の n 個の頂点がすべて辺で結ばれたものを complete bipartite graph（完全二部グラフ）といい、Kₘ,ₙ と書く。",
      "用例コーパスでは話し言葉は少なく（MIT OCW）、書き言葉はすべて MIT の講義ノート。Levin の本では 37 件（2.7 Matching in Bipartite Graphs 15 ほか）。"
    ],
    "related": [
      "graph-coloring",
      "vertex-graph",
      "graph-network"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "2部グラフ",
        "en": "Bipartite graph"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学特論 (3) 離散グラフ: 完全グラフや二部グラフ、木"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "bipartite graph": 4
      },
      "written": {
        "bipartite graph": 30
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "bisect",
    "ja": {
      "term": "二等分する",
      "reading": "にとうぶんする",
      "alt": []
    },
    "en": {
      "term": "bisect",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "線分や角を、大きさの等しい 2 つの部分に分ける。",
    "definition_en": "To cut a segment or an angle into two equal parts.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The diagonals of a parallelogram bisect each other, so O is the midpoint of both of them.",
        "ja": "平行四辺形の対角線は互いに他を二等分するから、O はどちらの対角線の中点にもなるね。",
        "register": "spoken"
      },
      {
        "en": "Ray OC bisects ∠AOB. If m∠AOB = (2x + 10)° and m∠AOC = (3x − 15)°, find x.",
        "ja": "半直線 OC は ∠AOB を二等分している。∠AOB = (2x + 10)°、∠AOC = (3x − 15)° のとき、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "bisect は動詞で、二等分する線は名詞 bisector で言う。角の二等分線は angle bisector（エントリ angle-bisector）、垂直二等分線は perpendicular bisector（エントリ perpendicular-bisector）。",
      "「互いに他を二等分する」は bisect each other。2 本の線分がそれぞれの中点で交わることと同じ。",
      "線分を二等分する点が中点（midpoint、エントリ midpoint）。「直線 ℓ が線分 AB を二等分する」は Line ℓ bisects segment AB.、「M は AB の中点」は M is the midpoint of AB. と書く。"
    ],
    "related": [
      "angle-bisector",
      "perpendicular-bisector",
      "midpoint",
      "construct",
      "segment"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.11 Congruent Angles and Angle Bisectors"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "bisect": 23
      },
      "written": {
        "bisect": 2
      },
      "sources": [
        "mit-18.02",
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "blinding",
    "ja": {
      "term": "盲検",
      "reading": "もうけん",
      "alt": [
        "盲検化"
      ]
    },
    "en": {
      "term": "blinding",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "実験の被験者（や評価する人）に、だれがどの処理を受けているかを知らせないようにすること。両方に知らせないものを二重盲検という。",
    "definition_en": "Keeping subjects (and possibly evaluators) from knowing who receives which treatment; keeping both from knowing is double-blinding.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's double-blind: neither the patients nor the doctors know who got the real drug.",
        "ja": "二重盲検なので、患者も医師もだれが本物の薬をもらったか知らない。",
        "register": "spoken"
      },
      {
        "en": "Explain how blinding could be used in this experiment.",
        "ja": "この実験で盲検をどう使えるか説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "double-blind",
        "ja": "二重盲検の"
      }
    ],
    "pitfalls": [
      "用例コーパスの blinding には語形変化で blind（見えない）も入る。"
    ],
    "related": [
      "placebo",
      "control-group",
      "treatment"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "blinding": 12,
        "double-blind": 3
      },
      "written": {
        "blinding": 15,
        "double-blind": 2
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "block-matrix",
    "ja": {
      "term": "ブロック行列",
      "reading": "ぶろっくぎょうれつ",
      "alt": [
        "区分行列"
      ]
    },
    "en": {
      "term": "block matrix",
      "alt": [
        "partitioned matrix"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（block matrix ／ partitioned matrix は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列を縦横に区切り、各部分（ブロック）を 1 つの成分のように扱って書いた行列。",
    "definition_en": "A matrix divided into smaller submatrices (blocks), each treated like a single entry.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Think of this as a two-by-two block matrix, and multiply the blocks like numbers.",
        "ja": "これを 2 × 2 のブロック行列とみて、ブロックを数のように掛けよう。",
        "register": "spoken"
      },
      {
        "en": "Compute the product using block matrix multiplication.",
        "ja": "ブロック行列の積を使って積を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ブロックどうしの掛け算は順番を入れ替えられない（行列の積なので）。"
    ],
    "related": [
      "matrix",
      "matrix-multiplication"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "区分行列",
        "en": "Block matrix"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "2.4 Matrix Inverses"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "block matrix": 1
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 block matrix（2.4 Matrix Inverses・2.5 Elementary Matrices・3.1 The Cofactor Expansion）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "blocking",
    "ja": {
      "term": "ブロック化",
      "reading": "ぶろっくか",
      "alt": []
    },
    "en": {
      "term": "blocking",
      "alt": [
        "block design",
        "randomized block design"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「ブロック化」は学習指導要領解説に無い（日本語版 Wikipedia「実験計画法」にはある）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "実験で、結果に影響しそうな性質が似た被験者をまとめてブロックに分け、ブロックごとに処理を無作為に割り当てること。",
    "definition_en": "In an experiment, grouping subjects that are alike in a way that may affect the response into blocks, and randomly assigning treatments within each block.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Block by gender first, then randomize within each block.",
        "ja": "まず性別でブロックに分け、それぞれのブロックの中で無作為に割り当てる。",
        "register": "spoken"
      },
      {
        "en": "Describe a randomized block design for this experiment.",
        "ja": "この実験のブロック化した計画（randomized block design）を述べよ。",
        "register": "written"
      },
      {
        "en": "Explain how blocking on gender reduces the variability in the comparison of the two treatments.",
        "ja": "性別でブロック化することが、2 つの処理の比較のばらつきをどう減らすかを説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP Statistics の CED は topic 1.13 で blocking を扱い、randomized block design の名前も出す。"
    ],
    "related": [
      "completely-randomized-design",
      "treatment",
      "control-group"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.13（blocking ／ block design ／ randomized block design）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「実験計画法」",
        "url": "https://ja.wikipedia.org/wiki/%E5%AE%9F%E9%A8%93%E8%A8%88%E7%94%BB%E6%B3%95",
        "note": "均質な群に分ける操作をブロック化と呼ぶ"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "block design | blocking variable": 9,
        "block design": 9
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満 ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは AP Statistics の CED の呼び方 block design | blocking variable（topic 1.13）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "both-sides",
    "ja": {
      "term": "両辺",
      "reading": "りょうへん",
      "alt": []
    },
    "en": {
      "term": "both sides",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "等式や不等式の左辺と右辺の両方。",
    "definition_en": "The left-hand side and the right-hand side of an equation or inequality, taken together.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Whatever happens to one side has to happen to the other, so let's subtract 4 from both sides.",
        "ja": "片方の辺にしたことは、もう片方にもしないといけない。だから両辺から 4 を引こう。",
        "register": "spoken"
      },
      {
        "en": "Divide both sides of the equation 6x = 42 by 6 to find the value of x.",
        "ja": "方程式 6x = 42 の両辺を 6 で割り、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「両辺に 3 を足す」は add 3 to both sides、「両辺から 3 を引く」は subtract 3 from both sides、「両辺を 2 で割る」は divide both sides by 2。操作によって前置詞が to ／ from ／ by と変わる。",
      "日本語の「移項する」に当たる操作も、英語では両辺に同じ数を足す・両辺から同じ数を引く、の言い方で言える（エントリ move-term-to-other-side）。"
    ],
    "related": [
      "left-hand-side",
      "right-hand-side",
      "move-term-to-other-side",
      "properties-of-equality",
      "equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "both sides": 2227
      },
      "written": {
        "both sides": 795
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "bound-estimate",
    "ja": {
      "term": "評価する",
      "reading": "ひょうかする",
      "alt": []
    },
    "en": {
      "term": "estimate",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "数学の「評価する」は、√10 がどの 2 つの整数の間にあるか（3 < √10 < 4）を調べるように、値を不等式ではさんで範囲を調べることを言う（日本語版 Wikipedia「不等式」は、a ≦ b となる b を求めることを「上から評価する」と書く）。OpenStax Prealgebra・Elementary Algebra はこれを Estimate Square Roots の節で扱い、どの 2 つの連続する整数の間にあるか（between two consecutive whole numbers）を答えさせる。日常語の「評価する」（ほめる・値打ちを決める）は evaluate や appreciate で、別の意味。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "√10 がどの 2 つの整数の間にあるか（3 < √10 < 4）を調べるように、値を不等式ではさんで、その範囲を調べる。",
    "definition_en": "To find a range that a value lies in, such as the two consecutive integers that a square root is between. For example, 3 < √10 < 4 because 9 < 10 < 16.",
    "latex": "3 < \\sqrt{10} < 4",
    "spoken_en": "three is less than the square root of ten, which is less than four",
    "examples": [
      {
        "en": "Let's estimate √10. 9 is 3 squared and 16 is 4 squared, so √10 has to be between 3 and 4.",
        "ja": "√10 を評価してみよう。9 は 3 の 2 乗、16 は 4 の 2 乗だから、√10 は 3 と 4 の間にあるはずだ。",
        "register": "spoken"
      },
      {
        "en": "Estimate √55 between two consecutive whole numbers.",
        "ja": "√55 を、連続する 2 つの整数ではさんで評価せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日常語の「評価する」（ほめる・値打ちを決める）は evaluate や appreciate で、別の意味。数学の evaluate は式の値を求めること（エントリ evaluate）で、範囲をはさむ「評価する」ではない。",
      "estimate は「おおよその値を見積もる」にも使う語。範囲を答えさせるときは、OpenStax Prealgebra・Elementary Algebra のように between two consecutive whole numbers（連続する 2 つの整数の間）と、何ではさむかを示す。",
      "統計の「推定」（エントリ estimation）も英語は estimate で、標本から母平均などを見積もる意味。範囲をはさむ「評価する」とは別の見出し。"
    ],
    "related": [
      "square-root-of-a-number",
      "consecutive-integers",
      "estimation",
      "evaluate",
      "approximate-value"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「不等式」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%8D%E7%AD%89%E5%BC%8F",
        "note": "a ≦ b となる b を求めることを a を b で上から評価するという"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "between two consecutive integers | between which two integers | between two consecutive whole numbers | estimate the square root | estimate the value of the square root": 2
      },
      "written": {
        "between two consecutive integers | between which two integers | between two consecutive whole numbers | estimate the square root | estimate the value of the square root": 18
      },
      "sources": [
        "khan-middle",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "boundary",
    "ja": {
      "term": "境界線",
      "reading": "きょうかいせん",
      "alt": [
        "境界"
      ]
    },
    "en": {
      "term": "boundary line",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Precalculus"
      ]
    },
    "definition_ja": "不等式の表す領域と、そうでない部分を分ける線。不等号に等号が入るとき、境界線は領域に含まれる。",
    "definition_en": "The line (or curve) that separates the region of an inequality from the rest; it belongs to the region when the inequality includes equality.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's strictly less than, so draw the boundary line dashed.",
        "ja": "等号が入らない「より小さい」なので、境界線は点線で描く。",
        "register": "spoken"
      },
      {
        "en": "Shade the region y > x² − 1. Is the boundary included?",
        "ja": "領域 y > x² − 1 を図示せよ。境界線は含むか。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "graph the boundary line",
        "ja": "境界線をかく"
      }
    ],
    "pitfalls": [
      "共通テストの問題文は、領域の図に「ただし，境界線を含まない」と添える（令和 8 年度 本試験 数学II，数学B，数学C 第 1 問。令和 6 年度 本試験 数学II 第 1 問は「境界（境界線）は含まない」）。IM Algebra 1（2.21）と OpenStax Elementary Algebra（4.7）は、境界線を含むかどうかをグラフの実線（solid line）と点線（dashed line）で描き分ける。言葉では the boundary is (not) included（OpenStax Algebra and Trigonometry 11.3）。",
      "用例コーパスの boundary line は話し言葉ではほとんどが Khan Academy（Algebra）。書き言葉のほうがずっと多く、ほとんどが OpenStax Elementary Algebra（1 次不等式のグラフ）。"
    ],
    "related": [
      "region",
      "sketch"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.3（the boundary is included）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "4.7 Graphs of Linear Inequalities（boundary line、solid ／ dashed）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和8年度 大学入学共通テスト 本試験 数学Ⅱ，数学Ｂ，数学Ｃ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=2144&f=abm00017550.pdf&n=2026_ol_18_sugaku2BC.pdf",
        "note": "第1問: ただし，境界線を含まない"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和6年度 大学入学共通テスト 本試験 数学Ⅱ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=666&f=abm00004686.pdf&n=2024_or_20_sugaku2.pdf",
        "note": "第1問: 境界（境界線）は含まない"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 1 2.21〜2.26（boundary line、dashed line）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "boundary line": 12
      },
      "written": {
        "boundary line": 84
      },
      "sources": [
        "khan-algebra",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "bounded",
    "ja": {
      "term": "有界",
      "reading": "ゆうかい",
      "alt": []
    },
    "en": {
      "term": "bounded",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "analysis",
      "sequences"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "数列や関数の値が、ある範囲に収まっていること（|aₙ| ≦ M となる M がある）。",
    "definition_en": "Of a sequence or function: having all its values within some fixed range, |aₙ| ≤ M.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The sequence is bounded: every term is between 0 and 1.",
        "ja": "この数列は有界。どの項も 0 と 1 の間にある。",
        "register": "spoken"
      },
      {
        "en": "Show that the sequence aₙ = n/(n + 1) is bounded and increasing.",
        "ja": "数列 aₙ = n/(n + 1) が有界で増加することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "the region bounded by the curves（曲線で囲まれた部分）の bounded（囲まれた）と同じ語。"
    ],
    "related": [
      "monotonic",
      "convergence",
      "monotone-convergence-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "有界",
        "en": "Bounded set"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "bounded sequence | sequence is bounded | bounded function | function is bounded": 4
      },
      "written": {
        "bounded sequence | sequence is bounded | bounded function | function is bounded": 18
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "box-plot",
    "ja": {
      "term": "箱ひげ図",
      "reading": "はこひげず",
      "alt": []
    },
    "en": {
      "term": "box plot",
      "alt": [
        "boxplot"
      ],
      "uk": null,
      "variants": [
        {
          "term": "box-and-whisker plot",
          "register": "spoken",
          "note": "話し言葉はほとんどが Khan Academy（AP Statistics）。書き言葉では box plot よりずっと少ない。 字幕の box and whisker(s) plot も同じ言い方として数えた。"
        },
        {
          "term": "box plot",
          "register": "both",
          "note": "話し言葉は Khan Academy の中学の講義と Professor Leonard。書き言葉はすべて OpenStax Introductory Statistics で、box-and-whisker plot よりずっと多い。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "最小値・第 1 四分位数・中央値・第 3 四分位数・最大値を、箱とその両側のひげで表した図。",
    "definition_en": "A graph of the five-number summary: a box from the first to the third quartile with a line at the median, and whiskers out to the minimum and maximum.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a box and whisker plot, the box covers the middle half of the data.",
        "ja": "箱ひげ図では、箱がデータの真ん中の半分を表す。",
        "register": "spoken"
      },
      {
        "en": "Draw box plots for the two data sets on the same scale and compare them.",
        "ja": "2 つのデータの箱ひげ図を同じ目盛りでかき、比べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉は box-and-whisker plot、書き言葉は box plot（どちらも 3 倍以上の差）。"
    ],
    "related": [
      "five-number-summary",
      "whisker",
      "quartile",
      "interquartile-range"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "箱ひげ図",
        "en": "Box plot"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "box-and-whisker plot": 19,
        "box plot": 14
      },
      "written": {
        "box plot": 89,
        "box-and-whisker plot": 6
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "can-be-integrated",
    "ja": {
      "term": "積分できる",
      "reading": "せきぶんできる",
      "alt": []
    },
    "en": {
      "term": "integrable",
      "alt": [],
      "variants": [
        {
          "term": "integrable",
          "register": "written",
          "note": "教科書の形容詞。書き言葉はすべて OpenStax Calculus。話し言葉にはほとんど出てこない。"
        },
        {
          "term": "can be integrated",
          "register": "written",
          "note": "話し言葉・書き言葉とも少ない（話し言葉は 3Blue1Brown、書き言葉は OpenStax Calculus と MIT の講義ノート）。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「積分できる」は、原始関数を式で書けること（計算できること）の意味で使うことがある。英語の integrable は定積分が存在すること（区間で連続なら integrable）を指す別の概念。「式で積分できない」は has no elementary antiderivative と言う。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Calculus I",
        "Calculus II"
      ]
    },
    "definition_ja": "関数の原始関数を求めたり、定積分の値を定めたりできること。",
    "definition_en": "Said of a function whose integral exists; in English, integrable means the definite integral exists, not that it has a formula.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "e to the minus x squared is integrable, but you can't write its antiderivative with the functions you know.",
        "ja": "e^{−x²} は積分可能だけど、知っている関数でその原始関数を書くことはできない。",
        "register": "spoken"
      },
      {
        "en": "If f is continuous on [a, b], then f is integrable on [a, b].",
        "ja": "f が [a, b] で連続ならば、f は [a, b] で積分可能である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「積分できない」を can't be integrated と言うと、「定積分が存在しない」と受け取られることがある。式で表せないと言いたいなら no elementary antiderivative。"
    ],
    "related": [
      "antiderivative",
      "definite-integral"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "integrable": 3,
        "can be integrated": 1
      },
      "written": {
        "integrable": 40,
        "can be integrated": 5
      },
      "sources": [
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cancel-out",
    "ja": {
      "term": "消える",
      "reading": "きえる",
      "alt": []
    },
    "en": {
      "term": "cancel out",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「消える」は項がなくなった結果を言い、英語の cancel out は 2 つの項が「打ち消し合う」ことを言う。加減法で y の項どうしが打ち消し合うとき（the y's cancel out）にも、約分で分子と分母の共通因数が消えるとき（cancel）にも使う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの項や因数が打ち消し合って、式からなくなること。3y と −3y のように符号だけが違う項を足したときや、分数の分子と分母の共通因数を約分したときに言う。",
    "definition_en": "Of two terms or factors: to undo each other so that both drop out of the expression. Opposite terms like 3y and −3y cancel out when added, and a common factor of the numerator and denominator cancels out when a fraction is simplified.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When you add these two equations, the 2y and the negative 2y cancel out, so you're left with just x's.",
        "ja": "2 つの式を足すと、2y と −2y が消えて x だけが残るね。",
        "register": "spoken"
      },
      {
        "en": "In 3x over 4x, the x on top and the x on the bottom cancel out, and you get 3 over 4.",
        "ja": "3x/4x では、分子の x と分母の x が消えて 3/4 になる。",
        "register": "spoken"
      },
      {
        "en": "When the equations x + 2y = 7 and 3x − 2y = 5 are added, the y-terms cancel out, giving 4x = 12.",
        "ja": "2 つの式 x + 2y = 7 と 3x − 2y = 5 を加えると、y の項が消えて 4x = 12 となる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "cancel out は「打ち消し合う」という意味で、主語は打ち消し合う項どうし（the y's cancel out、2y and −2y cancel out）。日本語の「消える」は、なくなった結果のほうを言う。",
      "「y を消去する」と解く人を主語にして言うなら eliminate y（エントリ eliminate）。",
      "約分で分子と分母の共通因数が消えるのも cancel と言う（エントリ reduce）。足して 0 になるときにも、割って 1 になるときにも使う。"
    ],
    "related": [
      "elimination",
      "eliminate",
      "reduce",
      "common-factor",
      "opposite"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cancel out": 628
      },
      "written": {
        "cancel out": 32
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "candidates-test",
    "ja": {
      "term": "候補点テスト",
      "reading": "こうほてんてすと",
      "alt": []
    },
    "en": {
      "term": "candidates test",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学II）は、導関数でグラフの概形を調べて区間が制限された関数の最大値や最小値を考察すると書くが、端点と臨界点での値を比べて最大値・最小値を求める方法の名前は出てこない。AP の CED（topic 5.5）はこの方法を Candidates Test と呼び、OpenStax Calculus Volume 1（4.3）は同じ手順を名前を付けずに示す（Locating Absolute Extrema over a Closed Interval）。見出しの「候補点テスト」は本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "閉区間で連続な関数の最大値・最小値を、臨界点と両端での値を比べて求める方法。",
    "definition_en": "Finding the absolute extrema of a continuous function on a closed interval by comparing its values at the critical points and at the endpoints.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For the candidates test, list the critical points and the endpoints, and plug them all in.",
        "ja": "候補点テストでは、臨界点と端点を並べて、すべて代入する。",
        "register": "spoken"
      },
      {
        "en": "Use the Candidates Test to find the absolute maximum of f(x) = x³ − 3x on [−2, 3].",
        "ja": "候補点テストを用いて、区間 [−2, 3] における f(x) = x³ − 3x の最大値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "absolute-extrema",
      "critical-point",
      "extreme-value-theorem"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.5（Candidates Test）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 微分の考え: グラフの概形を観察して区間が制限された関数の最大値や最小値を考察する（方法の名前は出てこない）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "4.3 Maxima and Minima（Locating Absolute Extrema over a Closed Interval。mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CED の呼び方 candidates test（topic 5.5）。register は主張しない。",
        "raised": "2026-10-09"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cardioid",
    "ja": {
      "term": "カージオイド",
      "reading": "かーじおいど",
      "alt": [
        "心臓形"
      ]
    },
    "en": {
      "term": "cardioid",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "極方程式 r = a(1 + cos θ) などで表される、心臓の形をした曲線。",
    "definition_en": "The heart-shaped curve with polar equation r = a(1 + cos θ) (or with sin θ).",
    "latex": "r = a(1 + \\cos\\theta)",
    "spoken_en": "r equals a times one plus cosine theta",
    "examples": [
      {
        "en": "r = 1 + cos θ is a cardioid — it has a cusp at the origin.",
        "ja": "r = 1 + cos θ はカージオイド。原点で尖る。",
        "register": "spoken"
      },
      {
        "en": "Sketch the cardioid r = 2 − 2 sin θ and find the area it encloses.",
        "ja": "カージオイド r = 2 − 2 sin θ をかき、囲む面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数III ／ 数C の教科書では「カージオイド（心臓形）」と書く。話し言葉はすべて Professor Leonard。"
    ],
    "related": [
      "limacon",
      "rose-curve",
      "polar-equation",
      "area-in-polar-coordinates"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "cardioid": 12
      },
      "written": {
        "cardioid": 44
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cartesian-product",
    "ja": {
      "term": "直積",
      "reading": "ちょくせき",
      "alt": [
        "直積集合",
        "デカルト積"
      ]
    },
    "en": {
      "term": "Cartesian product",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "set-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "集合 A, B に対し、a ∈ A, b ∈ B である順序対 (a, b) 全体の集合 A × B のこと。A, B の要素がそれぞれ m 個、n 個なら、A × B の要素は mn 個ある。",
    "definition_en": "For sets A and B, the set A × B of all ordered pairs (a, b) with a in A and b in B. If A has m elements and B has n elements, then A × B has mn elements.",
    "latex": "A \\times B = \\{(a, b) \\mid a \\in A,\\ b \\in B\\}",
    "spoken_en": "A cross B equals the set of all ordered pairs a comma b such that a is in A and b is in B",
    "examples": [
      {
        "en": "The xy-plane is really just the Cartesian product of the real line with itself, R cross R.",
        "ja": "xy 平面は、実は数直線とそれ自身の直積 R × R にほかならない。",
        "register": "spoken"
      },
      {
        "en": "Let A = {1, 2} and B = {x, y, z}. List the elements of the Cartesian product A × B.",
        "ja": "A = {1, 2}, B = {x, y, z} とする。直積 A × B の要素をすべて書き出せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "A × B と B × A はふつう等しくない。(1, x) と (x, 1) は別の順序対である。",
      "Descartes にちなむ名前なので、英語では Cartesian を大文字で書き始める（「デカルト積」）。",
      "用例コーパスでは書き言葉にわずか（OpenStax Calculus）で、話し言葉には出てこない。Levin の本では 5 件（5.1 Sets に 3 件ほか）。"
    ],
    "related": [
      "set",
      "coordinates",
      "relation"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Double Integrals over Rectangular Regions（見出しを決めた参照）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "Cartesian product": 1
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは OpenStax の呼び方 Cartesian product（本文 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "case-analysis",
    "ja": {
      "term": "場合分け",
      "reading": "ばあいわけ",
      "alt": []
    },
    "en": {
      "term": "case analysis",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスで case analysis は話し言葉には出てこず、書き言葉にもわずか（MIT の講義ノート））。英語では動詞句で consider two cases ／ break it into cases のように「場合に分ける」と言う（エントリ split-into-cases）。英語版 Wikipedia「Proof by exhaustion」は、証明法の別名として case analysis（proof by cases）を冒頭で挙げる（エントリ proof-by-cases）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "definition_ja": "文字の値などによって式の扱いが変わるとき、場合ごとに分けて調べ、最後に結果をまとめる方法。",
    "definition_en": "Splitting a problem into separate cases, such as x ≥ 0 and x < 0, when an expression behaves differently in each, solving each case, and then combining the results.",
    "latex": "|x| = \\begin{cases} x & (x \\ge 0) \\\\ -x & (x < 0) \\end{cases}",
    "spoken_en": "the absolute value of x is x when x is greater than or equal to zero, and negative x when x is less than zero",
    "examples": [
      {
        "en": "Let's consider two cases, depending on the sign of x − 1.",
        "ja": "x − 1 の符号で 2 つの場合に分けよう。",
        "register": "spoken"
      },
      {
        "en": "The proof uses a case analysis on whether n is even or odd.",
        "ja": "n が偶数か奇数かで場合分けして証明する。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "各場合で出た解は、その場合の条件（x ≧ 1 など）と共通部分をとってから、最後にすべての場合を合わせる。",
      "「場合分けする」という動詞は consider … cases ／ break … into cases（エントリ split-into-cases）。「x ≧ 1 のとき」は in the case x ≥ 1 か when x ≥ 1。",
      "証明で場合を尽くす形（n が偶数か奇数か）は proof by cases（エントリ proof-by-cases。en.alt に case analysis）。"
    ],
    "related": [
      "split-into-cases",
      "absolute-value-inequality",
      "absolute-value-equation",
      "proof",
      "proof-by-cases"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Proof_by_exhaustion",
        "note": "記事「Proof by exhaustion」（数学カテゴリから 2 段）の冒頭: case analysis と proof by cases を太字の別名として挙げる"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 数学と人間の活動: 余りで場合分け（1 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "case analysis": 1
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 1 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "categorical-variable",
    "ja": {
      "term": "質的データ",
      "reading": "しつてきでーた",
      "alt": [
        "カテゴリ変数",
        "質的変数"
      ]
    },
    "en": {
      "term": "qualitative data",
      "alt": [
        "categorical variable",
        "categorical data"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数量ではなく、種類や分類で表されるデータ（またはその変量）。血液型、好きな教科など。",
    "definition_en": "Data, or a variable, recorded as categories rather than numbers, such as blood type or favorite subject.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Eye color is qualitative data: you can count how many people have each color, but you can't average the colors.",
        "ja": "目の色は質的データ。色ごとの人数は数えられるが、色の平均はとれない。",
        "register": "spoken"
      },
      {
        "en": "Classify each variable as qualitative or quantitative.",
        "ja": "それぞれの変量が質的データか量的データかを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は qualitative data、AP Statistics の講義は categorical variable（Khan Academy）と呼ぶ。",
      "質的データは棒グラフや円グラフで表す。"
    ],
    "related": [
      "quantitative-variable",
      "bar-chart",
      "pie-chart"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "categorical variable": 3,
        "categorical data": 3
      },
      "written": {
        "categorical variable": 7,
        "qualitative data": 14,
        "categorical data": 2
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cauchy-schwarz-inequality",
    "ja": {
      "term": "コーシー・シュワルツの不等式",
      "reading": "こーしー・しゅわるつのふとうしき",
      "alt": [
        "コーシー＝シュワルツの不等式"
      ]
    },
    "en": {
      "term": "Cauchy-Schwarz inequality",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（CED・OpenStax・IM・CK-12 にも用例コーパスにも出てこない）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "(a² + b²)(x² + y²) ≧ (ax + by)² のように、2 組の実数について「平方の和の積」が「積の和の平方」以上になるという不等式。",
    "definition_en": "The inequality stating that the product of two sums of squares is at least the square of the sum of products, e.g. (a² + b²)(x² + y²) ≥ (ax + by)².",
    "latex": "(a^2 + b^2)(x^2 + y^2) \\ge (ax + by)^2",
    "spoken_en": "a squared plus b squared, times x squared plus y squared, is greater than or equal to a x plus b y, squared",
    "examples": [
      {
        "en": "The Cauchy-Schwarz inequality is really about the dot product: u dot v is at most the length of u times the length of v.",
        "ja": "コーシー・シュワルツの不等式は実は内積の話で、u·v は |u||v| 以下になる。",
        "register": "spoken"
      },
      {
        "en": "Prove that (a² + b²)(x² + y²) ≥ (ax + by)², and state when equality holds.",
        "ja": "(a² + b²)(x² + y²) ≧ (ax + by)² を証明し、等号が成り立つ場合を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパス（OpenStax 9 冊を含む）には出てこない。Nicholson（Linear Algebra with Applications）は、Rⁿ の |x · y| ≤ ‖x‖‖y‖ を Cauchy inequality（Theorem 5.3.2）、内積空間の形を Cauchy-Schwarz inequality（Theorem 10.1.4）と呼ぶ。OpenStax Calculus Volume 3（2.3 The Dot Product）は u · v = ‖u‖‖v‖ cos θ を扱うが、この不等式は書かない。"
    ],
    "related": [
      "am-gm-inequality",
      "prove",
      "triangle-inequality",
      "dot-product",
      "equality-holds"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "コーシー＝シュワルツの不等式",
        "en": "Cauchy–Schwarz inequality"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.3 Orthogonality（Theorem 5.3.2 Cauchy Inequality）・10.1 Inner Products and Norms（Theorem 10.1.4 Cauchy-Schwarz Inequality）・Exercise 4.2.44（the Cauchy-Schwarz inequality）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.3 The Dot Product"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "causation",
    "ja": {
      "term": "因果関係",
      "reading": "いんがかんけい",
      "alt": []
    },
    "en": {
      "term": "causal relationship",
      "alt": [
        "causation",
        "cause and effect"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Integrated Math 1",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "一方の変量の変化が、他方の変量の変化の原因になっている関係。",
    "definition_en": "A relationship in which a change in one variable produces a change in the other.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Correlation doesn't prove causation; there might be a lurking variable.",
        "ja": "相関は因果関係の証明にならない。隠れた変数があるかもしれない。",
        "register": "spoken"
      },
      {
        "en": "Does the study show a causal relationship? Explain your reasoning.",
        "ja": "この研究は因果関係を示しているか。理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "因果関係を示すには、観察研究（observational study）ではなく、無作為に処理を割り当てた実験（experiment）が要る。",
      "AP Statistics の CED（topic 1.10・1.13）は cause-and-effect conclusion と形容詞の形で使い、IM Algebra 1（3.9）は causal relationship と言う。"
    ],
    "related": [
      "correlation",
      "confounding-variable"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.10（causal relationship。見出しの根拠）・1.13（cause-and-effect conclusion）・5.2（correlation does not necessarily imply causation）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 1)",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "3.9 Causal Relationships（causal relationship 9 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "causal relationship": 6,
        "cause and effect": 1
      },
      "written": {
        "causation": 1,
        "cause and effect": 1
      },
      "sources": [
        "khan-ap-stats",
        "mit-notes",
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは AP Statistics の CED の呼び方 causal relationship（topic 1.10）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cavalieris-principle",
    "ja": {
      "term": "カヴァリエリの原理",
      "reading": "かゔぁりえりのげんり",
      "alt": []
    },
    "en": {
      "term": "Cavalieri's principle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 つの立体を底面に平行な平面で切ったとき、どの高さでも断面積が等しければ、2 つの立体の体積は等しいという原理。",
    "definition_en": "If two solids have the same height and every plane parallel to their bases cuts them in cross sections of equal area, then the two solids have the same volume.",
    "latex": "A_1(h) = A_2(h) \\text{ for all } h \\implies V_1 = V_2",
    "spoken_en": "if A one of h equals A two of h for every h, then V one equals V two",
    "examples": [
      {
        "en": "Think of a stack of coins. If you push it so it leans, every coin still has the same area, so by Cavalieri's principle the volume doesn't change.",
        "ja": "コインを積んだ山を思い浮かべよう。押して斜めにしても、どのコインの面積も変わらない。だからカヴァリエリの原理で体積は変わらない。",
        "register": "spoken"
      },
      {
        "en": "A right cylinder and an oblique cylinder both have radius 3 and height 10. Use Cavalieri's principle to explain why they have the same volume.",
        "ja": "半径 3、高さ 10 の直円柱と斜円柱がある。カヴァリエリの原理を用いて、2 つの体積が等しい理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では人名の所有格で Cavalieri's principle と書き、C は大文字。",
      "断面積を積分して体積を求める方法と同じ考え方で、断面積が等しければ積分の値（体積）も等しくなる。",
      "IM Geometry は glossary に入れ、5.10 Cross Sections and Volume ほかで 4 件使う。用例コーパスにはほとんど出てこない。"
    ],
    "related": [
      "cross-section",
      "volume-by-cross-sections",
      "cross-sectional-area"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "カヴァリエリの原理",
        "en": "Cavalieri's principle"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary・Geometry 5.10 Cross Sections and Volume"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 Cavalieri's principle（Geometry glossary・Geometry 5.10 Cross Sections and Volume・Geometry 5.11 Prisms Practice・計 4 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "ceiling-function",
    "ja": {
      "term": "天井関数",
      "reading": "てんじょうかんすう",
      "alt": []
    },
    "en": {
      "term": "ceiling function",
      "alt": [
        "least integer function"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "number-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "実数 x に対し、x 以上の最小の整数 ⌈x⌉ を対応させる関数のこと。たとえば ⌈2.3⌉ = 3、⌈−2.3⌉ = −2 である。",
    "definition_en": "The function that sends a real number x to the smallest integer greater than or equal to x, written ⌈x⌉. For example, ⌈2.3⌉ = 3 and ⌈−2.3⌉ = −2.",
    "latex": "\\lceil x \\rceil = \\min\\{n \\in \\mathbb{Z} \\mid n \\ge x\\}",
    "spoken_en": "the ceiling of x equals the minimum of the integers n with n greater than or equal to x",
    "examples": [
      {
        "en": "The ceiling function always rounds up, so the ceiling of 4.1 is 5, but the ceiling of 4 is just 4.",
        "ja": "天井関数はつねに切り上げる。だから ⌈4.1⌉ = 5 だが、⌈4⌉ は 4 のままだ。",
        "register": "spoken"
      },
      {
        "en": "A bus holds 40 students. Use the ceiling function to write the number of buses needed for n students, and evaluate it for n = 130.",
        "ja": "バス 1 台には生徒が 40 人乗れる。生徒 n 人に必要なバスの台数を天井関数で表し、n = 130 のときの値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校のガウス記号 [x] は x 以下の最大の整数（床関数 ⌊x⌋）で、天井関数ではない。⌈x⌉ と ⌊x⌋ の向きに注意。",
      "負の数では 0 に近いほうへ上がる。⌈−2.3⌉ = −2 で、−3 ではない。",
      "用例コーパスでは話し言葉（MIT OCW）にも書き言葉（OpenStax Calculus）にもほとんど出てこない。英語版 Wikipedia は床関数と 1 つの記事（Floor and ceiling functions）で扱う。"
    ],
    "related": [
      "floor-function",
      "integer-part"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "床関数と天井関数",
        "en": "Floor and ceiling functions"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "ceiling function": 2
      },
      "written": {
        "ceiling function": 1
      },
      "sources": [
        "mit-6.042",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは OpenStax の呼び方 ceiling function（本文 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "census",
    "ja": {
      "term": "全数調査",
      "reading": "ぜんすうちょうさ",
      "alt": []
    },
    "en": {
      "term": "census",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "調べたい集団のすべてを対象にして行う調査。国勢調査など。",
    "definition_en": "A survey that collects data from every member of the group being studied, such as a count of every resident of a country.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If we could ask every single person, that'd be a census, not a sample.",
        "ja": "全員に聞けるならそれは全数調査で、標本ではない。",
        "register": "spoken"
      },
      {
        "en": "Explain why a census is not practical for testing the lifetime of light bulbs.",
        "ja": "電球の寿命を調べるのに全数調査が現実的でない理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの census は、ほとんどが U.S. Census（米国の国勢調査・国勢調査局）や census.gov を指す（OpenStax Introductory Statistics）。全数調査の意味の census は AP Statistics の CED（topic 1.10）が扱う。"
    ],
    "related": [
      "sample-survey",
      "population",
      "survey"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.10（census）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "a census !bureau": 1
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは AP Statistics の CED の呼び方 a census !bureau（topic 1.10）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "center-of-dilation",
    "ja": {
      "term": "相似の中心",
      "reading": "そうじのちゅうしん",
      "alt": []
    },
    "en": {
      "term": "center of dilation",
      "alt": [
        "center of a dilation"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説の相似の位置は、対応する点を通る直線が 1 点を通ることで定め、中心が 2 つの図形の間にある場合も含む。日本語版 Wikipedia「相似の中心」は外相似点（相似比が正）と内相似点（負）に分け、英語版 Wikipedia「Homothetic center」も external と internal を挙げる。英語の center of dilation は、IM Grade 8 の glossary と CK-12 Geometry 7.16 の例が正の scale factor の dilation なので、外相似点に当たる。IM Geometry の発展問題は負の scale factor（−1・−2）の dilation も出し、その中心は内相似点に当たる。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "図形を拡大・縮小するときの基準になる点。もとの図形の各点と、それを移した点を結ぶ直線は、すべてこの点を通る。",
    "definition_en": "The fixed point that a figure is enlarged or shrunk from in a dilation. The line through any point and its image always passes through the center of dilation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Draw a ray from the center of dilation through each vertex, and then go out twice as far.",
        "ja": "相似の中心から各頂点を通る半直線を引いて、2 倍の距離のところに点をとろう。",
        "register": "spoken"
      },
      {
        "en": "Triangle ABC has vertices A(1, 2), B(3, 1), and C(2, 4). Dilate it by a scale factor of 3 using the origin as the center of dilation, and give the coordinates of A′, B′, and C′.",
        "ja": "△ABC の頂点は A(1, 2)、B(3, 1)、C(2, 4) である。原点を相似の中心として △ABC を 3 倍に拡大した △A′B′C′ の頂点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "dilation は拡大にも縮小にも使う（エントリ dilation）。center of dilation も、拡大・縮小どちらの中心にも使う。",
      "中学校学習指導要領解説の「相似の位置」にある 2 つの図形で、対応する点を結ぶ直線が集まる点（相似の中心）がこれに当たる。IM Grade 8（2.2〜2.5）は、一方の図形をこの点を中心に dilation するともう一方になる、と変換として見る（エントリ dilation）。",
      "中心は図形の外・内・周上のどこにとってもよい。中心の点だけは移しても動かない。"
    ],
    "related": [
      "dilation",
      "scale-factor",
      "similar",
      "similar-triangles",
      "scale-drawing"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "相似の中心",
        "en": "Homothetic center"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math / IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/",
        "note": "レッスンで center of dilation を使う（Grade 8 の glossary の見出しは center of a dilation）。IM Geometry の発展問題（Are you ready for more?）は負の scale factor の dilation も出す"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形: その点を相似の中心として、相似の位置にある"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「相似の中心」",
        "url": "https://ja.wikipedia.org/wiki/%E7%9B%B8%E4%BC%BC%E3%81%AE%E4%B8%AD%E5%BF%83",
        "note": "外相似点・内相似点"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Homothetic_center",
        "note": "記事「Homothetic center」（数学カテゴリから 4 段）: external と internal"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "7.16 Dilation in the Coordinate Plane: 正の scale factor だけ"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "center of dilation": 21
      },
      "sources": [
        "khan-middle"
      ],
      "counted": "2026-10-09"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "center-of-mass",
    "ja": {
      "term": "重心（質量中心）",
      "reading": "じゅうしん・しつりょうちゅうしん",
      "alt": []
    },
    "en": {
      "term": "center of mass",
      "alt": [],
      "uk": "centre of mass",
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "物体の質量の分布の中心となる点。板や棒の重心を、モーメントを質量で割って積分で求める。",
    "definition_en": "The point at which the mass of an object is balanced; for a plate or rod it is found by dividing moments by the mass, using integrals.",
    "latex": "\\bar{x} = \\frac{\\int_a^b x\\,\\rho(x)\\,dx}{\\int_a^b \\rho(x)\\,dx}",
    "spoken_en": "x bar equals the integral from a to b of x rho of x dx over the integral from a to b of rho of x dx",
    "examples": [
      {
        "en": "If the plate is uniform, the center of mass is the same as the centroid.",
        "ja": "板が一様なら、質量中心は幾何的な重心と一致する。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "三角形の重心のような幾何の重心は centroid。物体の質量の中心は center of mass。一様な板なら両者は一致する。",
      "英国の綴りは centre of mass。"
    ],
    "related": [
      "hydrostatic-force"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "重心",
        "en": "Center of mass"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences#re_er",
        "note": "記事「American and British English spelling differences」の節 -re, -er（英国の綴りは centre、米国は center）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "center of mass": 93
      },
      "written": {
        "center of mass": 165
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "center",
    "ja": {
      "term": "中心",
      "reading": "ちゅうしん",
      "alt": []
    },
    "en": {
      "term": "center",
      "alt": [],
      "uk": "centre",
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円や球で、周上（表面上）のどの点からも距離が等しい点。",
    "definition_en": "The point inside a circle or sphere whose distance to every point on it is the same.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Put the point of your compass on the center, and then draw the circle.",
        "ja": "コンパスの針を中心に置いて、円をかこう。",
        "register": "spoken"
      },
      {
        "en": "Point O is the center of the circle, and point A is on the circle. If OA = 6 cm, find the diameter of the circle.",
        "ja": "点 O は円の中心、点 A は円周上の点である。OA = 6 cm のとき、この円の直径を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英国の綴りは centre。",
      "「中心が (2, −1) の円」は a circle centered at (2, −1)。centered の後の前置詞は at。",
      "三角形の外心 circumcenter・内心 incenter も center を含む語。重心は centroid（エントリ centroid・circumcenter）。"
    ],
    "related": [
      "circle",
      "radius",
      "sphere",
      "centroid",
      "circumcenter"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "中心",
        "en": "Centre (geometry)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences#re_er",
        "note": "記事「American and British English spelling differences」の節 -re, -er（英国の綴りは centre、米国は center）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "center of the circle | center of a circle | center of the sphere | its center | the center is | centered at": 132
      },
      "written": {
        "center of the circle | center of a circle | center of the sphere | its center | the center is | centered at": 369
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "central-angle",
    "ja": {
      "term": "中心角",
      "reading": "ちゅうしんかく",
      "alt": []
    },
    "en": {
      "term": "central angle",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円の中心を頂点とし、2 本の半径を辺とする角。おうぎ形の弧の長さと面積は、中心角の大きさに比例する。",
    "definition_en": "An angle with its vertex at the center of a circle and two radii as its sides. Arc length and sector area grow in proportion to the central angle.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The central angle has its vertex right at the center, so here it's angle AOB.",
        "ja": "中心角は頂点がちょうど中心にある角だから、ここでは ∠AOB だね。",
        "register": "spoken"
      },
      {
        "en": "A sector has a radius of 9 cm and an arc length of 4π cm. Find its central angle.",
        "ja": "半径 9 cm、弧の長さ 4π cm のおうぎ形の中心角を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "円周上に頂点がある円周角は inscribed angle（エントリ inscribed-angle）。同じ弧に対する円周角は中心角の半分。",
      "中心角が a° のおうぎ形の弧の長さと面積は、同じ半径の円の円周と面積の a/360 になる。",
      "弧の度数（エントリ arc-measure）は、その弧に対する中心角の大きさで表す。弧の長さ（arc length）とは別の量。"
    ],
    "related": [
      "inscribed-angle",
      "sector",
      "arc",
      "arc-measure",
      "area-of-a-sector"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "central angle": 1
      },
      "written": {
        "central angle": 24
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "central-limit-theorem",
    "ja": {
      "term": "中心極限定理",
      "reading": "ちゅうしんきょくげんていり",
      "alt": []
    },
    "en": {
      "term": "central limit theorem",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "母集団の分布にかかわらず、標本の大きさ n が大きいとき、標本平均の分布は正規分布で近似できるという定理。",
    "definition_en": "The theorem that, for a large sample size n, the sample mean is approximately normally distributed no matter what shape the population has.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "By the central limit theorem, with n equals 50, the sample mean is approximately normal.",
        "ja": "中心極限定理により、n = 50 なら標本平均はほぼ正規分布に従う。",
        "register": "spoken"
      },
      {
        "en": "Use the central limit theorem to find the probability that the mean of 40 randomly selected values exceeds 102.",
        "ja": "中心極限定理を用いて、無作為に選んだ 40 個の値の平均が 102 を超える確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "略して CLT と書く（OpenStax Introductory Statistics、AP Statistics の CED topic 2.12）。",
      "標本の大きさの目安を AP Statistics の CED（topic 4.1）は n ≥ 30、OpenStax Introductory Statistics（7 章の導入）は at least 30 とする（CED は母集団の分布が極端にゆがんでいればもっと大きな n が要ると書く）。"
    ],
    "related": [
      "sampling-distribution-of-the-mean",
      "normal-distribution",
      "law-of-large-numbers",
      "sampling-distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "中心極限定理",
        "en": "Central limit theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "7 章 The Central Limit Theorem（導入の at least 30、7.1〜7.3、CLT）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.12（central limit theorem、CLT）・Topic 4.1（n ≥ 30）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "central limit theorem": 50
      },
      "written": {
        "central limit theorem": 72
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "centroid",
    "ja": {
      "term": "重心",
      "reading": "じゅうしん",
      "alt": [
        "重心の座標"
      ]
    },
    "en": {
      "term": "centroid",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A",
        "数II"
      ],
      "us": [
        "Geometry",
        "Calculus II"
      ]
    },
    "definition_ja": "三角形の 3 本の中線が交わる点。座標は 3 頂点の座標の平均。",
    "definition_en": "The point where the three medians of a triangle meet; its coordinates are the averages of the vertices' coordinates.",
    "latex": "\\left(\\frac{x_1 + x_2 + x_3}{3}, \\frac{y_1 + y_2 + y_3}{3}\\right)",
    "spoken_en": null,
    "examples": [
      {
        "en": "The centroid is just the average of the three vertices.",
        "ja": "重心は 3 頂点の平均をとるだけ。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinates of the centroid of the triangle with vertices (0, 0), (6, 0), and (3, 9).",
        "ja": "3 点 (0, 0), (6, 0), (3, 9) を頂点とする三角形の重心の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの centroid の多くは、OpenStax Calculus の「平面図形の重心（質量が一様な板の重心）」の意味。三角形の重心はその特別な場合。",
      "物理の重心（質量中心）は center of mass（center-of-mass を参照）。"
    ],
    "related": [
      "midpoint",
      "point-of-internal-division",
      "center-of-mass"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "三角形の重心（中線の交点）: 用語欄 the point of intersection of the medians in a triangle、1.6 Points that Partition Line Segments（重心の座標）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "centroid": 16
      },
      "written": {
        "centroid": 66
      },
      "sources": [
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "certain",
    "ja": {
      "term": "必ず起こる",
      "reading": "かならずおこる",
      "alt": []
    },
    "en": {
      "term": "certain",
      "alt": [],
      "uk": null
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "あることがらが、いつでも必ず起こること。必ず起こることがらの確率は 1 である。",
    "definition_en": "Sure to happen every time. An event that is certain has a probability of 1.",
    "latex": "P(A) = 1",
    "spoken_en": "P of A equals one",
    "examples": [
      {
        "en": "Rolling a number less than 7 on a regular die is certain, so the probability is 1.",
        "ja": "ふつうのさいころで 7 より小さい目が出るのは必ず起こることだから、確率は 1。",
        "register": "spoken"
      },
      {
        "en": "A bag contains only blue marbles. Explain why drawing a blue marble is certain, and give its probability.",
        "ja": "袋の中には青玉だけが入っている。青玉を取り出すことが必ず起こる理由を説明し、その確率を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a certain number（ある数）のように a certain を名詞の前に置くと「ある…」（特定の）の意味になる。a certain event も「ある事象」と読めるので、「必ず起こる事象」をはっきり言うときは an event that is certain to happen と言う。",
      "日本語の「必ず起こる」は動詞の形だが、英語の certain は形容詞で、The event is certain. ／ It is certain to happen. のように be 動詞と使う。",
      "確率の反対の端は、決して起こらないことがら（impossible、エントリ impossible）の確率 0。"
    ],
    "related": [
      "impossible",
      "likelihood",
      "probability",
      "event",
      "sample-space"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "!a certain event | event is certain | certain to happen | certain to occur": 1
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは OpenStax の呼び方 !a certain event | event is certain | certain to happen | certain to occur（本文 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cevas-theorem",
    "ja": {
      "term": "チェバの定理",
      "reading": "ちぇばのていり",
      "alt": []
    },
    "en": {
      "term": "Ceva's theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Ceva's theorem は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "definition_ja": "△ABC の頂点と 1 点 O を結ぶ直線 AO, BO, CO が対辺（またはその延長）と P, Q, R で交わるとき、(BP/PC)(CQ/QA)(AR/RB) = 1 が成り立つという定理。逆（P, Q, R が辺 BC, CA, AB 上にあり、積が 1 なら AP, BQ, CR は 1 点で交わる）も成り立ち、3 本の直線が 1 点で交わることの証明に使う。",
    "definition_en": "If lines through the vertices of triangle ABC and a common point O meet the opposite sides (or their extensions) at P, Q, and R, then (BP/PC)(CQ/QA)(AR/RB) = 1. The converse also holds when P, Q, and R lie on the sides themselves, so the theorem is a tool for proving that three such lines are concurrent.",
    "latex": "\\frac{BP}{PC}\\cdot\\frac{CQ}{QA}\\cdot\\frac{AR}{RB} = 1",
    "spoken_en": "BP over PC, times CQ over QA, times AR over RB, equals one",
    "examples": [
      {
        "en": "Go around the triangle in one direction, vertex to point to vertex, and by Ceva's theorem the three ratios multiply to one.",
        "ja": "三角形を一方向に、頂点→分点→頂点とたどると、チェバの定理より 3 つの比の積は 1 になる。",
        "register": "spoken"
      },
      {
        "en": "In triangle ABC, the segments AP, BQ, and CR meet at one point, with BP : PC = 2 : 3 and CQ : QA = 4 : 1. Use Ceva's theorem to find AR : RB.",
        "ja": "△ABC において、線分 AP, BQ, CR が 1 点で交わり、BP : PC = 2 : 3, CQ : QA = 4 : 1 である。チェバの定理を用いて AR : RB を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "比は三角形の周りを一方向にたどって書く。順番を崩すと積が 1 にならない。",
      "点 O が三角形の外部にあっても成り立つ（そのときは 2 つの分点が辺の延長上にくる）。",
      "英語版 Wikipedia「Cevian」は、頂点と対辺（またはその延長）上の点を結ぶ線分を cevian と呼ぶ。"
    ],
    "related": [
      "menelauss-theorem",
      "concurrent",
      "point-of-internal-division",
      "triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "チェバの定理",
        "en": "Ceva's theorem"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Cevian",
        "note": "記事「Cevian」の冒頭（cevian を太字で定義。数学カテゴリから 4 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 Ceva's theorem（英語版「Ceva's theorem」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "chain-rule",
    "ja": {
      "term": "合成関数の微分",
      "reading": "ごうせいかんすうのびぶん",
      "alt": [
        "連鎖律",
        "合成関数の微分法"
      ]
    },
    "en": {
      "term": "chain rule",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "合成関数の導関数 {f(g(x))}′ = f′(g(x))·g′(x) を与える公式。",
    "definition_en": "The rule for differentiating a composite function: (f(g(x)))′ = f′(g(x))·g′(x).",
    "latex": "\\frac{d}{dx} f(g(x)) = f'(g(x))\\,g'(x)",
    "spoken_en": "the derivative of f of g of x is f prime of g of x times g prime of x",
    "examples": [
      {
        "en": "Chain rule: take the derivative of the outside, leave the inside alone, then multiply by the derivative of the inside.",
        "ja": "合成関数の微分は、外側を微分して中身はそのまま、それに中身の微分を掛ける。",
        "register": "spoken"
      },
      {
        "en": "Use the chain rule to find the derivative of y = (3x² + 1)⁵.",
        "ja": "合成関数の微分を用いて y = (3x² + 1)⁵ の導関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "use the chain rule",
        "ja": "合成関数の微分を使う"
      }
    ],
    "pitfalls": [
      "学習指導要領解説（数学III）は「合成関数の微分法」と言い、「連鎖律」は出てこない（日本語版 Wikipedia の記事名は「連鎖律」）。AP の CED（topic 3.1）は chain rule と呼ぶ。"
    ],
    "related": [
      "composite-function",
      "product-rule",
      "quotient-rule",
      "implicit-differentiation",
      "multivariable-chain-rule"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "連鎖律",
        "en": "Chain rule"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 3.1（chain rule）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 微分法: 合成関数の微分法"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "chain rule": 616
      },
      "written": {
        "chain rule": 183
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "change-of-base-formula",
    "ja": {
      "term": "底の変換公式",
      "reading": "ていのへんかんこうしき",
      "alt": []
    },
    "en": {
      "term": "change-of-base formula",
      "alt": [
        "change-of-base rule"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "log_a b = log_c b / log_c a（a, c は 1 でない正の数、b > 0）。底の違う対数を同じ底にそろえる。",
    "definition_en": "log_a b = log_c b / log_c a (a, c > 0, a ≠ 1, c ≠ 1, b > 0), used to rewrite logarithms with a common base.",
    "latex": "\\log_a b = \\frac{\\log_c b}{\\log_c a}",
    "spoken_en": "log base a of b equals log base c of b over log base c of a",
    "examples": [
      {
        "en": "Your calculator only has log and ln, so use the change-of-base formula.",
        "ja": "電卓には log と ln しかないので、底の変換公式を使う。",
        "register": "spoken"
      },
      {
        "en": "Use the change-of-base formula to show that log₂ 3 · log₃ 4 = 2.",
        "ja": "底の変換公式を用いて log₂ 3 · log₃ 4 = 2 を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ハイフンなしの change of base formula とも書く（OpenStax Algebra and Trigonometry にも両方の書き方がある）。"
    ],
    "related": [
      "logarithm",
      "common-logarithm",
      "natural-logarithm",
      "write-with-the-same-base"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "change-of-base formula": 23
      },
      "written": {
        "change-of-base formula": 42,
        "change-of-base rule": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "change-of-basis",
    "ja": {
      "term": "基底の変換",
      "reading": "きていのへんかん",
      "alt": [
        "基底変換"
      ]
    },
    "en": {
      "term": "change of basis",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "ある基底での座標を別の基底での座標に書き換えること。その対応を表す行列を基底変換行列という。",
    "definition_en": "Rewriting coordinates relative to one basis as coordinates relative to another; the matrix that does it is the change-of-basis matrix.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A change of basis is just a translation between two people's coordinate systems.",
        "ja": "基底の変換は、2 人の座標系のあいだの翻訳のようなものだ。",
        "register": "spoken"
      },
      {
        "en": "Find the change-of-basis matrix from B to C.",
        "ja": "B から C への基底変換行列を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉は大半が 3Blue1Brown。"
    ],
    "related": [
      "basis",
      "coordinate-vector",
      "similar-matrices"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "9.1 The Matrix of a Linear Transformation・9.2 Operators and Similarity"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "change of basis": 17
      },
      "sources": [
        "mit-18.06",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "change-of-variables",
    "ja": {
      "term": "変数変換",
      "reading": "へんすうへんかん",
      "alt": []
    },
    "en": {
      "term": "change of variables",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "重積分で、変数を x = g(u, v)、y = h(u, v) と置きかえて計算すること。ヤコビアンの絶対値を掛ける。",
    "definition_en": "Rewriting a multiple integral with new variables x = g(u, v), y = h(u, v), multiplying by the absolute value of the Jacobian.",
    "latex": "\\iint_R f(x, y)\\,dA = \\iint_S f(g(u,v), h(u,v)) \\left| \\frac{\\partial(x, y)}{\\partial(u, v)} \\right| du\\,dv",
    "spoken_en": null,
    "examples": [
      {
        "en": "Polar coordinates are just one change of variables, with Jacobian r.",
        "ja": "極座標はヤコビアンが r の変数変換の 1 つにすぎない。",
        "register": "spoken"
      },
      {
        "en": "Use the change of variables u = x + y, v = x − y to evaluate ∬_R (x + y) dA.",
        "ja": "変数変換 u = x + y、v = x − y を用いて ∬_R (x + y) dA を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "1 変数の置換積分（u-substitution）も change of variables と呼ぶことがある（OpenStax Calculus Volume 1 に 10 件）。"
    ],
    "related": [
      "jacobian",
      "double-integral",
      "triple-integral",
      "integration-by-substitution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "change of variables": 35
      },
      "written": {
        "change of variables": 47
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "change-the-limits-of-integration",
    "ja": {
      "term": "積分区間を変える",
      "reading": "せきぶんくかんをかえる",
      "alt": []
    },
    "en": {
      "term": "change the limits of integration",
      "alt": [],
      "variants": [
        {
          "term": "change the bounds",
          "register": "spoken",
          "note": "授業での言い方。大学の講義（MIT OCW）に多い。書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "置換積分で、積分区間をもとの変数の値から新しい変数の値に書き換えること。",
    "definition_en": "In a substitution, to replace the limits of integration by the corresponding values of the new variable.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When x is zero, u is one, and when x is one, u is two, so change the bounds to one and two.",
        "ja": "x = 0 のとき u = 1、x = 1 のとき u = 2 だから、区間を 1 から 2 に変える。",
        "register": "spoken"
      },
      {
        "en": "Change the limits of integration: when x = 0, u = 1, and when x = 1, u = 2.",
        "ja": "積分区間を変える。x = 0 のとき u = 1、x = 1 のとき u = 2 である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「x | 0 → 1 ／ t | 1 → 2」のような対応表を英語で書くなら、when x = 0, u = 1 のように文で書く。"
    ],
    "related": [
      "substitution-in-a-definite-integral",
      "let-u-equal",
      "interval-of-integration"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "change the bounds": 12
      },
      "written": {
        "change the limits of integration": 11
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "change-together",
    "ja": {
      "term": "ともなって変わる",
      "reading": "ともなってかわる",
      "alt": [
        "y が x に伴って変わる"
      ]
    },
    "en": {
      "term": "change together",
      "alt": [
        "vary together"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。「伴って変わる」は中学校学習指導要領解説が関数を導入する言い方（「伴って変わる二つの数量」）で、英語では y changes as x changes、y depends on x、as x increases, y increases のように、文で言い表す。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの量のうち、一方の値が変わると、それにつれてもう一方の値も変わること。中 1 で関数を学ぶときの出発点になる言い方。",
    "definition_en": "Describes two quantities where a change in the value of one goes along with a change in the value of the other. It is the starting idea for functions.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Time and distance change together: the longer you walk, the farther you go.",
        "ja": "時間と道のりはともなって変わる。長く歩くほど、遠くまで進む。",
        "register": "spoken"
      },
      {
        "en": "The side length x cm and the perimeter y cm of a square change together. Express y in terms of x.",
        "ja": "正方形の 1 辺の長さ x cm と周の長さ y cm は、ともなって変わる。y を x の式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「伴って」を accompany と直訳しない。英語では as を使って y changes as x changes と言う。",
      "ともなって変わる 2 つの量が、いつも関数の関係になるとは限らない。x の値を決めると y の値がただ 1 つ決まるときに、y is a function of x（y は x の関数）と言う（エントリ function）。"
    ],
    "related": [
      "function",
      "variable",
      "quantity",
      "increase",
      "decrease"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "C 関数「伴って変わる二つの数量」"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "change together": 2
      },
      "sources": [
        "mit-18.03"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 2 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "change",
    "ja": {
      "term": "増加量",
      "reading": "ぞうかりょう",
      "alt": []
    },
    "en": {
      "term": "change",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説の「x の増加量」「y の増加量」（変化の割合は x の増加量に対する y の増加量の割合）は、英語で change in x、change in y と言い、Δx、Δy（delta x、delta y と読む）とも書く。「増加」の字が付いていても、英語の change は増えても減っても使い、減ったときは負の change になる。変化の割合は y の増加量 ÷ x の増加量（エントリ rate-of-change）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "x や y の値が変わったときの、変わった後の値から変わる前の値を引いた量。減ったときは負の数になる。",
    "definition_en": "How much a quantity goes up or down: the final value minus the starting value. It is negative when the quantity decreases.",
    "latex": "\\Delta y = y_2 - y_1",
    "spoken_en": "delta y equals y two minus y one",
    "examples": [
      {
        "en": "As x goes from 2 to 5, the change in x is 3 and the change in y is negative 6.",
        "ja": "x が 2 から 5 まで変わると、x の増加量は 3、y の増加量は −6。",
        "register": "spoken"
      },
      {
        "en": "For y = −2x + 1, find the change in y when x increases from 2 to 5.",
        "ja": "y = −2x + 1 で、x が 2 から 5 まで増加するときの y の増加量を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「増加量」も、減ったときは負の数になる（y が 6 減れば y の増加量は −6）。英語では同じ量を change in y と言い、増えても減っても change を使う。",
      "変化の割合 =（y の増加量）÷（x の増加量）は、英語で change in y over change in x。Δy/Δx と書き、delta y over delta x と読む（エントリ rate-of-change）。"
    ],
    "related": [
      "rate-of-change",
      "slope",
      "increase",
      "decrease",
      "linear-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 関数: x の増加量に対する y の増加量の割合（変化の割合）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "change in x | change in y | change in the x | change in the y": 744
      },
      "written": {
        "change in x | change in y | change in the x | change in the y": 38
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "changes-sign",
    "ja": {
      "term": "符号が変わる",
      "reading": "ふごうがかわる",
      "alt": []
    },
    "en": {
      "term": "changes sign",
      "alt": [],
      "variants": [
        {
          "term": "switches sign",
          "register": "spoken",
          "note": "Khan Academy の言い方。話し言葉はほとんどが Khan Academy。書き言葉は少ない（OpenStax Calculus だけ）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "式の値が正から負へ、または負から正へ変わる。",
    "definition_en": "Of a quantity: to go from positive to negative or from negative to positive.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "f prime switches sign at x = 3, so that's an extremum.",
        "ja": "x = 3 で f′ の符号が変わるので、そこは極値。",
        "register": "spoken"
      },
      {
        "en": "Since f′ changes sign from positive to negative at x = 0, f has a local maximum there.",
        "ja": "x = 0 で f′ の符号が正から負に変わるので、f はそこで極大となる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "f″(c) = 0 でも符号が変わらなければ変曲点ではない（x⁴ の x = 0）。"
    ],
    "related": [
      "first-derivative-test",
      "critical-point"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "changes sign": 34,
        "switches sign": 26
      },
      "written": {
        "changes sign": 16,
        "switches sign": 3
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "changing-the-order-of-integration",
    "ja": {
      "term": "積分順序の交換",
      "reading": "せきぶんじゅんじょのこうかん",
      "alt": []
    },
    "en": {
      "term": "changing the order of integration",
      "alt": [
        "reversing the order of integration",
        "switching the order of integration"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "累次積分の dx と dy の順序を入れかえること。積分範囲も領域に合わせて書き直す。",
    "definition_en": "Swapping the order of dx and dy in an iterated integral, and rewriting the limits to describe the same region.",
    "latex": "\\int_0^1 \\int_x^1 f\\,dy\\,dx = \\int_0^1 \\int_0^y f\\,dx\\,dy",
    "spoken_en": null,
    "examples": [
      {
        "en": "This integral is impossible as written, so let's switch the order of integration.",
        "ja": "このままでは積分できないから、積分順序を交換しよう。",
        "register": "spoken"
      },
      {
        "en": "Evaluate ∫₀¹ ∫ₓ¹ e^{y²} dy dx by changing the order of integration.",
        "ja": "積分順序を交換して ∫₀¹ ∫ₓ¹ e^{y²} dy dx を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "まず領域の図をかいて、範囲を読み直す。",
      "話し言葉ではどの言い方も少なく、switch ／ reverse ／ change がわずかずつ出てくるだけ。"
    ],
    "related": [
      "iterated-integral",
      "double-integral",
      "fubinis-theorem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "reversing the order of integration": 2,
        "changing the order of integration": 1,
        "switching the order of integration": 2
      },
      "written": {
        "switching the order of integration": 2,
        "changing the order of integration": 10,
        "reversing the order of integration": 4
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "characteristic-equation",
    "ja": {
      "term": "特性方程式",
      "reading": "とくせいほうていしき",
      "alt": []
    },
    "en": {
      "term": "characteristic equation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「数列」は、aₙ₊₁ = paₙ + q に対する x = px + q と、aₙ₊₂ = paₙ₊₁ + qaₙ に対する x² = px + q の両方を特性方程式と呼ぶ。Levin（4.4）は 2 階の同次線形漸化式の x² + αx + β = 0 を characteristic equation と呼び、aₙ₊₁ = paₙ + q の α = pα + q に当たる式にはこの名前を使わない。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "漸化式の一般項を求めるために立てる補助の方程式。aₙ₊₁ = paₙ + q では α = pα + q、aₙ₊₂ = paₙ₊₁ + qaₙ では x² = px + q。",
    "definition_en": "An auxiliary equation used to solve a recurrence: α = pα + q for aₙ₊₁ = paₙ + q, and x² = px + q for aₙ₊₂ = paₙ₊₁ + qaₙ.",
    "latex": "a_{n+2} = a_{n+1} + 2a_n \\;\\Rightarrow\\; x^2 = x + 2",
    "spoken_en": null,
    "examples": [
      {
        "en": "The roots of the characteristic equation are 2 and −1, so the general term is a combination of 2ⁿ and (−1)ⁿ.",
        "ja": "特性方程式の解が 2 と −1 だから、一般項は 2ⁿ と (−1)ⁿ の組み合わせになる。",
        "register": "spoken"
      },
      {
        "en": "Solve the recurrence aₙ₊₂ = 5aₙ₊₁ − 6aₙ using its characteristic equation.",
        "ja": "特性方程式を用いて漸化式 aₙ₊₂ = 5aₙ₊₁ − 6aₙ を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "微分方程式（MIT 18.03、OpenStax Calculus Volume 3）の特性方程式も同じ名前の同じ考え方。"
    ],
    "related": [
      "recurrence-relation",
      "solve-the-recurrence",
      "system-of-recurrences",
      "linear-recurrence-relation",
      "fibonacci-sequence"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "7.1 Second-Order Linear Equations（微分方程式の characteristic equation）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数列」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%88%97",
        "note": "漸化式の節（特性方程式 x = px + q・x² = px + q）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.4 Exponential Sequences（characteristic equation・characteristic roots）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "characteristic equation": 56
      },
      "written": {
        "characteristic equation": 59
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "characteristic-polynomial",
    "ja": {
      "term": "特性多項式",
      "reading": "とくせいたこうしき",
      "alt": [
        "固有多項式"
      ]
    },
    "en": {
      "term": "characteristic polynomial",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列 A について det(A − λI)（または det(λI − A)）を λ の多項式とみたもの。その根が固有値。",
    "definition_en": "det(A − λI) (or det(λI − A)) viewed as a polynomial in λ for a square matrix A; its roots are the eigenvalues.",
    "latex": "p(\\lambda) = \\det(A - \\lambda I)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Set the characteristic polynomial equal to zero and solve for lambda.",
        "ja": "特性多項式を 0 とおいて λ について解く。",
        "register": "spoken"
      },
      {
        "en": "Find the characteristic polynomial of the matrix.",
        "ja": "行列の特性多項式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "det(A − λI) = 0 のほうは characteristic equation（特性方程式・固有方程式）。微分方程式の特性方程式も同じ名前で大量に出てくるので、候補にしなかった。",
      "用例コーパスの話し言葉は大半が MIT 18.03（線形微分方程式の特性多項式）で、行列の意味は少ない。"
    ],
    "related": [
      "eigenvalue",
      "determinant"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "固有多項式",
        "en": "Characteristic polynomial"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.3 Diagonalization and Eigenvalues"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "characteristic polynomial": 50
      },
      "written": {
        "characteristic polynomial": 22
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "chebyshevs-inequality",
    "ja": {
      "term": "チェビシェフの不等式",
      "reading": "ちぇびしぇふのふとうしき",
      "alt": []
    },
    "en": {
      "term": "Chebyshev's theorem",
      "alt": [
        "Chebyshev's inequality",
        "Chebyshev's rule"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Discrete Math",
        "Intro Statistics"
      ]
    },
    "definition_ja": "どんな分布でも、平均から標準偏差の k 倍以上離れた値をとる確率は 1/k² 以下であるという不等式。言いかえると、平均から標準偏差の k 倍より近い範囲に、全体の 1 − 1/k² 以上が入る。",
    "definition_en": "The inequality stating that, for any distribution, the probability of a value at least k standard deviations away from the mean is at most 1/k². Equivalently, at least 1 − 1/k² of the values lie within k standard deviations of the mean.",
    "latex": "P(|X - \\mu| \\ge k\\sigma) \\le \\frac{1}{k^2}",
    "spoken_en": "the probability that the absolute value of X minus mu is greater than or equal to k sigma is less than or equal to one over k squared",
    "examples": [
      {
        "en": "Chebyshev's theorem works for any distribution, not just normal ones, but the bound it gives is pretty rough.",
        "ja": "チェビシェフの不等式は正規分布に限らずどんな分布にも使えるが、得られる評価はかなり粗い。",
        "register": "spoken"
      },
      {
        "en": "A data set has mean 50 and standard deviation 5. Use Chebyshev's theorem to find the minimum percentage of the data that lies between 40 and 60.",
        "ja": "平均 50、標準偏差 5 のデータがある。チェビシェフの不等式を用いて、40 より大きく 60 より小さい範囲にあるデータの割合の最小値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は「不等式」だが、見出しの Chebyshev's theorem は用例コーパスでは書き言葉（MIT の講義ノート）に多く、話し言葉（MIT OCW）には少ない。英語版 Wikipedia の記事名は Chebyshev's inequality、OpenStax Introductory Statistics は Chebyshev's rule と書く（1 件）。",
      "正規分布の 68–95–99.7 の目安（empirical rule）とは別物。チェビシェフの不等式はどんな分布でも成り立つ下限で、k = 2 なら 75% 以上としか言えない。",
      "Chebyshev's theorem という名前は、素数の分布についての別の定理を指すこともある。"
    ],
    "related": [
      "standard-deviation",
      "variance",
      "expected-value",
      "normal-distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "チェビシェフの不等式",
        "en": "Chebyshev's inequality"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学II (8) 統計的な推測（発展・拡充）: チェビシェフの不等式"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "Chebyshev's theorem": 3
      },
      "written": {
        "Chebyshev's rule": 1,
        "Chebyshev's theorem": 12
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "check-the-concavity",
    "ja": {
      "term": "凹凸を調べる",
      "reading": "おうとつをしらべる",
      "alt": []
    },
    "en": {
      "term": "determine the concavity",
      "alt": [
        "test for concavity"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "f″(x) の符号を調べて、曲線が下に凸・上に凸になる区間を求める。",
    "definition_en": "To find, from the sign of f″(x), where a graph is concave up and where it is concave down.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To determine the concavity, look at where f double prime is positive or negative.",
        "ja": "凹凸を調べるには、f″ が正か負かを見る。",
        "register": "spoken"
      },
      {
        "en": "Determine the concavity of f(x) = x³ − 3x and find any inflection points.",
        "ja": "f(x) = x³ − 3x の凹凸を調べ、変曲点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "concavity",
      "second-derivative",
      "inflection-point"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.6（determining concavity）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "determine the concavity": 2
      },
      "written": {
        "determine the concavity": 6,
        "test for concavity": 2
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは OpenStax の呼び方 determine the concavity（本文 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "check",
    "ja": {
      "term": "確かめる",
      "reading": "たしかめる",
      "alt": [
        "解の確かめ"
      ]
    },
    "en": {
      "term": "check",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "求めた答えが正しいかどうかを調べること。方程式では、解をもとの式に代入して、両辺が等しくなるかを見る。",
    "definition_en": "To make sure an answer is right. For an equation, you substitute the solution back into the original equation and see if both sides are equal.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's check: 2 times 4 plus 3 is 11, so x = 4 works.",
        "ja": "確かめよう。2 × 4 + 3 = 11 だから、x = 4 で合っているね。",
        "register": "spoken"
      },
      {
        "en": "Solve 5x − 2 = 3x + 6 and check your answer.",
        "ja": "方程式 5x − 2 = 3x + 6 を解き、解を確かめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "check は「チェックマークを付ける」「点検する」など意味が広い。答えを確かめるのは check your answer、代入して確かめるのは check by substituting。",
      "確かめるときは、途中で変形した式ではなく、もとの方程式に代入する。変形の途中で間違えていると、変形後の式に代入しても間違いに気づけない。",
      "数値を代入して確かめても、すべての値で成り立つことの証明（prove、エントリ prove）にはならない。"
    ],
    "related": [
      "substitute",
      "solution",
      "satisfy",
      "hold",
      "solve"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "check your answer | check the answer | check our answer | check your work | check the solution | check our solution | check by | check your solution": 119
      },
      "written": {
        "check your answer | check the answer | check our answer | check your work | check the solution | check our solution | check by | check your solution": 346
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "checking-whether-the-solution-makes-sense",
    "ja": {
      "term": "解の吟味",
      "reading": "かいのぎんみ",
      "alt": [
        "解の吟味をする"
      ]
    },
    "en": {
      "term": "check that the answer makes sense",
      "alt": [
        "check that the answer is reasonable"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「解の吟味」に当たる 1 語の名詞は用例コーパスと参照に出てこず、Does the answer make sense? ／ Is the answer reasonable? と問いの形で言うか、check that the answer makes sense のように動詞句で言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "中2",
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "文章題で求めた方程式の解が、問題の条件（個数や人数は自然数、長さは正の数など）に合っているかを確かめること。合わない解は問題の答えにしない。",
    "definition_en": "Checking whether a solution of the equation fits the situation in the word problem, for example that a number of people is a positive whole number or that a length is positive. A solution that does not fit is not used as the answer.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Before you box your answer, check that the answer makes sense: you can't have negative 3 students.",
        "ja": "答えを囲む前に解の吟味をしよう。生徒が −3 人なんてありえないよね。",
        "register": "spoken"
      },
      {
        "en": "We got 4.5 buses, so check that the answer is reasonable: can you rent half a bus?",
        "ja": "バスが 4.5 台と出たから、解の吟味をしよう。バスを半台だけ借りられる？",
        "register": "spoken"
      },
      {
        "en": "Let the side length of a square with area 49 cm² be x cm. Solve x² = 49 and check that the answer makes sense.",
        "ja": "面積が 49 cm² の正方形の 1 辺の長さを x cm とする。x² = 49 を解き、解の吟味をせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「吟味」に当たる 1 語の名詞は用例コーパスと参照に出てこない。Does the answer make sense? ／ Is the answer reasonable? のように問いの形で言う。",
      "解を元の式に代入して成り立つかを確かめる検算（エントリ check）とは別。吟味では、式の上では正しい解が問題の条件（人数は自然数、長さは正など）に合うかを見る。"
    ],
    "related": [
      "word-problem",
      "check",
      "solution",
      "natural-number",
      "solve"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A(3) 一元一次方程式の活用: 求めた解がもとの問題の答えとして適切かを調べる・第 3 学年 二次方程式の活用（level.jp）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "answer makes sense | solution makes sense | answer is reasonable | solution is reasonable | reasonable answer | makes sense in the context": 8
      },
      "written": {
        "answer makes sense | solution makes sense | answer is reasonable | solution is reasonable | reasonable answer | makes sense in the context": 34
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "chi-square-test",
    "ja": {
      "term": "カイ二乗検定",
      "reading": "かいにじょうけんてい",
      "alt": [
        "χ² 検定"
      ]
    },
    "en": {
      "term": "chi-square test",
      "alt": [
        "chi-squared test"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "観測度数と期待度数のずれを表す統計量 χ² がカイ二乗分布に従うことを使う検定の総称。適合度検定・独立性の検定・同質性の検定がある。",
    "definition_en": "Any test based on the statistic χ², which measures how far observed counts are from expected counts and follows a chi-square distribution; it includes goodness-of-fit, independence and homogeneity tests.",
    "latex": "\\chi^2 = \\sum \\frac{(O - E)^2}{E}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Big differences between observed and expected give you a big chi-square value.",
        "ja": "観測度数と期待度数の差が大きいと、カイ二乗の値も大きくなる。",
        "register": "spoken"
      },
      {
        "en": "Perform a chi-square test to determine whether the die is fair.",
        "ja": "さいころが公正かどうか、カイ二乗検定で判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の表記は chi-square（OpenStax・AP Statistics の CED）と chi-squared（Wikipedia の記事名）の両方がある。CED の chi-squared は 1 件。",
      "χ（chi）は英語では「カイ」と発音する（Merriam-Webster: ˈkī）。"
    ],
    "related": [
      "goodness-of-fit-test",
      "test-for-independence",
      "test-for-homogeneity",
      "expected-count",
      "observed-count",
      "degrees-of-freedom"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "カイ二乗検定",
        "en": "Chi-squared test"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.14・3.15"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「chi」",
        "url": "https://www.merriam-webster.com/dictionary/chi",
        "note": "発音 ˈkī（2026-09-26 に確かめた）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学II (8) 統計的な推測（発展・拡充）: t 検定やカイ二乗検定"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "chi-square test": 17
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "chinese-remainder-theorem",
    "ja": {
      "term": "中国剰余定理",
      "reading": "ちゅうごくじょうよていり",
      "alt": [
        "中国の剰余定理"
      ]
    },
    "en": {
      "term": "Chinese remainder theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Chinese remainder theorem は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": []
    },
    "definition_ja": "法 m₁, …, mₖ がどの 2 つも互いに素なとき、連立合同式 x ≡ aᵢ (mod mᵢ) は必ず解をもち、その解は m₁m₂⋯mₖ を法としてただ 1 つに定まるという定理。",
    "definition_en": "If the moduli m₁, …, mₖ are pairwise relatively prime, then the system x ≡ aᵢ (mod mᵢ) always has a solution. The solution is unique modulo the product m₁m₂⋯mₖ.",
    "latex": "x \\equiv 2 \\pmod{3},\\ x \\equiv 3 \\pmod{5} \\implies x \\equiv 8 \\pmod{15}",
    "spoken_en": "if x is congruent to 2 mod 3 and x is congruent to 3 mod 5, then x is congruent to 8 mod 15",
    "examples": [
      {
        "en": "Three and five are relatively prime, so the Chinese remainder theorem tells us there's exactly one answer mod fifteen.",
        "ja": "3 と 5 は互いに素だから、中国剰余定理により 15 を法とする答えはちょうど 1 つだ。",
        "register": "spoken"
      },
      {
        "en": "Use the Chinese remainder theorem to find all integers x such that x ≡ 1 (mod 4), x ≡ 2 (mod 3), and x ≡ 3 (mod 5).",
        "ja": "中国剰余定理を用いて、x ≡ 1 (mod 4)、x ≡ 2 (mod 3)、x ≡ 3 (mod 5) を満たす整数 x をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "法が互いに素でないと解が無いことがある。x ≡ 0 (mod 2) と x ≡ 1 (mod 4) を同時に満たす x は無い。",
      "日本の数A では「3 で割ると 2 余り、5 で割ると 3 余る整数」のような問題として、定理の名前を出さずに扱うことがある。"
    ],
    "related": [
      "remainder",
      "linear-diophantine-equation",
      "modulus",
      "relatively-prime",
      "modular-inverse",
      "congruence-modulo-n"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "中国の剰余定理",
        "en": "Chinese remainder theorem"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 Chinese remainder theorem（英語版「Chinese remainder theorem」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "choose",
    "ja": {
      "term": "選ぶ",
      "reading": "えらぶ",
      "alt": []
    },
    "en": {
      "term": "choose",
      "alt": [],
      "uk": null,
      "register": "both",
      "variants": [
        {
          "term": "pick",
          "register": "spoken",
          "note": "話し言葉では choose より多く使う（MIT OCW・Khan Academy・Professor Leonard）。書き言葉では choose のほうがずっと多い。"
        }
      ]
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "いくつかのものの中から、一部を取り出す。場合の数では、順序を考えずに r 個を取り出すことを言う。",
    "definition_en": "To pick some of the objects from a larger set; in counting, choosing r objects means taking them without regard to order.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We need to choose 3 of the 10 students, and order doesn't matter, so that's 10 choose 3.",
        "ja": "10 人の生徒から 3 人を選ぶ。順序は関係ないので ₁₀C₃ だ。",
        "register": "spoken"
      },
      {
        "en": "In how many ways can a committee of 4 be chosen from 7 men and 5 women if it must include at least one woman?",
        "ja": "男子 7 人、女子 5 人から 4 人の委員を選ぶとき、女子を少なくとも 1 人含む選び方は何通りあるか。",
        "register": "written"
      },
      {
        "en": "Choose δ = ε/3. Then |x − 2| < δ implies |3x − 6| < ε.",
        "ja": "δ = ε/3 と選ぶ。すると |x − 2| < δ ならば |3x − 6| < ε となる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "₁₀C₃ は ten choose three と読む（OpenStax Intermediate Algebra は (n r) を n choose r と読むと書く）。記号は C(10, 3) か二項係数の縦書き。n choose r は記号の読み方で、動詞 choose と同じ語。",
      "証明の「δ をこう選ぶ」「適当に選ぶ」も choose（Choose δ = … ／ choose any point）。場合の数以外でも普通に使う。"
    ],
    "related": [
      "combination",
      "binomial-coefficient",
      "select-at-random",
      "order-matters",
      "arrange"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.4 Binomial Theorem（n choose r）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "choose": 513,
        "pick !up !out": 960
      },
      "written": {
        "choose": 839,
        "pick !up !out": 222
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "chord",
    "ja": {
      "term": "弦",
      "reading": "げん",
      "alt": []
    },
    "en": {
      "term": "chord",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数A"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円周上の 2 点を結ぶ線分。中心を通る弦が直径で、弦の中で最も長い。",
    "definition_en": "A line segment whose two endpoints lie on a circle; a chord through the center is a diameter, the longest chord.",
    "latex": "2\\sqrt{r^2 - d^2}",
    "spoken_en": "two times the square root of r squared minus d squared (the length of a chord at distance d from the center)",
    "examples": [
      {
        "en": "A chord just connects two points on the circle, and if it goes through the center, it's a diameter.",
        "ja": "弦は円周上の 2 点を結んだだけの線分で、中心を通るならそれが直径だよ。",
        "register": "spoken"
      },
      {
        "en": "A chord of length 8 is 3 units from the center of a circle. Find the radius of the circle.",
        "ja": "円の中心からの距離が 3 である長さ 8 の弦がある。この円の半径を求めよ。",
        "register": "written"
      },
      {
        "en": "The perpendicular from the center of a circle to a chord bisects the chord.",
        "ja": "円の中心から弦に引いた垂線は、その弦を 2 等分する。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "length of the chord",
        "ja": "弦の長さ"
      }
    ],
    "pitfalls": [
      "用例コーパスには幾何の教科書が入っていないので、chord は話し言葉・書き言葉ともほとんど出てこない。IM Geometry では glossary の見出しで、レッスン（7.1・7.2 ほか）に 39 件、CK-12 Geometry では 89 件（6.13 Segments from Chords・6.12 Chords and Central Angle Arcs ほか）出てくる。",
      "綴りは chord（h が入る）。cord（ひも）と書き間違えない。弧は arc で、弦とは別の語。"
    ],
    "related": [
      "segment",
      "arc-length",
      "inscribed-angle",
      "cyclic-quadrilateral",
      "perpendicular-bisector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "弦 (数学)",
        "en": "Chord (geometry)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary・Geometry 7.1 Lines, Angles, and Curves・Geometry 7.2 Inscribed Angles"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.12 Chords and Central Angle Arcs・6.13 Segments from Chords・6.16 Angles On and Inside a Circle"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "chord": 3
      },
      "written": {
        "chord": 1
      },
      "sources": [
        "openstax-algtrig",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CK-12 の呼び方 chord（CK-12 Geometry 6.12 Chords and Central Angle Arcs・CK-12 Geometry 6.13 Segments from Chords・CK-12 Geometry 6.16 Angles On and Inside a Circle・計 89 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "circle-through-the-intersections-of-two-circles",
    "ja": {
      "term": "2 円の交点を通る円",
      "reading": "にえんのこうてんをとおるえん",
      "alt": []
    },
    "en": {
      "term": "circle through the intersections of two circles",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。一方の式に k を掛けてもう一方に足す考え方（kf + g = 0）は、米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない。英語版 Wikipedia には記事「Pencil (geometry)」がある（節 Pencil of circles。同じ 2 点を通る円の族を elliptic pencil と呼ぶ）。説明するなら the family of circles through the two intersection points。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": []
    },
    "definition_ja": "2 円 f = 0, g = 0 の 2 つの交点を通る円（または直線）を kf + g = 0 の形で表す考え方。",
    "definition_en": "The idea of writing a circle (or line) through the two intersection points of circles f = 0 and g = 0 as kf + g = 0 (every such circle except f = 0 itself).",
    "latex": "k(x^2 + y^2 - 4) + (x^2 + y^2 - 4x - 2y) = 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "Any curve of the form k times the first plus the second passes through both intersection points.",
        "ja": "「k × 1 つ目 ＋ 2 つ目」の形の曲線は、どれも 2 つの交点を通る。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of the circle that passes through the two intersection points of x² + y² = 4 and x² + y² − 4x − 2y = 0 and through the point (1, 1).",
        "ja": "2 円 x² + y² = 4, x² + y² − 4x − 2y = 0 の 2 つの交点と点 (1, 1) を通る円の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "k = −1 のときは 2 円の交点を通る直線（共通弦を含む直線）になる。",
      "kf + g = 0 は円 f = 0 自身を表せない（f = 0 も含めるなら kf + lg = 0 と 2 つの文字を使う）。"
    ],
    "related": [
      "equation-of-a-circle",
      "point-of-intersection",
      "relative-positions-of-two-circles"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Pencil_%28geometry%29",
        "note": "節 Pencil of circles（pencil of circles・coaxial system を太字、elliptic pencil は斜体）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「円束 (射影幾何学)」",
        "url": "https://ja.wikipedia.org/wiki/%E5%86%86%E6%9D%9F_%28%E5%B0%84%E5%BD%B1%E5%B9%BE%E4%BD%95%E5%AD%A6%29",
        "note": "英: pencil of circles。二円の交点を通る円（または直線）全体の成す族"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和8年度 大学入学共通テスト 本試験 数学Ⅱ，数学Ｂ，数学Ｃ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=2144&f=abm00017550.pdf&n=2026_ol_18_sugaku2BC.pdf",
        "note": "第1問: 2 円の式の差が 2 つの交点を通る直線"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "circle",
    "ja": {
      "term": "円",
      "reading": "えん",
      "alt": []
    },
    "en": {
      "term": "circle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "平面上で、1 つの点（中心）から等しい距離にある点の集まり。その距離を半径という。",
    "definition_en": "All the points in a plane at one fixed distance, the radius, from one fixed point, the center.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Every point on the circle is 5 units from the center.",
        "ja": "円の上のどの点も、中心からの距離は 5 だ。",
        "register": "spoken"
      },
      {
        "en": "Find the area of a circle with a radius of 3 cm.",
        "ja": "半径 3 cm の円の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の circle は周の曲線だけを指し、内部を含む「円板」は disk（エントリ disk）。ただし「円の面積」は英語でも the area of a circle と言う。",
      "円の周の長さ（円周）は circumference（エントリ circumference）。",
      "動詞の circle は「丸で囲む」。問題文の Circle the answer. は「答えを丸で囲め」で、図形の円とは関係ない。"
    ],
    "related": [
      "center",
      "radius",
      "diameter",
      "circumference",
      "disk"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円 (数学)",
        "en": "Circle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "circle": 1610
      },
      "written": {
        "circle": 1175
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "circular-permutation",
    "ja": {
      "term": "円順列",
      "reading": "えんじゅんれつ",
      "alt": []
    },
    "en": {
      "term": "circular permutation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "英語版 Wikipedia の Cyclic permutation（circular permutation のリダイレクト先）は群論の巡回置換（1 つの巡回だけからなる置換）で、円順列とは別の概念。円順列に当たるのは英語版 Wikipedia「Permutation」の節 Circular permutations で、円形に並べて回転で重なるものを同じとみなす並べ方を circular permutation と呼ぶ。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "いくつかのものを円形に並べる並べ方。回転して重なるものは同じとみなすので、異なる n 個の円順列は (n − 1)! 通り。",
    "definition_en": "An arrangement of objects around a circle, where arrangements that differ only by a rotation count as the same; n distinct objects have (n − 1)! circular permutations.",
    "latex": "(n-1)!",
    "spoken_en": "n minus one factorial",
    "examples": [
      {
        "en": "For a circular permutation, fix one person's seat first, then arrange the other four: that's 4 factorial.",
        "ja": "円順列では、まず 1 人の席を固定して、残りの 4 人を並べる。だから 4! 通りだ。",
        "register": "spoken"
      },
      {
        "en": "How many circular permutations of 6 people seated at a round table are there?",
        "ja": "6 人が円卓に座るときの円順列は何通りあるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数珠や首飾りのように裏返しても同じとみなす並べ方（じゅず順列）は、異なる n 個（n ≥ 3）なら円順列をさらに 2 で割って (n − 1)!/2 通りになる。",
      "OpenStax Calculus Volume 3（5.4 Triple Integrals）の circular permutation は、変数 x, y, z を順に入れ替える（巡回置換の）意味で、円順列ではない。円卓の問題は、Levin（3.4 の練習問題）のように seating arrangements … around a round table と並べ方を文で書く。"
    ],
    "related": [
      "permutation",
      "factorial",
      "arrange",
      "permutation-with-repetition"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Permutation",
        "note": "記事「Permutation」の節 Circular permutations（circular permutation を太字で定義。数学カテゴリから 4 段）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "5.4 Triple Integrals（circular permutation of the variables）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.4 Combinations and Permutations（練習問題: seating arrangements around a round table）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元 3 の前の修正」の、残った ③ を人間が決めた行）。en.term は circular permutation のまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "circumcenter",
    "ja": {
      "term": "外心",
      "reading": "がいしん",
      "alt": []
    },
    "en": {
      "term": "circumcenter",
      "alt": [],
      "uk": "circumcentre"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "三角形の 3 辺の垂直二等分線が交わる点。3 頂点から等しい距離にあり、外接円の中心になる。",
    "definition_en": "The single point shared by the perpendicular bisectors of a triangle's three sides. It is equally far from all three vertices, so it is the center of the circumscribed circle.",
    "latex": "OA = OB = OC",
    "spoken_en": null,
    "examples": [
      {
        "en": "The circumcenter is where the perpendicular bisectors meet, so it's the same distance from all three vertices.",
        "ja": "外心は垂直二等分線の交点なので、3 つの頂点から等しい距離にある。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinates of the circumcenter of the triangle with vertices (0, 0), (8, 0), and (0, 6).",
        "ja": "3 点 (0, 0), (8, 0), (0, 6) を頂点とする三角形の外心の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "外心は鋭角三角形では内部、直角三角形では斜辺の中点、鈍角三角形では三角形の外部にある。",
      "英語版 Wikipedia「Circumcircle」は、外接円を circumcircle（circumscribed circle）、その半径を circumradius と書く。英国式のつづりは circumcentre。"
    ],
    "related": [
      "circumscribed-circle",
      "perpendicular-bisector",
      "the-five-centers-of-a-triangle",
      "law-of-sines",
      "incenter",
      "orthocenter"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary・Geometry 7.5 Triangles in Circles・Geometry 7.6 A Special Point"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences#re_er",
        "note": "記事「American and British English spelling differences」の節 -re, -er（英国の綴りは centre、米国は center）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Circumcircle",
        "note": "記事「Circumcircle」の冒頭（circumcircle・circumradius・circumcenter を太字で定義。数学カテゴリから 5 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 circumcenter（Geometry glossary・Geometry 7.5 Triangles in Circles・Geometry 7.6 A Special Point・計 20 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "circumference",
    "ja": {
      "term": "円周",
      "reading": "えんしゅう",
      "alt": [
        "円周の長さ"
      ]
    },
    "en": {
      "term": "circumference",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円のまわりの線（円周）と、その長さ。長さは円周率 × 直径で、半径を r とすると 2πr。",
    "definition_en": "The distance around a circle; the word is also used for the curve itself. It equals pi times the diameter, or 2πr for a circle with radius r.",
    "latex": "C = 2\\pi r",
    "spoken_en": "C equals two pi r",
    "examples": [
      {
        "en": "Circumference is just the perimeter of a circle, the distance all the way around.",
        "ja": "円周の長さは円の周の長さのこと、ぐるっと一周した長さだよ。",
        "register": "spoken"
      },
      {
        "en": "A circle has a radius of 5 cm. Find its circumference. Use 3.14 for π.",
        "ja": "半径 5 cm の円の円周の長さを求めよ。円周率は 3.14 とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は線（円周）と長さ（円周の長さ）を言い分けるが、英語はどちらも circumference で言う。",
      "多角形などのまわりの長さは perimeter（エントリ perimeter）で、円のときだけ circumference と言う。",
      "発音は「サーカムフェレンス」に近く、2 つ目の音節（cum）に強勢がある（Merriam-Webster: sər-ˈkəm-fə-rəns）。"
    ],
    "related": [
      "perimeter",
      "diameter",
      "pi",
      "radius",
      "circle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円周",
        "en": "Circumference"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「circumference」",
        "url": "https://www.merriam-webster.com/dictionary/circumference",
        "note": "発音 sər-ˈkəm(p)-f(ə-)rən(t)s（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "circumference": 240
      },
      "written": {
        "circumference": 127
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "circumscribed-circle",
    "ja": {
      "term": "外接円",
      "reading": "がいせつえん",
      "alt": []
    },
    "en": {
      "term": "circumscribed circle",
      "alt": [
        "circumcircle"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "三角形（多角形）のすべての頂点を通る円。三角形の外接円の中心（外心）は 3 辺の垂直二等分線の交点で、半径 R は正弦定理から a/sin A = 2R で求まる。",
    "definition_en": "The circle that passes through all the vertices of a triangle or polygon. For a triangle, its center is the common point of the three perpendicular bisectors, and its radius R satisfies a/sin A = 2R.",
    "latex": "R = \\frac{a}{2\\sin A}",
    "spoken_en": "R equals a over two sine A",
    "examples": [
      {
        "en": "The circumscribed circle goes through all three vertices, and the law of sines tells you its radius: a over sin A is 2R.",
        "ja": "外接円は 3 つの頂点をすべて通る。その半径は正弦定理でわかる。a/sin A = 2R だ。",
        "register": "spoken"
      },
      {
        "en": "In triangle ABC, a = 6 and A = 30°. Find the radius R of the circumscribed circle.",
        "ja": "△ABC において a = 6, A = 30° のとき、外接円の半径 R を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語版 Wikipedia「Circumcircle」は外接円の半径を circumradius、中心（外心）を circumcenter と呼ぶ（IM Geometry・CK-12 Geometry は circumcenter だけで、circumradius は参照と用例コーパスに出てこない）。",
      "主語の向きに注意。「円が三角形に外接する」は the circle is circumscribed about the triangle、「三角形が円に内接する」は the triangle is inscribed in the circle。",
      "IM Geometry（7.4・7.5・7.7・7.13 ほか、21 件）と CK-12 Geometry（4.22 Concurrence and Constructions、13 件）は circumscribed circle と書き、英語版 Wikipedia の三角形の外接円の記事名は Circumcircle。"
    ],
    "related": [
      "law-of-sines",
      "inscribed-circle",
      "perpendicular-bisector",
      "inscribed-angle",
      "triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "外接円",
        "en": "Circumscribed circle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.4 Quadrilaterals in Circles（9 件）・7.5 Triangles in Circles（6）・7.7 Circles in Triangles（2）・7.13 Using Radians（2）ほか、計 21 件"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.22 Concurrence and Constructions: circumscribed circle（12 件）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Circumcircle",
        "note": "三角形の外接円の記事（数学カテゴリから 5 段）: circumradius・circumcenter"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 図形の性質: 外心・内心・重心の性質。数学I 図形と計量: 正弦定理の R は外接円の半径"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "circumcircle": 1
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 circumscribed circle（Geometry 7.4 Quadrilaterals in Circles・Geometry 7.5 Triangles in Circles・Geometry 7.13 Using Radians・計 18 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "circumscribed-polygon",
    "ja": {
      "term": "外接多角形",
      "reading": "がいせつたかくけい",
      "alt": []
    },
    "en": {
      "term": "circumscribed polygon",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Integrated Math 2"
      ]
    },
    "definition_ja": "すべての辺が 1 つの円に接している多角形。このとき、その円は多角形の内接円である。",
    "definition_en": "A polygon whose sides are all tangent to a single circle. That circle is the inscribed circle of the polygon.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a circumscribed polygon, the circle touches every side exactly once, so every side is tangent to the circle.",
        "ja": "円に外接する多角形では、円がどの辺にもちょうど 1 回ずつ触れる。つまりどの辺も円の接線になっている。",
        "register": "spoken"
      },
      {
        "en": "Quadrilateral ABCD is a circumscribed polygon about circle O. If AB = 7, BC = 9, and CD = 8, find AD.",
        "ja": "四角形 ABCD は円 O に外接している。AB = 7, BC = 9, CD = 8 のとき、AD の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "向きに注意。「多角形が円に外接する」は the polygon is circumscribed about the circle で、円が内側。外接円（circumscribed circle）では円が外側なので、circumscribed が付くものの位置が逆になる。",
      "円に外接する四角形では、向かい合う辺の和が等しい（AB + CD = BC + AD）。頂点から 2 つの接点までの接線の長さが等しいことから出る。",
      "用例コーパスにも参照（CED・OpenStax・IM・CK-12）にも出てこない。"
    ],
    "related": [
      "inscribed-polygon",
      "inscribed-circle",
      "tangent-segments-are-equal"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。そのまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "class-midpoint",
    "ja": {
      "term": "階級値",
      "reading": "かいきゅうち",
      "alt": []
    },
    "en": {
      "term": "class midpoint",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスに class midpoint は出てこない）。OpenStax Introductory Statistics は the midpoint of each interval と説明する（2 件）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "各階級の真ん中の値。度数分布表から平均値を求めるとき、その階級のデータの代わりに使う。",
    "definition_en": "The value in the middle of a class interval, used in place of the data in that class when the mean is found from a frequency table.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Use the midpoint of each class as its value, then weight it by the frequency.",
        "ja": "各階級の真ん中の値を代表として、度数で重みをつける。",
        "register": "spoken"
      },
      {
        "en": "Estimate the mean from the frequency table by using class midpoints.",
        "ja": "階級値を用いて、度数分布表から平均値を推定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "class",
      "frequency-table",
      "arithmetic-mean"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "2.5 Measures of the Center of the Data（the midpoint of each interval、2 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-28",
      "human": null
    },
    "updated": "2026-09-28",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "class-width",
    "ja": {
      "term": "階級の幅",
      "reading": "かいきゅうのはば",
      "alt": []
    },
    "en": {
      "term": "class width",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスには class width は出てこず、width of the class も書き言葉にほとんど出てこない）。説明するときは the width of each interval ／ class と言う。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "1 つの階級の区間の幅（上限と下限の差）。",
    "definition_en": "The width of one class interval: its upper limit minus its lower limit.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If the class width is 10, the classes go 40 to 50, 50 to 60, and so on.",
        "ja": "階級の幅が 10 なら、階級は 40〜50、50〜60 と続く。",
        "register": "spoken"
      },
      {
        "en": "Using a class width of 5, make a frequency table for the data.",
        "ja": "階級の幅を 5 として、度数分布表を作れ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ヒストグラムでは柱の横幅が階級の幅に当たる（the width of each bar）。"
    ],
    "related": [
      "class",
      "histogram"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "class",
    "ja": {
      "term": "階級",
      "reading": "かいきゅう",
      "alt": []
    },
    "en": {
      "term": "class interval",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語の class は授業・クラスの意味で使うことがずっと多く、階級の意味では class interval と言う（書き言葉はすべて OpenStax Introductory Statistics）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "度数分布表で、データを区切るそれぞれの区間。",
    "definition_en": "One of the intervals into which the data are divided in a frequency table.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each bar of the histogram stands for one class interval.",
        "ja": "ヒストグラムの 1 本の柱が 1 つの階級を表している。",
        "register": "spoken"
      },
      {
        "en": "Group the data into class intervals of width 5, starting at 40.",
        "ja": "40 から始めて、幅 5 の階級にデータを分けよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「データを階級に分ける」は group the data into classes ／ intervals。"
    ],
    "related": [
      "class-width",
      "class-midpoint",
      "frequency-table",
      "histogram"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "class interval": 17
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "classification-by-remainder",
    "ja": {
      "term": "余りによる分類",
      "reading": "あまりによるぶんるい",
      "alt": []
    },
    "en": {
      "term": "classification by remainder",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（大学の用語 remainder class・residue class も用例コーパスには話・書とも出てこない）。Nicholson（付録 B の練習問題のヒント）も、どの整数も 3k・3k + 1・3k + 2 のどれかの形になると場合を書き出して示す。Levin（6.2.2 Remainder Classes）は、余りで分けたそれぞれの組を remainder class modulo b（residue class とも）と呼ぶ。英語版 Wikipedia「Modular arithmetic」の節 Congruence classes は congruence class ／ residue class と呼ぶ。見出しの classification by remainder は本プロジェクトの説明的な訳。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "整数を、ある自然数 m で割ったときの余り 0, 1, …, m − 1 によって m 個の組に分けること。整数の性質を場合分けで証明するときに使う。",
    "definition_en": "Sorting the integers into m groups by the remainder 0, 1, …, m − 1 they leave when divided by m, so that a statement can be checked case by case.",
    "latex": "n = 3k,\\ 3k+1,\\ 3k+2",
    "spoken_en": "n equals 3k, 3k plus 1, or 3k plus 2",
    "examples": [
      {
        "en": "Every integer is 3k, 3k + 1, or 3k + 2, so let's just check each case.",
        "ja": "どの整数も 3k, 3k + 1, 3k + 2 のどれかだから、それぞれの場合を調べよう。",
        "register": "spoken"
      },
      {
        "en": "Let k be an integer. Consider the cases n = 3k, n = 3k + 1, and n = 3k + 2, and show that n² leaves a remainder of 0 or 1 when divided by 3.",
        "ja": "k を整数とする。n = 3k, 3k + 1, 3k + 2 の場合に分けて、n² を 3 で割った余りが 0 か 1 であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "答案では「k は整数」を必ず書き添える（where k is an integer）。",
      "3k + 2 の代わりに 3k − 1 と置いてもよい。どちらも 3 で割った余りが 2 の整数を表す。"
    ],
    "related": [
      "remainder",
      "division-algorithm",
      "split-into-cases",
      "modulus",
      "parity",
      "divisibility-rules"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2.2 Remainder Classes（remainder class modulo b、residue class とも）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "Appendix B（Exercise B.1 のヒント: 3k, 3k + 1, or 3k + 2）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Modular_arithmetic",
        "note": "記事「Modular arithmetic」の節 Congruence classes（congruence class・residue class。数学カテゴリから 4 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "clear-the-denominators",
    "ja": {
      "term": "分母を払う",
      "reading": "ぶんぼをはらう",
      "alt": [
        "係数を整数にする"
      ]
    },
    "en": {
      "term": "clear the fractions",
      "alt": [
        "clear the denominators",
        "clear the decimals"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "「払う」に当たる動詞は用例コーパスと参照に出てこず、「分数を消す」（clear the fractions）、「分母を消す」（clear the denominators）と言う。係数が小数のときも同じ言い方で、両辺に 10 や 100 を掛けることを clear the decimals と言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "分数を含む方程式の両辺に分母の最小公倍数を掛けて、係数を整数にすること。係数が小数のときは、両辺に 10 や 100 を掛ける。",
    "definition_en": "To multiply both sides of an equation by the LCD of its fractions so that the fractions disappear. With decimals, multiply both sides by 10, 100, and so on.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Multiply everything by 6 to clear the fractions, and you get 3x + 2x = 30.",
        "ja": "全部に 6 を掛けて分母を払うと、3x + 2x = 30 になるね。",
        "register": "spoken"
      },
      {
        "en": "Solve x/4 − 1 = x/6. First clear the fractions by multiplying both sides by the least common denominator.",
        "ja": "方程式 x/4 − 1 = x/6 を解け。まず両辺に分母の最小公倍数を掛けて、分母を払え。",
        "register": "written"
      },
      {
        "en": "Multiply both sides by 10 to clear the decimals, and then solve 0.3x + 1.2 = 0.5x.",
        "ja": "両辺に 10 を掛けて係数を整数にしてから、方程式 0.3x + 1.2 = 0.5x を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「払う」を pay と直訳しない。英語は clear（消す）を使う。",
      "両辺のすべての項に掛ける。分数でない項に掛け忘れやすい。x/4 − 1 = x/6 を 12 倍すると 3x − 12 = 2x で、−1 も −12 になる。",
      "分母を払えるのは方程式（等式）の両辺に同じ数を掛けるとき。x/2 + x/3 のような式を計算するときは分母を払えないので、通分する（エントリ find-a-common-denominator）。"
    ],
    "related": [
      "find-a-common-denominator",
      "least-common-multiple",
      "denominator",
      "both-sides",
      "properties-of-equality"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "clear the denominators": 4,
        "clear the fractions": 2
      },
      "written": {
        "clear the fractions": 50,
        "clear the denominators": 2,
        "clear the decimals": 6
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "clockwise",
    "ja": {
      "term": "負の向き",
      "reading": "ふのむき",
      "alt": [
        "時計回り"
      ]
    },
    "en": {
      "term": "clockwise",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "near",
    "mapping_note": "日本語の「負の向き」は角の回転の向きの約束で、中身は時計回り。英語では向きをそのまま clockwise と言う。回転の向きとしての「負の向き」は学習指導要領解説・共通テスト／センター試験・日本語版 Wikipedia には出てこない（日本語版 Wikipedia「時計回り・反時計回り」は、数学では反時計回りを正の向き、測量の方位角では時計回りを正の向きとする）。共通テストの問題文の「y 軸上を負の向きに」（令和6年度 本試験 数学I 第3問）は座標が減る向きのことで、英語は in the negative direction（clockwise ではない）。",
    "domains": [
      "trigonometry",
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus"
      ]
    },
    "definition_ja": "時計の針と同じ回転の向き。角の負の向き。",
    "definition_en": "Turning in the same direction as a clock's hands; the negative direction for angles.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A rotation of −90 degrees means 90 degrees clockwise.",
        "ja": "−90° の回転は、時計回りに 90° のこと。",
        "register": "spoken"
      },
      {
        "en": "Rotate the point (0, 3) clockwise by π/2 about the origin, and give the coordinates of the image.",
        "ja": "点 (0, 3) を原点のまわりに −π/2 だけ回転した点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "counterclockwise",
      "rotation",
      "bearing",
      "general-angle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Clockwise",
        "note": "記事「Clockwise」の冒頭の太字（clockwise。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「時計回り・反時計回り」",
        "url": "https://ja.wikipedia.org/wiki/%E6%99%82%E8%A8%88%E5%9B%9E%E3%82%8A%E3%83%BB%E5%8F%8D%E6%99%82%E8%A8%88%E5%9B%9E%E3%82%8A",
        "note": "時計回り（英: clockwise）。節「数学」: 平面角は反時計回りが正の向き。節「測量や航法などの分野」: 方位角は時計回りが正の向き"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和6年度 大学入学共通テスト 本試験 数学Ⅰ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=666&f=abm00004687.pdf&n=2024_or_18_sugaku1.pdf",
        "note": "第3問: x 軸上を正の向きに、y 軸上を負の向きに（座標軸の向き）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 1.2 Naming the Moves・1.8 Rotation Patterns（clockwise）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 1.13〜1.21（clockwise）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.10 Defining Rotation・8.12 Geometry Software and Graphing Rotations（clockwise）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "回転の向きとしての「負の向き」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "clockwise": 67
      },
      "written": {
        "clockwise": 60
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "closed-interval",
    "ja": {
      "term": "閉区間",
      "reading": "へいくかん",
      "alt": []
    },
    "en": {
      "term": "closed interval",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I",
        "数III"
      ],
      "us": [
        "Algebra 2",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "両端の数を含む区間。a ≦ x ≦ b を満たす実数 x 全体で、[a, b] と書く。",
    "definition_en": "An interval that includes both of its endpoints: all real numbers x with a ≤ x ≤ b, written [a, b].",
    "latex": "[a, b] = \\{\\, x \\mid a \\le x \\le b \\,\\}",
    "spoken_en": "the closed interval from a to b",
    "examples": [
      {
        "en": "On a closed interval, a continuous function always reaches a max and a min, so check the endpoints too.",
        "ja": "閉区間では連続関数は必ず最大値と最小値をとるから、端の点も調べておこう。",
        "register": "spoken"
      },
      {
        "en": "Find the maximum and minimum values of f(x) = x² − 4x + 1 on the closed interval [0, 3].",
        "ja": "閉区間 0 ≦ x ≦ 3 における f(x) = x² − 4x + 1 の最大値と最小値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "共通テスト・センター試験の数学I は定義域を 0 ≦ x ≦ 3 のように不等式で書き、学習指導要領解説で閉区間 [a, b] が出るのは数学III の積分だけ。OpenStax Calculus と AP Calculus の CED（topic 1.16・5.5）は on the closed interval [a, b] と書く。",
      "角括弧 [ ] は端を含む、丸括弧 ( ) は端を含まない。[0, 3) のように片方だけ含む区間は閉区間でも開区間でもない。"
    ],
    "related": [
      "open-interval",
      "interval",
      "endpoint",
      "extreme-value-theorem",
      "maximum",
      "interval-notation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "平成 30 年度 数学I: ア ≦ x ≦ イ における関数の最小値（区間の記法は無い）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 積分: 閉区間 [a, b]（1 件）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.16・5.5（closed interval）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Interval_(mathematics)",
        "note": "節 Open and closed intervals: closed interval を太字で定義（日本語版「区間 (数学)」も同じ）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "closed interval": 147
      },
      "written": {
        "closed interval": 44
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "openstax-calculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cluster-sampling",
    "ja": {
      "term": "集落抽出",
      "reading": "しゅうらくちゅうしゅつ",
      "alt": [
        "集落抽出法"
      ]
    },
    "en": {
      "term": "cluster sampling",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "母集団をいくつかの集まり（集落）に分け、無作為に選んだ集落の全員を標本とする方法。",
    "definition_en": "Dividing the population into groups (clusters), randomly choosing some clusters, and taking every member of the chosen clusters.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Pick five homerooms at random and survey everyone in them; that's cluster sampling.",
        "ja": "ホームルームを 5 つ無作為に選んで全員を調べる。それが集落抽出。",
        "register": "spoken"
      },
      {
        "en": "Explain the difference between stratified sampling and cluster sampling.",
        "ja": "層化抽出と集落抽出の違いを説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "層化抽出は「各層から何人か」、集落抽出は「選んだ集まりの全員」。"
    ],
    "related": [
      "stratified-sampling",
      "systematic-sampling",
      "sampling"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "cluster sampling": 2
      },
      "written": {
        "cluster sampling": 17
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coefficient-of-determination",
    "ja": {
      "term": "決定係数",
      "reading": "けっていけいすう",
      "alt": []
    },
    "en": {
      "term": "coefficient of determination",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "回帰直線で説明できる応答変数のばらつきの割合。相関係数 r の 2 乗 r²。",
    "definition_en": "The fraction of the variation in the response variable explained by the regression line; the square of the correlation coefficient, r².",
    "latex": "r^2",
    "spoken_en": "r squared",
    "examples": [
      {
        "en": "An r squared of 0.81 means 81 percent of the variation in y is explained by the line.",
        "ja": "r² が 0.81 なら、y のばらつきの 81% が直線で説明される。",
        "register": "spoken"
      },
      {
        "en": "Find the coefficient of determination and interpret it in context.",
        "ja": "決定係数を求め、問題の文脈で解釈せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "r² は r squared と読む。"
    ],
    "related": [
      "correlation-coefficient",
      "regression-line",
      "residual-plot"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "決定係数",
        "en": "Coefficient of determination"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "coefficient of determination": 4
      },
      "written": {
        "coefficient of determination": 15
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coefficient-of-variation",
    "ja": {
      "term": "変動係数",
      "reading": "へんどうけいすう",
      "alt": []
    },
    "en": {
      "term": "coefficient of variation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "OpenStax Introductory Statistics と AP Statistics の CED には出てこない（どちらも 0 件）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "標準偏差を平均値で割った値。平均の大きさが違うデータの散らばりを比べるのに使う。",
    "definition_en": "The standard deviation divided by the mean; it compares the spread of data sets whose means differ in size.",
    "latex": "CV = \\frac{s}{\\bar{x}}",
    "spoken_en": null,
    "examples": [
      {
        "en": "To compare spread when the means are so different, use the coefficient of variation.",
        "ja": "平均がこれだけ違うときは、変動係数で散らばりを比べる。",
        "register": "spoken"
      },
      {
        "en": "Compute the coefficient of variation for each data set.",
        "ja": "それぞれのデータの変動係数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉にわずかに出てくるだけで、大半が YouTube の The Organic Chemistry Tutor。",
      "略して CV と書く。"
    ],
    "related": [
      "standard-deviation",
      "arithmetic-mean",
      "spread"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "変動係数",
        "en": "Coefficient of variation"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "coefficient of variation": 6
      },
      "sources": [
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "coefficient",
    "ja": {
      "term": "係数",
      "reading": "けいすう",
      "alt": []
    },
    "en": {
      "term": "coefficient",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "文字を含む項で、文字にかけられている数の部分。3x の係数は 3、−x の係数は −1。",
    "definition_en": "The number that multiplies the variable in a term. In 3x the coefficient is 3, and in −x it is −1.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In 7 − 4x, the coefficient of x is negative 4, so keep the sign with it.",
        "ja": "7 − 4x で、x の係数は −4。符号も一緒に考えてね。",
        "register": "spoken"
      },
      {
        "en": "State the coefficient of x in the expression 5x² − x + 8.",
        "ja": "式 5x² − x + 8 における x の係数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "x の係数は 1、−x の係数は −1。1 は書かないが、係数は 1 として扱う。",
      "相関係数（correlation coefficient）や二項係数（binomial coefficient）は別の語（エントリ correlation-coefficient、binomial-coefficient）。",
      "発音は「コウエフィシェント」に近く、fi にアクセントを置く（Merriam-Webster: ˌkō-ə-ˈfi-shənt）。"
    ],
    "related": [
      "term",
      "constant-term",
      "like-terms",
      "correlation-coefficient",
      "binomial-coefficient"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "係数",
        "en": "Coefficient"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「coefficient」",
        "url": "https://www.merriam-webster.com/dictionary/coefficient",
        "note": "発音 ˌkō-ə-ˈfi-shənt（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!correlation !binomial coefficient": 911
      },
      "written": {
        "!correlation !binomial coefficient": 885
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cofactor-expansion",
    "ja": {
      "term": "余因子展開",
      "reading": "よいんしてんかい",
      "alt": []
    },
    "en": {
      "term": "cofactor expansion",
      "alt": [
        "Laplace expansion"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（cofactor expansion ／ Laplace expansion は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列式を、1 つの行（列）の各成分とその余因子の積の和として計算する方法。",
    "definition_en": "Computing a determinant as the sum of the entries of one row (or column) times their cofactors.",
    "latex": "\\det A = \\sum_{j} a_{ij} C_{ij}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Pick the row with the most zeros and do a cofactor expansion along it.",
        "ja": "0 がいちばん多い行を選んで、その行で余因子展開しよう。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the determinant by cofactor expansion along the second column.",
        "ja": "第 2 列に沿った余因子展開で行列式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉（MIT OCW）に Laplace expansion のほうが多く、cofactor expansion はほとんど出てこない。OpenStax Algebra and Trigonometry は expand by minors（小行列式で展開する）と書く。"
    ],
    "related": [
      "cofactor",
      "minor",
      "determinant"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "余因子展開",
        "en": "Laplace expansion"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.1 The Cofactor Expansion"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "Laplace expansion": 8,
        "cofactor expansion": 1
      },
      "sources": [
        "mit-18.02",
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 9 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 cofactor expansion（3.1 The Cofactor Expansion・3.6 Proof of the Cofactor Expansion Theorem・3.2 Determinants and Matrix Inverses）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cofactor",
    "ja": {
      "term": "余因子",
      "reading": "よいんし",
      "alt": []
    },
    "en": {
      "term": "cofactor",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列の (i, j) 成分に対し、第 i 行と第 j 列を除いた行列の行列式に符号 (−1)^{i+j} を掛けた値。",
    "definition_en": "For the (i, j) entry of a square matrix, the determinant of the matrix left after deleting row i and column j, multiplied by the sign (−1)^{i+j}.",
    "latex": "C_{ij} = (-1)^{i+j} M_{ij}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Don't forget the checkerboard of signs when you write down each cofactor.",
        "ja": "余因子を書くときは、市松模様の符号を忘れないように。",
        "register": "spoken"
      },
      {
        "en": "Compute the cofactors C₁₁, C₁₂ and C₁₃ of the matrix.",
        "ja": "行列の余因子 C₁₁, C₁₂, C₁₃ を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "符号を付ける前の行列式は小行列式（minor）。OpenStax Algebra and Trigonometry は余因子という名前を使わず expand by minors と書く。"
    ],
    "related": [
      "minor",
      "cofactor-expansion",
      "adjugate",
      "determinant"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "小行列式",
        "en": "Minor (linear algebra)"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.1 The Cofactor Expansion"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "cofactor": 176
      },
      "written": {
        "cofactor": 2
      },
      "sources": [
        "mit-18.02",
        "mit-18.06",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cofunction-identity",
    "ja": {
      "term": "90° − θ の三角比",
      "reading": "きゅうじゅうどまいなすしーたのさんかくひ",
      "alt": [
        "余角公式"
      ]
    },
    "en": {
      "term": "cofunction identity",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "高等学校学習指導要領解説（数学I 図形と計量）は三角比の基本的な相互関係として sin A = cos(90° − A)・cos A = sin(90° − A) を挙げるだけで名前を付けず、OpenStax Algebra and Trigonometry（Right Triangle Trigonometry の節）は cofunction identities と呼ぶ（15 件）。near なのは、OpenStax の cofunction identities が sin／cos・tan／cot・sec／csc の 3 組で、高等学校学習指導要領〔用語・記号〕の数学I が sin・cos・tan だけだから。見出しの「90° − θ の三角比」は学習指導要領解説に無い（日本語版 Wikipedia「三角関数」は余角公式）。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "θ の余角 90° − θ の三角比を、θ の三角比で表す公式。sin(90° − θ) = cos θ、cos(90° − θ) = sin θ、tan(90° − θ) = 1/tan θ。",
    "definition_en": "The identities that express the trigonometric ratios of 90° − θ in terms of those of θ, such as sin(90° − θ) = cos θ and cos(90° − θ) = sin θ.",
    "latex": "\\sin(90^\\circ - \\theta) = \\cos\\theta,\\quad \\cos(90^\\circ - \\theta) = \\sin\\theta,\\quad \\tan(90^\\circ - \\theta) = \\frac{1}{\\tan\\theta}",
    "spoken_en": "sine of ninety degrees minus theta equals cosine theta",
    "examples": [
      {
        "en": "Sine of 90 minus theta is just cosine theta. That's the cofunction identity.",
        "ja": "sin(90° − θ) は cos θ になる。これが 90° − θ の三角比の公式だ。",
        "register": "spoken"
      },
      {
        "en": "Use a cofunction identity to write sin 70° in terms of cos.",
        "ja": "90° − θ の三角比を用いて、sin 70° を cos で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "cosine の co- は complement（余角）の意味で、「余角の sine」が cosine。それで sine と cosine、tangent と cotangent を cofunction と呼ぶ。",
      "高等学校学習指導要領〔用語・記号〕（数学I 図形と計量）は sin・cos・tan だけで cot は無く、共通テストの問題文にも cot は出てこない。cot を使わなければ tan(90° − θ) = 1/tan θ になるが、英語の cofunction identity は tan(90° − θ) = cot θ の形になる。",
      "ラジアンでは 90° − θ が π/2 − θ になる。"
    ],
    "related": [
      "trigonometric-ratio",
      "sine",
      "cosine",
      "supplementary-angle-identity",
      "table-of-trigonometric-ratios"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 三角比の基本的な相互関係として sin A = cos(90° − A)・cos A = sin(90° − A) を挙げる（名前は無い。cot も無い）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）〔用語・記号〕",
        "note": "数I 図形と計量: 正弦・余弦・正接、sin・cos・tan（cot は無い）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0",
        "note": "節「負角・余角・補角公式」: 余角公式"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "問題文に cot は 0 件"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "cofunction identity": 2
      },
      "written": {
        "cofunction identity": 18
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coin",
    "ja": {
      "term": "硬貨",
      "reading": "こうか"
    },
    "en": {
      "term": "coin",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "金属でできた円形のお金。確率では表と裏の 2 通りの結果が出る道具として使い、公正な硬貨では表と裏が同様に確からしい。",
    "definition_en": "A flat, round piece of metal money; in probability, tossing a coin has two outcomes, heads and tails. For a fair coin, heads and tails are equally likely.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each time you flip a fair coin, heads and tails are still 50-50, no matter what came up before.",
        "ja": "公正な硬貨なら、前に何が出ていても、投げるたびに表と裏は五分五分のまま。",
        "register": "spoken"
      },
      {
        "en": "A coin is tossed three times. Find the probability of getting exactly two heads.",
        "ja": "1 枚の硬貨を 3 回投げる。表がちょうど 2 回出る確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "表は heads、裏は tails で、1 枚・1 回でも -s が付く（The coin came up heads.）。「表が出る」は land heads up ／ come up heads。",
      "公正な硬貨は a fair coin。「硬貨を投げる」は toss a coin ／ flip a coin（エントリ toss）。",
      "coin は数えられる名詞で、「硬貨を 2 枚投げる」は toss two coins、「1 枚の硬貨を 2 回投げる」は toss a coin twice。"
    ],
    "related": [
      "toss",
      "die",
      "equally-likely",
      "probability",
      "tree-diagram"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "coin": 266
      },
      "written": {
        "coin": 404
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coincide",
    "ja": {
      "term": "重なる",
      "reading": "かさなる",
      "alt": []
    },
    "en": {
      "term": "coincide",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語の「重なる」は、2 つの図形がぴったり一致する意味にも、一部だけが重なり合う意味にも使う。英語の coincide は「一致する」で、ぴったり重なるときだけに使い、一部が重なり合うのは overlap。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの図形や直線が、すべての点で一致して、ぴったり重なる。",
    "definition_en": "Of two figures or lines: to fit exactly on top of each other, so that every point of one is also a point of the other.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Fold the paper along the line, and watch — the two halves coincide exactly.",
        "ja": "この線で紙を折ってみて。ほら、両側がぴったり重なるね。",
        "register": "spoken"
      },
      {
        "en": "Triangle ABC is rotated 180° about point O so that it coincides with triangle DEF. Which vertex of triangle DEF does vertex A coincide with?",
        "ja": "△ABC を点 O を中心として 180° 回転移動すると、△DEF に重なる。頂点 A と重なる頂点を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「A が B に重なる」は A coincides with B（前置詞は with）、「A と B が重なる」は A and B coincide。coincide は目的語を直接取らない。",
      "2 つの円が一部重なっているような場面で coincide を使うと、2 つが完全に同じ円だという意味になる。一部が重なり合うのは overlap。",
      "合同（congruent、エントリ congruent）は「移動すれば重ねられる」という 2 つの図形の関係。coincide は実際に同じ位置でぴったり一致していることを言う。"
    ],
    "related": [
      "congruent",
      "translation",
      "rotation",
      "reflection",
      "image"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "coincide": 15
      },
      "written": {
        "coincide": 23
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "collinear",
    "ja": {
      "term": "一直線上にある",
      "reading": "いっちょくせんじょうにある",
      "alt": [
        "共線"
      ]
    },
    "en": {
      "term": "collinear",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数A",
        "数II"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "3 つ以上の点が、すべて同じ 1 本の直線の上にのっていること。",
    "definition_en": "Describes three or more points that all lie on a single straight line.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If A, B, and C are collinear, then B has to be on the line through A and C.",
        "ja": "A, B, C が一直線上にあるなら、B は A と C を通る直線の上にあるはずだ。",
        "register": "spoken"
      },
      {
        "en": "Show that the points (1, 2), (3, 6), and (5, 10) are collinear.",
        "ja": "3 点 (1, 2), (3, 6), (5, 10) は一直線上にあることを示せ。",
        "register": "written"
      },
      {
        "en": "Any three points that are not collinear determine exactly one plane.",
        "ja": "一直線上にない 3 点は、平面をただ 1 つに定める。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "collinear は形容詞。「3 点 A, B, C は一直線上にある」は The points A, B, C are collinear.、「共線の点」は collinear points と言う。",
      "ベクトルでは、A, B, C が一直線上にある ⇔ AC→ = k AB→ となる実数 k がある（collinearity-condition を参照）。"
    ],
    "related": [
      "collinearity-condition",
      "concurrent",
      "menelauss-theorem",
      "segment"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 B 図形（空間図形）: 平面は同一直線上にない 3 点で決まる（level.jp 中1）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成27年度 大学入試センター試験 本試験 数学Ⅱ（問題）",
        "url": "https://web.archive.org/web/20160601/https://www.dnc.ac.jp/albums/abm.php?f=abm00004812.pdf&n=2701-0501+suugaku2.pdf",
        "note": "第1問(2): 3 点 O, P, Q が一直線上にある（level.jp 数II）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "collinear": 3
      },
      "written": {
        "collinear": 19
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "collinearity-condition",
    "ja": {
      "term": "共線条件",
      "reading": "きょうせんじょうけん",
      "alt": []
    },
    "en": {
      "term": "condition for collinearity",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。英語は形容詞 collinear（一直線上にある。用例コーパスでは話し言葉・書き言葉とも少ない）で言い、条件に名前をつけない。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "3 点 A, B, C が一直線上にあるための条件。AC = k AB となる実数 k があること。",
    "definition_en": "The condition for three points A, B, C to lie on one line: AC = k AB for some real number k.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If AC is a multiple of AB, then A, B, and C are collinear.",
        "ja": "AC が AB の実数倍なら、A, B, C は一直線上にある。",
        "register": "spoken"
      },
      {
        "en": "Show that the points A(1, 2), B(3, 5), and C(7, 11) are collinear.",
        "ja": "3 点 A(1, 2), B(3, 5), C(7, 11) が一直線上にあることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "parallel-vectors",
      "scalar-multiplication",
      "coplanarity-condition"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "column-space",
    "ja": {
      "term": "列空間",
      "reading": "れつくうかん",
      "alt": [
        "像"
      ]
    },
    "en": {
      "term": "column space",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列の列ベクトルが張る空間。Ax = b が解をもつのは b が列空間に含まれるときに限る。",
    "definition_en": "The space spanned by the columns of a matrix; Ax = b has a solution exactly when b is in it.",
    "latex": "\\operatorname{Col} A",
    "spoken_en": null,
    "examples": [
      {
        "en": "Ax = b is solvable exactly when b is in the column space.",
        "ja": "Ax = b が解けるのは、b が列空間に入っているときだけだ。",
        "register": "spoken"
      },
      {
        "en": "Determine whether b lies in the column space of A.",
        "ja": "b が A の列空間に含まれるか判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "線形写像の像は英語で image または range。どちらも関数の値域などの意味でも使う。",
      "用例コーパスの話し言葉はほとんどが MIT OCW 18.06。",
      "「像」は図形の変換で移った点・図形（image）の意味でも使う。それは別の見出し image（像）。"
    ],
    "related": [
      "null-space",
      "row-space",
      "rank",
      "span",
      "image"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "列空間",
        "en": "Column space"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.4 Rank of a Matrix"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "column space": 291
      },
      "sources": [
        "mit-18.06",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "column",
    "ja": {
      "term": "列",
      "reading": "れつ",
      "alt": []
    },
    "en": {
      "term": "column",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列の縦の並び。左から第 1 列、第 2 列と数える。",
    "definition_en": "A vertical line of entries in a matrix, counted from the left as the first column, the second column, and so on.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A times the first column of B gives the first column of the answer.",
        "ja": "A に B の第 1 列を掛けると、答えの第 1 列になる。",
        "register": "spoken"
      },
      {
        "en": "Write the entries of the second column of the matrix.",
        "ja": "行列の第 2 列の成分を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数学用語の「列」は数列（sequence）の意味でも使うが、行列の列は column。"
    ],
    "related": [
      "row",
      "matrix"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "first column | second column | column of the matrix": 205
      },
      "written": {
        "first column | second column | column of the matrix": 42
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "combination-with-repetition",
    "ja": {
      "term": "重複組合せ",
      "reading": "ちょうふくくみあわせ",
      "alt": []
    },
    "en": {
      "term": "combination with repetition",
      "alt": [
        "stars and bars"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（combination with repetition は用例コーパスの話し言葉には出てこず、書き言葉も MIT の講義ノートにわずかにあるだけ）。英語版 Wikipedia には記事「Combination」の節 Number of combinations with repetition がある。日本語版 Wikipedia「重複組合せ」は総数を H の記号で書くが、高等学校学習指導要領〔用語・記号〕の数学A の記号は nPr・nCr・n! で H は無い。英語で書くときは ₙ₊ᵣ₋₁Cᵣ（二項係数）の形に直す。〇と仕切りを並べるこの数え方を、Levin の Discrete Mathematics は sticks and stones と呼び（3.5 Counting Multisets ほか）、stars and bars・balls and bins とも呼ぶと添える。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "異なる n 種類のものから、同じものを何度でも選んでよいとして、順序を考えずに r 個を選ぶ選び方。総数は ₙHᵣ = ₙ₊ᵣ₋₁Cᵣ 通り。",
    "definition_en": "A selection of r objects from n kinds, where a kind may be chosen more than once and order does not matter; there are C(n + r − 1, r) of them.",
    "latex": "{}_n\\mathrm{H}_r = {}_{n+r-1}\\mathrm{C}_r",
    "spoken_en": "n plus r minus one choose r",
    "examples": [
      {
        "en": "Picking 5 donuts from 3 flavors is a combination with repetition, so line up 5 stars and 2 bars.",
        "ja": "3 種類のドーナツから 5 個選ぶのは重複組合せなので、〇 5 個と仕切り 2 本を並べる。",
        "register": "spoken"
      },
      {
        "en": "Use a combination with repetition to count the nonnegative integer solutions of x + y + z = 7.",
        "ja": "重複組合せを用いて、x + y + z = 7 を満たす 0 以上の整数の組 (x, y, z) の個数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ₙHᵣ の H は、日本語版 Wikipedia「重複組合せ」が使う記号。Levin（3.5）は二項係数で書くので、英語で書くときは ₙ₊ᵣ₋₁Cᵣ か (n + r − 1 choose r) にする。",
      "「各文字を 1 個以上」の条件があるときは、先に 1 個ずつ配ってから重複組合せにする。条件を読み落とすと 0 個の場合まで数えてしまう。"
    ],
    "related": [
      "combination",
      "permutation-with-repetition",
      "binomial-coefficient",
      "counting"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.5 Counting Multisets（sticks and stones。stars and bars・balls and bins とも）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Combination",
        "note": "記事「Combination」の節 Number of combinations with repetition（数学カテゴリから 3 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「重複組合せ」",
        "url": "https://ja.wikipedia.org/wiki/%E9%87%8D%E8%A4%87%E7%B5%84%E5%90%88%E3%81%9B",
        "note": "総数を H の記号で書く"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）〔用語・記号〕",
        "note": "数学A 場合の数と確率: nPr、nCr、n!（H は無い）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "combination with repetition": 1
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 1 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "combination",
    "ja": {
      "term": "組合せ",
      "reading": "くみあわせ",
      "alt": []
    },
    "en": {
      "term": "combination",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "異なる n 個のものから、順序を考えずに r 個を選ぶ選び方。総数は ₙCᵣ = n!/(r!(n − r)!)。",
    "definition_en": "A selection of r objects from n distinct objects in which order does not matter; the number of them is n!/(r!(n − r)!).",
    "latex": "{}_n\\mathrm{C}_r = \\binom{n}{r} = \\frac{n!}{r!\\,(n-r)!}",
    "spoken_en": "n choose r equals n factorial over r factorial times n minus r factorial",
    "examples": [
      {
        "en": "Order doesn't matter when you pick a committee, so it's a combination, not a permutation.",
        "ja": "委員を選ぶときは順序が関係ないので、順列ではなく組合せだ。",
        "register": "spoken"
      },
      {
        "en": "Find the number of combinations of 3 books chosen from 8 different books.",
        "ja": "異なる 8 冊の本から 3 冊を選ぶ組合せの総数を求めよ。",
        "register": "written"
      },
      {
        "en": "By the combination formula, C(10, 2) = 10!/(2! 8!) = 45.",
        "ja": "組合せの公式より ₁₀C₂ = 10!/(2! 8!) = 45。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "combination は一次結合（linear combination）や混ぜ合わせの意味でも使う。",
      "高等学校学習指導要領〔用語・記号〕（数学A）は nCr と書く。OpenStax Algebra and Trigonometry（13.5 Counting Principles）は C(n, r) と書き、二項定理（13.6）では縦に並べた (n r) も使う。OpenStax Intermediate Algebra（12.4 Binomial Theorem）は (n r) を n choose r と読む。",
      "組合せ錠（combination lock）の combination は「番号の並び」で、順序が大事。数学の combination とは逆なので注意。"
    ],
    "related": [
      "permutation",
      "binomial-coefficient",
      "choose",
      "order-matters",
      "combination-with-repetition"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "組合せ (数学)",
        "en": "Combination"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles（C(n, r)）・13.6 Binomial Theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.4 Binomial Theorem（n choose r）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）〔用語・記号〕",
        "note": "数学A 場合の数と確率: nCr"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "number of combinations | permutations and combinations | combinations of n | combination formula | combinations of r | a combination of r | combinations of k": 2
      },
      "written": {
        "number of combinations | permutations and combinations | combinations of n | combination formula | combinations of r | a combination of r | combinations of k": 9
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "yt:nancypi"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 number of combinations | permutations and combinations | combinations of n | combination formula | combinations of r | a combination of r | combinations of k（本文 8 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "combine-like-terms",
    "ja": {
      "term": "同類項をまとめる",
      "reading": "どうるいこうをまとめる",
      "alt": []
    },
    "en": {
      "term": "combine like terms",
      "alt": [
        "collect like terms"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "文字の部分が同じ項（同類項）どうしを、係数を足したり引いたりして 1 つの項にまとめる。3x + 5x = 8x のように計算する。",
    "definition_en": "To add or subtract terms that have exactly the same variable part by adding or subtracting their coefficients, so that they become a single term. For example, 3x + 5x = 8x.",
    "latex": "ax + bx = (a + b)x",
    "spoken_en": "a x plus b x equals the quantity a plus b times x",
    "examples": [
      {
        "en": "In 4x + 7 + 2x − 3, combine like terms first: 4x plus 2x is 6x, and 7 minus 3 is 4.",
        "ja": "4x + 7 + 2x − 3 では、まず同類項をまとめよう。4x + 2x で 6x、7 − 3 で 4 だね。",
        "register": "spoken"
      },
      {
        "en": "Combine like terms to simplify 5a − 2b − 3a + 6b.",
        "ja": "同類項をまとめて、5a − 2b − 3a + 6b を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「同類項」は like terms（エントリ like-terms）。この like は「似た・同じ種類の」という意味の形容詞で、動詞の like（好き）ではない。",
      "まとめられるのは文字の部分がまったく同じ項だけ。x と x²、x と y は同類項ではないので、3x + 2x² や 3x + 2y はこれ以上まとめられない。"
    ],
    "related": [
      "like-terms",
      "coefficient",
      "term",
      "simplify",
      "distributive-property"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "combine like terms": 90,
        "collect like terms": 3
      },
      "written": {
        "combine like terms": 214,
        "collect like terms": 7
      },
      "sources": [
        "khan-algebra",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "common-difference",
    "ja": {
      "term": "公差",
      "reading": "こうさ",
      "alt": []
    },
    "en": {
      "term": "common difference",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "等差数列で、隣り合う項の差 aₙ₊₁ − aₙ（一定の値）。",
    "definition_en": "In an arithmetic sequence, the fixed difference aₙ₊₁ − aₙ between consecutive terms.",
    "latex": "d = a_{n+1} - a_n",
    "spoken_en": null,
    "examples": [
      {
        "en": "The common difference is negative three, so the terms go down by 3 each time.",
        "ja": "公差が −3 だから、項は 3 ずつ減っていく。",
        "register": "spoken"
      },
      {
        "en": "Find the common difference d of the arithmetic sequence −7, −3, 1, 5, ….",
        "ja": "等差数列 −7, −3, 1, 5, … の公差 d を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では少なく、書き言葉（OpenStax Algebra and Trigonometry・OpenStax Intermediate Algebra）で多い語。",
      "日本語版 Wikipedia「公差」は工学の許容差（tolerance）の記事で、数列の公差とは別の語。"
    ],
    "related": [
      "arithmetic-sequence",
      "common-ratio",
      "first-term",
      "consecutive-terms",
      "difference"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「公差」",
        "url": "https://ja.wikipedia.org/wiki/%E5%85%AC%E5%B7%AE",
        "note": "機械工学などで許容される差（tolerance）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "common difference": 7
      },
      "written": {
        "common difference": 156
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "common-factor",
    "ja": {
      "term": "共通因数",
      "reading": "きょうつういんすう",
      "alt": []
    },
    "en": {
      "term": "common factor",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2",
        "中3",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "式のどの項にも共通して含まれる因数。分配法則を逆に使って、かっこの外にくくり出せる。",
    "definition_en": "A factor shared by every term of an expression (or by every number in a list); by the distributive property it can be pulled out in front of parentheses.",
    "latex": "ab + ac = a(b + c)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Both terms have a common factor of 3x, so factor out the 3x first.",
        "ja": "どちらの項にも共通因数 3x があるから、まず 3x をくくり出そう。",
        "register": "spoken"
      },
      {
        "en": "Factor 6x²y − 9xy² by factoring out the common factor.",
        "ja": "6x²y − 9xy² を、共通因数をくくり出して因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「共通因数をくくり出す」は factor out the common factor。「くくる」を直訳しない。",
      "数の公約数も英語では common factor という。"
    ],
    "related": [
      "polynomial",
      "multiple",
      "prime-factorization"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!greatest common factor": 110
      },
      "written": {
        "!greatest common factor": 323
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "common-logarithm",
    "ja": {
      "term": "常用対数",
      "reading": "じょうようたいすう",
      "alt": []
    },
    "en": {
      "term": "common logarithm",
      "alt": [
        "common log",
        "base 10 logarithm"
      ],
      "variants": [
        {
          "term": "log base 10",
          "register": "spoken",
          "note": "底の読み。話し言葉では common logarithm・common log より多く、書き言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "10 を底とする対数 log₁₀ x。",
    "definition_en": "The logarithm to base 10, log₁₀ x.",
    "latex": "\\log_{10} x",
    "spoken_en": "log base ten of x",
    "examples": [
      {
        "en": "Log base 10 of 1000 is 3, because 10 cubed is 1000.",
        "ja": "log₁₀ 1000 は 3。10³ = 1000 だから。",
        "register": "spoken"
      },
      {
        "en": "Using log₁₀ 2 = 0.3010, find the common logarithm of 5.",
        "ja": "log₁₀ 2 = 0.3010 を用いて、5 の常用対数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry（6.3）は、底を書かない log x を常用対数とする（IM Algebra 2 の 4.9 も同じ）。高等学校学習指導要領解説（数学II）と共通テスト（令和 7 年度 本試験 数学II，数学B，数学C 第 2 問）は、底 10 を書いて log₁₀ と表す。",
      "話し言葉の log base 10 は大半が 3Blue1Brown。"
    ],
    "related": [
      "logarithm",
      "natural-logarithm",
      "number-of-digits",
      "logarithmic-scale",
      "leading-digit",
      "change-of-base-formula"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "常用対数",
        "en": "Common logarithm"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.3 Logarithmic Functions（Definition of the Common Logarithm）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 4.9 What is a Logarithm?（base 10 logarithm）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 指数関数・対数関数: 例は log₁₀ と底を書く"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和7年度 大学入学共通テスト 本試験 数学Ⅱ，数学Ｂ，数学Ｃ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=771&f=abm00005945.pdf&n=2025_op_30_sugaku2BC.pdf",
        "note": "第2問: log₁₀ と底を書く"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "log base 10": 67,
        "common logarithm": 1,
        "common log": 2
      },
      "written": {
        "common logarithm": 23,
        "common log": 8,
        "log base 10": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "common-ratio",
    "ja": {
      "term": "公比",
      "reading": "こうひ",
      "alt": []
    },
    "en": {
      "term": "common ratio",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus BC",
        "Integrated Math 1",
        "Algebra 1"
      ]
    },
    "definition_ja": "等比数列で、隣り合う項の比 aₙ₊₁ / aₙ（一定の値）。",
    "definition_en": "In a geometric sequence, the fixed ratio aₙ₊₁ / aₙ of consecutive terms.",
    "latex": "r = \\frac{a_{n+1}}{a_n}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The common ratio is negative three, so the signs alternate.",
        "ja": "公比が −3 だから符号が交互に変わる。",
        "register": "spoken"
      },
      {
        "en": "Find the common ratio of the geometric sequence 81, 27, 9, 3, ….",
        "ja": "等比数列 81, 27, 9, 3, … の公比を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉はほとんどが Khan Academy。",
      "IM Algebra 2（1.2）は公比を growth factor とも呼ぶ（指数関数の増え方の言い方と同じ）。"
    ],
    "related": [
      "geometric-sequence",
      "common-difference",
      "infinite-geometric-series",
      "sum-of-a-geometric-sequence",
      "first-term"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.3 Geometric Sequences（common ratio）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 1.2 Introducing Geometric Sequences（growth factor or common ratio）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "common ratio": 169
      },
      "written": {
        "common ratio": 158
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-intalg",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "common-tangent",
    "ja": {
      "term": "共通接線",
      "reading": "きょうつうせっせん",
      "alt": []
    },
    "en": {
      "term": "common tangent",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "2 つの円（または 2 つの曲線）の両方に接する直線。2 円が共通接線の同じ側にあれば共通外接線、反対側にあれば共通内接線という。",
    "definition_en": "A line tangent to each of two circles (or curves). It is an external tangent if both circles lie on the same side of it and an internal tangent if they lie on opposite sides.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These two circles are completely separate — they don't even touch — so they have four common tangents: two external ones and two internal ones that cross between the circles.",
        "ja": "この 2 円は離れているので、共通接線は 4 本ある。共通外接線が 2 本と、2 円の間で交わる共通内接線が 2 本。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of every common tangent to the circles x² + y² = 1 and (x − 4)² + y² = 1.",
        "ja": "2 円 x² + y² = 1, (x − 4)² + y² = 1 の共通接線の方程式をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語版 Wikipedia「Tangent lines to circles」は、共通外接線を external tangent line（節 Outer tangent）、共通内接線を internal tangent line（節 Inner tangent）と書く。日本語の「外接・内接」を circumscribed ／ inscribed と訳さない。",
      "本数は 2 円の位置関係で決まる。離れていれば 4 本、外接なら 3 本、2 点で交われば 2 本、内接なら 1 本、一方が内部にあれば 0 本。"
    ],
    "related": [
      "tangent-line",
      "relative-positions-of-two-circles",
      "tangent-segments-are-equal",
      "equation-of-a-circle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.2 Identify Circle Components"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Tangent_lines_to_circles",
        "note": "記事「Tangent lines to circles」の節 Outer tangent ／ Inner tangent（external ／ internal tangent line。数学カテゴリから 4 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 common tangent（CK-12 Geometry 6.2 Identify Circle Components・計 1 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "commutative-property",
    "ja": {
      "term": "交換法則",
      "reading": "こうかんほうそく",
      "alt": []
    },
    "en": {
      "term": "commutative property",
      "alt": [],
      "variants": [
        {
          "term": "commutative law",
          "register": "spoken",
          "note": "話し言葉で commutative property に次いで使う（ほとんどが Khan Academy）。OpenStax Prealgebra は commutative property と書く。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "2 つの数の和や積で、順番を入れかえても結果が変わらないという法則。a + b = b + a、ab = ba。",
    "definition_en": "The rule that changing the order of two numbers does not change their sum or their product: a + b = b + a and ab = ba.",
    "latex": "a + b = b + a, \\quad ab = ba",
    "spoken_en": "a plus b equals b plus a, and a b equals b a",
    "examples": [
      {
        "en": "8 plus 5 and 5 plus 8 are both 13 — that's the commutative property, so the order doesn't matter.",
        "ja": "8 + 5 も 5 + 8 もどちらも 13。これが交換法則で、順番は関係ない。",
        "register": "spoken"
      },
      {
        "en": "Use the commutative property of addition to rewrite x + 9.",
        "ja": "加法の交換法則を使って、x + 9 を書きかえよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "成り立つのは加法と乗法だけで、減法・除法では成り立たない（5 − 3 ≠ 3 − 5、8 ÷ 2 ≠ 2 ÷ 8）。",
      "OpenStax Prealgebra は law ではなく property と書き、どの演算の法則かを付けて commutative property of addition ／ of multiplication とも言う。話し言葉では commutative law とも言う。",
      "結合法則（エントリ associative-property）との区別: 交換法則は数の順番を入れかえる法則、結合法則は順番はそのままでかっこの位置を変える法則。"
    ],
    "related": [
      "associative-property",
      "distributive-property",
      "addition",
      "multiplication"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "交換法則",
        "en": "Commutative property"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "commutative law": 10,
        "commutative property": 21
      },
      "written": {
        "commutative property": 90,
        "commutative law": 1
      },
      "sources": [
        "khan-middle",
        "mit-18.03",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "compare-coefficients",
    "ja": {
      "term": "係数を比較する",
      "reading": "けいすうをひかくする",
      "alt": [
        "係数を比べる",
        "係数比較"
      ]
    },
    "en": {
      "term": "equate coefficients",
      "alt": [
        "compare … coefficients",
        "compare coefficients"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "恒等式の両辺で、同じ次数の項の係数が等しいとおいて未知の係数を決める。",
    "definition_en": "To find unknown coefficients by setting the coefficients of like powers on the two sides of an identity equal to each other.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Now equate coefficients: the x² terms give A + B = 3.",
        "ja": "では係数を比較しよう。x² の項から A + B = 3。",
        "register": "spoken"
      },
      {
        "en": "Find a, b, c so that x² + 2x + 3 = a(x − 1)² + b(x − 1) + c holds for all x, by equating coefficients.",
        "ja": "x² + 2x + 3 = a(x − 1)² + b(x − 1) + c が x についての恒等式となるように、係数を比較して a, b, c を定めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 2（3.4 Partial Fractions）は部分分数分解の係数を決める手順を equate coefficients（method of equating coefficients）と書き、OpenStax Precalculus（7.4 Partial Fractions）・Algebra and Trigonometry（11.4）は compare the coefficients of both sides と書く。用例コーパスでは話し言葉・書き言葉とも少ない。",
      "「係数比較法」と「数値代入法」は日本語の呼び名で、学習指導要領解説には出てこない（高等学校学習指導要領解説（数学II）は、恒等式の未定係数法を等式の証明とあわせて扱ってよいと書く）。英語はどちらも手順として言う（equate coefficients ／ plug in convenient values of x）。"
    ],
    "related": [
      "identity",
      "substituting-values",
      "partial-fraction-decomposition"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "3.4 Partial Fractions（method of equating coefficients）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "「係数比較法」「数値代入法」は出てこない"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式: 恒等式の未定係数法（等式の証明に関連して）。「係数比較法」「数値代入法」は出てこない"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "7.4 Partial Fractions（compare the coefficients of both sides。pitfalls[0]・en.alt）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.4 Partial Fractions（compare the coefficients of both sides。pitfalls[0]・en.alt）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "compare coefficients": 2,
        "equate coefficients": 3
      },
      "written": {
        "equate coefficients": 6,
        "compare … coefficients": 3
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "yt:patrickjmt"
      ],
      "counted": "2026-10-09"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 equate coefficients（本文 6 件）。register は主張しない。",
        "raised": "2026-10-09"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "compare",
    "ja": {
      "term": "比較する",
      "reading": "ひかくする",
      "alt": [
        "大小を比較する",
        "大小を比べる"
      ]
    },
    "en": {
      "term": "compare",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics",
      "algebra",
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "中2",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "AP Statistics"
      ]
    },
    "definition_ja": "2 つ以上の数・式の大小や、2 つ以上のデータの分布（代表値・散らばり）の違いを調べる。",
    "definition_en": "To determine which of two or more numbers or expressions is larger, or how two or more distributions differ in center and spread.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To compare 2^30 and 3^20, write both as powers with the same exponent.",
        "ja": "2³⁰ と 3²⁰ を比べるには、両方を同じ指数の累乗で表す。",
        "register": "spoken"
      },
      {
        "en": "Compare the sizes of log₂ 3, log₄ 5, and 3/2.",
        "ja": "log₂ 3, log₄ 5, 3/2 の大小を比較せよ。",
        "register": "written"
      },
      {
        "en": "Compare the two box plots: which class did better, and which had more spread?",
        "ja": "2 つの箱ひげ図を比べよう。どちらのクラスの成績がよく、どちらのほうが散らばりが大きいか。",
        "register": "spoken"
      },
      {
        "en": "Compare the distributions of the two data sets using their medians and interquartile ranges.",
        "ja": "2 つのデータの分布を、中央値と四分位範囲を使って比較せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「大小を比較せよ」は Compare …、または Which is larger? と聞く。並べる問いは Arrange … in increasing order。",
      "指数関数・対数関数の大小比較では、底が 1 より大きいか小さいかで向きが変わる（The base is greater than 1, so the function is increasing.）。",
      "分布の比較では、中心（center）と散らばり（spread）の両方に触れて比べる。"
    ],
    "related": [
      "exponential-function",
      "logarithmic-function",
      "box-plot",
      "distribution",
      "spread"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "compare": 395
      },
      "written": {
        "compare": 527
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "comparison-for-divergence",
    "ja": {
      "term": "追い出しの原理",
      "reading": "おいだしのげんり",
      "alt": []
    },
    "en": {
      "term": "comparison for divergence",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。数列の「aₙ ≧ bₙ で bₙ → ∞ なら aₙ → ∞」という性質は、参照に名前も定理の形も出てこない（同じ比較を、OpenStax Calculus Volume 2 は広義積分で A Comparison Theorem（3.7）、級数で Comparison Test（5.4）として示し、AP の CED（topic 10.6）も級数の comparison test を扱う）。数列で書くときは Since aₙ ≥ bₙ and bₙ → ∞, aₙ → ∞ as well. と理由を文で書く。見出しの英語は本プロジェクトの説明的な訳。日本語の「追い出しの原理」は大学受験で使われる名前（日本語版 Wikipedia「はさみうちの原理」）。",
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": []
    },
    "definition_ja": "すべての n で aₙ ≧ bₙ であり、bₙ が正の無限大に発散するなら、aₙ も正の無限大に発散するという性質。",
    "definition_en": "The property that if aₙ ≥ bₙ for every n and bₙ → ∞, then aₙ → ∞ as well.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Since a sub n is always bigger than n, and n goes to infinity, a sub n goes to infinity too.",
        "ja": "aₙ はいつも n より大きく、n が正の無限大に発散するので、aₙ も発散する。",
        "register": "spoken"
      },
      {
        "en": "Since 2ⁿ ≥ n for all n ≥ 1 and n → ∞, it follows that 2ⁿ → ∞.",
        "ja": "n ≧ 1 のとき 2ⁿ ≧ n であり n → ∞ なので、2ⁿ → ∞ である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "squeeze-theorem",
      "diverge-to-positive-infinity"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「はさみうちの原理」",
        "url": "https://ja.wikipedia.org/wiki/%E3%81%AF%E3%81%95%E3%81%BF%E3%81%86%E3%81%A1%E3%81%AE%E5%8E%9F%E7%90%86",
        "note": "追い出しの原理（日本の大学受験で使われる名前）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "3.7 Improper Integrals（A Comparison Theorem）・5.4 Comparison Tests（Comparison Test。mapping_note）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.6（comparison test。mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "comparison-test-for-integrals",
    "ja": {
      "term": "比較判定法（積分）",
      "reading": "ひかくはんていほう・せきぶん",
      "alt": []
    },
    "en": {
      "term": "comparison theorem",
      "alt": [
        "comparison test for integrals"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "0 ≦ f(x) ≦ g(x) のとき、g の広義積分が収束すれば f も収束し、f が発散すれば g も発散する、という判定法。",
    "definition_en": "If 0 ≤ f(x) ≤ g(x), then the improper integral of f converges when that of g does, and that of g diverges when that of f does.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Compare it with one over x squared: it's smaller, and that one converges, so ours converges too.",
        "ja": "1/x² と比べる。こちらのほうが小さくて、1/x² は収束するから、この積分も収束する。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "級数の比較判定法は comparison test（direct comparison test）。広義積分では comparison theorem と呼ぶ教科書があり、comparison test と呼ぶこともある。"
    ],
    "related": [
      "convergence-of-improper-integrals",
      "improper-integral",
      "comparison-test"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "comparison theorem": 5
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 comparison theorem（本文 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "comparison-test",
    "ja": {
      "term": "比較判定法",
      "reading": "ひかくはんていほう",
      "alt": []
    },
    "en": {
      "term": "comparison test",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "0 ≤ aₙ ≤ bₙ のとき、Σbₙ が収束すれば Σaₙ も収束し、Σaₙ が発散すれば Σbₙ も発散するという判定法。",
    "definition_en": "If 0 ≤ aₙ ≤ bₙ, then Σaₙ converges when Σbₙ does, and Σbₙ diverges when Σaₙ does.",
    "latex": "0 \\le a_n \\le b_n",
    "spoken_en": null,
    "examples": [
      {
        "en": "Compare it to 1/n²: it's smaller, and 1/n² converges, so this converges too.",
        "ja": "1/n² と比べよう。こちらのほうが小さく、1/n² は収束するから、これも収束する。",
        "register": "spoken"
      },
      {
        "en": "Use the comparison test to show that Σ 1/(n² + 5) converges.",
        "ja": "比較判定法を用いて Σ 1/(n² + 5) が収束することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "極限比較判定法（limit comparison test）と区別して direct comparison test と言うこともある（話し言葉にはほとんど出てこない）。"
    ],
    "related": [
      "limit-comparison-test",
      "p-series",
      "integral-test",
      "comparison-test-for-integrals"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "比較判定法",
        "en": "Direct comparison test"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.6（comparison tests）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "comparison test": 100
      },
      "written": {
        "comparison test": 57
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "compass",
    "ja": {
      "term": "コンパス",
      "reading": "こんぱす",
      "alt": []
    },
    "en": {
      "term": "compass",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "円や弧をかく道具。脚の開きを保ったまま動かして、長さを別の場所に写し取ることにも使う。",
    "definition_en": "A tool for drawing circles and arcs, with one pointed leg that stays at the center. It can also copy a length from one place to another.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Put the point of your compass on A and draw an arc that crosses the line twice.",
        "ja": "コンパスの針を A に置いて、直線と 2 か所で交わる弧をかいて。",
        "register": "spoken"
      },
      {
        "en": "Using a compass, draw a circle with center O and a radius of 4 cm.",
        "ja": "コンパスを使って、点 O を中心とする半径 4 cm の円をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "方位磁針も compass と言う。",
      "英語版 Wikipedia「Compass (drawing tool)」は a pair of compasses とも呼ぶ。",
      "角度を測る分度器は protractor で、compass とは別の道具。"
    ],
    "related": [
      "construction",
      "straightedge",
      "circle",
      "arc",
      "construct"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 1.8 Using Technology for Constructions・1.1 Build It・1.5 Construction Techniques 3"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Compass_(drawing_tool)",
        "note": "記事「Compass (drawing tool)」の冒頭: also commonly known as a pair of compasses（地域は書いていない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "compass": 6
      },
      "written": {
        "compass": 1
      },
      "sources": [
        "mit-6.042",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 6 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 compass（Geometry 1.8 Using Technology for Constructions・Geometry 1.1 Build It・Geometry 1.5 Construction Techniques 3: Perpendicular Lines and Angle Bisectors・計 85 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "complement",
    "ja": {
      "term": "補集合",
      "reading": "ほしゅうごう",
      "alt": []
    },
    "en": {
      "term": "complement",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "全体集合 U の要素のうち、集合 A に属さないもの全体の集合。",
    "definition_en": "The set of all elements of the universal set U that are not in A.",
    "latex": "\\overline{A}",
    "spoken_en": "A complement (or A bar)",
    "examples": [
      {
        "en": "The complement of A is just everything in U that's not in A.",
        "ja": "A の補集合は、U の中で A に入っていないもの全部だ。",
        "register": "spoken"
      },
      {
        "en": "Let U be the set of integers from 1 to 20 and A the set of multiples of 3 in U. Find the complement of A.",
        "ja": "1 から 20 までの整数全体を U、そのうち 3 の倍数全体を A とする。A の補集合を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高等学校学習指導要領解説（数学I）は集合の記号として Ā（A の補集合）を挙げ、共通テスト（令和 4 年度 数学I ほか）は「X の補集合を X̄ と表す」と書く。英語の本の記号は分かれる: OpenStax Introductory Statistics は A′、Algebra and Trigonometry は E′、Levin は Ā と書き、英語版 Wikipedia「Complement (set theory)」は Aᶜ も挙げる。",
      "同じ complement が余事象（complement of an event）にも使われ、complementary angle（余角）という形でも出てくる。",
      "つづりは complement（補う）。compliment（ほめ言葉）と取り違えない。"
    ],
    "related": [
      "universal-set",
      "union",
      "de-morgans-laws",
      "venn-diagram",
      "set",
      "intersection"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 集合と命題: 集合の記号 Ā（A の補集合）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "令和 4 年度 数学I ほか: X の補集合を X̄ と表す（8 件）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "Sets: Ā（complement）"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "補集合",
        "en": "Complement (set theory)"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "complement of": 14
      },
      "written": {
        "complement of": 38
      },
      "sources": [
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "complementary-angle",
    "ja": {
      "term": "余角",
      "reading": "よかく",
      "alt": []
    },
    "en": {
      "term": "complementary angles",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1",
        "Pre-Algebra"
      ]
    },
    "definition_ja": "和が 90° になる 2 つの角の、一方から見た他方の角。θ の余角は 90° − θ。",
    "definition_en": "Two angles are complementary when their measures add up to 90°; the complement of an angle θ is 90° − θ.",
    "latex": "\\alpha + \\beta = 90^\\circ",
    "spoken_en": null,
    "examples": [
      {
        "en": "The acute angles of a right triangle always add to 90, so they're complementary angles: if one is 30, the other is 60.",
        "ja": "直角三角形の 2 つの鋭角は足すと 90° になるので余角どうしだ。一方が 30° なら他方は 60° だ。",
        "register": "spoken"
      },
      {
        "en": "Two complementary angles differ by 18°. Find the measure of each angle.",
        "ja": "互いに余角である 2 つの角の差は 18° である。それぞれの角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「補角」は supplementary、「余角」は complementary。「補」の字から complement を補角と取り違えやすい。",
      "complement は「補集合」「余事象」にも使う（the complement of A）。三角関数の cosine の co- も、英語版 Wikipedia「Sine and cosine」（Etymology）によると余角（complement）の sine の意味（cofunction-identity）。",
      "complementary（補い合う）と complimentary（無料の、ほめる）はつづりが 1 字違い。"
    ],
    "related": [
      "supplementary-angle",
      "complement",
      "cofunction-identity",
      "angle",
      "adjacent-angles"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Sine_and_cosine",
        "note": "記事「Sine and cosine」の節 Etymology（cosine は complementi sinus から）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 7 7.2〜7.5（complementary、glossary）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "complementary angles": 4
      },
      "written": {
        "complementary angles": 26
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "complementary-event",
    "ja": {
      "term": "余事象",
      "reading": "よじしょう",
      "alt": []
    },
    "en": {
      "term": "complement of an event",
      "alt": [
        "complementary event"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "AP Statistics",
        "Integrated Math 2",
        "Intro Statistics"
      ]
    },
    "definition_ja": "事象 A に対して、「A が起こらない」という事象のこと。標本空間の根元事象のうち、A に含まれないもの全体からなる。",
    "definition_en": "The event that A does not happen: it contains exactly the outcomes of the sample space that A leaves out, and its probability is 1 − P(A).",
    "latex": "P(A^c) = 1 - P(A)",
    "spoken_en": "the probability of A complement equals one minus the probability of A",
    "examples": [
      {
        "en": "It's easier to find the probability of the complement of the event and subtract it from one.",
        "ja": "余事象の確率を求めて 1 から引くほうが楽だ。",
        "register": "spoken"
      },
      {
        "en": "Two dice are rolled. Let A be the event that the sum is at least 4. Describe the complement of the event A, and use it to find P(A).",
        "ja": "2 個のさいころを投げる。目の和が 4 以上である事象を A とする。A の余事象を述べ、それを用いて P(A) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "記号が違う。センター試験の問題文（平成 30 年度 数学Ⅰ・数学Ａ）は余事象を Ā と書き、学習指導要領解説も P(Ā) = 1 − P(A) と書く。OpenStax Introductory Statistics は A′、Algebra and Trigonometry は E′ と書く（教科書による）。",
      "互いに排反（mutually exclusive）な 2 つの事象でも、合わせて標本空間全体にならなければ余事象どうしではない。",
      "用例コーパスでは、書き言葉では complement of an event の形（OpenStax Introductory Statistics・Algebra and Trigonometry）のほうが多く、complementary event は話・書ともほとんど出てこない。"
    ],
    "related": [
      "complement",
      "sample-space",
      "union-of-events",
      "intersection-of-events",
      "mutually-exclusive-events"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成30年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20190601/https://www.dnc.ac.jp/albums/abm.php?f=abm00033141.pdf&n=h30+suugaku1+02suugaku1A.pdf",
        "note": "事象 A の余事象を Ā と表す"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 場合の数と確率: 余事象の確率 P(Ā) = 1 − P(A)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "The complement of an event E, denoted E′"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "complement of an event | complement of the event | complement of event": 1,
        "complementary event": 2
      },
      "written": {
        "complement of an event | complement of the event | complement of event": 13,
        "complementary event": 2
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "completely-randomized-design",
    "ja": {
      "term": "完全無作為化計画",
      "reading": "かんぜんむさくいかけいかく",
      "alt": []
    },
    "en": {
      "term": "completely randomized design",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「完全無作為化計画」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "実験で、すべての被験者を処理に完全に無作為に割り当てる計画。",
    "definition_en": "An experimental design in which all the subjects are assigned to the treatments completely at random.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a completely randomized design, you just randomly split everyone into the treatment groups.",
        "ja": "完全無作為化計画では、全員を無作為に処理の群に分けるだけ。",
        "register": "spoken"
      },
      {
        "en": "Describe a completely randomized design for comparing the two fertilizers.",
        "ja": "2 種類の肥料を比べる完全無作為化計画を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパス・OpenStax Introductory Statistics とも出てこない。AP Statistics の CED は topic 1.13 でこの名前を出す。"
    ],
    "related": [
      "blocking",
      "treatment",
      "control-group"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.13"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは AP Statistics の CED の呼び方 completely randomized design（topic 1.13）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "completing-the-square",
    "ja": {
      "term": "平方完成",
      "reading": "へいほうかんせい",
      "alt": [
        "平方完成する"
      ]
    },
    "en": {
      "term": "completing the square",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "二次式を a(x − p)² + q の形に変形すること。二次方程式を解くときや、二次関数の頂点・最大最小を求めるときに使う。",
    "definition_en": "Rewriting a quadratic expression as a(x − h)² + k. It is used to solve quadratic equations and to read off the vertex, or the maximum or minimum, of a quadratic function.",
    "latex": "x^2 + 6x + 5 = (x+3)^2 - 4",
    "spoken_en": "x squared plus six x plus five equals the quantity x plus three, squared, minus four",
    "examples": [
      {
        "en": "Complete the square to find the vertex.",
        "ja": "平方完成して頂点を求めなさい。",
        "register": "written"
      },
      {
        "en": "You could factor it, but completing the square is faster here.",
        "ja": "因数分解でもできるけど、ここは平方完成のほうが速い。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "complete the square",
        "ja": "平方完成する"
      },
      {
        "en": "in vertex form",
        "ja": "平方完成した形で"
      }
    ],
    "pitfalls": [
      "平方完成した形 a(x − h)² + k の呼び方は資料で分かれる。OpenStax Algebra and Trigonometry は standard form と呼び、vertex form とも言うと書く。OpenStax Intermediate Algebra は両方の名前を挙げ、IM Algebra 1 は 6.15 Vertex Form の題にする。用例コーパスの話し言葉の vertex form はほぼ YouTube の解説。"
    ],
    "related": [
      "quadratic-formula",
      "integration-by-completing-the-square"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "平方完成",
        "en": "Completing the square"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 6.15 Vertex Form"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "completing the square": 145,
        "in vertex form": 9
      },
      "written": {
        "completing the square": 280,
        "in vertex form": 2
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "complex-conjugate",
    "ja": {
      "term": "共役複素数",
      "reading": "きょうやくふくそすう",
      "alt": [
        "複素共役"
      ]
    },
    "en": {
      "term": "complex conjugate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II",
        "数C"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "複素数 a + bi に対する a − bi。複素数平面では実軸について対称な点になる。",
    "definition_en": "For a + bi, the number a − bi; in the complex plane it is the mirror image in the real axis.",
    "latex": "\\overline{a+bi} = a - bi",
    "spoken_en": "the conjugate of a plus b i is a minus b i",
    "examples": [
      {
        "en": "Multiply top and bottom by 3 + 4i, the complex conjugate of the denominator.",
        "ja": "分子と分母に、分母の共役複素数 3 + 4i を掛けよう。",
        "register": "spoken"
      },
      {
        "en": "Show that z is a real number if and only if z̄ = z.",
        "ja": "z が実数であることと z̄ = z が同値であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the complex conjugate of z",
        "ja": "z の共役複素数"
      }
    ],
    "pitfalls": [
      "記号 z̄ は z bar と読み、the complex conjugate of z とも言う（記号のエントリ complex-conjugate-bar と同じ）。ほかの文字も同じく bar を付けて読む（λ̄ は lambda bar）。",
      "「z が実数となる条件」は z̄ = z、「純虚数となる条件」は z̄ = −z かつ z ≠ 0。英語でも同じ式で書く。",
      "「共役をとる」は take the conjugate（口頭）。"
    ],
    "related": [
      "pure-imaginary-number",
      "complex-plane",
      "real-axis",
      "modulus-of-a-complex-number",
      "complex-number",
      "conjugate-roots"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "複素共役",
        "en": "Complex conjugate"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-28",
      "human": null
    },
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "complex conjugate": 38
      },
      "written": {
        "complex conjugate": 67
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "complex-fraction",
    "ja": {
      "term": "繁分数",
      "reading": "はんぶんすう",
      "alt": []
    },
    "en": {
      "term": "complex fraction",
      "alt": [],
      "uk": null,
      "register": "both",
      "variants": [
        {
          "term": "complex fraction",
          "register": "both",
          "note": "話し言葉ではほとんどがこの言い方。書き言葉では OpenStax Prealgebra（数の分数）と Elementary Algebra に多い。"
        },
        {
          "term": "complex rational expression",
          "register": "written",
          "note": "分子・分母が文字の式のときの言い方（OpenStax Elementary Algebra・Intermediate Algebra の節 Simplify Complex Rational Expressions、Algebra and Trigonometry の Rational Expressions）。話し言葉にはほとんど出てこない。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "arithmetic"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus"
      ]
    },
    "definition_ja": "分子や分母にさらに分数（分数式）を含む分数。",
    "definition_en": "A fraction whose numerator or denominator itself contains a fraction.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is a complex fraction, so multiply the top and bottom by x.",
        "ja": "これは繁分数なので、分子と分母に x を掛ける。",
        "register": "spoken"
      },
      {
        "en": "Simplify the complex fraction (1 + 1/x)/(1 − 1/x).",
        "ja": "繁分数 (1 + 1/x)/(1 − 1/x) を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "complex は「複素数の」ではなく「込み入った」の意味。複素数の分数ではない。"
    ],
    "related": [
      "rational-expression"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Multiply and Divide Mixed Numbers and Complex Fractions（complex fraction）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「分数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%88%86%E6%95%B0",
        "note": "節「繁分数」: 分子または分母が分数の分数（英: compound fraction, complex fraction）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Simplify Complex Rational Expressions（complex rational expression）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "complex fraction": 43,
        "complex rational expression": 2
      },
      "written": {
        "complex fraction": 56,
        "complex rational expression": 38
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "complex-number",
    "ja": {
      "term": "複素数",
      "reading": "ふくそすう",
      "alt": []
    },
    "en": {
      "term": "complex number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "実数 a, b と虚数単位 i を使って a + bi と表される数。a を実部、b を虚部という。",
    "definition_en": "A number of the form a + bi, with real numbers a and b and the imaginary unit i; a is called its real part, and b is called its imaginary part.",
    "latex": "a + bi",
    "spoken_en": "a plus b i",
    "examples": [
      {
        "en": "To divide complex numbers, multiply the top and bottom by the conjugate.",
        "ja": "複素数の割り算は、分子と分母に共役な複素数を掛ける。",
        "register": "spoken"
      },
      {
        "en": "Express (3 + 2i)/(1 − i) in the form a + bi.",
        "ja": "(3 + 2i)/(1 − i) を a + bi の形で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は a + bi の形を standard form と呼び、極形式（polar form）と比べるときは rectangular form と呼ぶ。IM Algebra 2 は the form a + bi と書く。"
    ],
    "related": [
      "imaginary-number",
      "imaginary-unit",
      "real-part",
      "imaginary-part",
      "complex-conjugate",
      "complex-plane",
      "equality-of-complex-numbers",
      "square-root-of-a-negative-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "複素数",
        "en": "Complex number"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 3.12〜3.18（the form a + bi）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "complex number": 319
      },
      "written": {
        "complex number": 280
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "complex-plane",
    "ja": {
      "term": "複素数平面",
      "reading": "ふくそすうへいめん",
      "alt": []
    },
    "en": {
      "term": "complex plane",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "複素数 a + bi を点 (a, b) に対応させた平面。横軸が実軸、縦軸が虚軸。",
    "definition_en": "The plane in which the complex number a + bi is plotted as the point (a, b), with the real axis horizontal and the imaginary axis vertical.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In the complex plane, multiplying by i rotates everything by 90 degrees.",
        "ja": "複素数平面では、i を掛けるとすべてが 90° 回転する。",
        "register": "spoken"
      },
      {
        "en": "Plot the complex numbers 3 − 2i and −1 + 4i in the complex plane.",
        "ja": "複素数 3 − 2i と −1 + 4i を複素数平面上に図示せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "plot … in the complex plane",
        "ja": "複素数平面上に図示する"
      }
    ],
    "pitfalls": [
      "数C の「複素数平面」を、英語では complex plane と言う（number は入らない）。",
      "2 点 z と w の距離は |z − w|。英語でも the distance between z and w is the modulus of z − w と言える。"
    ],
    "related": [
      "real-axis",
      "imaginary-axis",
      "polar-form",
      "complex-conjugate",
      "locus",
      "rotation",
      "modulus-of-a-complex-number",
      "argument-of-a-complex-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "複素平面",
        "en": "Complex plane"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "complex plane": 36
      },
      "written": {
        "complex plane": 30
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "component-form",
    "ja": {
      "term": "成分表示",
      "reading": "せいぶんひょうじ",
      "alt": [
        "成分形"
      ]
    },
    "en": {
      "term": "component form",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトルを x 成分と y 成分（空間では z 成分も）の組 (a₁, a₂) で表すこと。",
    "definition_en": "Writing a vector by its x- and y-components (and z in space) as ⟨a₁, a₂⟩.",
    "latex": "\\vec{a} = (a_1, a_2)",
    "spoken_en": null,
    "examples": [
      {
        "en": "In component form, you just subtract the initial point from the terminal point.",
        "ja": "成分表示では、終点から始点を引くだけ。",
        "register": "spoken"
      },
      {
        "en": "Find the component form of the vector from P(1, 4) to Q(−2, 6).",
        "ja": "P(1, 4) から Q(−2, 6) へのベクトルを成分表示せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "米国の教科書は成分表示を山かっこ ⟨a₁, a₂⟩ で書き、点の座標 (a₁, a₂) と区別する（OpenStax Calculus Volume 3）。日本の教科書は丸かっこ。"
    ],
    "related": [
      "vector",
      "standard-unit-vectors",
      "components",
      "initial-point",
      "terminal-point"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "component form": 1
      },
      "written": {
        "component form": 75
      },
      "sources": [
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "components",
    "ja": {
      "term": "ベクトルの成分",
      "reading": "べくとるのせいぶん",
      "alt": [
        "x 成分",
        "y 成分"
      ]
    },
    "en": {
      "term": "components",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "座標平面（空間）でベクトルを表す数の組 (a₁, a₂) の、それぞれの数。a₁ を x 成分、a₂ を y 成分という。",
    "definition_en": "The numbers a₁ and a₂ in the representation (a₁, a₂) of a vector in the coordinate plane (or space): the x-component and the y-component.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Just subtract the coordinates to get the components of the vector.",
        "ja": "座標を引き算すれば、ベクトルの成分が出る。",
        "register": "spoken"
      },
      {
        "en": "Find the horizontal and vertical components of a force of 20 N acting at 30° above the horizontal.",
        "ja": "水平から 30° 上向きにはたらく 20 N の力の水平成分と垂直成分を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "行列の「成分」は英語では entry（element）と言い、components とは言わない。"
    ],
    "related": [
      "component-form",
      "vector",
      "decomposition-of-a-vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "components of the vector | component of the vector | horizontal component | vertical component | x-component": 20
      },
      "written": {
        "components of the vector | component of the vector | horizontal component | vertical component | x-component": 61
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "compose",
    "ja": {
      "term": "合成する",
      "reading": "ごうせいする",
      "alt": []
    },
    "en": {
      "term": "compose",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus I"
      ]
    },
    "definition_ja": "2 つの関数を、一方の出力を他方の入力にしてつなぎ、合成関数をつくる。",
    "definition_en": "To combine two functions by feeding the output of one into the other.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you compose f with its inverse, you get x back.",
        "ja": "f とその逆関数を合成すると x に戻る。",
        "register": "spoken"
      },
      {
        "en": "Compose f(x) = x² with g(x) = 3x − 1 to find f(g(x)).",
        "ja": "f(x) = x² と g(x) = 3x − 1 を合成して f(g(x)) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「f と g を合成する」は compose f with g。できた関数は composite function（または the composition of f and g）。"
    ],
    "related": [
      "composite-function",
      "find-the-inverse"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "compose": 70
      },
      "written": {
        "compose": 29
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "composite-figure",
    "ja": {
      "term": "複合図形",
      "reading": "ふくごうずけい",
      "alt": [
        "合成図形",
        "組み合わせた図形"
      ]
    },
    "en": {
      "term": "composite figure",
      "alt": [
        "composite shape"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の教科書は「いくつかの図形を組み合わせた図形」と説明することが多く、決まった名前を使わない。英語の composite figure は、面積や体積を部分に分けて求める図形の名前。CK-12 Geometry は composite figure と呼ぶ（5.18 Area and Perimeter of Composite Shapes ほか、11 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "小学校"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "長方形・三角形・円などの基本的な図形を組み合わせてできた図形。面積や体積は、基本的な図形に分けて足したり引いたりして求める。",
    "definition_en": "A figure made up of two or more basic shapes, such as rectangles, triangles, and circles. Its area or volume is found by splitting it into those shapes and adding or subtracting.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Break the composite figure into a rectangle and a semicircle, find each area, and then add them up.",
        "ja": "この図形は長方形と半円に分けて、それぞれの面積を求めてから足そう。",
        "register": "spoken"
      },
      {
        "en": "The composite figure is a 10 cm by 6 cm rectangle with a semicircle attached to one of the 6 cm sides. Find its area. Use 3.14 for π.",
        "ja": "縦 6 cm、横 10 cm の長方形の 6 cm の辺に半円をつなげた図形がある。この図形の面積を求めよ。ただし、円周率は 3.14 とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "CK-12 Geometry は composite figure（11 件）と composite shape（10 件）をほぼ同じくらい使う。節の名前は 5.18 Area and Perimeter of Composite Shapes。",
      "周の長さ（perimeter）を求めるときは、図形どうしがくっついている内側の辺を数えない。"
    ],
    "related": [
      "area"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.18 Area and Perimeter of Composite Shapes"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 composite figure（CK-12 Geometry 6.7 Areas of Combined Figures Involving Semicircles・CK-12 Geometry 5.18 Area and Perimeter of Composite Shapes・CK-12 Geometry 1.2 Shapes・計 11 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "composite-function",
    "ja": {
      "term": "合成関数",
      "reading": "ごうせいかんすう",
      "alt": [
        "関数の合成"
      ]
    },
    "en": {
      "term": "composite function",
      "alt": [],
      "variants": [
        {
          "term": "composition of functions",
          "register": "both",
          "note": "合成する操作の名前。OpenStax Algebra and Trigonometry の節の名前でもある。話し言葉では composite function に次いで使われ（MIT OCW に多い）、書き言葉では composite function のほうが多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "関数 g の値を関数 f に入れてできる関数 f(g(x))。",
    "definition_en": "The function f(g(x)) formed by feeding the output of g into f.",
    "latex": "(f \\circ g)(x) = f(g(x))",
    "spoken_en": "f composed with g of x equals f of g of x",
    "examples": [
      {
        "en": "f of g of x means you do g first, then f.",
        "ja": "f(g(x)) は、先に g、次に f を施す。",
        "register": "spoken"
      },
      {
        "en": "Find the composite function (f ∘ g)(x) for f(x) = x² and g(x) = x + 1.",
        "ja": "f(x) = x²、g(x) = x + 1 のとき、合成関数 (f ∘ g)(x) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "f ∘ g は f composed with g（または f circle g）と読む。(f ∘ g)(x) の書き方は日本語でも同じ（日本語版 Wikipedia「写像の合成」）。"
    ],
    "related": [
      "rational-function",
      "radical-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "写像の合成",
        "en": "Function composition"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 3.1（composite function）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「写像の合成」",
        "url": "https://ja.wikipedia.org/wiki/%E5%86%99%E5%83%8F%E3%81%AE%E5%90%88%E6%88%90",
        "note": "g ∘ f の記法"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "composition of functions": 18,
        "composite function": 40
      },
      "written": {
        "composite function": 87,
        "composition of functions": 26
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "composite-number",
    "ja": {
      "term": "合成数",
      "reading": "ごうせいすう",
      "alt": []
    },
    "en": {
      "term": "composite number",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math",
        "Pre-Algebra"
      ]
    },
    "definition_ja": "1 より大きい整数のうち、素数でないもの。1 とそれ自身以外にも正の約数をもつ。",
    "definition_en": "An integer greater than 1 that is not prime, so it has a positive divisor other than 1 and itself.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Nine is a composite number because it's 3 times 3.",
        "ja": "9 は 3 × 3 だから合成数だ。",
        "register": "spoken"
      },
      {
        "en": "Show that n² + 2n is a composite number for every integer n ≥ 2.",
        "ja": "2 以上のすべての整数 n について、n² + 2n は合成数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "1 は素数でも合成数でもない（1 is neither prime nor composite）。",
      "composite だけだと composite function（合成関数）の composite と紛れる。数のときは composite number か、補語で is composite と言う。"
    ],
    "related": [
      "prime-factorization",
      "number-of-divisors",
      "multiple",
      "composite-function",
      "prime-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "合成数",
        "en": "Composite number"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "2.4 Find Multiples and Factors・2.5 Prime Factorization and the Least Common Multiple"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「合成数」は出てこない（数学A 数学と人間の活動は素数・素因数分解を扱う。level.jp 数A は素数の対概念として）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "composite number": 4
      },
      "written": {
        "composite number": 39
      },
      "sources": [
        "khan-middle",
        "mit-notes",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "composition-of-transformations",
    "ja": {
      "term": "合成変換",
      "reading": "ごうせいへんかん",
      "alt": [
        "変換の合成",
        "関数の変換の合成"
      ]
    },
    "en": {
      "term": "combining transformations",
      "alt": [
        "composition of transformations"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "functions",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Linear Algebra",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "平行移動・対称移動・拡大縮小などの変換を続けて行うこと。順番によって結果が変わることがある。",
    "definition_en": "Carrying out transformations such as shifts, reflections, and stretches one after another; the order can change the result.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Do the horizontal stretch before the shift, or you'll get a different graph.",
        "ja": "横の拡大を平行移動より先にしないと、別のグラフになる。",
        "register": "spoken"
      },
      {
        "en": "Describe the combination of transformations that takes the graph of y = √x to the graph of y = −2√(x + 3).",
        "ja": "y = √x のグラフを y = −2√(x + 3) のグラフに移す変換の組合せを説明せよ。",
        "register": "written"
      },
      {
        "en": "Doing T first and then S is the same as multiplying by the matrix BA, in that order.",
        "ja": "T のあとに S を行うのは、行列 BA を掛けるのと同じ（この順番で）。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は関数のグラフの変換をまとめて行うことを combining transformations と書く（本文 3 件）。線形代数では線形変換の合成（composition）を指す。",
      "線形代数では、行列 A の変換のあとに行列 B の変換を行う合成は積 BA に対応し、掛ける順番は行う順番と逆になる（Nicholson 2.3・7.3 は composition と呼ぶ）。"
    ],
    "related": [
      "stretch-vertically",
      "compress-horizontally",
      "reflection",
      "phase-shift",
      "linear-transformation",
      "matrix-multiplication"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "7.3 Isomorphisms and Composition"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "combining transformations": 3
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは OpenStax の呼び方 combining transformations（本文 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "compound-interest",
    "ja": {
      "term": "複利",
      "reading": "ふくり",
      "alt": []
    },
    "en": {
      "term": "compound interest",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "利息を元金に加え、次の期間はその合計にも利息がつくようにする利息の計算。元金 P、年利率 r、1 年に利息をつける回数 n のとき、t 年後の金額は A = P(1 + r/n)ⁿᵗ。",
    "definition_en": "Interest that is added to the principal, so that in the next period you earn interest on the new, larger balance. The balance after t years is A = P(1 + r/n)^(nt).",
    "latex": "A = P\\left(1 + \\frac{r}{n}\\right)^{nt}",
    "spoken_en": "A equals P times the quantity one plus r over n, to the n t",
    "examples": [
      {
        "en": "With compound interest, you earn interest on your interest, too.",
        "ja": "複利だと、利息にもさらに利息がつく。",
        "register": "spoken"
      },
      {
        "en": "$1,000 is invested at 4% annual interest, compounded quarterly. Use the compound interest formula to find the balance after 5 years, to the nearest cent.",
        "ja": "1,000 ドルを年利率 4 %、年 4 回の複利で預ける。複利の公式を用いて、5 年後の残高をセントの位まで求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "compounded annually は年 1 回（n = 1）、compounded monthly は毎月（n = 12）、compounded quarterly は年 4 回（n = 4）。問題文のこの語句から n を読み取る。",
      "単利（simple interest、エントリ simple-interest）は元金にだけ利息がつく計算で、複利とは別。",
      "連続複利（continuous compounding）では e を使う式になる。"
    ],
    "related": [
      "simple-interest",
      "exponential-growth",
      "growth-rate",
      "base-of-the-natural-logarithm",
      "percent"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "compound interest": 26
      },
      "written": {
        "compound interest": 48
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "compress-horizontally",
    "ja": {
      "term": "横に縮める",
      "reading": "よこにちぢめる",
      "alt": []
    },
    "en": {
      "term": "compress … horizontally",
      "alt": [
        "compress horizontally"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "グラフの各点の x 座標を 1/b 倍（b > 1）にして、x 軸方向に縮める。y = f(x) は y = f(bx) になる。",
    "definition_en": "To multiply every x-coordinate of a graph by 1/b (b > 1), squeezing it toward the y-axis; y = f(x) becomes y = f(bx).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The 3 inside compresses it horizontally, so the period shrinks from 2π to 2π/3.",
        "ja": "中の 3 で横に縮むので、周期は 2π から 2π/3 になる。",
        "register": "spoken"
      },
      {
        "en": "The graph of y = sin 2x is the graph of y = sin x compressed horizontally by a factor of 1/2. Explain why.",
        "ja": "y = sin 2x のグラフは y = sin x のグラフを x 軸方向に 1/2 倍に縮めたものである。その理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "中の係数は外の係数と逆に働く（f(2x) は縦ではなく横に 1/2 倍）。",
      "名詞の言い方 horizontal compression は書き言葉（OpenStax）に多い。動詞の形は書き言葉でも少ない。",
      "f(bx) の横の縮小を何倍と書くかは資料で逆になる。IM Algebra 2（5.9 Scaling the Inputs）は f(2t) を compressed horizontally by a factor of 1/2、OpenStax Algebra and Trigonometry（Transformation of Functions）は |−4x| を horizontally compressed by a factor of 1/4 と書き、OpenStax Calculus Volume 1（1.3 Trigonometric Functions）は sin(4x) を compressed horizontally by a factor of 4 と書く。"
    ],
    "related": [
      "stretch-vertically",
      "period",
      "transformations-of-functions"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Transformation of Functions（horizontally compressed by a factor of 1/4）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.3 Trigonometric Functions（compressed horizontally by a factor of 4）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 5.9 Scaling the Inputs（compressed horizontally by a factor of 1/2）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "compress … horizontally": 7
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 compress horizontally（本文 5 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "compute",
    "ja": {
      "term": "計算する",
      "reading": "けいさんする",
      "alt": []
    },
    "en": {
      "term": "calculate",
      "alt": [],
      "variants": [
        {
          "term": "compute",
          "register": "spoken",
          "note": "話し言葉では calculate と同じくらい使うが、大半が MIT OCW。書き言葉では calculate のほうがずっと多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "数や式について、足し算・引き算・掛け算・割り算などを行い、結果を出す。",
    "definition_en": "To find a result by carrying out operations such as adding, subtracting, multiplying, and dividing.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Try to calculate 15% of 80 in your head before you grab a calculator.",
        "ja": "電卓を使う前に、80 の 15% を暗算で計算してみよう。",
        "register": "spoken"
      },
      {
        "en": "Compute (−3)² − 4 · 2 · (−5).",
        "ja": "(−3)² − 4 × 2 × (−5) を計算せよ。",
        "register": "written"
      },
      {
        "en": "Calculate the area of a circle with a radius of 3 cm. Leave your answer in terms of π.",
        "ja": "半径 3 cm の円の面積を求めよ。答えは π を使って表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "calculate と compute はどちらも使い、意味は同じ（話し言葉では compute も多いが、講義（MIT OCW）に偏る）。名詞はそれぞれ calculation、computation。",
      "figure out や work out は「（答えを）出す・考えて分かる」の広い意味の句動詞で、計算に限らない（figure out which plan is cheaper はどちらのプランが安いかを考えて出す）。",
      "文字に数を代入して式の値を出すのは evaluate（エントリ evaluate）。"
    ],
    "related": [
      "evaluate",
      "the-four-operations",
      "value-of-the-expression",
      "substitute"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "calculate": 1311,
        "compute": 1138
      },
      "written": {
        "calculate": 1696,
        "compute": 552
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "concavity",
    "ja": {
      "term": "曲線の凹凸",
      "reading": "きょくせんのおうとつ",
      "alt": [
        "凹凸"
      ]
    },
    "en": {
      "term": "concavity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線が下に凸か上に凸かという性質。f″(x) の符号で調べる。",
    "definition_en": "Whether a graph bends upward (concave up) or downward (concave down); found from the sign of f″(x).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A positive second derivative means concave up, like a cup.",
        "ja": "第 2 次導関数が正なら下に凸（カップの形）。",
        "register": "spoken"
      },
      {
        "en": "Determine the intervals of concavity of f(x) = x³ − 3x².",
        "ja": "f(x) = x³ − 3x² の凹凸を調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の「下に凸」は concave up、「上に凸」は concave down。直訳の convex upward は用例コーパスに出てこない。",
      "f″(x) > 0 は f double prime of x is positive と読む。"
    ],
    "related": [
      "second-derivative",
      "inflection-point",
      "check-the-concavity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.6（concavity）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "concavity": 62
      },
      "written": {
        "concavity": 40
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "conclusion",
    "ja": {
      "term": "結論",
      "reading": "けつろん",
      "alt": []
    },
    "en": {
      "term": "conclusion",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "命題「p ならば q」の q の部分。証明では、仮定から出発して最後に示すことがらを指す。",
    "definition_en": "The \"then\" part q of a statement \"if p, then q.\" In a proof, it is what you must show, starting from the hypothesis.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The part after \"if\" is the hypothesis, and the part after \"then\" is the conclusion.",
        "ja": "「〜ならば」の前の部分が仮定で、後ろの部分が結論だよ。",
        "register": "spoken"
      },
      {
        "en": "Identify the conclusion of the statement \"If a number is divisible by 6, then it is divisible by 3.\"",
        "ja": "命題「6 の倍数ならば 3 の倍数である」の結論を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "仮定 hypothesis（エントリ hypothesis）と組で使う。「p ならば q」の p が hypothesis、q が conclusion。",
      "仮定と結論を入れかえた命題が逆（エントリ converse）。もとの命題が正しくても、逆が正しいとは限らない。",
      "統計の検定の「結論を述べよ」も state your conclusion と言う（エントリ hypothesis-testing）。命題の結論とは別の使い方。"
    ],
    "related": [
      "hypothesis",
      "converse",
      "conditional-statement",
      "proof",
      "theorem"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "conclusion": 213
      },
      "written": {
        "conclusion": 293
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "concurrent",
    "ja": {
      "term": "1 点で交わる",
      "reading": "いってんでまじわる",
      "alt": [
        "共点"
      ]
    },
    "en": {
      "term": "concurrent",
      "alt": [],
      "uk": null
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": "concurrent という名前は、米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（4 つとも 0 件）。概念は、IM Geometry（7.5 Triangles in Circles・7.6 A Special Point）が meet at a single point、CK-12 Geometry（4.22 Concurrence and Constructions・4.23 Medians）が intersect at one point と言って扱う。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Linear Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "3 本以上の直線が、すべて同じ 1 点を通ること。",
    "definition_en": "Describes three or more lines that all pass through one common point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The three medians of a triangle are concurrent, and the point where they meet is the centroid.",
        "ja": "三角形の 3 本の中線は 1 点で交わり、その交点が重心だ。",
        "register": "spoken"
      },
      {
        "en": "Prove that the three altitudes of triangle ABC are concurrent.",
        "ja": "三角形 ABC の 3 本の垂線は 1 点で交わることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "concurrent は形容詞で、「3 直線が 1 点で交わる」は The three lines are concurrent. と言う。英語版 Wikipedia「Concurrent lines」は、交わる点を the point of concurrency と呼ぶ（節 Triangles）。",
      "日常英語の concurrent は「同時に起こる」の意味。幾何の concurrent は「1 点を共有する」で、時間の意味はない。"
    ],
    "related": [
      "collinear",
      "cevas-theorem",
      "the-five-centers-of-a-triangle",
      "centroid"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "4.1 Vectors and Lines（練習問題 4.1.35: 四面体の 4 本の中線）・Supplementary Exercises for Chapter 4（三角形の中線・高さ）"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "共点",
        "en": "Concurrent lines"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.5 Triangles in Circles・7.6 A Special Point（meet at a single point）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.22 Concurrence and Constructions・4.23 Medians（intersect at one point）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "concurrent": 1
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 concurrent（4.5 An Application to Computer Graphics・4.1 Vectors and Lines）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "concyclic",
    "ja": {
      "term": "同一円周上にある",
      "reading": "どういつえんしゅうじょうにある",
      "alt": [
        "4 点が同一円周上にある"
      ]
    },
    "en": {
      "term": "lie on the same circle",
      "alt": [],
      "uk": null
    },
    "pos": "adjective",
    "mapping": "near",
    "mapping_note": "IM Geometry（7.4 Quadrilaterals in Circles）は、いくつかの点が同一円周上にあることを lie on the same circle のように文で言う。英語版 Wikipedia には記事「Concyclic points」があり、冒頭で 1 語の形容詞 concyclic（A, B, C, and D are concyclic）を太字で定義する（用例コーパスと参照には出てこない）。4 つの頂点が同一円周上にある四角形は cyclic quadrilateral（エントリ cyclic-quadrilateral）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3",
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "いくつかの点が、すべて 1 つの円の周上にあること。4 点 A、B、C、D が同一円周上にあるとき、四角形 ABCD は円に内接する。",
    "definition_en": "Said of points that all lie on the same circle, so that one circle passes through every one of them.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Opposite angles add up to 180, so all four vertices lie on the same circle.",
        "ja": "対角の和が 180° だから、4 つの頂点は同一円周上にある。",
        "register": "spoken"
      },
      {
        "en": "In quadrilateral ABCD, ∠A = 83° and ∠C = 97°. Show that A, B, C, and D lie on the same circle.",
        "ja": "四角形 ABCD で、∠A = 83°、∠C = 97° である。4 点 A、B、C、D が同一円周上にあることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「同一円周上にある」は円周の上にあることで、円の内部にあることではない。英語でも on the circle（周の上）と inside the circle（内部）を分ける（IM Geometry）。",
      "4 点が同一円周上にあることを示すには、円周角の定理の逆（エントリ converse-of-the-inscribed-angle-theorem）か、四角形の対角の和が 180° であること（エントリ cyclic-quadrilateral）を使う。",
      "一直線上にない 3 点は、必ず 1 つの円周上にある（その円が外接円、エントリ circumscribed-circle）。同一円周上にあるかどうかが問題になるのは 4 点以上のとき。"
    ],
    "related": [
      "cyclic-quadrilateral",
      "converse-of-the-inscribed-angle-theorem",
      "circle",
      "inscribed-angle-theorem",
      "circumscribed-circle"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.4 Quadrilaterals in Circles"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Concyclic_points",
        "note": "記事「Concyclic points」（数学カテゴリから 4 段）の冒頭: concyclic を太字で定義"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 lie on the same circle（Geometry 7.4 Quadrilaterals in Circles・Algebra 2 6.8 Rising and Falling・計 3 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "condition",
    "ja": {
      "term": "条件",
      "reading": "じょうけん",
      "alt": []
    },
    "en": {
      "term": "condition",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Precalculus",
        "Discrete Math"
      ]
    },
    "definition_ja": "変数を含み、変数の値によって真にも偽にもなる文や式。「x > 2」のように、成り立つかどうかが x しだいで決まるもの。",
    "definition_en": "A statement or relation involving a variable whose truth depends on the value of the variable, such as x > 2.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "So the condition here is x > 2, and we want to know which x satisfy it.",
        "ja": "ここでの条件は x > 2 で、それを満たす x がどれかを知りたいわけだ。",
        "register": "spoken"
      },
      {
        "en": "Let P be the set of real numbers x that satisfy the condition x² < 4. Write P as an interval.",
        "ja": "条件 x² < 4 を満たす実数 x 全体の集合を P とする。P を区間で表せ。",
        "register": "written"
      },
      {
        "en": "Under what condition on k does the equation have two distinct real solutions?",
        "ja": "方程式が異なる 2 つの実数解をもつための k の条件を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "センター試験（平成 30 年度 数学I・A）は「実数 x に関する次の条件 p, q, r, s を考える」、共通テスト（令和 3 年度 第1日程 数学I）は「条件 p, q を次のように定める」と、変数を含み真偽が変わる文を「条件」と呼ぶ。高等学校学習指導要領解説（数学I）の「条件」は「命題の条件や結論」（仮定）の意味。英語の condition はもっと広く、初期条件（initial condition。AP Calculus の CED（Unit 7）・OpenStax Calculus）なども言い、変数を含み真偽が変わる文の論理の用語は Levin の predicate（1.1.3 Quantifiers and Predicates）。",
      "「条件 p を満たす」は satisfy the condition p。「〜のための条件」は the condition for …、「k に関する条件」は a condition on k と on で言える。"
    ],
    "related": [
      "proposition",
      "necessary-condition",
      "sufficient-condition",
      "set",
      "initial-condition"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 集合と命題: 命題の条件や結論に着目"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "平成 30 年度 数学I・A: 実数 x に関する次の条件 p, q, r, s。令和 3 年度 第1日程 数学I 第1問〔2〕: 条件 p, q を次のように定める（数学Ⅰ・数学Ａ には無い）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 7（initial condition）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1.3 Quantifiers and Predicates: predicate（変数を含み真偽が変わる文）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "condition": 839
      },
      "written": {
        "condition": 506
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "conditional-convergence",
    "ja": {
      "term": "条件収束",
      "reading": "じょうけんしゅうそく",
      "alt": []
    },
    "en": {
      "term": "conditional convergence",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "級数 Σaₙ は収束するが、Σ|aₙ| は発散すること。",
    "definition_en": "The case in which a series Σaₙ converges but Σ|aₙ| diverges.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The alternating harmonic series is the standard example of conditional convergence.",
        "ja": "交代調和級数が条件収束の典型例。",
        "register": "spoken"
      },
      {
        "en": "Show that Σ (−1)ⁿ⁺¹/n converges conditionally.",
        "ja": "Σ (−1)ⁿ⁺¹/n が条件収束することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞の形 converges conditionally、形容詞は conditionally convergent。用例コーパスでは名詞も含めてどれも少ない。"
    ],
    "related": [
      "absolute-convergence",
      "alternating-series",
      "harmonic-series"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "条件収束",
        "en": "Conditional convergence"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.9（conditional convergence）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "conditional convergence": 2
      },
      "written": {
        "conditional convergence": 7
      },
      "sources": [
        "openstax-calculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは CED の呼び方 conditional convergence（topic 10.9）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "conditional-probability",
    "ja": {
      "term": "条件付き確率",
      "reading": "じょうけんつきかくりつ",
      "alt": []
    },
    "en": {
      "term": "conditional probability",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics",
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2",
        "Intro Statistics",
        "AP Statistics"
      ]
    },
    "definition_ja": "事象 A が起こったという条件のもとで、事象 B が起こる確率。P(A ∩ B) を P(A) で割って求める。",
    "definition_en": "The probability that event B happens given that event A has happened, found by dividing P(A ∩ B) by P(A).",
    "latex": "P(B \\mid A) = \\frac{P(A \\cap B)}{P(A)}",
    "spoken_en": "the probability of B given A equals the probability of A and B over the probability of A",
    "examples": [
      {
        "en": "We already know the card is a face card, so this is a conditional probability: the probability of a king given a face card.",
        "ja": "引いたカードが絵札だとわかっているので、これは条件付き確率で、絵札であるときにキングである確率だ。",
        "register": "spoken"
      },
      {
        "en": "A family has two children, and at least one is a girl. Find the conditional probability that both children are girls.",
        "ja": "ある家庭に子どもが 2 人いて、少なくとも 1 人は女の子である。2 人とも女の子である条件付き確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（数学A）は P_A(B) と書くが、AP Statistics の CED（topic 2.6）と OpenStax Introductory Statistics は P(B | A) の形で書く。OpenStax Introductory Statistics（3.1）はこれを the probability of B given A の形で読み、CED は事象 A が起こったときに事象 B が起こる確率と説明する。条件の A が縦棒の右に来るので、文字の順が逆になる。",
      "P(B | A) と P(A | B) は一般に違う。入れ替えるにはベイズの定理を使う。"
    ],
    "related": [
      "general-multiplication-rule",
      "bayes-theorem",
      "intersection-of-events",
      "independence-of-events",
      "tree-diagram",
      "conditional-relative-frequency",
      "sampling-without-replacement"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "条件付き確率",
        "en": "Conditional probability"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（The conditional probability of A given B is written P(A | B)）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A (2) 条件付き確率: PA(B)"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.6（conditional probability、P(A | B)）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "conditional probability": 31
      },
      "written": {
        "conditional probability": 57
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "conditional-relative-frequency",
    "ja": {
      "term": "条件付き相対度数",
      "reading": "じょうけんつきそうたいどすう",
      "alt": []
    },
    "en": {
      "term": "conditional relative frequency",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Integrated Math 1"
      ]
    },
    "definition_ja": "二元分割表で、ある行（または列）の中だけで計算した相対度数。度数をその行（列）の合計で割って求める。",
    "definition_en": "In a two-way table, a relative frequency computed within just one row or one column: the count divided by that row's or column's total.",
    "latex": "\\frac{\\text{cell count}}{\\text{row total}}",
    "spoken_en": "cell count over row total",
    "examples": [
      {
        "en": "Just look at the girls: 12 out of 30 of them like soccer, so the conditional relative frequency is 0.4.",
        "ja": "女子だけを見ると、30 人中 12 人がサッカー好きだから、条件付き相対度数は 0.4 だ。",
        "register": "spoken"
      },
      {
        "en": "Use the two-way table to find the conditional relative frequency of walking to school among the seventh graders.",
        "ja": "二元分割表を用いて、7 年生のうち徒歩で通学する生徒の条件付き相対度数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "割る数に注意。条件付き相対度数はその行（列）の合計で割る。表全体の合計で割ると joint relative frequency、行や列の合計の相対度数は marginal relative frequency。",
      "「女子のうちサッカー好きの割合」と「サッカー好きのうち女子の割合」は割る合計が違い、値も違う。",
      "相対度数（エントリ relative-frequency）の一種。AP Statistics の CED は、1 つのカテゴリに限って計算した relative frequency として定義する。"
    ],
    "related": [
      "relative-frequency",
      "contingency-table",
      "conditional-probability",
      "frequency-table",
      "categorical-variable"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは AP Statistics の CED の呼び方 conditional relative frequency（topic 2.2）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "conditional-statement",
    "ja": {
      "term": "含意",
      "reading": "がんい",
      "alt": [
        "条件文",
        "「p ならば q」の形の命題"
      ]
    },
    "en": {
      "term": "implication",
      "alt": [
        "conditional statement",
        "if-then statement"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "高等学校学習指導要領解説（数学I）は命題を「x > 2 ならば x > 0 である。」のように「ならば」で書き（記号の「p → q」は理数に関する学科の科目「理数数学I」の発展・拡充の内容に出てくる）、この形の命題そのものに名前を付けない（含意は論理学の語）。英語では Levin の離散数学が implication、CK-12 Geometry（2.11 If Then Statements）が conditional statement（33 件）・if-then statement（11 件）と呼ぶ。見出しの「含意」は学習指導要領解説に無い（日本語版 Wikipedia「論理包含」「必要条件と十分条件」にはある）。",
    "domains": [
      "proof",
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "2 つの命題 p, q を「p ならば q」の形に結んだ命題。p が真で q が偽のときだけ偽になる。",
    "definition_en": "A statement of the form \"if p, then q,\" built from two statements p and q. It is false only when p is true and q is false.",
    "latex": "p \\Rightarrow q",
    "spoken_en": "p implies q",
    "examples": [
      {
        "en": "An implication is false in only one case: the hypothesis is true and the conclusion is false.",
        "ja": "含意が偽になるのは 1 通りだけ。仮定が真で結論が偽のときだ。",
        "register": "spoken"
      },
      {
        "en": "Let's rewrite this as an if-then statement: if two angles are vertical angles, then they're congruent.",
        "ja": "これを「ならば」の形に書き直そう。2 つの角が対頂角ならば、その 2 つの角は等しい。",
        "register": "spoken"
      },
      {
        "en": "Write the converse of the conditional statement \"If x = 3, then x² = 9,\" and determine whether the converse is true.",
        "ja": "命題「x = 3 ならば x² = 9」の逆を書き、その真偽を調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日常英語の implication は「含み・影響」の意味でも使う。",
      "「p ならば q」の p を hypothesis、q を conclusion と呼ぶ。"
    ],
    "related": [
      "hypothesis",
      "inverse",
      "contrapositive",
      "biconditional",
      "necessary-condition",
      "sufficient-condition",
      "proposition"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.2 Implications"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 集合と命題: 命題「x>2 ならば x>0 である。」（「p→q」は理数数学I の発展・拡充）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.11 If Then Statements: conditional statement（33 件）・if-then statement（11 件）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「論理包含」",
        "url": "https://ja.wikipedia.org/wiki/%E8%AB%96%E7%90%86%E5%8C%85%E5%90%AB",
        "note": "含意"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "implication": 29
      },
      "written": {
        "conditional statement": 1,
        "implication": 37,
        "if-then statement": 1
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "conditions-for-a-parallelogram",
    "ja": {
      "term": "平行四辺形になる条件",
      "reading": "へいこうしへんけいになるじょうけん",
      "alt": []
    },
    "en": {
      "term": "conditions for a parallelogram",
      "alt": [
        "prove that a quadrilateral is a parallelogram",
        "parallelogram theorems"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。中学校学習指導要領解説（第 2 学年）は「平行四辺形になるための条件」を平行四辺形の性質と並べて挙げる（本エントリの定義の 5 つ）が、用例コーパスと参照にはそれらをまとめた名前が出てこず、「四角形が平行四辺形であることを示す」（prove that a quadrilateral is a parallelogram）のように、示すことを文で言う。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "四角形が平行四辺形になるための条件。2 組の対辺がそれぞれ平行、2 組の対辺がそれぞれ等しい、2 組の対角がそれぞれ等しい、対角線がそれぞれの中点で交わる、1 組の対辺が平行でその長さが等しい、のどれか 1 つが成り立てば平行四辺形である。",
    "definition_en": "The facts that let you conclude that a quadrilateral is a parallelogram: both pairs of opposite sides are parallel, both pairs of opposite sides are equal, both pairs of opposite angles are equal, the diagonals bisect each other, or one pair of opposite sides is both parallel and equal. Any one of these is enough.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To prove that a quadrilateral is a parallelogram, you don't need everything; one condition is enough, like showing both pairs of opposite sides are equal.",
        "ja": "四角形が平行四辺形であることを示すのに、全部は要らない。条件が 1 つあれば十分で、たとえば 2 組の対辺がそれぞれ等しいことを示せばいい。",
        "register": "spoken"
      },
      {
        "en": "Prove that a quadrilateral is a parallelogram if its diagonals bisect each other.",
        "ja": "対角線がそれぞれの中点で交わる四角形は平行四辺形であることを証明せよ。",
        "register": "written"
      },
      {
        "en": "State two conditions for a parallelogram, and use one of them to prove that ABCD is a parallelogram.",
        "ja": "平行四辺形になる条件を 2 つ述べ、そのうちの 1 つを使って四角形 ABCD が平行四辺形であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「対角線がそれぞれの中点で交わる」は英語で the diagonals bisect each other（エントリ bisect）。",
      "「1 組の対辺が平行でその長さが等しい」は、同じ 1 組の辺が平行で、しかも長さが等しいという条件。1 組が平行で別の 1 組が等しいだけなら、平行四辺形とは限らない（等脚台形がその例）。",
      "平行四辺形の性質（平行四辺形ならば対辺が等しい）と、平行四辺形になる条件（対辺が等しければ平行四辺形）は、仮定と結論を入れ替えた逆（エントリ converse）の関係。"
    ],
    "related": [
      "parallelogram",
      "opposite-side",
      "opposite-angle",
      "diagonal",
      "converse"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 B 図形: 平行四辺形になるための条件"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "conditions-that-determine-a-triangle",
    "ja": {
      "term": "三角形の決定条件",
      "reading": "さんかくけいのけっていじょうけん",
      "alt": []
    },
    "en": {
      "term": "determine a unique triangle",
      "alt": [
        "conditions that determine a triangle"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "名詞の「決定条件」に当たる語は用例コーパスと参照に出てこない。IM Grade 7（7.9・7.10 Drawing Triangles）は「どの測定値の組が三角形をただ 1 つに決めるか」を determine one unique triangle と言い、IM Geometry（2.4・2.11・2.12）は define a unique triangle と言う。IM Geometry・CK-12 Geometry は同じ事実を合同条件 SSS・SAS・ASA として扱う。高等学校学習指導要領解説（数学I）は正弦定理・余弦定理を三角形の決定条件と関連付けて理解すると書き、中学校学習指導要領解説（第 2 学年）は合同条件を三角形の決定条件を基に認めると書く。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "三角形の形と大きさがただ 1 つに決まるための条件。3 辺の長さ、2 辺とその間の角、1 辺とその両端の角のいずれかが分かればよい。",
    "definition_en": "The sets of given parts that fix a triangle's shape and size uniquely: three sides, two sides and the included angle, or one side and the two angles at its ends.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "So what are the conditions that determine a triangle? Three sides, two sides and the angle between them, or one side and the two angles on its ends.",
        "ja": "では、三角形の決定条件は何だろう。3 辺、2 辺とその間の角、1 辺とその両端の角だ。",
        "register": "spoken"
      },
      {
        "en": "State the conditions that determine a triangle, and explain why two sides and a non-included angle are not enough.",
        "ja": "三角形の決定条件を述べ、2 辺とその間にない角では足りない理由を説明せよ。",
        "register": "written"
      },
      {
        "en": "Which of the given sets of measurements determine a unique triangle?",
        "ja": "与えられた測定値の組のうち、三角形がただ 1 つに決まるのはどれか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 辺とその間にない角（SSA）では、三角形が 2 通りできることがある（正弦定理の ambiguous case）。決定条件にも合同条件にも入らない。",
      "3 つの角（AAA）だけでは大きさが決まらず、相似な三角形しか決まらない。"
    ],
    "related": [
      "triangle",
      "congruent",
      "similar-triangles",
      "law-of-sines",
      "law-of-cosines",
      "congruence-criteria"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "7.9・7.10 Drawing Triangles: determine one unique triangle（3 件）、unique triangle（10 件）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "2.4・2.11・2.12: define a unique triangle（4 件）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 正弦定理や余弦定理について三角形の決定条件や三平方の定理と関連付けて理解（4 件）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 図形: 合同条件を三角形の決定条件を基に認める（1 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.13 SSS・4.14 SAS・4.15 ASA and AAS: 三角形の合同条件"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 determine a unique triangle | determine one unique triangle | define a unique triangle（Geometry 2.11 Side-Side-Angle (Sometimes) Congruence・Grade 7 7.9 Drawing Triangles (Part 1)・Grade 7 7.10 Drawing Triangles (Part 2)・計 4 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cone",
    "ja": {
      "term": "円錐",
      "reading": "えんすい",
      "alt": []
    },
    "en": {
      "term": "cone",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "円の底面と、底面の外にある 1 つの頂点を結んでできる、曲がった側面をもつ立体。",
    "definition_en": "A solid with a circular base and a curved surface that narrows to a single point called the vertex.",
    "latex": "V = \\frac{1}{3}\\pi r^2 h",
    "spoken_en": "V equals one third pi r squared h",
    "examples": [
      {
        "en": "The slant height runs along the side of the cone, but the height goes straight down from the tip to the center of the base.",
        "ja": "母線は円錐の側面に沿った線だけど、高さは頂点から底面の中心へまっすぐ下ろした線だよ。",
        "register": "spoken"
      },
      {
        "en": "The base of a cone has a radius of 3 cm, and the cone is 8 cm tall. Find its volume in terms of π.",
        "ja": "底面の半径が 3 cm、高さが 8 cm の円錐の体積を求めよ。ただし、円周率は π とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "母線は slant height（エントリ slant-height）。高さ（height）は頂点から底面に下ろした垂線の長さで、母線の長さとは別。",
      "円錐の頂点は vertex（apex とも）。",
      "cone は日常語で、アイスクリームのコーンや道路のコーン（traffic cone）も指す。"
    ],
    "related": [
      "pyramid",
      "cylinder",
      "slant-height",
      "volume",
      "circle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円錐",
        "en": "Cone"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cone": 116
      },
      "written": {
        "cone": 291
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "confidence-interval",
    "ja": {
      "term": "信頼区間",
      "reading": "しんらいくかん",
      "alt": []
    },
    "en": {
      "term": "confidence interval",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "区間推定で、母数を指定した信頼度で含むように求めた区間。",
    "definition_en": "In interval estimation, an interval computed so that it contains the population parameter with a stated level of confidence.",
    "latex": "\\bar{x} \\pm z^{*}\\frac{\\sigma}{\\sqrt{n}}",
    "spoken_en": "x bar plus or minus z star times sigma over the square root of n",
    "examples": [
      {
        "en": "We're 95% confident the true mean is somewhere in this confidence interval.",
        "ja": "真の平均がこの信頼区間のどこかにあると 95% の信頼度で言える。",
        "register": "spoken"
      },
      {
        "en": "Construct a 95% confidence interval for the population mean and interpret it.",
        "ja": "母平均の 95% 信頼区間を求め、解釈せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "construct a confidence interval",
        "ja": "信頼区間を求める"
      }
    ],
    "pitfalls": [
      "「95% の確率で母平均がこの区間に入る」とは言わない。同じ大きさの標本を繰り返し取って同じ方法で区間を作ると、そのおよそ 95% が母平均を含む（AP Statistics の CED topic 4.3）。この意味で 95% confident と言う。",
      "z* は z star と読む（用例コーパスの話し言葉ではすべて Khan Academy）。"
    ],
    "related": [
      "confidence-level",
      "interval-estimation",
      "point-estimate",
      "error-bound",
      "standard-error"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "信頼区間",
        "en": "Confidence interval"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 4.3（confidence level）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "confidence interval": 127
      },
      "written": {
        "confidence interval": 377
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "confidence-level",
    "ja": {
      "term": "信頼度",
      "reading": "しんらいど",
      "alt": []
    },
    "en": {
      "term": "confidence level",
      "alt": [
        "level of confidence"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "信頼区間を作る方法が、母数を含む区間を与える割合。95% や 99% がよく使われる。",
    "definition_en": "The proportion of intervals made by the method that contain the population parameter, often 95% or 99%.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you raise the confidence level, the interval gets wider.",
        "ja": "信頼度を上げると、区間は広くなる。",
        "register": "spoken"
      },
      {
        "en": "What happens to the margin of error if the confidence level is increased from 95% to 99%?",
        "ja": "信頼度を 95% から 99% に上げると、誤差の範囲はどうなるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の教科書では「信頼度 95%」と言い、英語では 95% confidence level または at the 95% level と言う。"
    ],
    "related": [
      "confidence-interval",
      "significance-level"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "confidence level": 66,
        "level of confidence": 8
      },
      "written": {
        "confidence level": 74,
        "level of confidence": 17
      },
      "sources": [
        "khan-ap-stats",
        "mit-notes",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "confounding-variable",
    "ja": {
      "term": "交絡変数",
      "reading": "こうらくへんすう",
      "alt": []
    },
    "en": {
      "term": "confounding variable",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "説明変数と応答変数の両方に関わり、その影響を説明変数の影響と区別できなくする変数。",
    "definition_en": "A variable related to both the explanatory and the response variable, whose effect cannot be told apart from that of the explanatory variable.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Maybe it's not the coffee but age that's doing it; age could be a confounding variable.",
        "ja": "コーヒーではなく年齢のせいかもしれない。年齢が交絡変数になりうる。",
        "register": "spoken"
      },
      {
        "en": "Identify a possible confounding variable in this observational study.",
        "ja": "この観察研究で考えられる交絡変数を挙げよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は lurking variable（潜在変数）という語を使う。用例コーパスの話し言葉の confounding variable はすべて Khan Academy。"
    ],
    "related": [
      "causation",
      "explanatory-variable",
      "response-variable"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "交絡",
        "en": "Confounding"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "confounding variable": 13
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "congruence-criteria-for-right-triangles",
    "ja": {
      "term": "直角三角形の合同条件",
      "reading": "ちょっかくさんかくけいのごうどうじょうけん",
      "alt": []
    },
    "en": {
      "term": "right triangle congruence",
      "alt": [
        "congruence of right triangles",
        "right triangle congruence theorems"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。CK-12 Geometry は、日本の 2 つの条件のうち「斜辺と他の 1 辺がそれぞれ等しい」を Hypotenuse-Leg (HL) Congruence Theorem と呼んで定理として扱う（4.16 HL Triangle Congruence）。「斜辺と 1 つの鋭角がそれぞれ等しい」は AAS で示せる。2 つをまとめた名前は用例コーパスと参照に出てこない。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの直角三角形が合同になるための条件。「斜辺と 1 つの鋭角がそれぞれ等しい」「斜辺と他の 1 辺がそれぞれ等しい」の 2 つがある。",
    "definition_en": "The two tests for congruent right triangles: equal hypotenuses together with one pair of equal acute angles, or equal hypotenuses together with one pair of equal legs.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For right triangle congruence, there are two shortcuts: the hypotenuse and a leg, which is HL, or the hypotenuse and an acute angle, which is really just AAS.",
        "ja": "直角三角形の合同には近道が 2 つある。斜辺と他の 1 辺（HL）か、斜辺と 1 つの鋭角で、こっちは実は AAS だ。",
        "register": "spoken"
      },
      {
        "en": "In right triangles ABC and DEF, ∠C = ∠F = 90°, AB = DE, and ∠A = ∠D. Prove that △ABC ≅ △DEF, and state which condition for right triangle congruence you used.",
        "ja": "∠C = ∠F = 90° の直角三角形 ABC と DEF で、AB = DE、∠A = ∠D のとき、△ABC ≡ △DEF であることを証明せよ。また、用いた直角三角形の合同条件を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「斜辺と他の 1 辺がそれぞれ等しい」は HL（hypotenuse-leg、エントリ hl-congruence）。英語の leg は直角をはさむ辺で、日本語の「他の 1 辺」に当たる。",
      "「斜辺と 1 つの鋭角がそれぞれ等しい」は、直角と合わせると 2 組の角とその間にない 1 辺が等しくなるので、AAS（エントリ aas-congruence）で示せる。",
      "答案では、2 つの三角形に直角があること（∠C = ∠F = 90°）を先に書く。HL も AAS も直角の組を使う。"
    ],
    "related": [
      "hl-congruence",
      "aas-congruence",
      "congruence-criteria",
      "right-triangle",
      "hypotenuse"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.16 HL Triangle Congruence（Hypotenuse-Leg (HL) Congruence Theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "congruence-criteria",
    "ja": {
      "term": "合同条件",
      "reading": "ごうどうじょうけん",
      "alt": []
    },
    "en": {
      "term": "congruence theorems",
      "alt": [
        "congruence criteria",
        "triangle congruence criteria",
        "congruence postulates"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説の三角形の合同条件は 3 つ（対応する 3 組の辺、2 組の辺とその間の角、1 組の辺とその両端の角）で、「合同条件」とまとめて呼ぶ。CK-12 Geometry は SSS・SAS・ASA を公準（postulate、4.13〜4.15）とし、AAS と直角三角形の HL を定理（theorem、4.15・4.16）として加え、それぞれを名前で呼ぶ（エントリ sss-congruence、sas-congruence、asa-congruence、aas-congruence、hl-congruence）。IM Geometry は 2.6 Side-Angle-Side Triangle Congruence、2.7 Angle-Side-Angle Triangle Congruence のように 3 つを 1 つずつ扱い、AAS・HL の名前は出てこない。見出しの congruence theorems は IM の言い方。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの三角形が合同であるといえるための条件。中学校学習指導要領解説（第 2 学年）では「3 組の辺」「2 組の辺とその間の角」「1 組の辺とその両端の角」がそれぞれ等しい、の 3 つ。",
    "definition_en": "The sets of matching parts that are enough to prove two triangles congruent without checking every side and angle. Each one is named by the parts that match, such as SSS, SAS, and ASA.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Which of the congruence theorems can we use here? We've got two sides and the angle between them, so it's SAS.",
        "ja": "ここで使える合同条件はどれかな。2 組の辺とその間の角が分かっているから、「2 組の辺とその間の角がそれぞれ等しい」だね。",
        "register": "spoken"
      },
      {
        "en": "In quadrilateral ABCD, AB = DC and m∠ABC = m∠DCB. Which of the congruence theorems shows that △ABC ≅ △DCB? Explain.",
        "ja": "四角形 ABCD で、AB = DC、∠ABC = ∠DCB である。△ABC ≡ △DCB を示すには、どの合同条件を用いればよいか。理由も説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AAS（エントリ aas-congruence）は学習指導要領解説の合同条件に無い。日本語の証明で言うなら、残りの角も等しいことを示して「1 組の辺とその両端の角」にする。HL（エントリ hl-congruence）は、解説では直角三角形の合同条件（エントリ congruence-criteria-for-right-triangles）として別に扱う。",
      "英語では同じ条件を theorem（定理）とも postulate（公準、エントリ postulate）とも呼ぶ。congruence theorems と congruence postulates は同じものを指す。IM Geometry は合同条件を rigid transformation から証明するので theorem、CK-12 Geometry は証明せずに認めるので postulate と呼ぶ。",
      "2 辺とその間にない角（SSA）や 3 つの角（AAA）が等しくても、合同とはいえない。AAA で分かるのは相似（エントリ aa-similarity）まで。"
    ],
    "related": [
      "sss-congruence",
      "sas-congruence",
      "asa-congruence",
      "congruence-criteria-for-right-triangles",
      "congruent"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 2.6 Side-Angle-Side Triangle Congruence・2.7 Angle-Side-Angle Triangle Congruence・2.10 Practicing Proofs"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 B 図形: 三角形の合同条件（対応する３組の辺、２組の辺とその間の角、１組の辺とその両端の角）、直角三角形の合同条件"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.13 SSS、4.14 SAS、4.15 ASA and AAS、4.16 HL（SSS・SAS・ASA は Postulate、AAS・HL は Theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 congruence theorems（Geometry 2.7 Angle-Side-Angle Triangle Congruence・Geometry 2.6 Side-Angle-Side Triangle Congruence・Geometry 2.10 Practicing Proofs・計 45 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "congruence-modulo-n",
    "ja": {
      "term": "合同式",
      "reading": "ごうどうしき",
      "alt": [
        "〜を法として合同"
      ]
    },
    "en": {
      "term": "congruence modulo n",
      "alt": [
        "modular arithmetic"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（congruent modulo ／ congruence modulo の形と modular arithmetic は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "整数 a, b の差 a − b が n で割り切れるとき、a と b は n を法として合同であるといい、a ≡ b (mod n) と書く。この関係を表す式を合同式という。",
    "definition_en": "Two integers a and b are congruent modulo n when n divides a − b, that is, when they leave the same remainder on division by n. This is written a ≡ b (mod n).",
    "latex": "a \\equiv b \\pmod{n}",
    "spoken_en": "a is congruent to b mod n",
    "examples": [
      {
        "en": "With congruence modulo n, all we care about is the remainder, so mod 3, 17 and 5 act the same.",
        "ja": "合同式では余りだけを見るので、3 を法とすると 17 と 5 は同じに扱える。",
        "register": "spoken"
      },
      {
        "en": "Prove that congruence modulo n is an equivalence relation on the integers.",
        "ja": "n を法とする合同は、整数全体における同値関係であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a ≡ b (mod n) は a is congruent to b mod n と読む。mod は modulo の略。",
      "英語の congruent は図形の合同（congruent）と同じ語。日本語の「合同」と同じく、1 語が 2 つの意味を持つ。",
      "Levin の本では 6.2 Introduction to Number Theory に 13 件。用例コーパスでは congruent modulo などの形も modular arithmetic も話・書とも少なく、どれも MIT（MIT OCW の講義と MIT の講義ノート）。"
    ],
    "related": [
      "remainder",
      "modular-inverse",
      "chinese-remainder-theorem",
      "fermats-little-theorem",
      "congruent",
      "equivalence-relation",
      "modulus"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "整数の合同",
        "en": "Congruence (integers)"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2 Introduction to Number Theory"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学A 数学と人間の活動: 整数の合同"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "congruent modulo | congruent mod | congruence modulo | congruence mod": 2,
        "modular arithmetic": 3
      },
      "written": {
        "congruent modulo | congruent mod | congruence modulo | congruence mod": 3,
        "modular arithmetic": 7
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満）。見出しは Levin の呼び方 congruent modulo | congruent mod | congruence modulo | congruence mod（6.2 Introduction to Number Theory）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "congruent-arcs",
    "ja": {
      "term": "等しい弧",
      "reading": "ひとしいこ",
      "alt": []
    },
    "en": {
      "term": "congruent arcs",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "同じ円、または合同な円で、長さの等しい弧。等しい弧に対する中心角・円周角は、それぞれ等しい。",
    "definition_en": "Arcs in the same circle, or in congruent circles, that have the same length. Central angles on congruent arcs are equal, and so are inscribed angles on them.",
    "latex": "\\overset{\\frown}{AB} \\cong \\overset{\\frown}{CD}",
    "spoken_en": "arc AB is congruent to arc CD",
    "examples": [
      {
        "en": "In the same circle, both central angles are 70 degrees, so the arcs they cut off are congruent arcs.",
        "ja": "同じ円で、どちらの中心角も 70° だから、それに対する弧は等しい弧だ。",
        "register": "spoken"
      },
      {
        "en": "In circle O, arc AB and arc CD are congruent arcs, and ∠AOB = 48°. Find ∠COD.",
        "ja": "円 O で、弧 AB = 弧 CD、∠AOB = 48° のとき、∠COD の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "CK-12 Geometry 6.9 Arcs in Circles は合同の語を使って congruent arcs と呼び、弧どうしを ≅ で結ぶ（6.12 Chords and Central Angle Arcs も）。本プロジェクトの例文では、日本語は 弧 AB = 弧 CD、英語は arc AB ≅ arc CD と書く。",
      "比べるのは同じ円か合同な円の弧。半径の違う円の弧は、長さが同じでも曲がり方が違うので congruent arcs ではない。",
      "同じ円の中では、弧の長さが等しいことと、弧の度数（エントリ arc-measure）や中心角が等しいことは同じ。",
      "中学校学習指導要領解説は「同じ弧に対する円周角」と書き、「等しい弧」という語は出てこない。CK-12 Geometry 6.9 は congruent arcs と呼ぶ。"
    ],
    "related": [
      "arc",
      "congruent",
      "central-angle",
      "inscribed-angle",
      "arc-measure"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.9 Arcs in Circles"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形: 「同じ弧に対する円周角」（「等しい弧」は 0 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 congruent arcs（CK-12 Geometry 6.9 Arcs in Circles・計 2 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "congruent",
    "ja": {
      "term": "合同",
      "reading": "ごうどう",
      "alt": [
        "合同な図形"
      ]
    },
    "en": {
      "term": "congruent",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの図形が、平行移動・回転・裏返しによってぴったり重なること。形も大きさも同じ。",
    "definition_en": "Having exactly the same size and shape, so that one figure can be moved onto the other by translations, rotations, and reflections.",
    "latex": "\\triangle ABC \\cong \\triangle DEF",
    "spoken_en": "triangle ABC is congruent to triangle DEF",
    "examples": [
      {
        "en": "Congruent means same size and same shape: you can slide, turn, or flip one onto the other.",
        "ja": "合同というのは、大きさも形も同じということ。ずらしたり回したり裏返したりして重ねられる。",
        "register": "spoken"
      },
      {
        "en": "Two sides and the angle between them match, so these triangles are congruent.",
        "ja": "2 辺とその間の角がそれぞれ等しいから、この 2 つの三角形は合同だ。",
        "register": "spoken"
      },
      {
        "en": "Prove that triangle ABM is congruent to triangle ACM.",
        "ja": "△ABM ≡ △ACM であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "合同の記号は日本では ≡（中学校学習指導要領の〔用語・記号〕）、英語では ≅（CK-12 Geometry・IM）。英語で ≡ は恒等式や整数の合同（a ≡ b (mod n)）に使う。",
      "「合同」は名詞でも使うが、英語の congruent は形容詞（are congruent ／ is congruent to）。名詞は congruence。",
      "congruent は整数の合同（mod）にも使う。"
    ],
    "related": [
      "triangle",
      "proof",
      "isosceles-triangle",
      "translation",
      "rotation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）〔用語・記号〕",
        "note": "中2 B 図形: ≡"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.25 Corresponding Parts of Congruent Figures ほか（≅）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 2.1 Congruent Parts, Part 1 ほか（≅）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "congruent triangles | congruent angles | congruent sides | congruent segments | congruent figures | congruent shapes | congruent polygons": 24
      },
      "written": {
        "congruent triangles | congruent angles | congruent sides | congruent segments | congruent figures | congruent shapes | congruent polygons": 5
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "conic-section",
    "ja": {
      "term": "二次曲線",
      "reading": "にじきょくせん",
      "alt": [
        "円錐曲線"
      ]
    },
    "en": {
      "term": "conic section",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "放物線・楕円・双曲線の総称。円錐を平面で切った切り口として得られる。x, y の 2 次方程式で表される。",
    "definition_en": "A parabola, ellipse, or hyperbola: a curve cut from a cone by a plane, given by a second-degree equation in x and y.",
    "latex": "Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "All three conic sections come from slicing a cone at different angles.",
        "ja": "3 種類の二次曲線は、どれも円錐を違う角度で切るとできる。",
        "register": "spoken"
      },
      {
        "en": "Identify the conic section 4x² + 9y² − 16x = 20 and write its equation in standard form.",
        "ja": "4x² + 9y² − 16x = 20 はどの二次曲線か答え、標準形で表せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "identify the conic section",
        "ja": "二次曲線の種類を判別する"
      }
    ],
    "pitfalls": [
      "日本の数C は「二次曲線」、英語は円錐の切り口として conic section と呼ぶ。",
      "二次曲線と直線の共有点は、連立して得た 2 次方程式の判別式で調べる。英語でも discriminant を使う。"
    ],
    "related": [
      "ellipse",
      "focus",
      "directrix",
      "eccentricity",
      "standard-form-of-a-conic",
      "polar-equation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円錐曲線",
        "en": "Conic section"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "conic section": 18
      },
      "written": {
        "conic section": 148
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "conjugate-roots",
    "ja": {
      "term": "共役な解",
      "reading": "きょうやくなかい",
      "alt": [
        "共役複素数解"
      ]
    },
    "en": {
      "term": "complex conjugate roots",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus"
      ]
    },
    "definition_ja": "実数係数の方程式の、互いに共役な 2 つの虚数解 a + bi と a − bi。係数が実数なら、a + bi が解のとき a − bi も解になる。",
    "definition_en": "Two roots a + bi and a − bi (a and b real, b ≠ 0) of an equation with real coefficients, complex conjugates of each other; with real coefficients, if a + bi is a root, so is a − bi.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Since the coefficients are real, the complex roots come in conjugate pairs: 1 + 2i and 1 − 2i are complex conjugate roots.",
        "ja": "係数が実数なので、虚数解は共役な組で出てくる。1 + 2i と 1 − 2i は共役な解だ。",
        "register": "spoken"
      },
      {
        "en": "The cubic equation x³ + ax² + bx + 10 = 0 with real a and b has 1 + 2i as a solution. Find a, b, and the other solutions.",
        "ja": "a, b を実数とする 3 次方程式 x³ + ax² + bx + 10 = 0 が 1 + 2i を解にもつ。a, b の値と他の解を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "係数が実数であることが前提（with real coefficients）。書き落とすと成り立たない。",
      "実数係数の多項式が a + bi を零点にもてば a − bi も零点である、という性質を、OpenStax Algebra and Trigonometry（5.5）・Precalculus（3.6）の Zeros of Polynomial Functions は Complex Conjugate Theorem と呼ぶ。",
      "OpenStax Elementary Algebra（6.4 Special Products）・Intermediate Algebra（5.3 Multiply Polynomials）の conjugate pair は、多くが 2 項式の組 (a − b)(a + b) の意味。"
    ],
    "related": [
      "complex-conjugate",
      "imaginary-solution",
      "characteristic-equation"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "7.1 Second-Order Linear Equations（complex conjugate roots）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "5.5 Zeros of Polynomial Functions（Complex Conjugate Theorem）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "6.4 Special Products（conjugate pair）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "5.3 Multiply Polynomials（conjugate pair）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和5年度 大学入学共通テスト 本試験 数学Ⅱ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=432&f=abm00003620.pdf&n=2023_ot_20_sugaku2.pdf",
        "note": "第4問: 二つの虚数解は互いに共役な複素数"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「四次方程式」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%9B%E6%AC%A1%E6%96%B9%E7%A8%8B%E5%BC%8F",
        "note": "共役複素数解"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "3.6 Zeros of Polynomial Functions（Complex Conjugate Theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "complex conjugate roots": 11
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "connected",
    "ja": {
      "term": "連結",
      "reading": "れんけつ",
      "alt": [
        "連結グラフ"
      ]
    },
    "en": {
      "term": "connected",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフのどの 2 頂点の間にも経路があるという性質のこと。そのようなグラフを連結グラフという。",
    "definition_en": "Describes a graph in which there is a path between every pair of vertices. Such a graph is called a connected graph.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If a graph isn't connected, it falls apart into pieces, and there's no way to get from one piece to another.",
        "ja": "グラフが連結でないと、いくつかのかたまりに分かれて、かたまりの間を行き来する道がない。",
        "register": "spoken"
      },
      {
        "en": "Prove that a connected graph with n vertices has at least n − 1 edges.",
        "ja": "頂点が n 個の連結グラフは、少なくとも n − 1 本の辺をもつことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "連結でないグラフは disconnected（not connected とも言う）。分かれた 1 つ 1 つのかたまりを connected component（連結成分）、または単に component という。",
      "形容詞なので、英語では a connected graph ／ the graph is connected の形で使う。日本語の「連結グラフ」は connected graph。",
      "Levin の Discrete Mathematics は connected を 28 件使う（2.2 Trees 13 ほか）。"
    ],
    "related": [
      "path-in-a-graph",
      "spanning-tree",
      "eulerian-path",
      "graph-network",
      "tree"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.2 Trees・2.1 Problems and Definitions"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "connected graph | graph is connected": 16
      },
      "written": {
        "connected graph | graph is connected": 30
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "consecutive-integers",
    "ja": {
      "term": "連続する整数",
      "reading": "れんぞくするせいすう",
      "alt": []
    },
    "en": {
      "term": "consecutive integers",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "1 ずつ違う整数を小さい順に並べたもの。最も小さい数を n とすると、n, n + 1, n + 2, … と表せる。",
    "definition_en": "Integers that come one right after another, each 1 more than the one before. If the smallest one is n, they are n, n + 1, n + 2, and so on.",
    "latex": "n,\\ n + 1,\\ n + 2",
    "spoken_en": "n, n plus one, n plus two",
    "examples": [
      {
        "en": "For three consecutive integers, call the first one n, and then the next two are n plus 1 and n plus 2.",
        "ja": "連続する 3 つの整数なら、最初を n とおいて、次の 2 つは n + 1 と n + 2 だね。",
        "register": "spoken"
      },
      {
        "en": "The sum of three consecutive integers is 72. Find the three integers.",
        "ja": "連続する 3 つの整数の和が 72 である。この 3 つの整数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "連続する偶数は consecutive even integers（n, n + 2, n + 4, …）、連続する奇数は consecutive odd integers。どちらも隣どうしの差は 2。",
      "並びの「連続する」は consecutive で、関数の「連続」は continuous（エントリ continuous）。別の語。",
      "「連続する 2 つの整数」は two consecutive integers と、個数を consecutive の前に置く。数列の隣り合う項は consecutive terms（エントリ consecutive-terms）。"
    ],
    "related": [
      "integer",
      "even-number",
      "odd-number",
      "consecutive-terms",
      "word-problem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "consecutive integers": 3
      },
      "written": {
        "consecutive integers": 85
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "consecutive-terms",
    "ja": {
      "term": "隣り合う 2 項",
      "reading": "となりあうにこう",
      "alt": []
    },
    "en": {
      "term": "consecutive terms",
      "alt": [],
      "variants": [
        {
          "term": "successive terms",
          "register": "written",
          "note": "AP の CED（topic 10.2）の言い方（a constant ratio between successive terms）。Levin（4.2）も使う。each ／ every successive term（次々の項）は 2 項の組の意味ではない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "数列で、aₙ と aₙ₊₁ のように番号が 1 つ違う 2 つの項。",
    "definition_en": "Two terms of a sequence next to each other, such as aₙ and aₙ₊₁.",
    "latex": "a_{n+1} - a_n",
    "spoken_en": null,
    "examples": [
      {
        "en": "Take the ratio of successive terms and see if it's constant.",
        "ja": "隣り合う 2 項の比をとって、一定か確かめよう。",
        "register": "spoken"
      },
      {
        "en": "The difference between any two consecutive terms is 6.",
        "ja": "隣り合う 2 項の差はどれも 6 である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "common-difference",
      "common-ratio",
      "sequence-of-differences",
      "consecutive-integers"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.2 Arithmetic Sequences・13.3 Geometric Sequences（consecutive terms）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.2（successive terms）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.2（successive terms）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "consecutive terms": 7,
        "!each !every !next successive terms": 1
      },
      "written": {
        "consecutive terms": 53,
        "!each !every !next successive terms": 3
      },
      "sources": [
        "khan-ap-calc",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "conservative-vector-field",
    "ja": {
      "term": "保存場",
      "reading": "ほぞんば",
      "alt": [
        "保存ベクトル場"
      ]
    },
    "en": {
      "term": "conservative vector field",
      "alt": [],
      "variants": [
        {
          "term": "conservative vector field",
          "register": "written",
          "note": "OpenStax Calculus の節の名前（Conservative Vector Fields）。話し言葉にはほとんど出てこず（MIT OCW だけ）、書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "conservative field",
          "register": "written",
          "note": "vector を省いた言い方。書き言葉はすべて OpenStax Calculus。話し言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "ある関数 f の勾配 ∇f として表せるベクトル場。線積分が経路によらない。",
    "definition_en": "A vector field that is the gradient ∇f of some function f; its line integrals are independent of path.",
    "latex": "\\mathbf{F} = \\nabla f",
    "spoken_en": "F equals del f",
    "examples": [
      {
        "en": "If the field is conservative, the line integral only depends on the endpoints.",
        "ja": "場が保存場なら、線積分は端点だけで決まる。",
        "register": "spoken"
      },
      {
        "en": "Determine whether F(x, y) = ⟨2xy, x² + 3⟩ is a conservative vector field.",
        "ja": "F(x, y) = ⟨2xy, x² + 3⟩ が保存場かどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉ではどの言い方もほとんど出てこない。",
      "平面の場 ⟨P, Q⟩ では、∂Q/∂x = ∂P/∂y が保存場であることの目安になる（領域の条件つき）。"
    ],
    "related": [
      "potential-function",
      "gradient",
      "line-integral",
      "curl-vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "conservative vector field": 2
      },
      "written": {
        "conservative vector field": 41,
        "conservative field": 15
      },
      "sources": [
        "mit-18.02",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "constant-function",
    "ja": {
      "term": "定数関数",
      "reading": "ていすうかんすう",
      "alt": []
    },
    "en": {
      "term": "constant function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "x がどんな値をとっても、y がいつも同じ値 c になる関数 y = c。グラフは x 軸に平行な直線。",
    "definition_en": "A function whose output is the same number for every input, y = c. Its graph is a horizontal line with a slope of 0.",
    "latex": "y = c",
    "spoken_en": "y equals c",
    "examples": [
      {
        "en": "y = 4 is a constant function: whatever x is, y is always 4.",
        "ja": "y = 4 は定数関数。x が何でも、y はいつも 4。",
        "register": "spoken"
      },
      {
        "en": "Graph the constant function y = −2 and state its slope.",
        "ja": "定数関数 y = −2 のグラフをかき、その傾きを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "グラフは x 軸に平行な直線で、英語では horizontal line（水平な線）と言う。傾きは 0。",
      "x = 3 のような y 軸に平行な直線（縦の直線）は、x の関数ではない。傾きも定義されない（undefined）。",
      "日本語版 Wikipedia「一次関数」は、一次関数を y = ax + b（a ≠ 0）とするときは定数関数を含めず、a = 0 も含める流儀もあると書く（エントリ linear-function）。"
    ],
    "related": [
      "linear-function",
      "slope",
      "constant",
      "function",
      "x-axis"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "定数関数",
        "en": "Constant function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「一次関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%80%E6%AC%A1%E9%96%A2%E6%95%B0",
        "note": "a ≠ 0 とする場合と、a = 0 を含める場合"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "constant function": 23
      },
      "written": {
        "constant function": 38
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "constant-multiple-rule",
    "ja": {
      "term": "定数倍の法則",
      "reading": "ていすうばいのほうそく",
      "alt": [
        "定数倍"
      ]
    },
    "en": {
      "term": "constant multiple rule",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「定数倍の法則」は学習指導要領解説に無い（日本語版 Wikipedia「積の微分法則」にはある）。学習指導要領解説には {kf(x)}′ = kf′(x) の性質の名前が出てこない。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "定数倍の導関数は導関数の定数倍になる、{kf(x)}′ = kf′(x) という性質。",
    "definition_en": "The rule that the derivative of a constant times a function is the constant times the derivative.",
    "latex": "(kf)' = kf'",
    "spoken_en": "the derivative of k f is k times f prime",
    "examples": [
      {
        "en": "Constant multiple rule: pull the 5 out front and differentiate x squared.",
        "ja": "定数倍の法則で、5 を前に出して x² を微分する。",
        "register": "spoken"
      },
      {
        "en": "By the constant multiple rule, the derivative of 5x² is 10x.",
        "ja": "定数倍の法則より、5x² の導関数は 10x である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "derivative-of-a-sum",
      "power-rule"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「積の微分法則」",
        "url": "https://ja.wikipedia.org/wiki/%E7%A9%8D%E3%81%AE%E5%BE%AE%E5%88%86%E6%B3%95%E5%89%87",
        "note": "積の法則の特別な場合として「定数倍の法則」"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「定数倍の法則」は出てこない・{kf(x)}′ = kf′(x) の性質の名前が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "constant multiple rule": 10
      },
      "written": {
        "constant multiple rule": 15
      },
      "sources": [
        "mit-18.01",
        "openstax-calculus",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "constant-of-integration",
    "ja": {
      "term": "積分定数",
      "reading": "せきぶんていすう",
      "alt": [
        "プラス C"
      ]
    },
    "en": {
      "term": "constant of integration",
      "alt": [],
      "variants": [
        {
          "term": "plus C",
          "register": "spoken",
          "note": "話し言葉（講義）では積分定数そのものを指して plus C と言う。特定のソースに偏らない。話し言葉では constant of integration よりずっと多く、書き言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "不定積分に加える任意の定数。ふつう C で表し、答案では末尾に + C と書く。",
    "definition_en": "The arbitrary constant added to an indefinite integral, usually written as + C at the end.",
    "latex": "\\int 2x\\,dx = x^2 + C",
    "spoken_en": "the integral of two x dx equals x squared plus C",
    "examples": [
      {
        "en": "Every antiderivative of 2x is x squared plus some number, and that number is your constant of integration.",
        "ja": "2x の原始関数はどれも x² にある数を足したもの。その数が積分定数。",
        "register": "spoken"
      },
      {
        "en": "The initial condition y(0) = 3 gives the constant of integration C = 3.",
        "ja": "初期条件 y(0) = 3 から、積分定数は C = 3 と決まる。",
        "register": "written"
      },
      {
        "en": "You lost a point because there's no plus C.",
        "ja": "+C がないから 1 点減点ね。",
        "register": "spoken"
      },
      {
        "en": "Each answer should end with + C.",
        "ja": "どの答えにも + C を付けること。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「ただし C は積分定数」に当たる英語は where C is a constant。OpenStax Calculus の解答はほとんど添えない（慣習差 constant-of-integration-remark）。",
      "定積分では C は F(b) − F(a) で打ち消し合うので書かない。",
      "口頭では plus a constant と言うこともある。"
    ],
    "related": [
      "indefinite-integral",
      "antiderivative"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "plus C": 336,
        "constant of integration": 28
      },
      "written": {
        "constant of integration": 11,
        "plus C": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "constant-of-proportionality",
    "ja": {
      "term": "比例定数",
      "reading": "ひれいていすう",
      "alt": []
    },
    "en": {
      "term": "constant of proportionality",
      "alt": [],
      "variants": [
        {
          "term": "constant of variation",
          "register": "written",
          "note": "Algebra の教科書の呼び方（OpenStax Elementary Algebra・Algebra and Trigonometry の Variation の節）。書き言葉ではこちらが多い。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "比例 y = ax や反比例 y = a/x の式にある定数 a。比例では y ÷ x、反比例では x × y を計算すると求められる。",
    "definition_en": "The constant a in a direct variation y = ax or an inverse variation y = a/x. It equals y divided by x in a direct variation and x times y in an inverse variation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Divide each y by its x: you get 5 every time, so the constant of proportionality is 5.",
        "ja": "それぞれの y をその x でわってみて。毎回 5 になるから、比例定数は 5 だね。",
        "register": "spoken"
      },
      {
        "en": "y varies directly with x, and y = 12 when x = 3. Find the constant of variation.",
        "ja": "y は x に比例し、x = 3 のとき y = 12 である。比例定数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "見出しは中学の教材（IM Grade 7・Grade 8 の glossary）と授業（Khan Academy の中学）の constant of proportionality にした。Algebra の教科書（OpenStax Elementary Algebra・Algebra and Trigonometry）は direct variation ／ inverse variation の定数として constant of variation と呼ぶ。",
      "反比例 y = a/x の a も比例定数と呼ぶ（エントリ inverse-proportion）。",
      "proportionality は 6 音節の長い語で、強勢は -al- に置く（Merriam-Webster: prəˌpōrshəˈnalətē）。"
    ],
    "related": [
      "direct-proportion",
      "inverse-proportion",
      "ratio",
      "value-of-a-ratio",
      "constant"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Grade 7・Grade 8 glossary: constant of proportionality"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「proportionality」",
        "url": "https://www.merriam-webster.com/dictionary/proportionality",
        "note": "発音 prəˌpōrshəˈnalətē（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "constant of proportionality": 73,
        "constant of variation": 9
      },
      "written": {
        "constant of proportionality": 4,
        "constant of variation": 31
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 3 の準備の前の修正」の 2）。en.term は constant of proportionality（IM Grade 7・8 の glossary、Khan Academy の中学）、constant of variation は register written の variant（Algebra の教科書）。見出しの register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "constant-term",
    "ja": {
      "term": "定数項",
      "reading": "ていすうこう",
      "alt": []
    },
    "en": {
      "term": "constant term",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "文字を含まない、数だけの項。3x − 5 の定数項は −5。",
    "definition_en": "A term that has no variable, only a number. In 3x − 5, the constant term is −5.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In 3x − 5, the constant term is negative 5, not 5; the sign goes with it.",
        "ja": "3x − 5 の定数項は 5 じゃなくて −5。符号も含めるんだよ。",
        "register": "spoken"
      },
      {
        "en": "Find the constant term of the expression 2x² + 6x − 9.",
        "ja": "式 2x² + 6x − 9 の定数項を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "定数項は符号を含めて答える。3x − 5 の定数項は 5 ではなく −5。",
      "y = mx + b の定数項 b は、グラフの y 切片（y-intercept）になる。"
    ],
    "related": [
      "constant",
      "coefficient",
      "term",
      "linear-expression",
      "polynomial"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "constant term": 105
      },
      "written": {
        "constant term": 71
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "constant",
    "ja": {
      "term": "定数",
      "reading": "ていすう",
      "alt": []
    },
    "en": {
      "term": "constant",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "値が変わらない、決まった数。a や k のような文字でも、1 つの決まった値を表すものは定数という。",
    "definition_en": "A quantity whose value does not change, such as 5 or −2. A letter like k is also a constant when it stands for one fixed number.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In y = kx, k is a constant; it stays the same while x and y change.",
        "ja": "y = kx の k は定数。x と y が変わっても、k は変わらない。",
        "register": "spoken"
      },
      {
        "en": "In y = ax + 3, a is a constant. If the graph passes through the point (2, 11), find the value of a.",
        "ja": "y = ax + 3 のグラフが点 (2, 11) を通るとき、定数 a の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "比例定数は constant of proportionality、積分定数は constant of integration（エントリ constant-of-integration）。どちらも constant を使う。",
      "英語の constant は形容詞にもなり、「一定の」の意味で使う（a constant rate = 一定の割合、constant speed = 一定の速さ）。"
    ],
    "related": [
      "variable",
      "constant-term",
      "coefficient",
      "constant-of-integration"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "定数",
        "en": "Constant (mathematics)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "a constant !of !function !rate !speed !term | the constant !of !function !rate !speed !term": 1050
      },
      "written": {
        "a constant !of !function !rate !speed !term | the constant !of !function !rate !speed !term": 655
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "constraint",
    "ja": {
      "term": "制約条件",
      "reading": "せいやくじょうけん",
      "alt": [
        "制約"
      ]
    },
    "en": {
      "term": "constraint",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "大学"
      ],
      "us": [
        "Precalculus",
        "Calculus I",
        "Calculus III"
      ]
    },
    "definition_ja": "最適化問題で、変数が満たさなければならない条件（等式や不等式）。",
    "definition_en": "A condition (an equation or inequality) that the variables of an optimization problem must satisfy.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The constraint says x plus y is 10, so solve it for y and substitute.",
        "ja": "制約条件は x + y = 10 なので、y について解いて代入する。",
        "register": "spoken"
      },
      {
        "en": "Find the maximum of f(x, y) = xy subject to the constraint x + 2y = 8.",
        "ja": "制約条件 x + 2y = 8 のもとで f(x, y) = xy の最大値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "subject to the constraint",
        "ja": "〜という条件のもとで"
      }
    ],
    "pitfalls": [
      "問題文の「x, y が … を満たすとき」は、英語では subject to the constraint(s) … と書ける。"
    ],
    "related": [
      "objective-function",
      "feasible-region",
      "linear-programming",
      "lagrange-multipliers"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "制約 (数学)",
        "en": "Constraint (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "constraint": 356
      },
      "written": {
        "constraint": 114
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "construct",
    "ja": {
      "term": "作図する",
      "reading": "さくずする",
      "alt": []
    },
    "en": {
      "term": "construct",
      "alt": [],
      "uk": null
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "目盛りを使わない定規とコンパスだけを使って、条件に合う図形をかく。",
    "definition_en": "To draw a figure exactly using only a compass and a straightedge, without measuring any lengths or angles.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To construct the perpendicular bisector, put your compass on A, open it past the middle, and draw an arc.",
        "ja": "垂直二等分線を作図するには、コンパスの針を A に置いて、真ん中より広めに開いて弧をかこう。",
        "register": "spoken"
      },
      {
        "en": "Construct the bisector of ∠AOB using a compass and a straightedge.",
        "ja": "定規とコンパスを使って、∠AOB の二等分線を作図せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "名詞の「作図」は construction（エントリ construction）。「作図せよ」は Construct … で始める。",
      "construct は図形に限らず「つくる」一般にも使う。統計の construct a confidence interval（信頼区間をつくる、エントリ confidence-interval）は定規やコンパスとは関係がない。",
      "作図の道具は compass（エントリ compass）と straightedge（エントリ straightedge）。目盛りのある ruler で長さを測ってかくのは作図ではない。"
    ],
    "related": [
      "construction",
      "compass",
      "straightedge",
      "perpendicular-bisector",
      "angle-bisector"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 1.1 Build It ほか"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "construct a perpendicular | construct the perpendicular | construct an angle | construct a line | construct the bisector | construct a triangle | construct a circle | construct a square | construct a hexagon | construct a copy | construct the angle | construct the line | construct a parallel": 5
      },
      "written": {
        "construct a perpendicular | construct the perpendicular | construct an angle | construct a line | construct the bisector | construct a triangle | construct a circle | construct a square | construct a hexagon | construct a copy | construct the angle | construct the line | construct a parallel": 9
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは IM の呼び方 construct a perpendicular | construct the perpendicular | construct an angle | construct a line | construct the bisector | construct a triangle | construct a circle | construct a square | construct a hexagon | construct a copy | construct the angle | construct the line | construct a parallel（Geometry 1.7 Construction Techniques 5: Squares・Geometry 1.9 Speedy Delivery・Geometry 1.11 Defining Reflections・計 71 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "construction",
    "ja": {
      "term": "作図",
      "reading": "さくず",
      "alt": []
    },
    "en": {
      "term": "construction",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "定規とコンパスだけを使って、条件に合う図をかくこと。定規は直線を引くためだけに使い、長さは測らない。",
    "definition_en": "Drawing a geometric figure using only a compass and a straightedge. The straightedge is used only to draw straight lines, not to measure lengths.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's a construction, so no measuring: just your compass and straightedge.",
        "ja": "作図だから測らないよ。使うのはコンパスと定規だけ。",
        "register": "spoken"
      },
      {
        "en": "Using only a compass and straightedge, construct the bisector of ∠AOB. Do not erase your construction marks.",
        "ja": "コンパスと定規だけを使って、∠AOB の二等分線を作図せよ。作図に用いた線は消さないこと。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞の「作図する」は construct（エントリ construct）。「〜を作図せよ」は Construct … で始める。",
      "日常語の construction は「建設・工事」の意味。",
      "証明の中の by construction は「そうなるように作ったので」という理由の言い方で、作図の問題とは別の使い方。"
    ],
    "related": [
      "construct",
      "compass",
      "straightedge",
      "perpendicular-bisector",
      "angle-bisector"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 1.1 Build It ほか"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!by construction": 22
      },
      "written": {
        "!by construction": 19
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "contingency-table",
    "ja": {
      "term": "分割表",
      "reading": "ぶんかつひょう",
      "alt": [
        "二元分割表",
        "クロス集計表"
      ]
    },
    "en": {
      "term": "contingency table",
      "alt": [
        "two-way table"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Intro Statistics",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つのカテゴリ変数の値の組ごとに度数を数えて、行と列に並べた表。",
    "definition_en": "A table of counts for every combination of the values of two categorical variables, arranged in rows and columns.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Read across the row to get the total for that group.",
        "ja": "その群の合計は、行を横に読んでいけば分かる。",
        "register": "spoken"
      },
      {
        "en": "Use the contingency table to find the probability that a randomly chosen student is a senior and plays a sport.",
        "ja": "分割表を用いて、無作為に選んだ生徒が最上級生でスポーツをしている確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は contingency table（節の名前 Contingency Tables）、AP Statistics の CED（2026 年版）と Khan Academy は two-way table と呼ぶ。"
    ],
    "related": [
      "categorical-variable",
      "test-for-independence",
      "expected-count"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "分割表",
        "en": "Contingency table"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.1・2.2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "two-way table": 5
      },
      "written": {
        "contingency table": 21
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "continuity-correction",
    "ja": {
      "term": "連続修正",
      "reading": "れんぞくしゅうせい",
      "alt": [
        "半整数補正"
      ]
    },
    "en": {
      "term": "continuity correction",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "離散型の分布を連続型の正規分布で近似するとき、整数値 k を区間 k − 0.5 から k + 0.5 に置き換える補正。",
    "definition_en": "When a discrete distribution is approximated by the continuous normal distribution, replacing an integer value k by the interval from k − 0.5 to k + 0.5.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "With the continuity correction, P of X at least 60 becomes P of Y greater than 59.5.",
        "ja": "連続修正をすると、P(X ≥ 60) は P(Y > 59.5) になる。",
        "register": "spoken"
      },
      {
        "en": "Use the normal approximation with a continuity correction to find P(X ≤ 45).",
        "ja": "連続修正をした正規近似で P(X ≤ 45) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "normal-approximation-to-the-binomial",
      "binomial-distribution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "continuity correction": 2
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは OpenStax の呼び方 continuity correction（本文 2 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "continuity",
    "ja": {
      "term": "連続性",
      "reading": "れんぞくせい",
      "alt": []
    },
    "en": {
      "term": "continuity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数が連続であること。x = a で連続とは、lim_{x→a} f(x) = f(a) が成り立つこと。",
    "definition_en": "The property of being continuous; f is continuous at a when lim_{x→a} f(x) = f(a).",
    "latex": "\\lim_{x \\to a} f(x) = f(a)",
    "spoken_en": "the limit as x approaches a of f of x equals f of a",
    "examples": [
      {
        "en": "For continuity at a point you need three things: f of a exists, the limit exists, and they're equal.",
        "ja": "ある点で連続であるには 3 つ必要。f(a) がある、極限がある、両者が等しい。",
        "register": "spoken"
      },
      {
        "en": "Discuss the continuity of f(x) = (x² − 1)/(x − 1) at x = 1.",
        "ja": "f(x) = (x² − 1)/(x − 1) の x = 1 における連続性を調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "形容詞は continuous（continuous を参照）。「連続である」は is continuous、「連続性を調べよ」は Discuss the continuity of … ／ Determine whether … is continuous。"
    ],
    "related": [
      "continuous",
      "continuous-function",
      "removable-discontinuity",
      "intermediate-value-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "連続 (数学)",
        "en": "Continuity (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "continuity": 157
      },
      "written": {
        "continuity": 109
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "continuous-compounding",
    "ja": {
      "term": "連続複利",
      "reading": "れんぞくふくり",
      "alt": []
    },
    "en": {
      "term": "continuous compounding",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の高校数学では連続複利を扱わない（e は数III で極限として導入する）。OpenStax Algebra and Trigonometry は、e を使う連続的な増加の例として interest compounded continuously の問題を扱う。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "複利で、利息をつける回数を限りなく多くしたもの。元金 P、年利率 r、t 年後の元利合計は A = P eʳᵗ になる。",
    "definition_en": "Compound interest in which the number of compounding periods per year grows without bound. The amount after t years is A = Pe^(rt), where P is the principal and r is the annual interest rate.",
    "latex": "A = Pe^{rt}",
    "spoken_en": "A equals P e to the r t",
    "examples": [
      {
        "en": "With continuous compounding, the interest is added every instant, so we use A equals P e to the rt.",
        "ja": "連続複利では利息が一瞬ごとにつくから、A = P eʳᵗ の式を使う。",
        "register": "spoken"
      },
      {
        "en": "An account earns 5% annual interest with continuous compounding. Find the balance after 10 years if $1,000 is deposited.",
        "ja": "年利率 5 % の連続複利の口座に 1,000 ドルを預ける。10 年後の残高を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語は名詞の continuous compounding より、interest compounded continuously ／ continuously compounded interest の形で言うことが多い（教科書は compounded continuously、授業では continuously compounded の語順が多い）。",
      "年利率 r は小数で入れる（5 % なら r = 0.05）。e は約 2.718 の定数で、文字ではない（エントリ base-of-the-natural-logarithm）。",
      "回数を決めた複利（エントリ compound-interest）の式 A = P(1 + r/n)ⁿᵗ と取り違えない。連続複利の式に n は無い。"
    ],
    "related": [
      "compound-interest",
      "base-of-the-natural-logarithm",
      "exponential-growth",
      "exponential-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "continuous compounding": 1
      },
      "written": {
        "continuous compounding": 5
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 continuous compounding（本文 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "continuous-function",
    "ja": {
      "term": "連続関数",
      "reading": "れんぞくかんすう",
      "alt": []
    },
    "en": {
      "term": "continuous function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "定義域（または考える区間）のすべての点で連続な関数。",
    "definition_en": "A function that is continuous at every point of its domain, or of the interval in question.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Polynomials are continuous functions everywhere.",
        "ja": "多項式はどこでも連続な関数だ。",
        "register": "spoken"
      },
      {
        "en": "A continuous function on [a, b] attains a maximum and a minimum.",
        "ja": "[a, b] で連続な関数は最大値と最小値をとる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "continuous",
      "intermediate-value-theorem",
      "extreme-value-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "連続写像",
        "en": "Continuous function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.16（continuous function）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "continuous function": 84
      },
      "written": {
        "continuous function": 113
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "continuous-random-variable",
    "ja": {
      "term": "連続型確率変数",
      "reading": "れんぞくがたかくりつへんすう",
      "alt": []
    },
    "en": {
      "term": "continuous random variable",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "ある区間のすべての実数値をとりうる確率変数。確率は確率密度関数のグラフと x 軸の間の面積で表す。",
    "definition_en": "A random variable that can take any real value in an interval; probabilities are areas under its probability density function.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For a continuous random variable, the probability of hitting exactly one value is zero.",
        "ja": "連続型確率変数では、ちょうど 1 つの値をとる確率は 0。",
        "register": "spoken"
      },
      {
        "en": "The waiting time X is a continuous random variable. Find P(2 < X < 5).",
        "ja": "待ち時間 X は連続型確率変数である。P(2 < X < 5) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "discrete-random-variable",
      "probability-density-function",
      "random-variable",
      "uniform-distribution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "5.1 Continuous Probability Functions（continuous random variable）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.11（continuous random variable）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "continuous random variable": 2
      },
      "written": {
        "continuous random variable": 33
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "openstax-calculus",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "continuous",
    "ja": {
      "term": "連続",
      "reading": "れんぞく",
      "alt": []
    },
    "en": {
      "term": "continuous",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数 f が x = a で lim_{x→a} f(x) = f(a) を満たすこと。グラフがそこで途切れない。",
    "definition_en": "Of a function at x = a: having lim_{x→a} f(x) = f(a), so the graph has no break there.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "You can draw it without lifting your pencil, so it's continuous.",
        "ja": "鉛筆を離さずにかけるから、連続だ。",
        "register": "spoken"
      },
      {
        "en": "Show that f is continuous at x = 2.",
        "ja": "f が x = 2 で連続であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「x = a で連続」は continuous at x = a、「区間で連続」は continuous on [a, b]。"
    ],
    "related": [
      "continuous-function",
      "discontinuous",
      "differentiable",
      "intermediate-value-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "連続 (数学)",
        "en": "Continuity (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.12（continuous）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "continuous": 881
      },
      "written": {
        "continuous": 860
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "contradiction",
    "ja": {
      "term": "矛盾",
      "reading": "むじゅん",
      "alt": []
    },
    "en": {
      "term": "contradiction",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "ある命題とその否定が両方とも導かれること（両方が成り立つことはない）。背理法では、仮定から矛盾を導いて仮定が誤りだと結論する。",
    "definition_en": "A situation in which a statement and its negation have both been derived, which cannot actually be the case; in a proof by contradiction, reaching one shows that the starting assumption was false.",
    "latex": "p \\land \\neg p",
    "spoken_en": "p and not p",
    "examples": [
      {
        "en": "But we said n was odd, and now it's even. That's a contradiction.",
        "ja": "でも n は奇数だと言ったのに、今度は偶数になった。これは矛盾だ。",
        "register": "spoken"
      },
      {
        "en": "This is a contradiction, since p and q were assumed to have no common factor.",
        "ja": "p と q は互いに素と仮定したので、これは矛盾である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞の「矛盾する」は別エントリ（lead-to-a-contradiction：arrive at ／ lead to ／ get a contradiction）。「仮定に矛盾する」は contradicts the assumption と動詞 contradict を使う。",
      "用例コーパスでは話し言葉はすべて MIT OCW で、背理法（proof by contradiction）の説明の中で出てくる。",
      "OpenStax Elementary Algebra・Intermediate Algebra は、解の無い方程式（すべての値で偽になる方程式）も contradiction と呼び、identity・conditional equation と並べる。Algebra and Trigonometry・Precalculus は連立方程式を消去して出る 3 = 0 のような文を contradiction と呼ぶ。"
    ],
    "related": [
      "proof-by-contradiction",
      "lead-to-a-contradiction",
      "negation",
      "assume",
      "tautology"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "矛盾",
        "en": "Contradiction"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "解の無い方程式を contradiction と呼ぶ（32 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "解の無い方程式を contradiction と呼ぶ（36 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "contradiction": 79
      },
      "written": {
        "contradiction": 111
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-precalculus"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "contrapositive",
    "ja": {
      "term": "対偶",
      "reading": "たいぐう",
      "alt": []
    },
    "en": {
      "term": "contrapositive",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "命題「p ならば q」に対して、結論の否定を仮定に、仮定の否定を結論にした命題「q でないならば p でない」。元の命題と真偽が一致する。",
    "definition_en": "For a conditional statement \"if p, then q,\" the statement \"if not q, then not p.\" It is always true or false together with the original statement.",
    "latex": "\\neg q \\Rightarrow \\neg p",
    "spoken_en": "if not q, then not p",
    "examples": [
      {
        "en": "The contrapositive of \"if n² is even, then n is even\" is \"if n is odd, then n² is odd.\"",
        "ja": "「n² が偶数ならば n は偶数」の対偶は「n が奇数ならば n² は奇数」である。",
        "register": "written"
      },
      {
        "en": "Since a statement and its contrapositive are logically equivalent, it suffices to prove the contrapositive.",
        "ja": "命題とその対偶は同値なので、対偶を証明すれば十分である。",
        "register": "written"
      },
      {
        "en": "This direction is hard, so let's look at the contrapositive instead.",
        "ja": "こっちの向きは難しいから、代わりに対偶を考えよう。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "逆（converse）・裏（inverse）と混同しない。元の命題と必ず真偽が一致するのは対偶だけ。",
      "用例コーパスでは話し言葉は MIT OCW だけ、書き言葉は大半が MIT の講義ノート。"
    ],
    "related": [
      "inverse",
      "negation",
      "proof-by-contrapositive",
      "equivalent",
      "proposition",
      "converse"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "対偶 (論理学)",
        "en": "Contraposition"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "contrapositive": 2
      },
      "written": {
        "contrapositive": 15
      },
      "sources": [
        "mit-18.03",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "control-group",
    "ja": {
      "term": "対照群",
      "reading": "たいしょうぐん",
      "alt": []
    },
    "en": {
      "term": "control group",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "実験で、比べるために処理を与えない（またはプラセボを与える）群。",
    "definition_en": "In an experiment, the group that receives no treatment (or a placebo), used for comparison.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Without a control group, you can't tell whether the drug did anything.",
        "ja": "対照群がなければ、薬に効果があったのかわからない。",
        "register": "spoken"
      },
      {
        "en": "Explain the purpose of the control group in this experiment.",
        "ja": "この実験における対照群の目的を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて Khan Academy。"
    ],
    "related": [
      "treatment",
      "placebo",
      "blinding"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "control group": 24
      },
      "written": {
        "control group": 7
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "convenience-sample",
    "ja": {
      "term": "便宜的抽出",
      "reading": "べんぎてきちゅうしゅつ",
      "alt": [
        "便宜的標本"
      ]
    },
    "en": {
      "term": "convenience sample",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「便宜的抽出」は学習指導要領解説に無い（日本語版 Wikipedia「標本調査」には節「便宜的抽出」がある）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "調べやすい人や物だけを選んで標本にする方法。偏りが生じやすい。",
    "definition_en": "A sample made of the individuals that are easiest to reach; it is likely to be biased.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Just asking your friends is a convenience sample, and it's probably biased.",
        "ja": "友だちに聞くだけなのは便宜的抽出で、たぶん偏っている。",
        "register": "spoken"
      },
      {
        "en": "Explain why a convenience sample may not represent the population.",
        "ja": "便宜的抽出の標本が母集団を代表しないことがある理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "random-sampling",
      "is-biased",
      "voluntary-response-bias"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「標本調査」",
        "url": "https://ja.wikipedia.org/wiki/%E6%A8%99%E6%9C%AC%E8%AA%BF%E6%9F%BB",
        "note": "節「便宜的抽出」"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "convenience sample": 4
      },
      "written": {
        "convenience sample": 17
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "converge",
    "ja": {
      "term": "収束する",
      "reading": "しゅうそくする",
      "alt": []
    },
    "en": {
      "term": "converge",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "数列・級数・関数の値が、ある有限の値にいくらでも近づく。",
    "definition_en": "Of a sequence, series, or function: to approach a finite value as closely as you like.",
    "latex": "\\lim_{n \\to \\infty} a_n = \\alpha",
    "spoken_en": "the limit as n goes to infinity of a sub n equals alpha",
    "examples": [
      {
        "en": "This sequence converges to zero — the terms get as close to zero as you like.",
        "ja": "項がいくらでも 0 に近づくから、この数列は 0 に収束する。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the series Σ 1/n² converges or diverges.",
        "ja": "級数 Σ 1/n² が収束するか発散するかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "converge to 0",
        "ja": "0 に収束する"
      }
    ],
    "pitfalls": [
      "「〜に収束する」は converge to。名詞は convergence（convergence を参照）。"
    ],
    "related": [
      "convergence",
      "diverge",
      "limit-of-a-sequence",
      "infinite-series",
      "monotone-convergence-theorem"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 10、Topic 10.1–10.9（converge ／ diverge）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "converge": 623
      },
      "written": {
        "converge": 675
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "convergence-of-improper-integrals",
    "ja": {
      "term": "広義積分の収束",
      "reading": "こうぎせきぶんのしゅうそく",
      "alt": []
    },
    "en": {
      "term": "the integral converges",
      "alt": [
        "convergence of an improper integral"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "広義積分を定める極限が有限の値になること。有限でなければ発散するという。",
    "definition_en": "An improper integral converges when the limit that defines it is a finite number, and diverges otherwise.",
    "latex": "\\int_1^{\\infty} \\frac{dx}{x^p} \\text{ converges} \\iff p > 1",
    "spoken_en": "the integral from one to infinity of one over x to the p dx converges if and only if p is greater than one",
    "examples": [
      {
        "en": "One over x squared converges from one to infinity, but one over x diverges, even though both go to zero.",
        "ja": "1 から ∞ まで、1/x² の積分は収束するが、1/x は発散する。どちらも 0 に近づくのに。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "related": [
      "improper-integral",
      "comparison-test-for-integrals"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the integral converges": 6
      },
      "written": {
        "the integral converges": 5
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 6 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 the integral converges（本文 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "convergence",
    "ja": {
      "term": "収束",
      "reading": "しゅうそく",
      "alt": [
        "数列の収束"
      ]
    },
    "en": {
      "term": "convergence",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "数列や関数の値が、ある一定の値に限りなく近づくこと。",
    "definition_en": "Getting arbitrarily close to a single finite value, as a sequence or function does when it has a limit.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Convergence means the terms settle down to one number.",
        "ja": "収束とは、項が 1 つの数に落ち着くこと。",
        "register": "spoken"
      },
      {
        "en": "Determine the convergence or divergence of the sequence aₙ = (2n + 1)/n.",
        "ja": "数列 aₙ = (2n + 1)/n の収束・発散を調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞は converge（収束する）。「数列 {aₙ} は 2 に収束する」は The sequence converges to 2."
    ],
    "related": [
      "divergence",
      "oscillate",
      "limit",
      "infinite-series",
      "converge",
      "limit-of-a-sequence"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 10（convergence）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "convergence": 224
      },
      "written": {
        "convergence": 231
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "converse-of-the-inscribed-angle-theorem",
    "ja": {
      "term": "円周角の定理の逆",
      "reading": "えんしゅうかくのていりのぎゃく",
      "alt": []
    },
    "en": {
      "term": "converse of the inscribed angle theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の教材（CED・OpenStax・IM・CK-12）には出てこない（4 つとも 0 件）。CK-12 Geometry（6.14 Inscribed Angles in Circles）と IM Geometry（7.4 Quadrilaterals in Circles・7.5 Triangles in Circles）は円周角（inscribed angle）と円周角の定理を扱うが、その逆（2 点が直線の同じ側にあって角が等しければ 4 点は同じ円周上）は出てこない。見出しの英語は本プロジェクトの訳。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 点 P、Q が直線 AB について同じ側にあって ∠APB = ∠AQB ならば、4 点 A、B、P、Q は 1 つの円周上にあるという定理。",
    "definition_en": "The theorem that if points P and Q are on the same side of line AB and ∠APB = ∠AQB, then A, B, P, and Q all lie on one circle.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "P and Q are on the same side of AB, and both angles are 40 degrees, so by the converse of the inscribed angle theorem, A, B, P, and Q are all on one circle.",
        "ja": "P と Q は AB について同じ側にあって、どちらの角も 40° だから、円周角の定理の逆より、A、B、P、Q は 1 つの円周上にある。",
        "register": "spoken"
      },
      {
        "en": "In quadrilateral ABCD, ∠BAC = ∠BDC = 35°. Use the converse of the inscribed angle theorem to prove that A, B, C, and D are on one circle.",
        "ja": "四角形 ABCD で、∠BAC = ∠BDC = 35° である。円周角の定理の逆を用いて、4 点 A、B、C、D が 1 つの円周上にあることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "条件の「2 点 P、Q が直線 AB について同じ側にある」を落とさない。P、Q が反対側にあるときは、∠APB + ∠AQB = 180° なら 4 点が同じ円周上にある（エントリ cyclic-quadrilateral）。",
      "「逆」は converse（エントリ converse）で、仮定と結論を入れかえた命題。元の定理が正しくても逆が正しいとは限らないが、円周角の定理では逆も成り立つ。",
      "4 点が同じ円周上にあることはエントリ concyclic、4 つの頂点が 1 つの円周上にある四角形はエントリ cyclic-quadrilateral。"
    ],
    "related": [
      "inscribed-angle-theorem",
      "concyclic",
      "converse",
      "cyclic-quadrilateral",
      "inscribed-angle"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.14 Inscribed Angles in Circles（円周角の定理。逆は出てこない）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.4 Quadrilaterals in Circles・7.5 Triangles in Circles（円周角。逆は出てこない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元 3 の前の修正」の 3）。en.term は converse of the inscribed angle theorem のまま。米国の教材（CED・OpenStax・IM・CK-12）には出てこない。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "converse-of-the-pythagorean-theorem",
    "ja": {
      "term": "三平方の定理の逆",
      "reading": "さんへいほうのていりのぎゃく",
      "alt": []
    },
    "en": {
      "term": "converse of the Pythagorean theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "三角形の 3 辺の長さ a、b、c について a² + b² = c² が成り立てば、長さ c の辺の向かいの角は直角であるという定理。三平方の定理の仮定と結論を入れかえたもの。",
    "definition_en": "If the side lengths a, b, and c of a triangle satisfy a² + b² = c², then the angle opposite the side of length c is a right angle. It is used to test whether a triangle is a right triangle.",
    "latex": "a^2 + b^2 = c^2 \\Longrightarrow \\angle C = 90^\\circ",
    "spoken_en": "if a squared plus b squared equals c squared, then angle C is ninety degrees",
    "examples": [
      {
        "en": "Check 7, 24, 25: 49 plus 576 is 625, and that's 25 squared, so by the converse of the Pythagorean theorem it's a right triangle.",
        "ja": "7、24、25 を調べよう。49 + 576 = 625 で、これは 25 の 2 乗。だから三平方の定理の逆により直角三角形だ。",
        "register": "spoken"
      },
      {
        "en": "The sides of a triangle are 9, 12, and 15. Use the converse of the Pythagorean theorem to determine whether the triangle is a right triangle.",
        "ja": "3 辺の長さが 9、12、15 の三角形がある。三平方の定理の逆を用いて、この三角形が直角三角形かどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "c には最も長い辺を入れて調べる。短い辺を c にすると a² + b² = c² が成り立たず、直角三角形を見落とす。",
      "CK-12 Geometry は 4.29 Pythagorean Theorem to Classify Triangles と 4.36 Distance and Triangle Classification Using the Pythagorean Theorem で、最も長い辺を c として c² < a² + b² なら鋭角三角形、c² > a² + b² なら鈍角三角形と判定する（4.36 は逆定理を広げる形で導入する。エントリ acute-triangle、obtuse-triangle）。",
      "converse は「逆」（エントリ converse）。the converse of … の形で、もとの定理の名前につけて呼ぶ。"
    ],
    "related": [
      "pythagorean-theorem",
      "converse",
      "right-triangle",
      "acute-triangle",
      "obtuse-triangle"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.31 Converse of the Pythagorean Theorem（鋭角・鈍角の判定は 4.29・4.36）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 converse of the Pythagorean theorem（CK-12 Geometry 4.31 Converse of the Pythagorean Theorem・CK-12 Geometry 4.33 Pythagorean Theorem and its Converse・CK-12 Geometry 4.36 Distance and Triangle Classification Using the Pythagorean Theorem・計 16 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "converse",
    "ja": {
      "term": "逆",
      "reading": "ぎゃく",
      "alt": []
    },
    "en": {
      "term": "converse",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "命題「p ならば q」の仮定と結論を入れかえた命題「q ならば p」。もとの命題が正しくても、逆が正しいとは限らない。",
    "definition_en": "For a statement \"if p, then q,\" the statement \"if q, then p,\" made by switching the hypothesis and the conclusion. The converse of a true statement is not always true.",
    "latex": "q \\Rightarrow p",
    "spoken_en": "if q, then p",
    "examples": [
      {
        "en": "The converse just flips it around. \"If it's a multiple of 4, it's even\" becomes \"if it's even, it's a multiple of 4,\" and that one's false, because 6 is even but it's not a multiple of 4.",
        "ja": "逆は入れかえるだけ。「4 の倍数ならば偶数」が「偶数ならば 4 の倍数」になる。こっちは正しくない。6 は偶数だけど 4 の倍数じゃないからね。",
        "register": "spoken"
      },
      {
        "en": "Write the converse of the statement \"If two angles are vertical angles, then they are equal.\" Is the converse true? If not, give a counterexample.",
        "ja": "「2 つの角が対頂角ならば、それらは等しい」の逆を書け。逆は正しいか。正しくなければ反例を 1 つあげよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "論理の逆・裏・対偶は converse・inverse・contrapositive（エントリ inverse、contrapositive）。英語の inverse は論理では「裏」のことで、「逆」ではない。",
      "逆関数（inverse function）や逆数（reciprocal）の「逆」も converse ではない（エントリ inverse-function、reciprocal）。日常語の「逆」（反対）は opposite や reverse で、converse は命題の逆にだけ使う。",
      "「〜の定理の逆」は the converse of the … theorem（二等辺三角形の定理の逆は the converse of the base angles theorem、エントリ isosceles-triangle-theorem）。"
    ],
    "related": [
      "inverse",
      "contrapositive",
      "conditional-statement",
      "counterexample",
      "proposition"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "逆",
        "en": "Converse (logic)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "converse": 2
      },
      "written": {
        "converse": 15
      },
      "sources": [
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "convert",
    "ja": {
      "term": "換算する",
      "reading": "かんさんする",
      "alt": [
        "変換する"
      ]
    },
    "en": {
      "term": "convert",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語の「換算」は主に単位や通貨について言い、分数を小数にするのは「直す」「変換する」と言う。英語はどれも convert。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある単位で表された量を、別の単位で表し直す。分数を小数に直すなど、数の表し方を変えることにも使う。",
    "definition_en": "To change a measurement from one unit to another. It is also used for changing how a number is written, such as from a fraction to a decimal.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To convert feet to inches, multiply by 12, since there are 12 inches in a foot.",
        "ja": "フィートをインチに換算するには 12 をかける。1 フィートは 12 インチだからね。",
        "register": "spoken"
      },
      {
        "en": "Convert 3.5 hours to minutes.",
        "ja": "3.5 時間を分に換算せよ。",
        "register": "written"
      },
      {
        "en": "Convert 3/8 to a decimal.",
        "ja": "3/8 を小数に直せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "形は convert A to B ／ convert A into B。「A を B に」の「に」は to か into。",
      "単位の換算は unit conversion とも言う（名詞は conversion）。",
      "度をラジアンに直すのも convert（convert 60° to radians）。"
    ],
    "related": [
      "unit",
      "decimal",
      "fraction",
      "percent",
      "radian-measure"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "convert … to | convert … into": 229
      },
      "written": {
        "convert … to | convert … into": 628
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coordinate-axes",
    "ja": {
      "term": "座標軸",
      "reading": "ざひょうじく",
      "alt": []
    },
    "en": {
      "term": "coordinate axes",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "座標平面の x 軸と y 軸をまとめた呼び名。2 本の軸は原点で垂直に交わる。",
    "definition_en": "The x-axis and the y-axis taken together. The two axes are perpendicular and meet at the origin.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "First draw your coordinate axes, then label them x and y.",
        "ja": "まず座標軸をかいて、x と y の名前を書き入れよう。",
        "register": "spoken"
      },
      {
        "en": "The graph of y = 6/x gets closer and closer to the coordinate axes but never touches them.",
        "ja": "y = 6/x のグラフは座標軸に限りなく近づくが、座標軸とは交わらない。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "axes は axis の複数形で、発音はアクシーズ（Merriam-Webster: ˈak-ˌsēz）。x 軸と y 軸の 2 本を指すので coordinate axes と複数形で言う。",
      "軸の上の点（原点を含む）はどの象限にも入らない（エントリ quadrant）。"
    ],
    "related": [
      "x-axis",
      "y-axis",
      "origin",
      "quadrant",
      "coordinate-plane"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "座標軸",
        "en": "Coordinate axis"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「axis」",
        "url": "https://www.merriam-webster.com/dictionary/axis",
        "note": "発音 複数形 axes ˈak-ˌsēz（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "coordinate axes": 16
      },
      "written": {
        "coordinate axes": 46
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coordinate-plane",
    "ja": {
      "term": "座標平面",
      "reading": "ざひょうへいめん",
      "alt": []
    },
    "en": {
      "term": "coordinate plane",
      "alt": [
        "coordinate grid",
        "Cartesian plane"
      ],
      "variants": [
        {
          "term": "xy-plane",
          "register": "spoken",
          "note": "話し言葉では coordinate plane より多いが、大半が MIT OCW（18.02 の多変数の微積分）。中学・Algebra の講義と教科書は coordinate plane。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "原点で垂直に交わる x 軸と y 軸を定めて、点の位置を 2 つの数の組 (x, y) で表せるようにした平面。",
    "definition_en": "A flat surface with a horizontal x-axis and a vertical y-axis that cross at the origin, so that every point can be named by an ordered pair (x, y).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Plot (2, 5) on the coordinate plane: start at the origin, go right 2, then up 5.",
        "ja": "(2, 5) を座標平面にとってみよう。原点から右に 2、それから上に 5。",
        "register": "spoken"
      },
      {
        "en": "Plot the points A(−3, 1) and B(4, −2) on the coordinate plane, and name the quadrant that each point lies in.",
        "ja": "座標平面上に点 A(−3, 1)、B(4, −2) をとり、それぞれの点がどの象限にあるかを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "点の座標 (x, y) は ordered pair（順序対）とも呼ぶ。順番に意味があり、(2, 5) と (5, 2) は別の点。",
      "plane（平面）は、発音が同じ plain（平らな・飾りのない。Merriam-Webster はどちらも ˈplān）と綴りを取り違えやすい。"
    ],
    "related": [
      "origin",
      "coordinates",
      "coordinate-axes",
      "quadrant",
      "plot-a-point"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「plane」",
        "url": "https://www.merriam-webster.com/dictionary/plane",
        "note": "発音 ˈplān。plain も ˈplān（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "xy-plane": 52,
        "coordinate plane": 34,
        "coordinate grid": 5
      },
      "written": {
        "coordinate plane": 100,
        "Cartesian plane": 18,
        "xy-plane": 21,
        "coordinate grid": 14
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coordinate-proof",
    "ja": {
      "term": "座標を用いた証明",
      "reading": "ざひょうをもちいたしょうめい",
      "alt": [
        "座標証明"
      ]
    },
    "en": {
      "term": "coordinate proof",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（coordinate proof は用例コーパスには話・書とも出てこず、参照でも IM Geometry のレッスン名 6.14 Coordinate Proof にあるだけ）。日本の数II では「座標を用いて証明せよ」として、図形を座標平面に置き、2 点間の距離や中点の座標で性質を示す。",
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "図形を座標平面上に置き、2 点間の距離・中点・傾きなどを計算して図形の性質を示す証明。",
    "definition_en": "A proof that places a figure on the coordinate plane and uses distances, midpoints, or slopes to show that the figure has a certain property.",
    "latex": "AB^2 + AC^2 = 2(AM^2 + BM^2)",
    "spoken_en": "A B squared plus A C squared equals 2 times the quantity A M squared plus B M squared",
    "examples": [
      {
        "en": "For a coordinate proof, put the figure somewhere convenient, like one vertex at the origin and one side on the x-axis.",
        "ja": "座標を使って証明するときは、図形を計算しやすい場所に置こう。たとえば頂点の 1 つを原点に、1 辺を x 軸上に置く。",
        "register": "spoken"
      },
      {
        "en": "Write a coordinate proof that the diagonals of a parallelogram bisect each other.",
        "ja": "平行四辺形の 2 本の対角線が互いに他を 2 等分することを、座標を用いて証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の名詞 coordinate proof は用例コーパスに話・書とも出てこない。IM でも Geometry 6.14 Coordinate Proof の 1 件だけ。",
      "座標の置き方で計算の量が変わる。一般性を失わないように、頂点を (0, 0), (a, 0), (b, c) のように文字で置く。",
      "数II では中線定理 AB² + AC² = 2(AM² + BM²) の証明が代表的な例。"
    ],
    "related": [
      "prove",
      "two-column-proof",
      "midpoint-formula",
      "distance",
      "midpoint"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "coordinate-rule",
    "ja": {
      "term": "座標の規則",
      "reading": "ざひょうのきそく",
      "alt": []
    },
    "en": {
      "term": "mapping rule",
      "alt": [
        "coordinate rule",
        "coordinate notation"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「座標の規則」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書は平行移動を「x 軸方向に 3、y 軸方向に −2 だけ平行移動」と言葉で書き、(x, y) → (x + 3, y − 2) の形の記法を使わない。CK-12 Geometry はこの書き方を mapping rule と呼ぶ（8.9 Translation Notation ほか、18 件）。",
    "domains": [
      "geometry",
      "notation"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "変換で点 (x, y) がどの点に移るかを、(x, y) → (x + 3, y − 2) のように座標の式で表したもの。",
    "definition_en": "A rule that describes a transformation by showing where a general point (x, y) goes, such as (x, y) → (x + 3, y − 2).",
    "latex": "(x, y) \\to (x + 3, y - 2)",
    "spoken_en": "x, y maps to x plus 3, y minus 2",
    "examples": [
      {
        "en": "Every point moves 3 to the right and 2 down, so the mapping rule is x, y goes to x plus 3, y minus 2.",
        "ja": "どの点も右に 3、下に 2 動くから、点 (x, y) は (x + 3, y − 2) に移る。",
        "register": "spoken"
      },
      {
        "en": "Write a mapping rule for a reflection across the y-axis. Then use it to find the image of P(4, −1).",
        "ja": "y 軸に関する対称移動で点 (x, y) が移る点の座標を x, y で表せ。また、この移動で点 P(4, −1) が移る点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "呼び方は教材で分かれる。CK-12 Geometry は mapping rule（18 件）と coordinate notation（3 件）、IM は coordinate rule（Geometry 6.2 ほか 3 件）と呼ぶ。",
      "回転も同じ形で書ける。原点のまわりの反時計回り 90° の回転は (x, y) → (−y, x)。",
      "用例コーパスの mapping rule は話・書とも少なく、すべて MIT（MIT OCW の講義と MIT の講義ノート）。高校幾何の変換の記法の意味とは限らない。"
    ],
    "related": [
      "translation",
      "image",
      "rigid-motion",
      "reflection",
      "rotation"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.9 Translation Notation・8.18 Notation for Composite Transformations・8.11 Rotation Rules"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "mapping rule": 5
      },
      "written": {
        "mapping rule": 6
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは CK-12 の呼び方 mapping rule（CK-12 Geometry 8.9 Translation Notation・CK-12 Geometry 8.18 Notation for Composite Transformations・CK-12 Geometry 8.11 Rotation Rules・計 16 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "coordinate-vector",
    "ja": {
      "term": "座標ベクトル",
      "reading": "ざひょうべくとる",
      "alt": []
    },
    "en": {
      "term": "coordinate vector",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（coordinate vector は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "基底を決めたとき、ベクトルを基底の 1 次結合で表す係数を並べたベクトル。",
    "definition_en": "Given a basis, the vector of coefficients that express a vector as a linear combination of the basis vectors.",
    "latex": "[\\vec{x}]_{B}",
    "spoken_en": null,
    "examples": [
      {
        "en": "In this basis the coordinate vector is (2, −1): two of the first, minus one of the second.",
        "ja": "この基底での座標ベクトルは (2, −1)。1 つ目を 2 つ、2 つ目を −1 つだ。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinate vector of x relative to the basis B.",
        "ja": "基底 B に関する x の座標ベクトルを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉にわずかに出てくるだけ。"
    ],
    "related": [
      "basis",
      "change-of-basis"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "9.1 The Matrix of a Linear Transformation"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "coordinate vector": 2
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 coordinate vector（9.1 The Matrix of a Linear Transformation・4.2 Projections and Planes・4.3 More on the Cross Product）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "coordinates-in-space",
    "ja": {
      "term": "空間座標",
      "reading": "くうかんざひょう",
      "alt": [
        "三次元座標系"
      ]
    },
    "en": {
      "term": "three-dimensional coordinate system",
      "alt": [
        "coordinates in space"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "原点で直交する x 軸・y 軸・z 軸を使って、空間の点を (x, y, z) の 3 つの数で表す座標。",
    "definition_en": "A system that locates each point in space by three numbers (x, y, z), measured along three mutually perpendicular axes through the origin.",
    "latex": "P(x, y, z)",
    "spoken_en": null,
    "examples": [
      {
        "en": "In three dimensions, the z-axis comes straight out toward you.",
        "ja": "3 次元では、z 軸はこちらに向かってまっすぐ出てくる。",
        "register": "spoken"
      },
      {
        "en": "Plot the point (2, −1, 3) in a three-dimensional coordinate system.",
        "ja": "空間座標で点 (2, −1, 3) をとれ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは three-dimensional coordinate system は書き言葉（OpenStax Calculus Volume 3）に少しあるだけ。授業では in three dimensions とも言う（話し言葉によく出てくる）。",
      "xyz 座標の向きは右手系（right-handed system）。"
    ],
    "related": [
      "equation-of-a-plane",
      "cylindrical-coordinates",
      "spherical-coordinates",
      "equation-of-a-sphere"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "three-dimensional coordinate system": 6
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 three-dimensional coordinate system（本文 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "coordinates",
    "ja": {
      "term": "座標",
      "reading": "ざひょう",
      "alt": [
        "座標の考え方"
      ]
    },
    "en": {
      "term": "coordinates",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "number-theory"
    ],
    "level": {
      "jp": [
        "中1",
        "数A"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Discrete Math",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "基準の点（原点）と座標軸を決めたとき、点の位置を表す数の組。平面では (x, y)、空間では (x, y, z) の形で書く。",
    "definition_en": "The ordered set of numbers that locates a point relative to a fixed origin and axes, written (x, y) in the plane and (x, y, z) in space.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "What are the coordinates of this point? Go over 3, down 2, so it's (3, −2).",
        "ja": "この点の座標は？ 右に 3、下に 2 だから (3, −2) だね。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinates of the point where the line y = 2x − 6 crosses the x-axis.",
        "ja": "直線 y = 2x − 6 が x 軸と交わる点の座標を求めよ。",
        "register": "written"
      },
      {
        "en": "In space, the point P has coordinates (1, 2, 3), so its distance from the origin is √14.",
        "ja": "空間で点 P の座標は (1, 2, 3) なので、原点からの距離は √14 である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "coordinates of the point",
        "ja": "点の座標"
      },
      {
        "en": "find the coordinates",
        "ja": "座標を求める"
      }
    ],
    "pitfalls": [
      "1 つの点の座標は x と y の組なので複数形 coordinates。1 つずつを指すときは the x-coordinate、the y-coordinate と単数になる。",
      "数Aの「座標の考え方」で扱う空間の点の位置も、英語では coordinates in space（(x, y, z)）で言う。"
    ],
    "related": [
      "number-line",
      "x-intercept",
      "coordinates-in-space",
      "rectangular-coordinates",
      "distance"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "座標",
        "en": "Coordinate"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "coordinates": 1030
      },
      "written": {
        "coordinates": 1530
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coplanar",
    "ja": {
      "term": "同一平面上にある",
      "reading": "どういつへいめんじょうにある",
      "alt": [
        "共面"
      ]
    },
    "en": {
      "term": "coplanar",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数A",
        "数C"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "いくつかの点や直線が、すべて同じ 1 つの平面上にあること。3 点はつねに同一平面上にあるが、4 点はそうとは限らない。",
    "definition_en": "Lying in the same plane. Any three points are coplanar, but four points may not be.",
    "latex": "\\overrightarrow{AD} = s\\overrightarrow{AB} + t\\overrightarrow{AC}",
    "spoken_en": "vector A D equals s times vector A B plus t times vector A C",
    "examples": [
      {
        "en": "Any three points are coplanar, because you can always fit a plane through three points.",
        "ja": "3 点はいつでも同一平面上にある。3 点を通る平面は必ずとれるからだ。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the points A(1, 0, 0), B(0, 1, 0), C(0, 0, 1), and D(2, −1, 1) are coplanar.",
        "ja": "4 点 A(1, 0, 0), B(0, 1, 0), C(0, 0, 1), D(2, −1, 1) が同一平面上にあるかどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は「同一平面上にある」と句で言うが、英語は形容詞 1 語。「4 点は同一平面上にない」は The four points are not coplanar. と書く。",
      "直線の場合の対になる語は collinear（同一直線上にある）。",
      "用例コーパスで coplanar は書き言葉に少しあり（大半が OpenStax Calculus）、話し言葉には出てこない。CK-12 Geometry は 1.1 Geometry Terms を中心に 11 件使う。"
    ],
    "related": [
      "collinear",
      "collinearity-condition",
      "equation-of-a-plane",
      "coplanarity-condition",
      "skew-lines"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.1 Geometry Terms（coplanar）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「共面」",
        "url": "https://ja.wikipedia.org/wiki/%E5%85%B1%E9%9D%A2",
        "note": "共面（英: coplanar）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Coplanarity",
        "note": "記事「Coplanarity」の冒頭（coplanar を太字で定義。数学カテゴリから 3 段）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.4 The Cross Product（coplanar）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 B 図形（空間図形）: その二つの直線は同一平面上にある"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "coplanar": 10
      },
      "sources": [
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coplanarity-condition",
    "ja": {
      "term": "共面条件",
      "reading": "きょうめんじょうけん",
      "alt": []
    },
    "en": {
      "term": "condition for coplanarity",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。英語は形容詞 coplanar（同じ平面上にある。書き言葉では OpenStax Calculus Volume 3 などに出てくる）で言い、条件に名前をつけない。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "4 点 A, B, C, P が同じ平面上にあるための条件。AP = sAB + tAC となる実数 s, t があること。",
    "definition_en": "The condition for four points A, B, C, P to lie in one plane: AP = sAB + tAC for some real numbers s and t.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If AP is a combination of AB and AC, then P is in the plane through A, B, and C.",
        "ja": "AP が AB と AC の一次結合なら、P は A, B, C を通る平面上にある。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the points A(1, 0, 0), B(0, 1, 0), C(0, 0, 1), and P(1, 1, −1) are coplanar.",
        "ja": "4 点 A(1, 0, 0), B(0, 1, 0), C(0, 0, 1), P(1, 1, −1) が同じ平面上にあるか判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 3 は、3 つのベクトルのスカラー三重積（triple scalar product）が 0 かどうかで coplanar を判定する。"
    ],
    "related": [
      "collinearity-condition",
      "linear-combination",
      "scalar-triple-product"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "corner-nondifferentiable",
    "ja": {
      "term": "角（微分不可能点）",
      "reading": "かど",
      "alt": []
    },
    "en": {
      "term": "sharp corner",
      "alt": [],
      "variants": [
        {
          "term": "sharp turn",
          "register": "spoken",
          "note": "Khan Academy の言い方。話し言葉ではすべて Khan Academy、書き言葉には出てこない。"
        },
        {
          "term": "corner point",
          "register": "written",
          "note": "少数の言い方。話し言葉にはほとんど出てこず、書き言葉は主に OpenStax（Algebra and Trigonometry・Precalculus・Calculus）。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「角（微分不可能点）」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "グラフが折れ曲がっていて、左右の接線の傾きが異なるため微分可能でない点（y = |x| の原点など）。",
    "definition_en": "A point where a graph bends sharply, so the slopes from the left and right differ and the function is not differentiable there (as for y = |x| at the origin).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "At x = 0, the absolute value graph makes a sharp turn, so there's no tangent line there.",
        "ja": "x = 0 で |x| のグラフは折れ曲がるので、そこには接線がない。",
        "register": "spoken"
      },
      {
        "en": "The graph of f(x) = |x − 2| has a sharp corner at x = 2.",
        "ja": "f(x) = |x − 2| のグラフは x = 2 で折れ曲がっている。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では単に corner とも言う（f has a corner at x = 0）。"
    ],
    "related": [
      "not-differentiable",
      "cusp",
      "vertical-tangent",
      "differentiable"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「角（微分不可能点）」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "sharp corner": 2,
        "sharp turn": 10,
        "corner point": 1
      },
      "written": {
        "sharp corner": 12,
        "corner point": 9
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "corollary",
    "ja": {
      "term": "系",
      "reading": "けい",
      "alt": []
    },
    "en": {
      "term": "corollary",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "証明したばかりの定理から、ほとんど手間をかけずに導かれる命題。",
    "definition_en": "A statement that follows directly from a theorem that has just been proved, with little or no extra work.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is really just a corollary of the mean value theorem.",
        "ja": "これは実は平均値の定理の系にすぎない。",
        "register": "spoken"
      },
      {
        "en": "Corollary: If f′(x) = 0 for every x in (a, b), then f is constant on (a, b).",
        "ja": "系：区間 (a, b) のすべての x で f′(x) = 0 ならば、f は (a, b) で定数関数である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の見出しは 1 字の「系」なので見落としやすい。英文で Corollary とあれば「系」、Lemma なら「補題」。",
      "米国式の発音は第 1 音節に強勢（COR-uh-lair-ee。Merriam-Webster: ˈkȯr-ə-ˌler-ē、英国式は kə-ˈrä-lə-rē）。"
    ],
    "related": [
      "lemma",
      "axiom",
      "proof",
      "mean-value-theorem",
      "theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "系 (数学)",
        "en": "Corollary"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「corollary」",
        "url": "https://www.merriam-webster.com/dictionary/corollary",
        "note": "発音 ˈkȯr-ə-ˌler-ē（British kə-ˈrä-lə-rē）（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "corollary": 52
      },
      "written": {
        "corollary": 84
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "correlation-coefficient",
    "ja": {
      "term": "相関係数",
      "reading": "そうかんけいすう",
      "alt": []
    },
    "en": {
      "term": "correlation coefficient",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Integrated Math 1",
        "Intro Statistics"
      ]
    },
    "definition_ja": "2 つの変量の直線的な相関の強さと向きを表す −1 から 1 までの値 r。共分散を 2 つの標準偏差の積で割って求める。",
    "definition_en": "A number r between −1 and 1. Its sign tells whether the two variables tend to rise together or move in opposite directions, and its size tells how closely the points follow a line. It equals the covariance divided by the product of the two standard deviations.",
    "latex": "r = \\frac{s_{xy}}{s_x s_y}",
    "spoken_en": null,
    "examples": [
      {
        "en": "An r of negative 0.9 is a strong negative correlation, and r close to zero means almost no linear relationship.",
        "ja": "r が −0.9 なら強い負の相関、r が 0 に近ければ直線的な関係はほとんどない。",
        "register": "spoken"
      },
      {
        "en": "Find the correlation coefficient r and interpret it in context.",
        "ja": "相関係数 r を求め、問題の文脈で解釈せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "記号で r とも呼ぶ（r is close to 1）。決定係数 r²（coefficient of determination）とは別。"
    ],
    "related": [
      "correlation",
      "coefficient-of-determination",
      "scatter-plot"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "相関係数",
        "en": "Correlation coefficient"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "correlation coefficient": 11
      },
      "written": {
        "correlation coefficient": 113
      },
      "sources": [
        "khan-ap-stats",
        "openstax-algtrig",
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "correlation",
    "ja": {
      "term": "相関",
      "reading": "そうかん",
      "alt": [
        "相関関係"
      ]
    },
    "en": {
      "term": "correlation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Integrated Math 1",
        "Intro Statistics"
      ]
    },
    "definition_ja": "2 つの変量の間で、一方が増えると他方も増える（または減る）という直線的な関係の傾向。",
    "definition_en": "A linear tendency between two variables for one to increase (or decrease) as the other increases.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "There's a strong correlation, but that doesn't mean one causes the other.",
        "ja": "強い相関はあるが、一方が他方の原因だということにはならない。",
        "register": "spoken"
      },
      {
        "en": "Is there a correlation between hours studied and test score? Use the scatter plot to explain.",
        "ja": "勉強時間とテストの得点に相関はあるか。散布図を使って説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "strong correlation",
        "ja": "強い相関"
      },
      {
        "en": "weak correlation",
        "ja": "弱い相関"
      }
    ],
    "pitfalls": [
      "「相関がない」は there is no correlation（no linear relationship）。",
      "相関があっても因果関係（causation）があるとは限らない。",
      "AP Statistics の CED は、散布図に見える傾向を association（topic 5.1）、相関係数 r を correlation（topic 5.2）と呼び分ける。"
    ],
    "related": [
      "correlation-coefficient",
      "positive-correlation",
      "negative-correlation",
      "scatter-plot",
      "causation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "相関",
        "en": "Correlation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 5.1（association）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "correlation": 58
      },
      "written": {
        "correlation": 179
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.06",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "correspond",
    "ja": {
      "term": "対応する",
      "reading": "たいおうする",
      "alt": []
    },
    "en": {
      "term": "correspond",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの集まりの間で、一方のものに、もう一方のものが決まって組になること。関数では、x の値を 1 つ決めると、それに対応する y の値がただ 1 つ決まる。",
    "definition_en": "To be matched or paired with something in another set. In a function, each x-value corresponds to exactly one y-value.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In this table, x = 2 corresponds to y = 10, and x = 3 corresponds to y = 15.",
        "ja": "この表では、x = 2 に y = 10 が、x = 3 に y = 15 が対応している。",
        "register": "spoken"
      },
      {
        "en": "Determine whether y is a function of x by checking if each x-value corresponds to exactly one y-value.",
        "ja": "x の値 1 つに y の値がただ 1 つ対応するかを調べて、y が x の関数であるかどうかを判定せよ。",
        "register": "written"
      },
      {
        "en": "Triangle ABC is congruent to triangle DEF. Which side of triangle DEF corresponds to side BC?",
        "ja": "△ABC ≡ △DEF である。辺 BC に対応する △DEF の辺を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「x の値に y の値がただ 1 つ対応する」は、英語では x を主語にして each x-value corresponds to exactly one y-value と言う。前置詞は to。",
      "合同な図形の「対応する辺・角」は形容詞 corresponding を使う（corresponding sides ／ corresponding angles）。corresponding angles は平行線の「同位角」（エントリ corresponding-angles-postulate）の意味にもなる。",
      "名詞は correspondence。「1 対 1 の対応」は one-to-one correspondence（エントリ bijection）。"
    ],
    "related": [
      "function",
      "table-of-values",
      "one-to-one",
      "bijection",
      "corresponding-angles-postulate"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "correspond": 773
      },
      "written": {
        "correspond": 712
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "corresponding-angles-of-congruent-figures",
    "ja": {
      "term": "対応する角",
      "reading": "たいおうするかく",
      "alt": []
    },
    "en": {
      "term": "corresponding angles",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "合同や相似な 2 つの図形で、一方を動かして（相似なら拡大・縮小もして）重ねたときに重なり合う角どうし。対応する角の大きさは等しい。",
    "definition_en": "In two congruent or similar figures, a pair of angles that are in matching positions, so they would line up if one figure were moved (and resized, for similar figures) onto the other. Corresponding angles of congruent or similar figures have equal measures.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The triangles are congruent, so their corresponding angles are equal. That means angle D is 50 degrees too.",
        "ja": "三角形が合同だから、対応する角は等しい。だから ∠D も 50° だ。",
        "register": "spoken"
      },
      {
        "en": "Given △ABC ≅ △DEF, list all three pairs of corresponding angles.",
        "ja": "△ABC ≡ △DEF である。対応する角の組を 3 つすべて答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の corresponding angles は、2 直線に 1 本の直線が交わってできる「同位角」（エントリ corresponding-angles）の意味にもなる。合同・相似な図形の話か、平行線の話かで区別する。",
      "△ABC ≅ △DEF のように頂点を対応する順に書くので、∠A と ∠D、∠B と ∠E、∠C と ∠F が対応する角になる。中学校学習指導要領の〔用語・記号〕は合同の記号に ≡ を挙げ、英語（CK-12 Geometry・IM）では ≅ を使う（エントリ congruent）。",
      "合同な三角形の対応する部分（角と辺）が等しいという事実は CPCTC（エントリ cpctc）。"
    ],
    "related": [
      "corresponding-sides",
      "congruent",
      "similar-triangles",
      "cpctc",
      "corresponding-angles"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）〔用語・記号〕",
        "note": "中2 B 図形: ≡"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.25 Corresponding Parts of Congruent Figures ほか（≅）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 2.1 Congruent Parts, Part 1 ほか（≅）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "corresponding angle measures | corresponding angles have the same measure | corresponding angles are equal in measure": 3
      },
      "written": {
        "corresponding angle measures | corresponding angles have the same measure | corresponding angles are equal in measure": 7
      },
      "sources": [
        "khan-middle",
        "openstax-elemalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 corresponding angle measures | corresponding angles have the same measure | corresponding angles are equal in measure（本文 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "corresponding-angles-postulate",
    "ja": {
      "term": "同位角の公準",
      "reading": "どういかくのこうじゅん",
      "alt": [
        "平行線の同位角"
      ]
    },
    "en": {
      "term": "corresponding angles postulate",
      "alt": [
        "corresponding angles theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「同位角の公準」は学習指導要領解説に無い、本プロジェクトの訳語。日本の中2 では「平行線の同位角は等しい」を平行線の性質として扱う。米国の教科書では公準（postulate）とするものと定理（theorem）とするものがある。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "平行な 2 直線に 1 本の直線が交わるとき、同位角は等しい、という公準。",
    "definition_en": "The postulate that when a transversal crosses two parallel lines, each pair of corresponding angles is congruent.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Angle 1 and angle 5 sit in the same spot at each intersection, so they're congruent by the corresponding angles postulate.",
        "ja": "∠1 と ∠5 はそれぞれの交点で同じ位置にある角だから、同位角で等しい。",
        "register": "spoken"
      },
      {
        "en": "Lines p and q are parallel and are cut by transversal t. ∠1 and ∠5 are corresponding angles, and m∠1 = 112°. Find m∠5 and the measure of the angle that forms a linear pair with ∠5.",
        "ja": "直線 p, q は平行で、直線 t がこれらと交わっている。∠1 と ∠5 は同位角で、∠1 = 112° である。∠5 と、∠5 と一直線をなす角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "逆の「同位角が等しければ 2 直線は平行」は converse of the corresponding angles postulate（theorem）と呼ぶ。",
      "これを公準として立てる場合、錯角が等しいこと（alternate interior angles theorem）は、この公準と対頂角から証明する。",
      "用例コーパスでも IM でも corresponding angles postulate は出てこない。CK-12 Geometry は 3.4 Corresponding Angles ほかで 8 件使う。"
    ],
    "related": [
      "transversal",
      "alternate-interior-angles-theorem",
      "postulate",
      "parallel-lines"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "3.4 Corresponding Angles・2.15 Proofs- Angle Pairs and Segments・3.8 Angles and Perpendicular Lines"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 corresponding angles postulate（CK-12 Geometry 3.4 Corresponding Angles・CK-12 Geometry 2.15 Proofs- Angle Pairs and Segments・CK-12 Geometry 3.8 Angles and Perpendicular Lines・計 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "corresponding-angles",
    "ja": {
      "term": "同位角",
      "reading": "どういかく",
      "alt": []
    },
    "en": {
      "term": "corresponding angles",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 本の直線に 1 本の直線が交わってできる角のうち、それぞれの交点で同じ位置にある 2 つの角。2 直線が平行なら、同位角は等しい。",
    "definition_en": "A pair of angles formed when a transversal crosses two lines, sitting in the same position at each intersection. If the two lines are parallel, corresponding angles are congruent.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This angle is in the upper right at the first intersection, and this one's in the upper right at the second, so they're corresponding angles.",
        "ja": "この角は 1 つ目の交点で右上にあって、こっちは 2 つ目の交点で右上にある。だから同位角だ。",
        "register": "spoken"
      },
      {
        "en": "Lines m and n are parallel and are cut by transversal t. A pair of corresponding angles measure (2x + 15)° and (4x − 25)°. Find x.",
        "ja": "直線 m と n は平行で、直線 t がこれらと交わっている。同位角の組の大きさが (2x + 15)°、(4x − 25)° のとき、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の corresponding angles は、合同・相似な図形の「対応する角」（エントリ corresponding-angles-of-congruent-figures）の意味でも使う。平行線と交わる直線の話か、2 つの図形の話かで区別する。",
      "同位角の「同じ位置」は、2 直線に交わる 1 本の直線（transversal、エントリ transversal）に対する位置で決まる。",
      "同位角が等しいのは 2 直線が平行なときだけ（エントリ corresponding-angles-postulate）。平行でなければ同位角でも大きさは違う。"
    ],
    "related": [
      "transversal",
      "corresponding-angles-postulate",
      "parallel-lines",
      "alternate-interior-angles",
      "corresponding-angles-of-congruent-figures"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "同位角",
        "en": "Corresponding angles"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "3.4 Corresponding Angles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "are corresponding angles | pair of corresponding angles | pairs of corresponding angles | corresponding angles theorem | corresponding angles converse": 5
      },
      "sources": [
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 are corresponding angles | pair of corresponding angles | pairs of corresponding angles | corresponding angles theorem | corresponding angles converse（Geometry 2.7 Angle-Side-Angle Triangle Congruence・Geometry 3.7 Reasoning about Similarity with Transformations・Grade 7 1.4 Scaled Relationships・計 25 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "corresponding-sides",
    "ja": {
      "term": "対応する辺",
      "reading": "たいおうするへん",
      "alt": []
    },
    "en": {
      "term": "corresponding sides",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "合同な図形や相似な図形で、重ね合わせたときに重なり合う辺どうし。合同なら長さが等しく、相似なら長さの比がすべて等しい。",
    "definition_en": "In two congruent or similar figures, the sides that match up when one figure is placed on the other. Corresponding sides of congruent figures are equal in length, and corresponding sides of similar figures are proportional.",
    "latex": "\\frac{AB}{DE} = \\frac{BC}{EF} = \\frac{CA}{FD}",
    "spoken_en": "AB over DE equals BC over EF equals CA over FD",
    "examples": [
      {
        "en": "Match the letters in order: A goes with D and B goes with E, so AB and DE are corresponding sides.",
        "ja": "文字を順番に対応させよう。A と D、B と E が対応するから、AB と DE が対応する辺だ。",
        "register": "spoken"
      },
      {
        "en": "Given △ABC ≅ △DEF, name all three pairs of corresponding sides.",
        "ja": "△ABC ≡ △DEF である。対応する辺を 3 組すべて答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "どの辺どうしが対応するかは、△ABC ≡ △DEF のような記号の頂点の順番で決まる（AB と DE、BC と EF、CA と FD）。図の中での位置や向きで決めない。",
      "相似な図形の対応する辺は、長さが等しいのではなく、長さの比が等しい。英語では corresponding sides are proportional と言う。",
      "「合同な三角形の対応する部分（辺と角）は等しい」は、英語の略語 CPCTC（エントリ cpctc）で表す。"
    ],
    "related": [
      "corresponding-angles-of-congruent-figures",
      "congruent",
      "similar-triangles",
      "cpctc",
      "scale-factor"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "corresponding sides": 59
      },
      "written": {
        "corresponding sides": 19
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cosecant",
    "ja": {
      "term": "コセカント",
      "reading": "こせかんと",
      "alt": []
    },
    "en": {
      "term": "cosecant",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の高校では csc を使わず 1/sin θ と書く。米国の教科書（OpenStax Algebra and Trigonometry）は csc・sec・cot を sin・cos・tan の逆数として定義し、微積分の公式にも csc・sec・cot が出てくる（d/dx cot x = −csc²x）。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "sin の逆数で表される三角関数。csc θ = 1/sin θ で、直角三角形では 斜辺 ÷ 対辺。",
    "definition_en": "The trigonometric function that is the reciprocal of sine: csc θ = 1/sin θ. In a right triangle, it is the hypotenuse divided by the opposite side.",
    "latex": "\\csc\\theta = \\frac{1}{\\sin\\theta}",
    "spoken_en": "cosecant theta equals 1 over sine theta",
    "examples": [
      {
        "en": "Cosecant is just the reciprocal of sine, so if sine is 3/5, cosecant is 5/3.",
        "ja": "コセカントはサインの逆数だから、サインが 3/5 ならコセカントは 5/3。",
        "register": "spoken"
      },
      {
        "en": "In right triangle ABC, angle C is a right angle, AB = 13, and BC = 5. Find the cosecant of angle A.",
        "ja": "∠C = 90° の直角三角形 ABC で、AB = 13、BC = 5 である。csc A（= 1/sin A）の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "名前に co が付くが、cosecant は cos ではなく sin の逆数。cos の逆数は secant（エントリ secant）。",
      "csc は cosecant と読む。csc θ（= 1/sin θ）と sin⁻¹ θ（arcsin、エントリ arcsine）を取り違えない。"
    ],
    "related": [
      "sine",
      "reciprocal",
      "secant",
      "cotangent",
      "trigonometric-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "cosecant": 192
      },
      "written": {
        "cosecant": 44
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cosine",
    "ja": {
      "term": "余弦",
      "reading": "よげん",
      "alt": [
        "コサイン"
      ]
    },
    "en": {
      "term": "cosine",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "角 θ に対して、直角三角形では（隣辺）/（斜辺）、座標では単位円上の点の x 座標で定める三角比・三角関数 cos θ。",
    "definition_en": "The trigonometric function cos θ: in a right triangle, the adjacent side over the hypotenuse; on the unit circle, the x-coordinate of the point at angle θ.",
    "latex": "\\cos\\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}}",
    "spoken_en": "cosine theta equals adjacent over hypotenuse",
    "examples": [
      {
        "en": "Cosine is adjacent over hypotenuse, so cosine of theta here is 12 over 13.",
        "ja": "cos は隣辺 / 斜辺だから、ここでの cos θ は 12/13 だ。",
        "register": "spoken"
      },
      {
        "en": "Find cos 120° and tan 120°.",
        "ja": "cos 120° と tan 120° の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "cos θ は \"cosine theta\" と読む。cos²θ は (cos θ)² のことで \"cosine squared theta\"。",
      "cos⁻¹ x は 1/cos x ではなく逆関数 arccos x（arccosine）。",
      "余弦定理は英語で law of cosines。"
    ],
    "related": [
      "sine",
      "tangent",
      "adjacent-side",
      "trigonometric-ratio",
      "arccosine"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "cosine": 3887
      },
      "written": {
        "cosine": 611
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cotangent",
    "ja": {
      "term": "コタンジェント",
      "reading": "こたんじぇんと",
      "alt": []
    },
    "en": {
      "term": "cotangent",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の高校では cot を使わず 1/tan θ と書く。米国の教科書（OpenStax Algebra and Trigonometry）は cot を tan の逆数として定義し、sin・cos・tan と並べて扱う。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "tan の逆数で表される三角関数。cot θ = 1/tan θ = cos θ/sin θ で、直角三角形では 隣辺 ÷ 対辺。",
    "definition_en": "The trigonometric function that is the reciprocal of tangent: cot θ = 1/tan θ = cos θ/sin θ. In a right triangle, it is the adjacent side divided by the opposite side.",
    "latex": "\\cot\\theta = \\frac{\\cos\\theta}{\\sin\\theta}",
    "spoken_en": "cotangent theta equals cosine theta over sine theta",
    "examples": [
      {
        "en": "Remember, cotangent is cosine over sine, so it's undefined wherever sine is zero.",
        "ja": "思い出そう。コタンジェントはコサイン ÷ サインだから、サインが 0 になるところでは定義されない。",
        "register": "spoken"
      },
      {
        "en": "Given that tan θ = 3/4, find the cotangent of θ.",
        "ja": "tan θ = 3/4 のとき、cot θ（= 1/tan θ）の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "cot は cotangent と読む（「コット」とは読まない）。",
      "cot θ（= 1/tan θ）と逆関数の arccot（cot⁻¹ θ）を取り違えない。tan⁻¹ θ も 1/tan θ ではなく arctan（エントリ arctangent）。",
      "θ = 90° では tan θ が定義されないが、cot θ = cos θ/sin θ = 0。1/tan θ より cos θ/sin θ の形で考えると安全。"
    ],
    "related": [
      "tangent",
      "reciprocal",
      "cosecant",
      "secant",
      "trigonometric-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "cotangent": 128
      },
      "written": {
        "cotangent": 50
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "coterminal-angle",
    "ja": {
      "term": "共終角",
      "reading": "きょうしゅうかく",
      "alt": []
    },
    "en": {
      "term": "coterminal angle",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「共終角」は学習指導要領解説に無い、本プロジェクトの訳語。日本の数II では一般角 θ + 360°n（動径が同じ角）として扱い、この関係の角に名前を付けない。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "標準の位置（始線が x 軸の正の向き）にかいたとき、動径が同じ位置になる角どうし。互いに 360° の整数倍だけ違う。",
    "definition_en": "Angles in standard position that share the same terminal side. They differ by a whole-number multiple of 360°, or 2π radians.",
    "latex": "\\theta + 360^\\circ n",
    "spoken_en": "theta plus 360 degrees times n",
    "examples": [
      {
        "en": "30 degrees and 390 degrees are coterminal angles, since you just go around one more full turn.",
        "ja": "30° と 390° は動径が同じ角。ちょうどもう 1 回転しただけだから。",
        "register": "spoken"
      },
      {
        "en": "Find a positive coterminal angle and a negative coterminal angle for 75°.",
        "ja": "75° と動径が同じ角を、正の角と負の角で 1 つずつ求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の一般角（エントリ general-angle）は θ + 360°n という角の表し方で、coterminal angle は動径が同じ角どうしの関係を指す。",
      "動径は英語で terminal side（エントリ terminal-side）。始線は initial side（エントリ initial-side）。",
      "弧度法では 2π の整数倍だけ違う角が coterminal angles になる。"
    ],
    "related": [
      "general-angle",
      "terminal-side",
      "initial-side",
      "reference-angle",
      "radian"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "coterminal angle": 9
      },
      "written": {
        "coterminal angle": 31
      },
      "sources": [
        "openstax-algtrig",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "countable",
    "ja": {
      "term": "可算",
      "reading": "かさん",
      "alt": [
        "可付番"
      ]
    },
    "en": {
      "term": "countable",
      "alt": [
        "countably infinite"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "set-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "集合の要素に 1 番目、2 番目、… と自然数で番号を付けられることをいう。有限集合も含める流儀と、可算無限（countably infinite）の集合だけを指す流儀がある。",
    "definition_en": "Describes a set whose elements can be listed as a first, second, third, and so on, that is, matched one-to-one with a subset of the natural numbers. Some books include finite sets, while others use the word only for countably infinite sets.",
    "latex": "\\mathbb{Z} = \\{0, 1, -1, 2, -2, \\dots\\}",
    "spoken_en": "Z is the set zero, one, negative one, two, negative two, and so on",
    "examples": [
      {
        "en": "The integers are countable, because I can list them: zero, one, negative one, two, negative two, and so on.",
        "ja": "整数全体は可算だ。0, 1, −1, 2, −2, … と並べられるからだ。",
        "register": "spoken"
      },
      {
        "en": "Prove that the set of all positive rational numbers is countable.",
        "ja": "正の有理数全体の集合が可算であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "反対は uncountable（非可算）。実数全体は非可算である（対角線論法）。",
      "有限集合を countable に含めるかは本による。無限であることをはっきり言うときは countably infinite を使う。",
      "用例コーパスでは countable は書き言葉に多く（ほとんどが MIT の講義ノート）、話し言葉にはほとんど出てこない。countably infinite は話・書とも少ない。AP Statistics の CED にも countable が 3 件（1.2 ほか）ある。"
    ],
    "related": [
      "bijection",
      "set",
      "one-to-one"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "countable": 3,
        "countably infinite": 1
      },
      "written": {
        "countable": 37,
        "countably infinite": 7
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "counterclockwise",
    "ja": {
      "term": "正の向き",
      "reading": "せいのむき",
      "alt": [
        "反時計回り"
      ]
    },
    "en": {
      "term": "counterclockwise",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "near",
    "mapping_note": "日本語の「正の向き」は角の回転の向きの約束で、中身は反時計回り。英語では角の向きをそのまま counterclockwise と言う。the positive direction は用例コーパスでは主に数直線上の運動の向き（Khan Academy）。英国などの anticlockwise（英語版 Wikipedia「Clockwise」は Commonwealth English の言い方とする）は用例コーパスに出てこない。共通テスト・センター試験の問題文の「x 軸上を正の向きに」（令和6年度 本試験 数学I 第3問）、「数直線上を…正の向きに」（平成28年度 本試験 数学II・数学B 第5問）は座標が増える向きのことで、英語は in the positive direction（counterclockwise ではない）。",
    "domains": [
      "trigonometry",
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus"
      ]
    },
    "definition_ja": "時計の針と逆の回転の向き。角の正の向き。",
    "definition_en": "Turning in the direction opposite to a clock's hands; the positive direction for angles.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Positive angles go counterclockwise, negative angles go clockwise.",
        "ja": "正の角は反時計回り、負の角は時計回り。",
        "register": "spoken"
      },
      {
        "en": "Find the point obtained by rotating (2, 0) counterclockwise by π/3 about the origin.",
        "ja": "点 (2, 0) を原点のまわりに π/3 だけ回転した点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "clockwise",
      "general-angle",
      "rotation"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Clockwise",
        "note": "記事「Clockwise」の冒頭: 反時計回りは Commonwealth English で anticlockwise、North American English で counterclockwise"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0",
        "note": "反時計回りを正の向きとして角を測る"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「時計回り・反時計回り」",
        "url": "https://ja.wikipedia.org/wiki/%E6%99%82%E8%A8%88%E5%9B%9E%E3%82%8A%E3%83%BB%E5%8F%8D%E6%99%82%E8%A8%88%E5%9B%9E%E3%82%8A",
        "note": "節「数学」: 平面角は反時計回りが正の向き"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和6年度 大学入学共通テスト 本試験 数学Ⅰ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=666&f=abm00004687.pdf&n=2024_or_18_sugaku1.pdf",
        "note": "第3問: x 軸上を正の向きに、y 軸上を負の向きに（座標軸の向き）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成28年度 大学入試センター試験 本試験 数学Ⅱ・数学Ｂ（問題）",
        "url": "https://web.archive.org/web/20170601/https://www.dnc.ac.jp/albums/abm.php?f=abm00007097.pdf&n=2801-0502+suugaku2B.pdf",
        "note": "第5問: 数直線上を正の向きに 3、負の向きに 1 だけ移動"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 1.2 Naming the Moves〜1.6 Describing Transformations（counterclockwise）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 1.13〜1.15（1.14 Defining Rotations ほか。counterclockwise）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.10 Defining Rotation・8.12 Geometry Software and Graphing Rotations（counterclockwise）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "counterclockwise": 86
      },
      "written": {
        "counterclockwise": 155
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "counterexample",
    "ja": {
      "term": "反例",
      "reading": "はんれい",
      "alt": []
    },
    "en": {
      "term": "counterexample",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "logic"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1",
        "Discrete Math"
      ]
    },
    "definition_ja": "仮定を満たすのに結論が成り立たない具体的な例。1 つあれば、その命題が偽であることが示せる。",
    "definition_en": "A specific case that satisfies the hypothesis of a statement but not its conclusion; one is enough to show the statement is false.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Give a counterexample to show that the statement \"if x² = 4, then x = 2\" is false.",
        "ja": "反例を挙げて、「x² = 4 ならば x = 2」が偽であることを示せ。",
        "register": "written"
      },
      {
        "en": "The function f(x) = |x| is a counterexample: it is continuous at 0 but not differentiable there.",
        "ja": "関数 f(x) = |x| が反例である。x = 0 で連続だが、そこで微分可能ではない。",
        "register": "written"
      },
      {
        "en": "Negative two is a counterexample, right? Its square is four, but it isn't two.",
        "ja": "−2 が反例だよね。2 乗すると 4 だけど、2 ではない。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "give a counterexample",
        "ja": "反例を挙げる"
      },
      {
        "en": "find a counterexample",
        "ja": "反例を見つける"
      }
    ],
    "pitfalls": [
      "counterexample は 1 語で書く（counter example と離さない）。",
      "反例は 1 つで命題を偽と示せるが、成り立つ例をいくつ挙げても「すべての〜について」の証明にはならない。"
    ],
    "related": [
      "proof",
      "prove",
      "all",
      "exist"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "反例",
        "en": "Counterexample"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "counterexample": 5
      },
      "written": {
        "counterexample": 22
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "counting",
    "ja": {
      "term": "場合の数",
      "reading": "ばあいのかず",
      "alt": []
    },
    "en": {
      "term": "counting",
      "alt": [
        "number of ways"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "分野名としての「場合の数」は counting（OpenStax Algebra and Trigonometry の章 Sequences, Probability and Counting Theory、節 Counting Principles）。個数としての「場合の数」は the number of ways（to …）で、「何通り」は How many ways …? と言う。用例コーパスと参照の number of cases は症例数や箱の数の意味で、場合の数の意味では出てこない（日本語版 Wikipedia「場合の数」は英語を number of cases と書く）。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "ある事柄の起こり方が全部で何通りあるか、またそれを漏れなく重複なく数える方法を扱う分野。",
    "definition_en": "The study of how many ways something can happen, and the methods for counting them without missing or repeating any.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is really a counting problem: how many ways can five people line up?",
        "ja": "これは場合の数の問題だ。5 人が 1 列に並ぶ並び方は何通りあるか。",
        "register": "spoken"
      },
      {
        "en": "Find the number of ways to choose a president and a vice president from a class of 30 students.",
        "ja": "30 人のクラスから委員長と副委員長を 1 人ずつ選ぶ選び方の数を求めよ。",
        "register": "written"
      },
      {
        "en": "The multiplication principle is one of the basic tools of counting.",
        "ja": "積の法則は場合の数の基本的な道具の 1 つである。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "和の法則・積の法則は addition principle ／ multiplication principle。"
    ],
    "related": [
      "enumeration",
      "permutation",
      "combination",
      "tree-diagram",
      "inclusion-exclusion-principle",
      "addition-principle",
      "multiplication-principle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13 Sequences, Probability and Counting Theory、13.5 Counting Principles"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「場合の数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%A0%B4%E5%90%88%E3%81%AE%E6%95%B0",
        "note": "冒頭（英: number of cases）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "counting principle | counting problem | counting theory | counting technique | counting argument | counting method | art of counting | counting rule": 8,
        "!a number of ways": 9
      },
      "written": {
        "!a number of ways": 36,
        "counting principle | counting problem | counting theory | counting technique | counting argument | counting method | art of counting | counting rule": 47
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "covariance",
    "ja": {
      "term": "共分散",
      "reading": "きょうぶんさん",
      "alt": []
    },
    "en": {
      "term": "covariance",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校・初年次統計（OpenStax Introductory Statistics・Khan Academy の AP Statistics・AP Statistics の CED）には出てこない。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "2 つの変量の偏差の積の平均値。正なら正の相関、負なら負の相関の傾向を表す。",
    "definition_en": "The mean of the products of the deviations of two variables; positive for a positive association and negative for a negative one.",
    "latex": "s_{xy} = \\frac{1}{n}\\sum_{i=1}^{n}(x_i - \\bar{x})(y_i - \\bar{y})",
    "spoken_en": null,
    "examples": [
      {
        "en": "The covariance is positive here because big x's tend to come with big y's.",
        "ja": "x が大きいと y も大きい傾向があるので、ここでは共分散が正になる。",
        "register": "spoken"
      },
      {
        "en": "Compute the covariance of x and y, and then the correlation coefficient.",
        "ja": "x と y の共分散を計算し、次に相関係数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "線形代数の Nicholson は 8.11 で、確率変数の共分散（期待値による定義）として使う（3 件）。"
    ],
    "related": [
      "correlation-coefficient",
      "variance",
      "deviation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "共分散",
        "en": "Covariance"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Unit 5（covariance は 0 件）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.11 An Application to Statistical Principal Component Analysis: covariance（3 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元 2 の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cpctc",
    "ja": {
      "term": "CPCTC",
      "reading": "しーぴーしーてぃーしー",
      "alt": [
        "合同な三角形の対応する部分は等しい"
      ]
    },
    "en": {
      "term": "CPCTC",
      "alt": [
        "corresponding parts of congruent triangles are congruent"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「CPCTC」は英語の略語をそのまま使う（学習指導要領解説に無い）。日本の証明では「合同な図形の対応する辺（角）は等しい」と文で書き、略語はない。",
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "「合同な三角形の対応する部分（辺と角）は合同である」という事実を表す英語の略語。三角形の合同を示したあと、対応する辺や角が等しいことを言う理由として証明に書く。",
    "definition_en": "An abbreviation for \"corresponding parts of congruent triangles are congruent.\" It is cited as the reason when a proof uses two congruent triangles to conclude that a pair of corresponding sides or angles are congruent.",
    "latex": "\\triangle ABC \\cong \\triangle DEF \\implies AB = DE,\\ \\angle A = \\angle D",
    "spoken_en": "triangle ABC is congruent to triangle DEF, so AB equals DE and angle A equals angle D",
    "examples": [
      {
        "en": "Once we know the triangles are congruent, we get angle A equals angle D for free. That's CPCTC.",
        "ja": "三角形が合同だと分かれば、∠A = ∠D はそのまま言える。合同な三角形の対応する角は等しいからだ。",
        "register": "spoken"
      },
      {
        "en": "First prove that △ABM ≅ △DCM. Then use CPCTC to show that AB ≅ DC.",
        "ja": "まず △ABM ≡ △DCM を証明し、合同な図形の対応する辺は等しいことから AB = DC を示せ。",
        "register": "written"
      },
      {
        "en": "Statement: ∠B ≅ ∠E. Reason: CPCTC.",
        "ja": "∠B = ∠E（合同な図形の対応する角は等しい）",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "by CPCTC",
        "ja": "合同な図形の対応する辺（角）は等しいから"
      }
    ],
    "pitfalls": [
      "CPCTC は三角形の合同を示したあとにしか使えない。合同を示す理由（SSS・SAS・ASA など）と混同しない。",
      "読むときは C-P-C-T-C と文字を 1 つずつ言う。",
      "用例コーパスでは話し言葉に少しあるだけで（すべて The Organic Chemistry Tutor）、書き言葉には出てこない。IM は略語を使わず、corresponding parts の形で書く。CK-12 Geometry は略語の CPCTC を 11 件（4.9 CPCTC ほか）、corresponding parts of congruent triangles are congruent を 2 件使う。"
    ],
    "related": [
      "congruent",
      "sss-congruence",
      "sas-congruence",
      "two-column-proof"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.9 CPCTC・4.4 Isosceles Triangles・4.15 ASA and AAS"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "CPCTC": 4,
        "corresponding parts of congruent triangles are congruent": 4
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 8 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 CPCTC（CK-12 Geometry 4.9 CPCTC・CK-12 Geometry 4.4 Isosceles Triangles・CK-12 Geometry 4.15 ASA and AAS・計 11 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cramers-rule",
    "ja": {
      "term": "クラメルの公式",
      "reading": "くらめるのこうしき",
      "alt": []
    },
    "en": {
      "term": "Cramer's rule",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "係数の行列式が 0 でない連立 1 次方程式の解を、行列式の比で表す公式。",
    "definition_en": "The formula expressing the solution of a system of linear equations with nonzero coefficient determinant as ratios of determinants.",
    "latex": "x = \\frac{D_x}{D},\\quad y = \\frac{D_y}{D}",
    "spoken_en": "x equals D sub x over D, y equals D sub y over D",
    "examples": [
      {
        "en": "With Cramer's rule, replace the x column with the constants and divide by the main determinant.",
        "ja": "クラメルの公式では、x の列を定数に置きかえて、係数の行列式で割る。",
        "register": "spoken"
      },
      {
        "en": "Use Cramer's rule to solve 2x + 3y = 7, x − y = 1.",
        "ja": "クラメルの公式を用いて 2x + 3y = 7, x − y = 1 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校の学習指導要領には含まれない。米国では Precalculus と Intermediate Algebra で扱う（書き言葉は大半が OpenStax Algebra and Trigonometry と OpenStax Intermediate Algebra）。"
    ],
    "related": [
      "augmented-matrix",
      "gaussian-elimination",
      "system-of-three-equations"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "クラメルの公式",
        "en": "Cramer's rule"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "Cramer's rule": 12
      },
      "written": {
        "Cramer's rule": 87
      },
      "sources": [
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "critical-point",
    "ja": {
      "term": "臨界点",
      "reading": "りんかいてん",
      "alt": []
    },
    "en": {
      "term": "critical point",
      "alt": [],
      "variants": [
        {
          "term": "critical number",
          "register": "both",
          "note": "少数の言い方（Khan Academy・patrickJMT など）。話し言葉・書き言葉とも critical point よりずっと少なく、書き言葉はすべて OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "AP の CED（topic 5.2・5.5）は f′ が 0 か存在しない点を critical point と呼ぶ。OpenStax Calculus は定義では c を critical number、点 (c, f(c)) を critical point と呼び分けるが、2 つの語はしばしば同じ意味で使われると注記し、本文では x の値も critical point と呼ぶ。どちらも極値の候補を探すのに使う。学習指導要領解説には f′(x) = 0 となる x の名前が出てこない。見出しの「臨界点」も解説に無いが、日本語版 Wikipedia「極値」は停留点（f′ = 0 の点）の別名として臨界点を挙げる。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "定義域内の点 c で、f′(c) = 0 となるか、f′(c) が存在しないもの。",
    "definition_en": "A number c in the domain of f where f′(c) = 0 or f′(c) does not exist.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Set f prime equal to zero to find the critical points, and don't forget where it's undefined.",
        "ja": "f′ = 0 とおいて臨界点を求める。f′ が存在しない点も忘れずに。",
        "register": "spoken"
      },
      {
        "en": "Find the critical points of f(x) = ∛(x²)(x − 5).",
        "ja": "f(x) = ∛(x²)(x − 5) の臨界点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "f′(c) が存在しない点（グラフが尖っているところなど）も critical point に含める。f′(c) = 0 だけを探すと極値を見落とす。",
      "統計の critical value（臨界値）とは別の語。"
    ],
    "related": [
      "local-extremum",
      "first-derivative-test",
      "find-the-local-extrema"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.2・5.5（critical point）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「極値」",
        "url": "https://ja.wikipedia.org/wiki/%E6%A5%B5%E5%80%A4",
        "note": "停留点（stationary point）・臨界点（critical point）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "f′(x) = 0 となる x の名前が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "critical point": 562,
        "critical number": 36
      },
      "written": {
        "critical point": 226,
        "critical number": 5
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:blackpenredpen",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "critical-value",
    "ja": {
      "term": "臨界値",
      "reading": "りんかいち",
      "alt": []
    },
    "en": {
      "term": "critical value",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "仮説検定や信頼区間で、棄却域の境目や信頼区間の幅を決める値。95% 信頼区間の z* = 1.96 など。",
    "definition_en": "In a hypothesis test or a confidence interval, the value that marks the edge of the rejection region or sets the width of the interval, such as z* = 1.96 for 95% confidence.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For 95 percent confidence, the critical value is 1.96.",
        "ja": "信頼度 95% なら、臨界値は 1.96。",
        "register": "spoken"
      },
      {
        "en": "Find the critical value t* for a 90% confidence interval with 15 degrees of freedom.",
        "ja": "自由度 15 の 90% 信頼区間の臨界値 t* を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "微積分でも f′(x) = 0 となる値を critical value（critical point）と呼ぶことがあり、用例コーパスの文脈にもその意味のものが一部あった。"
    ],
    "related": [
      "rejection-region",
      "confidence-interval",
      "test-statistic",
      "z-score"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "critical value": 77
      },
      "written": {
        "critical value": 62
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cross-method",
    "ja": {
      "term": "たすき掛け",
      "reading": "たすきがけ",
      "alt": [
        "たすきがけ"
      ]
    },
    "en": {
      "term": "trial and error",
      "alt": [],
      "variants": [
        {
          "term": "guess and check",
          "register": "spoken",
          "note": "話し言葉で使う言い方。因数分解の場面では trial and error のほうが多く、講義や解説動画では、方程式の解や原始関数を当てて確かめるときにも言う。IM Algebra 1 7.10 は因数の組を探す手順を guessing and checking と書く。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "OpenStax Elementary Algebra 7.3 は、ax² + bx + c の因数を、a の因数の組と c の因数の組を ( )( ) に当てはめ、外側と内側の積の和が中央の項 bx になるかを確かめて探す手順を trial and error と呼び（factor … using trial and error。Intermediate Algebra 6.2 も同じ）、IM Algebra 1 7.10 は同じ探し方を guessing and checking と書く。手順はたすき掛けと同じだが、斜めに掛ける図に当たる名前（cross method など）は用例コーパスと参照（CED・OpenStax・IM・CK-12）に出てこない。日本語版 Wikipedia「因数分解」には節「たすきがけ」がある。OpenStax Elementary Algebra 7.3・Intermediate Algebra 6.2 の “ac” method は、積が ac、和が b になる 2 つの数で bx を 2 つの項に分け、factoring by grouping（エントリ factoring-by-grouping）でくくる別の手順。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "ax² + bx + c（a ≠ 1）を因数分解するとき、a と c をそれぞれ 2 つの数の積に分けて並べ、斜めに掛けた積の和が b になる組み合わせを探す方法。",
    "definition_en": "A method for factoring ax² + bx + c when a ≠ 1: list factor pairs of a and of c, multiply them diagonally, and look for the arrangement whose two diagonal products add up to b.",
    "latex": "(px + q)(rx + s) = prx^2 + (ps + qr)x + qs",
    "spoken_en": "the quantity p x plus q times the quantity r x plus s equals p r x squared plus the quantity p s plus q r times x, plus q s",
    "examples": [
      {
        "en": "For 3x² + 7x + 2, let's use trial and error. Try (3x + 1)(x + 2): the outer terms give 6x, the inner terms give x, and 6x plus x is 7x, so that's it.",
        "ja": "3x² + 7x + 2 をたすき掛けで因数分解しよう。(3x + 1)(x + 2) なら、外側の積が 6x、内側の積が x で、足すと 7x になるから、これで合っている。",
        "register": "spoken"
      },
      {
        "en": "Factor 3x² + 7x + 2 using trial and error.",
        "ja": "たすき掛けを用いて、3x² + 7x + 2 を因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "直訳の cross multiplication は、比例式 a/b = c/d から ad = bc を作る cross-multiply（エントリ cross-multiply）の意味になり、たすき掛けの意味にはならない。",
      "trial and error・guess and check は因数分解だけの名前ではなく、答えの候補を当てはめて確かめるやり方一般を言う。たすき掛けの意味で言うときは、factor 3x² + 7x + 2 using trial and error のように因数分解と一緒に言う。",
      "ac method は、積が ac、和が b になる 2 つの数で x の項を 2 つに書き分け、factoring by grouping（エントリ factoring-by-grouping）で因数分解する方法。3x² + 7x + 2 = 3x² + x + 6x + 2 = x(3x + 1) + 2(3x + 1) = (x + 2)(3x + 1) のように、斜めに掛けるたすき掛けとは手順が違うが、答えは同じになる。",
      "a = 1 のとき（x² + 5x + 6 など）は、掛けて c、足して b になる 2 つの数を探せばよく、たすき掛けは要らない。"
    ],
    "related": [
      "factoring",
      "factoring-by-grouping",
      "cross-multiply",
      "trinomial",
      "leading-coefficient"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「因数分解」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%A0%E6%95%B0%E5%88%86%E8%A7%A3",
        "note": "節「たすきがけ」"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "7.3 Factor Trinomials of the Form ax²+bx+c: Trial and Error、“ac” method（mapping_note）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "6.2 Factor Trinomials: trial and error、“ac” method（mapping_note）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 7.10: guessing and checking"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 数と式: (ax + b)(cx + d) の公式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "guess and check": 10,
        "trial and error": 16
      },
      "written": {
        "trial and error": 19,
        "guess and check": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-10-09"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cross-multiply",
    "ja": {
      "term": "内項と外項の積は等しい",
      "reading": "ないこうとがいこうのせきはひとしい",
      "alt": [
        "内項の積",
        "外項の積"
      ]
    },
    "en": {
      "term": "cross-multiply",
      "alt": [
        "cross multiplication"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「内項の積と外項の積は等しい」（学習指導要領解説には内項・外項の語は無く、日本語版 Wikipedia「比例式」にある）に当たる性質を、OpenStax Algebra and Trigonometry（2.2）・Intermediate Algebra（7.5）は名前で呼ばず、比例式 a/b = c/d を cross-multiply する（ad = bc）と手順を動詞で言い、CK-12 Geometry はこの解き方を Cross-Multiplication Theorem と呼ぶ（7.2）。IM には cross-multiply の語が出てこない。内項・外項に当たる名前（means・extremes）は用例コーパスにも参照にも出てこない。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "比例式 a : b = c : d で、内側の 2 つの項（内項）b, c の積と外側の 2 つの項（外項）a, d の積が等しい（ad = bc）という性質。比例式を解くときに使う。",
    "definition_en": "In a proportion a/b = c/d, multiplying across the diagonals gives equal products: ad = bc. To cross-multiply is to use this step to rewrite a proportion as an equation with no fractions.",
    "latex": "\\frac{a}{b} = \\frac{c}{d} \\Rightarrow ad = bc",
    "spoken_en": "a over b equals c over d implies a d equals b c",
    "examples": [
      {
        "en": "3/4 = x/12 is a proportion, so just cross-multiply: 3 times 12 is 36, and that has to equal 4x.",
        "ja": "3/4 = x/12 は比例式だから、斜めに掛け合わせよう。3 × 12 = 36 が 4x と等しくなるね。",
        "register": "spoken"
      },
      {
        "en": "Cross-multiply to solve the proportion 5/x = 15/6.",
        "ja": "分母を払って、比例式 5/x = 15/6 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "因数分解の「たすき掛け」を直訳して cross multiplication と言うと、英語では比例式のこの操作の意味になる。たすき掛けと同じ手順の英語の言い方は、エントリ cross-method（trial and error）にある。",
      "ベクトルの外積は cross product（エントリ cross-product）で、cross-multiply とは別の語。",
      "cross-multiply できるのは、a/b = c/d のように両辺がそれぞれ 1 つの分数のとき。a/b + 1 = c/d のように他の項があるときは、そのままでは使えない。"
    ],
    "related": [
      "proportion",
      "ratio",
      "value-of-a-ratio",
      "denominator",
      "solve"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「比例式」",
        "url": "https://ja.wikipedia.org/wiki/%E6%AF%94%E4%BE%8B%E5%BC%8F",
        "note": "内項の積と外項の積は等しい"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "内項・外項の語は出てこない"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "7.2 Ratio and Proportion in Similar Figures（Cross-Multiplication Theorem）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "2.2 Linear Equations in One Variable（cross-multiplication）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "7.5 Solve Applications with Rational Equations（cross-multiply）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "cross-multiply は IM 6–8・IM 9–12 とも 0 件"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "cross-multiply | cross-multiplying | cross-multiplied | cross-multiplies | cross-multiplication": 49
      },
      "written": {
        "cross-multiply | cross-multiplying | cross-multiplied | cross-multiplies | cross-multiplication": 6
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cross-product",
    "ja": {
      "term": "外積",
      "reading": "がいせき",
      "alt": [
        "ベクトル積"
      ]
    },
    "en": {
      "term": "cross product",
      "alt": [
        "vector product"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "空間の 2 つのベクトル a, b から作る、両方に垂直で大きさが |a||b| sin θ のベクトル a × b。",
    "definition_en": "For two vectors a and b in space, the vector a × b perpendicular to both, with magnitude |a||b| sin θ.",
    "latex": "\\vec{a}\\times\\vec{b}",
    "spoken_en": "a cross b",
    "examples": [
      {
        "en": "The cross product gives you a vector perpendicular to both, and its length is the area of the parallelogram.",
        "ja": "外積は両方に垂直なベクトルで、長さは平行四辺形の面積。",
        "register": "spoken"
      },
      {
        "en": "Find a × b for a = ⟨1, 2, 0⟩ and b = ⟨0, 1, 3⟩.",
        "ja": "a = ⟨1, 2, 0⟩, b = ⟨0, 1, 3⟩ について a × b を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校の数C では外積を扱わない。米国では Calculus III（多変数の微積分）で扱う（OpenStax Calculus Volume 3）。",
      "a × b は a cross b と読み、内積 a · b（a dot b）と区別する。"
    ],
    "related": [
      "dot-product",
      "scalar-triple-product",
      "normal-vector",
      "area-of-a-triangle-using-vectors"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "cross product": 68
      },
      "written": {
        "cross product": 145,
        "vector product": 4
      },
      "sources": [
        "mit-18.02",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cross-section",
    "ja": {
      "term": "断面",
      "reading": "だんめん",
      "alt": [
        "切り口"
      ]
    },
    "en": {
      "term": "cross section",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry",
        "AP Calculus AB",
        "Integrated Math 3",
        "Calculus I"
      ]
    },
    "definition_ja": "立体を平面で切ったときにできる切り口の図形。微積分では、断面の面積を積分して立体の体積を求める。",
    "definition_en": "The two-dimensional shape you get when a plane slices through a solid. In calculus, the areas of cross sections are integrated to find a volume.",
    "latex": "V = \\int_a^b A(x)\\,dx",
    "spoken_en": "V equals the integral from a to b of A of x d x",
    "examples": [
      {
        "en": "If you slice the cylinder parallel to its base, the cross section is a circle, but if you slice it straight down through the middle, you get a rectangle.",
        "ja": "円柱を底面に平行に切ると断面は円になるけど、まん中を縦にまっすぐ切ると長方形になる。",
        "register": "spoken"
      },
      {
        "en": "A solid sits on the region under y = √x for 0 ≤ x ≤ 4, and each cross section taken perpendicular to the x-axis is a square. Find its volume.",
        "ja": "ある立体の底面は、x = 0 から x = 4 までの曲線 y = √x の下の部分である。x 軸に垂直な平面で切った断面はどれも正方形である。この立体の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「断面」「切り口」はどちらも cross section。断面の面積（断面積）は cross-sectional area と形容詞の形になる。ハイフン付きの cross-section と書くこともある。",
      "AP Calculus の CED では topic 8.7・8.8（Volumes with Cross Sections）に出て、問題文は x 軸に垂直な断面の形（正方形、半円など）を指定する。IM では Grade 7 の 7.11 Slicing Solids から出てくる。",
      "用例コーパスで cross section（ハイフン付きを含む）は、話し言葉では Khan Academy と Professor Leonard が大半、書き言葉では OpenStax Calculus が半分以上。"
    ],
    "related": [
      "cavalieris-principle",
      "volume-by-cross-sections",
      "cross-sectional-area"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "断面",
        "en": "Cross section (geometry)"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "topic 8.7・8.8 Volumes with Cross Sections"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Grade 7 7.11 Slicing Solids（cross section）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cross section | cross-section": 108
      },
      "written": {
        "cross section | cross-section": 46
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cross-sectional-area",
    "ja": {
      "term": "断面積",
      "reading": "だんめんせき",
      "alt": []
    },
    "en": {
      "term": "cross-sectional area",
      "alt": [
        "area of the cross section"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "立体を平面で切ったときの切り口の面積。",
    "definition_en": "The area of the slice you get when a solid is cut by a plane.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Write the cross-sectional area as a function of x, then integrate it from a to b.",
        "ja": "断面積を x の関数で表して、a から b まで積分する。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 1 は断面積を A(x) と書く（このエントリの日本語の定義で S(x) と書いたもの）。"
    ],
    "related": [
      "volume-by-cross-sections",
      "volume-of-a-solid-of-revolution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "断面",
        "en": "Cross section (geometry)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cross-sectional area": 9,
        "area of the cross section": 6
      },
      "written": {
        "cross-sectional area": 15
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cryptography",
    "ja": {
      "term": "暗号",
      "reading": "あんごう",
      "alt": [
        "暗号理論",
        "暗号学"
      ]
    },
    "en": {
      "term": "cryptography",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "情報を第三者に読めない形に変え（暗号化）、正しい相手だけが元に戻せる（復号）ようにする方法とその理論。数学では合同式や素数の性質を使う RSA 暗号などを扱う。",
    "definition_en": "The study of methods for encoding information so that only the intended receiver can decode it. In math courses it usually appears through modular arithmetic and primes, as in RSA.",
    "latex": "c \\equiv m^{e} \\pmod{n}",
    "spoken_en": "c is congruent to m to the e mod n",
    "examples": [
      {
        "en": "Today we'll see why primes matter in cryptography: RSA relies on the fact that multiplying two big primes is easy, but factoring the product is hard.",
        "ja": "今日は素数が暗号でなぜ大事かを見よう。RSA は、大きな素数 2 つをかけるのは簡単でも、その積を素因数分解するのは難しいことを利用している。",
        "register": "spoken"
      },
      {
        "en": "In public-key cryptography, the encryption key may be made public, but the decryption key must be kept secret.",
        "ja": "公開鍵暗号では、暗号化の鍵は公開してよいが、復号の鍵は秘密にしておかなければならない。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「暗号」1 語は、分野としての cryptography にも、個々の方式（cipher。Caesar cipher など）にも当たる。",
      "暗号化は encryption（動詞 encrypt）、復号は decryption（動詞 decrypt）。",
      "用例コーパスでは話し言葉・書き言葉とも少なく、大半が MIT（MIT OCW の講義と MIT の講義ノート）。Levin（6.2 Introduction to Number Theory）と Nicholson（8.8 An Application to Linear Codes over Finite Fields）には各 1 件。"
    ],
    "related": [
      "fermats-little-theorem",
      "modular-inverse",
      "modulus",
      "prime-factorization",
      "congruence-modulo-n"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "cryptography": 9
      },
      "written": {
        "cryptography": 11
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cube-solid",
    "ja": {
      "term": "立方体",
      "reading": "りっぽうたい",
      "alt": [
        "正六面体"
      ]
    },
    "en": {
      "term": "cube",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "6 つの面がすべて合同な正方形でできている立体。縦・横・高さがすべて等しい直方体ともいえる。",
    "definition_en": "A solid with six faces that are all congruent squares. It is a rectangular prism whose length, width, and height are all equal.",
    "latex": "V = s^3",
    "spoken_en": "V equals s cubed",
    "examples": [
      {
        "en": "A cube is a rectangular prism with all its edges the same length, so its volume is s cubed.",
        "ja": "立方体は、辺の長さが全部等しい直方体。だから体積は s³ になる。",
        "register": "spoken"
      },
      {
        "en": "A cube has an edge length of 4 cm. Find its volume and its surface area.",
        "ja": "1 辺の長さが 4 cm の立方体の体積と表面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "3 乗（立方、エントリ cube）と同じ cube という単語。1 辺 s の立方体の体積が s³ なので、s³ を s cubed と読む。",
      "米国の問題文（IM ほか）の number cube は「さいころ」のこと。",
      "立方体の面は face（6 つ）、辺は edge（12 本）、頂点は vertex（8 つ。複数形は vertices）。1 辺の長さは edge length や side length と言う。"
    ],
    "related": [
      "cube",
      "rectangular-prism",
      "regular-polyhedron",
      "volume",
      "net"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "number cube（さいころ。Grade 6〜8 で 28 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "a cube !root !roots | the cube !root !roots !of | unit cube": 140
      },
      "written": {
        "a cube !root !roots | the cube !root !roots !of | unit cube": 86
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cube",
    "ja": {
      "term": "立方",
      "reading": "りっぽう",
      "alt": [
        "3 乗"
      ]
    },
    "en": {
      "term": "cube",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "同じ数を 3 個掛け合わせること、またはその結果。a の立方（a の 3 乗）を a³ と書く。",
    "definition_en": "The result of using a number as a factor three times, written with an exponent of 3, as in a³. As a verb, to cube a number means to raise it to the third power.",
    "latex": "a^3 = a \\times a \\times a",
    "spoken_en": "a cubed equals a times a times a",
    "examples": [
      {
        "en": "The cube of negative 2 is negative 8, because you're multiplying three negatives.",
        "ja": "−2 の 3 乗は −8。負の数を 3 個掛けるからね。",
        "register": "spoken"
      },
      {
        "en": "Find the cube of −0.2.",
        "ja": "−0.2 の 3 乗を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "cube は「立方体」と同じ語（1 辺 x の立方体の体積が x³）。x³ は x cubed と読み、立方根は cube root。",
      "形容詞は cubic で、cubic function（3 次関数）や cubic centimeter（cm³）に使う。cube と cubic を取り違えない。"
    ],
    "related": [
      "power",
      "exponent",
      "square",
      "base-of-a-power",
      "nth-root"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the cube of | cubes of | cube the | cube both": 26
      },
      "written": {
        "the cube of | cubes of | cube the | cube both": 33
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cubic-equation",
    "ja": {
      "term": "三次方程式",
      "reading": "さんじほうていしき",
      "alt": [
        "3 次方程式"
      ]
    },
    "en": {
      "term": "cubic equation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "未知数の最高次数が 3 の方程式 ax³ + bx² + cx + d = 0（a ≠ 0）。",
    "definition_en": "An equation ax³ + bx² + cx + d = 0 (a ≠ 0) whose highest power of the unknown is 3.",
    "latex": "ax^3 + bx^2 + cx + d = 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "A cubic equation with real coefficients always has at least one real root.",
        "ja": "実数係数の 3 次方程式は必ず実数解を少なくとも 1 つもつ。",
        "register": "spoken"
      },
      {
        "en": "Solve the cubic equation x³ − 2x² − 5x + 6 = 0.",
        "ja": "3 次方程式 x³ − 2x² − 5x + 6 = 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "講義では名詞の a cubic（3 次式・3 次方程式）だけで言う例もある（MIT OCW 18.01、Professor Leonard）。a cubic は a cubic foot のように体積の単位にも使う。"
    ],
    "related": [
      "polynomial-equation-of-higher-degree",
      "quartic-equation",
      "cubic-function",
      "factor-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "三次方程式",
        "en": "Cubic equation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Zeros of Polynomial Functions（cubic equation）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "cubic equation": 3
      },
      "written": {
        "cubic equation": 1
      },
      "sources": [
        "mit-18.01",
        "mit-18.06",
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは OpenStax の呼び方 cubic equation（本文 1 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cubic-function",
    "ja": {
      "term": "3 次関数",
      "reading": "さんじかんすう",
      "alt": [
        "三次関数"
      ]
    },
    "en": {
      "term": "cubic function",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "3 次式 y = ax³ + bx² + cx + d（a ≠ 0）で表される関数。",
    "definition_en": "A function given by a polynomial of degree 3, y = ax³ + bx² + cx + d with a ≠ 0.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A cubic function can have two turning points, or none at all.",
        "ja": "3 次関数のグラフは、山と谷を 1 つずつもつか、まったくもたないかのどちらか。",
        "register": "spoken"
      },
      {
        "en": "Sketch the graph of the cubic function f(x) = x³ − 3x² + 4.",
        "ja": "3 次関数 f(x) = x³ − 3x² + 4 のグラフの概形をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "local-extremum",
      "sketch-the-graph",
      "number-of-real-solutions"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cubic function": 3
      },
      "written": {
        "cubic function": 18
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cubic-units",
    "ja": {
      "term": "立方単位",
      "reading": "りっぽうたんい",
      "alt": []
    },
    "en": {
      "term": "cubic units",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「立方単位」は学習指導要領解説に無い、本プロジェクトの訳語。日本では cm³・m³ のように決まった単位で答える。英語の問題では単位を決めずに、1 辺 1 の立方体いくつ分かを cubic units で答えさせることがある。用例コーパスでは話・書とも cubic units のほうが units cubed より多い（書き言葉では OpenStax Calculus が中心）。",
    "domains": [
      "geometry",
      "arithmetic"
    ],
    "level": {
      "jp": [
        "小学校"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "1 辺の長さが 1 単位の立方体を 1 つと数えて体積を表すときの単位。長さの単位が決まっていない問題で、体積を「いくつ分」で答えるのに使う。",
    "definition_en": "The unit used to measure volume by counting cubes whose edges are 1 unit long. It is used to give a volume when no specific unit of length, such as centimeters, is stated.",
    "latex": "V = 2 \\cdot 3 \\cdot 4 = 24 \\text{ units}^3",
    "spoken_en": "V equals 2 times 3 times 4, which is 24 cubic units",
    "examples": [
      {
        "en": "Count the little cubes in each layer and multiply by the number of layers. There are 24 of them, so the volume is 24 cubic units.",
        "ja": "1 段に並んだ小さな立方体の数に段の数をかけよう。全部で 24 個だから、体積は 24 立方単位だ。",
        "register": "spoken"
      },
      {
        "en": "A rectangular prism has length 5, width 3, and height 2. Find its volume in cubic units.",
        "ja": "縦 5、横 3、高さ 2 の直方体がある。この直方体の体積を求めよ（単位は立方単位）。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の問題では体積を cm³ や m³ で答え、「立方単位」とは言わない。英語の問題文が in cubic units と指定していたら、答えにも cubic units を付ける。",
      "記号では units³ と書き、units cubed と読むこともある。面積の square units と取り違えない。",
      "用例コーパスで cubic units は話・書とも使い、話し言葉はほとんどが Khan Academy（多くは中学の講義）、書き言葉は OpenStax Calculus と OpenStax Prealgebra が中心。IM では 99 件（Geometry 5.7 The Root of the Problem 18 ほか）。"
    ],
    "related": [
      "square-units",
      "area"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cubic units": 12
      },
      "written": {
        "cubic units": 17
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cumulative-frequency",
    "ja": {
      "term": "累積度数",
      "reading": "るいせきどすう",
      "alt": []
    },
    "en": {
      "term": "cumulative frequency",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "最初の階級からその階級までの度数を足し合わせた値。",
    "definition_en": "The sum of the frequencies of a class and all the classes before it.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Add up the frequencies as you go down the table to get the cumulative frequency.",
        "ja": "表を下へ進みながら度数を足していくと累積度数になる。",
        "register": "spoken"
      },
      {
        "en": "Add a cumulative frequency column to the table.",
        "ja": "表に累積度数の欄を加えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "cumulative-relative-frequency",
      "frequency"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cumulative frequency": 5
      },
      "written": {
        "cumulative frequency": 9
      },
      "sources": [
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 cumulative frequency（本文 9 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cumulative-relative-frequency",
    "ja": {
      "term": "累積相対度数",
      "reading": "るいせきそうたいどすう",
      "alt": []
    },
    "en": {
      "term": "cumulative relative frequency",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Intro Statistics"
      ]
    },
    "definition_ja": "最初の階級からその階級までの相対度数を足し合わせた値。最後の階級で 1 になる。",
    "definition_en": "The sum of the relative frequencies of a class and all the classes before it; it reaches 1 at the last class.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The cumulative relative frequency hits 0.5 somewhere in this class, so that's where the median is.",
        "ja": "累積相対度数がこの階級のどこかで 0.5 になるので、中央値はそこにある。",
        "register": "spoken"
      },
      {
        "en": "Use the cumulative relative frequency column to find the percentage of students who slept fewer than 7 hours.",
        "ja": "累積相対度数の欄を使って、睡眠時間が 7 時間未満の生徒の割合を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "累積相対度数の表から、ある値以下のデータの割合（パーセンタイル）が読める。"
    ],
    "related": [
      "cumulative-frequency",
      "relative-frequency",
      "median"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cumulative relative frequency": 9
      },
      "written": {
        "cumulative relative frequency": 58
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "curl-vector",
    "ja": {
      "term": "回転（ベクトル解析）",
      "reading": "かいてん・べくとるかいせき",
      "alt": []
    },
    "en": {
      "term": "curl",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトル場 F の各点でのまわり具合を表すベクトル ∇ × F。",
    "definition_en": "The vector ∇ × F that measures how a vector field F circulates around each point.",
    "latex": "\\operatorname{curl}\\mathbf{F} = \\nabla \\times \\mathbf{F}",
    "spoken_en": "curl F equals del cross F",
    "examples": [
      {
        "en": "If the curl is zero everywhere, there's no swirling, and the field might be conservative.",
        "ja": "回転がどこでも 0 なら渦がなく、保存場かもしれない。",
        "register": "spoken"
      },
      {
        "en": "Find the curl of F(x, y, z) = ⟨yz, xz, xy⟩.",
        "ja": "F(x, y, z) = ⟨yz, xz, xy⟩ の回転を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "書くときは curl F とも ∇ × F とも書く（curl F は書き言葉によく出てくる）。"
    ],
    "related": [
      "divergence-vector",
      "vector-field",
      "stokes-theorem",
      "greens-theorem",
      "conservative-vector-field"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "curl of": 51
      },
      "written": {
        "curl of": 52
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "curvature",
    "ja": {
      "term": "曲率",
      "reading": "きょくりつ",
      "alt": []
    },
    "en": {
      "term": "curvature",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "曲線の曲がり具合を表す量 κ。単位接ベクトルが弧長あたりに向きを変える速さ。",
    "definition_en": "A measure κ of how sharply a curve bends: how fast the unit tangent vector turns per unit of arc length.",
    "latex": "\\kappa = \\left\\| \\frac{d\\mathbf{T}}{ds} \\right\\| = \\frac{\\|\\mathbf{r}'(t) \\times \\mathbf{r}''(t)\\|}{\\|\\mathbf{r}'(t)\\|^3}",
    "spoken_en": null,
    "examples": [
      {
        "en": "A circle of radius R has constant curvature 1 over R.",
        "ja": "半径 R の円の曲率は一定で 1/R。",
        "register": "spoken"
      },
      {
        "en": "Find the curvature of the parabola y = x² at the point (1, 1).",
        "ja": "放物線 y = x² の点 (1, 1) における曲率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "曲率の逆数は radius of curvature（曲率半径）。"
    ],
    "related": [
      "unit-tangent-vector",
      "unit-normal-vector",
      "space-curve",
      "arc-length-of-a-curve"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "曲率",
        "en": "Curvature"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "curvature": 17
      },
      "written": {
        "curvature": 87
      },
      "sources": [
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "curve-sketching",
    "ja": {
      "term": "曲線の概形",
      "reading": "きょくせんのがいけい",
      "alt": []
    },
    "en": {
      "term": "curve sketching",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "導関数の符号・凹凸・漸近線などを調べて、曲線のおおまかな形をかくこと。",
    "definition_en": "Drawing the general shape of a graph from its derivatives, concavity, and asymptotes.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Curve sketching puts it all together: intercepts, critical points, concavity, and asymptotes.",
        "ja": "曲線の概形をかくには、切片・臨界点・凹凸・漸近線をすべて使う。",
        "register": "spoken"
      },
      {
        "en": "Use calculus to sketch the curve y = x⁴ − 4x³.",
        "ja": "微分を用いて曲線 y = x⁴ − 4x³ の概形をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は topic 5.9 で、f・f′・f″ のグラフどうしの関係を読み取る（connecting a function, its first derivative, and its second derivative）。"
    ],
    "related": [
      "sketch-the-graph",
      "concavity",
      "increasing-and-decreasing"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.9（connecting a function, its first derivative, and its second derivative）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "curve sketching": 4
      },
      "written": {
        "curve sketching": 3
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは OpenStax の呼び方 curve sketching（本文 2 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cusp",
    "ja": {
      "term": "尖点",
      "reading": "せんてん",
      "alt": []
    },
    "en": {
      "term": "cusp",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線が鋭く尖っている点。左右から近づくときの接線の傾きが +∞ と −∞ に発散する（y = x^(2/3) の原点など）。",
    "definition_en": "A sharp point on a graph where the slopes from the left and from the right go to +∞ and −∞, as for y = x^(2/3) at the origin.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "y equals x to the two-thirds has a cusp at the origin, so it's not differentiable there.",
        "ja": "y = x^(2/3) は原点で尖点をもつので、そこでは微分可能でない。",
        "register": "spoken"
      },
      {
        "en": "The graph of f(x) = x^(2/3) has a cusp at (0, 0).",
        "ja": "f(x) = x^(2/3) のグラフは点 (0, 0) で尖っている。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "微分可能でない点のうち、左右の接線の傾きが有限で違うものを corner、±∞ に発散するものを cusp と呼び分けることがある。"
    ],
    "related": [
      "not-differentiable",
      "corner-nondifferentiable",
      "vertical-tangent"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "尖点",
        "en": "Cusp (singularity)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cusp": 4
      },
      "written": {
        "cusp": 19
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cyclic-quadrilateral",
    "ja": {
      "term": "円に内接する四角形",
      "reading": "えんにないせつするしかくけい",
      "alt": []
    },
    "en": {
      "term": "cyclic quadrilateral",
      "alt": [
        "inscribed quadrilateral"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3",
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "4 つの頂点がすべて 1 つの円周上にある四角形。向かい合う角の和は 180° で、逆に対角の和が 180° の四角形は円に内接する。",
    "definition_en": "A quadrilateral whose four vertices all lie on one circle. Its opposite angles are supplementary, and conversely any quadrilateral with supplementary opposite angles is cyclic.",
    "latex": "\\angle A + \\angle C = 180^\\circ",
    "spoken_en": "angle A plus angle C equals 180 degrees",
    "examples": [
      {
        "en": "In a cyclic quadrilateral, opposite angles add up to 180, so this one is 180 minus 75, which is 105.",
        "ja": "円に内接する四角形では対角の和が 180° なので、この角は 180° − 75° = 105°。",
        "register": "spoken"
      },
      {
        "en": "ABCD is a cyclic quadrilateral with ∠A = 2∠C. Find ∠A.",
        "ja": "四角形 ABCD は円に内接し、∠A = 2∠C である。∠A の大きさを求めよ。",
        "register": "written"
      },
      {
        "en": "Prove that ABCD is a cyclic quadrilateral by showing that ∠B + ∠D = 180°.",
        "ja": "∠B + ∠D = 180° を示して、四角形 ABCD が円に内接することを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "opposite angles of a cyclic quadrilateral",
        "ja": "円に内接する四角形の対角"
      }
    ],
    "pitfalls": [
      "主語の向きに注意。the quadrilateral is inscribed in the circle は「四角形が円に内接する」、the circle is circumscribed about the quadrilateral は「円が四角形に外接する」で、同じ図を逆から言っている。",
      "四角形が円に内接する条件は、対角の和が 180° であること、または 1 つの外角がそれと隣り合わない内角（内対角）に等しいこと。",
      "用例コーパスには cyclic quadrilateral・inscribed quadrilateral とも出てこない（コーパスに Geometry の教科書が無い）。IM Geometry は glossary の見出しで、7.4・7.5 に 10 件。CK-12 Geometry は 6.15 Inscribed Quadrilaterals in Circles で inscribed quadrilateral と呼び（9 件）、cyclic quadrilateral は 2 件。"
    ],
    "related": [
      "inscribed-angle",
      "angle",
      "vertex",
      "chord"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円に内接する四角形",
        "en": "Cyclic quadrilateral"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary・Geometry 7.4 Quadrilaterals in Circles・Geometry 7.5 Triangles in Circles"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.15 Inscribed Quadrilaterals in Circles（inscribed quadrilateral）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 cyclic quadrilateral（Geometry glossary・Geometry 7.4 Quadrilaterals in Circles・Geometry 7.5 Triangles in Circles・計 10 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "cycloid",
    "ja": {
      "term": "サイクロイド",
      "reading": "さいくろいど",
      "alt": []
    },
    "en": {
      "term": "cycloid",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "円が直線上を滑らずに転がるとき、円周上の定点が描く曲線。",
    "definition_en": "The curve traced by a point on the rim of a circle as the circle rolls along a line without slipping.",
    "latex": "x = a(\\theta - \\sin\\theta), \\quad y = a(1 - \\cos\\theta)",
    "spoken_en": "x equals a times theta minus sine theta, y equals a times one minus cosine theta",
    "examples": [
      {
        "en": "Picture a dot on a bicycle wheel — that path is a cycloid.",
        "ja": "自転車の車輪についた点を思い浮かべて。その道筋がサイクロイド。",
        "register": "spoken"
      },
      {
        "en": "Find the length of one arch of the cycloid x = θ − sin θ, y = 1 − cos θ.",
        "ja": "サイクロイド x = θ − sin θ、y = 1 − cos θ の 1 つのアーチの長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉ではほとんど出てこない。書き言葉は OpenStax Calculus（ほとんどが Volume 2 の媒介変数表示の節）と MIT の講義ノート。"
    ],
    "related": [
      "parametric-equations",
      "arc-length-of-a-curve"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "サイクロイド",
        "en": "Cycloid"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "cycloid": 3
      },
      "written": {
        "cycloid": 33
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cylinder",
    "ja": {
      "term": "円柱",
      "reading": "えんちゅう",
      "alt": []
    },
    "en": {
      "term": "cylinder",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "合同な 2 つの円を底面とし、曲がった側面でつながった立体。",
    "definition_en": "A solid with two congruent, parallel circular bases joined by a curved surface.",
    "latex": "V = \\pi r^2 h",
    "spoken_en": "V equals pi r squared h",
    "examples": [
      {
        "en": "A cylinder is kind of like a prism with a circle for a base, so it's still area of the base times height.",
        "ja": "円柱は底面が円の角柱みたいなものだから、やっぱり底面積 × 高さだよ。",
        "register": "spoken"
      },
      {
        "en": "The radius of a cylinder is 4 cm and its height is 9 cm. Find the volume in terms of π.",
        "ja": "底面の半径が 4 cm、高さが 9 cm の円柱の体積を求めよ。ただし、円周率は π とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a cylinder with a radius of 4 の radius は底面の円の半径のこと。",
      "円柱の側面は曲面なので、角柱の lateral face ではなく lateral surface と言う（エントリ lateral-face）。",
      "cylinder は日常語で、エンジンのシリンダーやガスボンベ（gas cylinder）のような筒形の物も指す。"
    ],
    "related": [
      "prism",
      "cone",
      "circle",
      "volume",
      "lateral-area"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円柱 (数学)",
        "en": "Cylinder"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cylinder": 241
      },
      "written": {
        "cylinder": 284
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cylindrical-coordinates",
    "ja": {
      "term": "円柱座標",
      "reading": "えんちゅうざひょう",
      "alt": [
        "円筒座標"
      ]
    },
    "en": {
      "term": "cylindrical coordinates",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "xy 平面の極座標 (r, θ) に高さ z を加えて、空間の点を (r, θ, z) で表す座標。",
    "definition_en": "Coordinates (r, θ, z) for space: polar coordinates in the xy-plane together with the height z.",
    "latex": "x = r\\cos\\theta, \\; y = r\\sin\\theta, \\; z = z",
    "spoken_en": null,
    "examples": [
      {
        "en": "Anything with circular symmetry around the z-axis is easier in cylindrical coordinates.",
        "ja": "z 軸のまわりに対称なものは、円柱座標のほうが楽。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the triple integral ∭ (x² + y²) dV over the cylinder x² + y² ≤ 4, 0 ≤ z ≤ 3 using cylindrical coordinates.",
        "ja": "円柱 x² + y² ≤ 4、0 ≤ z ≤ 3 上の三重積分 ∭ (x² + y²) dV を円柱座標を用いて計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "体積要素は dV = r dz dr dθ（r を忘れない）。"
    ],
    "related": [
      "spherical-coordinates",
      "polar-coordinates",
      "triple-integral",
      "coordinates-in-space"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円筒座標系",
        "en": "Cylindrical coordinate system"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "cylindrical coordinates": 37
      },
      "written": {
        "cylindrical coordinates": 122
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "cylindrical-shell",
    "ja": {
      "term": "円柱の殻",
      "reading": "えんちゅうのから",
      "alt": []
    },
    "en": {
      "term": "cylindrical shell",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「円柱の殻」は英語の cylindrical shell を訳した言い方。学習指導要領解説には、この薄い円筒の名前も、それを使う体積の求め方の名前も出てこない。OpenStax Calculus Volume 1 は shell method の説明で、厚さ Δx の薄い円筒の皮を cylindrical shell と呼ぶ。見出しの「円柱の殻」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "半径 x、高さ f(x)、厚さ dx の薄い円筒の皮。体積はおよそ 2πx f(x) dx。",
    "definition_en": "A thin hollow cylinder of radius x, height f(x), and thickness dx, with volume about 2πx f(x) dx.",
    "latex": "V = 2\\pi \\int_a^b x\\,f(x)\\,dx",
    "spoken_en": "V equals two pi times the integral from a to b of x times f of x dx",
    "examples": [
      {
        "en": "Unroll one cylindrical shell and it's basically a thin slab: two pi x by f of x by dx.",
        "ja": "円柱の殻を 1 枚広げると、ほぼ 2πx × f(x) × dx の薄い板になる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 1 は方法の名前を複数形で the method of cylindrical shells と書く（本文と節の名前）。"
    ],
    "related": [
      "shell-method",
      "revolve-around-the-y-axis"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "この薄い円筒の名前も、それを使う体積の求め方の名前も出てこない・「円柱の殻」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "cylindrical shell": 29
      },
      "written": {
        "cylindrical shell": 27
      },
      "sources": [
        "mit-18.01",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "data",
    "ja": {
      "term": "データ",
      "reading": "でーた",
      "alt": []
    },
    "en": {
      "term": "data",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "AP Statistics",
        "Integrated Math 1",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "調べたり測ったりして集めた数値や記録の集まり。",
    "definition_en": "A collection of numbers or observations gathered by measuring, counting, or asking.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's look at the data before we compute anything.",
        "ja": "計算する前にデータを見てみよう。",
        "register": "spoken"
      },
      {
        "en": "The data below show the number of hours 20 students slept last night.",
        "ja": "次のデータは、生徒 20 人の昨夜の睡眠時間である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "data set",
        "ja": "データの集まり"
      },
      {
        "en": "data value",
        "ja": "データの値"
      }
    ],
    "pitfalls": [
      "英語の data は本来 datum の複数形。用例コーパスの書き言葉では the data are のほうがやや多く（ほとんどが OpenStax Introductory Statistics）、話し言葉では the data is しか出てこない。",
      "1 つ 1 つの値は、話し言葉では data point、書き言葉では data value が多い。"
    ],
    "related": [
      "distribution",
      "frequency-table",
      "histogram"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "data": 1025
      },
      "written": {
        "data": 2329
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "de-moivres-theorem",
    "ja": {
      "term": "ド・モアブルの定理",
      "reading": "ど・もあぶるのていり",
      "alt": []
    },
    "en": {
      "term": "De Moivre's theorem",
      "alt": [
        "De Moivre's formula"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "整数 n について (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ が成り立つという定理。",
    "definition_en": "The theorem that (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for every integer n.",
    "latex": "(\\cos\\theta + i\\sin\\theta)^n = \\cos n\\theta + i\\sin n\\theta",
    "spoken_en": "cosine theta plus i sine theta, to the n, equals cosine n theta plus i sine n theta",
    "examples": [
      {
        "en": "By De Moivre's theorem, raising to the fifth power just multiplies the angle by five.",
        "ja": "ド・モアブルの定理より、5 乗すると角が 5 倍になるだけ。",
        "register": "spoken"
      },
      {
        "en": "Use De Moivre's theorem to compute (1 + i)⁸.",
        "ja": "ド・モアブルの定理を用いて (1 + i)⁸ を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉には出てこず、書き言葉はすべて OpenStax Algebra and Trigonometry。Wikipedia の記事名は De Moivre's formula。"
    ],
    "related": [
      "polar-form",
      "nth-roots-of-unity",
      "nth-root",
      "eulers-formula"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ド・モアブルの定理",
        "en": "De Moivre's formula"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "De Moivre's theorem": 11
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "de-morgans-laws",
    "ja": {
      "term": "ド・モルガンの法則",
      "reading": "どもるがんのほうそく",
      "alt": []
    },
    "en": {
      "term": "De Morgan's laws",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "和集合の補集合は補集合どうしの共通部分に、共通部分の補集合は補集合どうしの和集合に等しいという法則。命題では「p かつ q」の否定が「p でない または q でない」になる形で使う。",
    "definition_en": "The rules that the complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements; in logic, not (p and q) is equivalent to (not p) or (not q), and not (p or q) to (not p) and (not q).",
    "latex": "\\overline{A \\cup B} = \\overline{A} \\cap \\overline{B}, \\quad \\overline{A \\cap B} = \\overline{A} \\cup \\overline{B}",
    "spoken_en": "the complement of A union B is A complement intersect B complement, and the complement of A intersect B is A complement union B complement",
    "examples": [
      {
        "en": "By De Morgan's laws, the negation of \"x > 0 and y > 0\" is \"x ≤ 0 or y ≤ 0.\"",
        "ja": "ド・モルガンの法則より、「x > 0 かつ y > 0」の否定は「x ≦ 0 または y ≦ 0」だ。",
        "register": "spoken"
      },
      {
        "en": "Use De Morgan's laws to find the complement of A ∪ B.",
        "ja": "ド・モルガンの法則を用いて、A ∪ B の補集合を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Levin（定理 1.3.7）と英語版 Wikipedia は 2 つの式をまとめて De Morgan's laws と複数形で呼び、CK-12 Geometry 2.10 は 2 つをまとめて単数の De Morgan's Law と呼ぶ。片方を使うとき Levin は De Morgan's law と単数で書く。",
      "否定すると「かつ」と「または」が入れ替わる。not (p and q) を not p and not q としてしまうのが典型的な誤り。",
      "CK-12 Geometry（2.10 Negative Statements、6 件）は単数の De Morgan's Law と書き、Levin（1.3 Rules of Logic）と英語版 Wikipedia の記事名は複数形の De Morgan's laws。"
    ],
    "related": [
      "complement",
      "union",
      "negation",
      "set",
      "proposition",
      "intersection"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ド・モルガンの法則",
        "en": "De Morgan's laws"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.3 Rules of Logic"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.10 Negative Statements"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 De Morgan's laws（CK-12 Geometry 2.10 Negative Statements・計 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "decimal-system",
    "ja": {
      "term": "10 進法",
      "reading": "じっしんほう",
      "alt": [
        "十進法"
      ]
    },
    "en": {
      "term": "base ten",
      "alt": [
        "decimal system",
        "decimal number system"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory",
      "arithmetic"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Pre-Algebra",
        "Discrete Math"
      ]
    },
    "definition_ja": "0 から 9 までの 10 個の数字を使い、位が 1 つ上がるごとに位の大きさが 10 倍になる数の表し方。",
    "definition_en": "The way of writing numbers with the ten digits 0 through 9, where each place is worth ten times the place to its right.",
    "latex": "307 = 3 \\times 10^2 + 0 \\times 10^1 + 7 \\times 10^0",
    "spoken_en": "three hundred seven equals three times ten squared plus zero times ten to the first plus seven times ten to the zero",
    "examples": [
      {
        "en": "In base ten, once the ones place goes past 9, you carry one over to the tens place.",
        "ja": "10 進法では、一の位が 9 を超えると十の位に 1 繰り上がる。",
        "register": "spoken"
      },
      {
        "en": "Write the binary number 1011₂ in base ten.",
        "ja": "2 進法で表された数 1011₍₂₎ を 10 進法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「10 進法で表せ」は in base ten と言う。base ten の形は、用例コーパスの書き言葉ではほとんどが OpenStax Prealgebra。IM では base ten の形が 43 件。decimal number system は IM に 1 件、decimal system は Nicholson に 1 件だけ。",
      "英語の decimal は単独では「小数」を指すことが多い（IM Grade 6 の Adding and Subtracting Decimals など）。decimals を 10 進法と読まない。"
    ],
    "related": [
      "base-n",
      "binary",
      "numeral-system",
      "place",
      "hexadecimal",
      "digit"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "1.1 Introduction to Whole Numbers（base-10 blocks）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 Unit 5（Adding and Subtracting Decimals）ほか: base ten"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "decimal system（1 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "base-ten !logarithmic | base ten system | base ten number | base ten numeral | base ten place | in base ten": 10,
        "decimal system": 4,
        "decimal number system": 5
      },
      "written": {
        "base-ten !logarithmic | base ten system | base ten number | base ten numeral | base ten place | in base ten": 19
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "decimal",
    "ja": {
      "term": "小数",
      "reading": "しょうすう",
      "alt": []
    },
    "en": {
      "term": "decimal",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "小学校",
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "小数点を使い、1 より小さい部分を 10 分の 1 の位、100 分の 1 の位…の数字で表した数。0.25 や 3.7 など。",
    "definition_en": "A number written with a decimal point, where the digits to the right of the point stand for tenths, hundredths, and so on, like 0.25 or 3.7.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To turn 3/8 into a decimal, just divide 3 by 8, and you get 0.375.",
        "ja": "3/8 を小数に直すには、3 を 8 で割ればいい。0.375 になる。",
        "register": "spoken"
      },
      {
        "en": "Write 7/20 as a decimal.",
        "ja": "7/20 を小数で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "小数点は decimal point。0.25 は zero point two five とも point two five とも読む。",
      "循環小数は repeating decimal（エントリ repeating-decimal）。",
      "decimal は「小数」のほかに「十進法の」という形容詞にもなる（decimal system、エントリ decimal-system）。"
    ],
    "related": [
      "fraction",
      "repeating-decimal",
      "place",
      "round",
      "decimal-system"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "小数",
        "en": "Decimal representation"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "decimal": 934
      },
      "written": {
        "decimal": 1431
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "decompose-into-partial-fractions",
    "ja": {
      "term": "部分分数に分解する",
      "reading": "ぶぶんぶんすうにぶんかいする",
      "alt": []
    },
    "en": {
      "term": "decompose into partial fractions",
      "alt": [
        "split into partial fractions"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "calculus"
    ],
    "level": {
      "jp": [
        "数B",
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "分数式を部分分数の和に書き直すこと。",
    "definition_en": "To rewrite a rational expression as a sum of partial fractions.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Factor the bottom first, then break it up into partial fractions.",
        "ja": "まず分母を因数分解して、それから部分分数に分解する。",
        "register": "spoken"
      },
      {
        "en": "Decompose 3/(x² − x − 2) into partial fractions.",
        "ja": "3/(x² − x − 2) を部分分数に分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "授業では do partial fractions（部分分数分解をする）と名詞で言うこともある。"
    ],
    "related": [
      "partial-fraction-decomposition"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "decompose into partial fractions": 2,
        "split into partial fractions": 2
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは OpenStax の呼び方 decompose into partial fractions（本文 2 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "decomposition-of-a-vector",
    "ja": {
      "term": "ベクトルの分解",
      "reading": "べくとるのぶんかい",
      "alt": [
        "ベクトルを分解する"
      ]
    },
    "en": {
      "term": "decomposition of a vector",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスには decomposition of a vector は出てこず、resolve … into components もほとんど出てこない）。平行でない 2 つのベクトルで表すことは write as a linear combination と言い、成分に分けることは break ／ resolve into components と説明する。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトルを、平行でない 2 つのベクトルの実数倍の和 sa + tb の形に表すこと。表し方はただ 1 通り。",
    "definition_en": "Writing a vector as a sum sa + tb of multiples of two nonparallel vectors; the expression is unique.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Break the force into a horizontal part and a vertical part.",
        "ja": "力を水平方向と垂直方向に分解しよう。",
        "register": "spoken"
      },
      {
        "en": "Express p as a linear combination of a and b.",
        "ja": "p を a と b の一次結合で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "linear-combination",
      "linearly-independent",
      "components"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "decrease",
    "ja": {
      "term": "減少する",
      "reading": "げんしょうする",
      "alt": []
    },
    "en": {
      "term": "decrease",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数量が小さくなること。関数で、x の値が増えるにつれて y の値が小さくなるとき、「y は減少する」という。",
    "definition_en": "To become smaller in value. In a function, y decreases when its values get smaller as x gets larger.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Look at y = 12/x: when x goes from 2 to 3, y decreases from 6 to 4.",
        "ja": "y = 12/x を見て。x が 2 から 3 になると、y は 6 から 4 に減少する。",
        "register": "spoken"
      },
      {
        "en": "For y = −2x + 7, how much does y decrease when x increases from 1 to 4?",
        "ja": "y = −2x + 7 で、x が 1 から 4 まで増加するとき、y はいくつ減少するか。",
        "register": "written"
      },
      {
        "en": "The temperature decreased by 6 degrees to −2 degrees. What was the temperature at first?",
        "ja": "気温が 6 度下がって −2 度になった。はじめの気温は何度だったか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "decrease by 6 は「6 減る」（減った分）、decrease to 6 は「6 になる」（減った後の値）。",
      "「関数がある範囲で減少する」（グラフが右下がり）は is decreasing の形で言う（エントリ decreasing）。",
      "似た語 reduce は、数学では「分数を約分する」（reduce the fraction）の意味でも使う。値が小さくなる意味の「減少する」は decrease。"
    ],
    "related": [
      "increase",
      "decreasing",
      "increasing-and-decreasing",
      "inverse-proportion",
      "rate-of-change"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "decrease": 739
      },
      "written": {
        "decrease": 633
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "decreasing",
    "ja": {
      "term": "単調に減少する",
      "reading": "たんちょうにげんしょうする",
      "alt": [
        "単調減少",
        "右下がり"
      ]
    },
    "en": {
      "term": "is decreasing",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "見出しの「単調に減少する」は学習指導要領解説に無い（日本語版 Wikipedia には「単調減少」がある）。「右下がり」（1 次関数のグラフ）にも当たるが、英語ではグラフの形ではなく関数の性質（is decreasing）か傾きの符号で言う。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2",
        "数II",
        "数III"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Calculus I",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "x が増えると、関数の値が必ず減る。1 次関数ではグラフが右下がりになること（傾きが負）。",
    "definition_en": "Of a function: its value goes down whenever x goes up; for a linear function, the graph falls from left to right (negative slope).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Since the base is between 0 and 1, the function is decreasing.",
        "ja": "底が 0 と 1 の間なので、この関数は減少する。",
        "register": "spoken"
      },
      {
        "en": "Show that f(x) = (1/2)^x is decreasing on the whole real line.",
        "ja": "f(x) = (1/2)^x が実数全体で減少することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "is decreasing on",
        "ja": "〜で減少する"
      }
    ],
    "pitfalls": [
      "「右下がり」は英語では falls from left to right、または the slope is negative と言う。",
      "教室では monotonically を付けずに is decreasing と言うのがふつう（is-monotonically-increasing と同じ）。"
    ],
    "related": [
      "is-monotonically-increasing",
      "increasing-and-decreasing",
      "exponential-function"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「一次関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%80%E6%AC%A1%E9%96%A2%E6%95%B0",
        "note": "傾きが負ならグラフは右下がり"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "「単調に減少する」は出てこない"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「単調に減少する」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "is decreasing": 147
      },
      "written": {
        "is decreasing": 109
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "deductive-reasoning",
    "ja": {
      "term": "演繹的推論",
      "reading": "えんえきてきすいろん",
      "alt": [
        "演繹"
      ]
    },
    "en": {
      "term": "deductive reasoning",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof",
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "正しいと認めた事柄（定義・公理・定理・仮定）から、論理の規則に従って結論を導く考え方。前提が正しければ、結論も必ず正しい。",
    "definition_en": "Reasoning that starts from accepted facts such as definitions, postulates, theorems, and given information, and reaches a conclusion by rules of logic. If the starting facts are true, the conclusion must be true.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a proof, every step has to come from deductive reasoning. \"It looks that way in the picture\" isn't a reason.",
        "ja": "証明では、どのステップも演繹的推論で導かないといけない。「図でそう見える」は理由にならない。",
        "register": "spoken"
      },
      {
        "en": "Use deductive reasoning to show that the sum of any two odd integers is even.",
        "ja": "演繹的推論を用いて、2 つの奇数の和は偶数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスで deductive reasoning は話し言葉にほとんど出てこず（Khan Academy だけ）、書き言葉には出てこない。IM にも出てこない。CK-12 Geometry では 2.4 Reasoning Types・2.7 Deductive Reasoning に 17 件。",
      "数学的帰納法（proof by induction）は、名前に反して演繹的推論による証明である。",
      "law of detachment と law of syllogism は、演繹的推論の代表的な形に付いた名前。"
    ],
    "related": [
      "inductive-reasoning",
      "law-of-detachment",
      "law-of-syllogism",
      "proof"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.7 Deductive Reasoning・2.4 Reasoning Types"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "deductive reasoning": 2
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 deductive reasoning（CK-12 Geometry 2.7 Deductive Reasoning・CK-12 Geometry 2.4 Reasoning Types・計 17 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "definite-integral",
    "ja": {
      "term": "定積分",
      "reading": "ていせきぶん",
      "alt": []
    },
    "en": {
      "term": "definite integral",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数 f(x) を区間 a ≦ x ≦ b で積分した値。原始関数 F を使って F(b) − F(a) で求まる。",
    "definition_en": "The number you get by integrating f from x = a to x = b; it equals F(b) − F(a) for any antiderivative F.",
    "latex": "\\int_a^b f(x)\\,dx = F(b) - F(a)",
    "spoken_en": "the integral from a to b of f of x dx equals F of b minus F of a",
    "examples": [
      {
        "en": "Remember, a definite integral is just a number, so there's no plus C at the end.",
        "ja": "定積分はただの数だから、最後に +C は付かないよ。",
        "register": "spoken"
      },
      {
        "en": "The definite integral ∫₀² x dx equals 2.",
        "ja": "定積分 ∫₀² x dx の値は 2 である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "set up the definite integral",
        "ja": "定積分の式を立てる"
      }
    ],
    "pitfalls": [
      "学習指導要領解説（数学II）は、面積 S(t) の導関数が f(t) になることから定積分が面積を表すことを導く扱いと、区分求積法の考えで定義する扱いに触れる。OpenStax Calculus は Riemann sum の極限で定義し、F(b) − F(a) で計算できることを微積分学の基本定理として示す。",
      "OpenStax Calculus は [F(x)]ₐᵇ のほかに、F(x) の右に縦線を引いて上下に b と a を書く形も使う。どちらも F of x evaluated from a to b と読む。"
    ],
    "related": [
      "indefinite-integral",
      "integrand",
      "interval-of-integration",
      "fundamental-theorem-of-calculus",
      "riemann-sum",
      "evaluate-the-integral",
      "integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 微分・積分の考え: 面積 S(t) と定積分、区分求積法の考え"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.3・6.7（definite integral）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "definite integral": 476
      },
      "written": {
        "definite integral": 216
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "definition",
    "ja": {
      "term": "定義",
      "reading": "ていぎ",
      "alt": []
    },
    "en": {
      "term": "definition",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "用語の意味を、ほかの意味にとれないようにはっきり決めた文。定義は約束ごとなので、証明はしない。",
    "definition_en": "A statement that tells exactly what a word or symbol means. A definition is agreed on, not proved.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "You don't prove a definition. By definition, an isosceles triangle has two equal sides, so for that step your reason is just \"by definition.\"",
        "ja": "定義は証明しない。定義より、二等辺三角形は 2 辺が等しいから、そこの理由は「定義より」と書けばいい。",
        "register": "spoken"
      },
      {
        "en": "State the definition of an isosceles triangle.",
        "ja": "二等辺三角形の定義を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「定義より」は by definition（冠詞を付けない）。何の定義かを言うときは by the definition of a midpoint のように the を付ける。",
      "定義は証明するものではない。定義・公準をもとに証明して正しいとわかった事柄が定理（エントリ theorem）。",
      "二等辺三角形の定義は「2 辺が等しい三角形」で、「2 つの底角が等しい」は定義から証明する定理（エントリ isosceles-triangle-theorem）。証明の理由を書くとき、定義と定理を取り違えない。"
    ],
    "related": [
      "theorem",
      "postulate",
      "undefined-terms",
      "proof",
      "isosceles-triangle-theorem"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "definition": 936
      },
      "written": {
        "definition": 1128
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "degree-measure",
    "ja": {
      "term": "度数法",
      "reading": "どすうほう",
      "alt": []
    },
    "en": {
      "term": "degree measure",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3",
        "Geometry"
      ]
    },
    "definition_ja": "1 回転を 360° として角の大きさを表す方法。",
    "definition_en": "Measuring angles with one full turn as 360°.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To go from radians to degrees, multiply by 180 over pi.",
        "ja": "ラジアンから度に直すには 180/π を掛ける。",
        "register": "spoken"
      },
      {
        "en": "Express 5π/6 in degree measure.",
        "ja": "5π/6 を度数法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の degree measure は、角の大きさを度で表した値にも使う（IM Grade 7 の the degree measure of an angle、OpenStax Calculus Volume 1（1.3）の a degree measure of 180°）。「度で」は in degrees と言う。"
    ],
    "related": [
      "radian-measure",
      "radian",
      "angle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.3 Trigonometric Functions（a degree measure of 180°）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Grade 7 7.1 Relationships of Angles・Grade 7 7.2 Adjacent Angles・Grade 7 7.7 Building Polygons (Part 2)"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.13 Using Radians（find the degree measure of each angle）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "degree measure": 6
      },
      "written": {
        "degree measure": 3
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 6 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは IM の呼び方 degree measure（Grade 7 7.1 Relationships of Angles・Grade 7 7.2 Adjacent Angles・Grade 7 7.7 Building Polygons (Part 2)・計 8 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "degree-vertex",
    "ja": {
      "term": "頂点の次数",
      "reading": "ちょうてんのじすう",
      "alt": [
        "次数"
      ]
    },
    "en": {
      "term": "degree of a vertex",
      "alt": [
        "vertex degree"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（degree of a vertex ／ degree of the vertex ／ vertex degree は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "数C",
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフの頂点に接している辺の本数のこと。ループ（同じ頂点に戻る辺）は 2 本と数える。",
    "definition_en": "The number of edges that meet at a vertex of a graph. A loop at the vertex is counted twice.",
    "latex": "\\deg(v)",
    "spoken_en": "the degree of v",
    "examples": [
      {
        "en": "To find the degree of a vertex, just count how many edges touch it, and remember a loop counts twice.",
        "ja": "頂点の次数は、その頂点に接している辺を数えればいい。ループは 2 と数えるのを忘れずに。",
        "register": "spoken"
      },
      {
        "en": "Find the degree of each vertex in the graph below, and check that the sum of the degrees is twice the number of edges.",
        "ja": "下のグラフの各頂点の次数を求め、次数の和が辺の数の 2 倍になることを確かめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「次数」は多項式の次数（degree of a polynomial、エントリ degree）と同じ degree。グラフでは degree of a vertex、deg(v) と言って区別する。",
      "次数が奇数の頂点を odd vertex（奇頂点）、偶数の頂点を even vertex（偶頂点）と呼ぶことがある。一筆書き（Euler trail）の判定に使う。"
    ],
    "related": [
      "degree",
      "vertex-graph",
      "handshaking-lemma",
      "eulerian-path",
      "graph-network",
      "edge-of-a-graph"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.1 Problems and Definitions"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "degree of a vertex | degree of the vertex | degree of each vertex | degree of vertex": 2
      },
      "written": {
        "degree of a vertex | degree of the vertex | degree of each vertex | degree of vertex": 2
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは Levin の呼び方 degree of a vertex | degree of the vertex | degree of each vertex | degree of vertex（2.1 Problems and Definitions・1.5 Proofs about Discrete Structures）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "degree",
    "ja": {
      "term": "次数",
      "reading": "じすう",
      "alt": []
    },
    "en": {
      "term": "degree",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "単項式では、かけ合わされている文字の個数（3x²y の次数は 3）。多項式では、各項の次数のうちで最も大きいもの。",
    "definition_en": "For a monomial, the number of variable factors multiplied together, which is the sum of the exponents; the degree of 3x²y is 3. For a polynomial, it is the greatest degree among its terms.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "x squared y has degree 3: add the exponents, 2 for the x and 1 for the y.",
        "ja": "x²y の次数は 3。指数を足すんだ。x が 2 で y が 1 だからね。",
        "register": "spoken"
      },
      {
        "en": "State the degree of the polynomial 4x³ − 2x²y² + 7.",
        "ja": "多項式 4x³ − 2x²y² + 7 の次数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "角の大きさの度（エントリ degree-measure）や、グラフの頂点の次数（エントリ degree-vertex）も同じ degree。「3 次の多項式」は a polynomial of degree 3、「3 度」は 3 degrees。",
      "多項式の次数は、いちばん大きい指数ではなく、項ごとの指数の和のうち最大のもの。4x³ − 2x²y² の次数は 3 ではなく 4。",
      "0 でない定数項（7 など）の次数は 0。特定の文字だけに注目した次数は、degree in x のように in を付けて言う。"
    ],
    "related": [
      "monomial",
      "polynomial",
      "quadratic-expression",
      "degree-measure",
      "degree-vertex"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "degree of the polynomial | degree of a polynomial | degree of the term | degree of a term | degree of the monomial | degree of each term | highest degree | degree two polynomial | degree three polynomial | degree n polynomial": 52
      },
      "written": {
        "degree of the polynomial | degree of a polynomial | degree of the term | degree of a term | degree of the monomial | degree of each term | highest degree | degree two polynomial | degree three polynomial | degree n polynomial": 65
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "degrees-of-freedom",
    "ja": {
      "term": "自由度",
      "reading": "じゆうど",
      "alt": []
    },
    "en": {
      "term": "degrees of freedom",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "t 分布やカイ二乗分布の形を決める数。1 標本の t 分布では標本の大きさから 1 を引いた n − 1。",
    "definition_en": "The number that fixes the shape of a t- or chi-square distribution; for a one-sample t procedure it is the sample size minus one, n − 1.",
    "latex": "df = n - 1",
    "spoken_en": null,
    "examples": [
      {
        "en": "Our sample size is 12, so we have 11 degrees of freedom.",
        "ja": "標本の大きさが 12 なので、自由度は 11。",
        "register": "spoken"
      },
      {
        "en": "State the degrees of freedom of the chi-square test statistic.",
        "ja": "カイ二乗検定の統計量の自由度を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "略記は df（degrees of freedom）。微分の df（d f）と同じ綴りになる。",
      "台帳の langlink（自由度 → Degrees of freedom (physics and chemistry)）は物理の自由度で、出典にしなかった。",
      "用例コーパスの話し言葉には、統計でない「動かせるパラメータの数」の意味も混ざる（3Blue1Brown・MIT 18.01）。書き言葉はほとんどが OpenStax Introductory Statistics。"
    ],
    "related": [
      "t-distribution",
      "chi-square-test"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.14・4.2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "degrees of freedom": 41
      },
      "written": {
        "degrees of freedom": 105
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.01",
        "mit-notes",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "denominator",
    "ja": {
      "term": "分母",
      "reading": "ぶんぼ",
      "alt": []
    },
    "en": {
      "term": "denominator",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "小学校",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "分数の横線の下にある数・式。0 になってはいけない。",
    "definition_en": "The number or expression below the fraction bar; it cannot be zero.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The denominator can't be zero, so x can't be 2.",
        "ja": "分母は 0 になれないので、x は 2 になれない。",
        "register": "spoken"
      },
      {
        "en": "Rationalize the denominator of 1/(√3 − 1).",
        "ja": "1/(√3 − 1) の分母を有理化せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "common denominator",
        "ja": "共通の分母"
      }
    ],
    "pitfalls": [
      "「分母を払う」を、OpenStax Elementary Algebra（Solve Rational Equations ほか）は clear the fractions と書く（エントリ clear-the-denominators）。講義では clear the denominators とも言う。"
    ],
    "related": [
      "numerator",
      "find-a-common-denominator",
      "rational-expression",
      "clear-the-denominators",
      "rationalizing-the-denominator",
      "fraction"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Solve Rational Equations（clear the fractions）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "denominator": 2340
      },
      "written": {
        "denominator": 1544
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "density-curve",
    "ja": {
      "term": "密度曲線",
      "reading": "みつどきょくせん",
      "alt": []
    },
    "en": {
      "term": "density curve",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "分布の形を表す、x 軸より上にあって下の面積が 1 の曲線。面積が相対度数（確率）を表す。",
    "definition_en": "A curve on or above the x-axis with total area 1 under it that describes the shape of a distribution; areas under it give proportions (probabilities).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The area under the density curve between two values is the proportion of the data in that range.",
        "ja": "密度曲線の下の、2 つの値の間の面積が、その範囲のデータの割合。",
        "register": "spoken"
      },
      {
        "en": "Sketch a density curve that is skewed left and mark the mean and the median.",
        "ja": "左に歪んだ密度曲線をかき、平均値と中央値の位置を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて Khan Academy。"
    ],
    "related": [
      "probability-density-function",
      "normal-distribution",
      "histogram"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "density curve": 48
      },
      "written": {
        "density curve": 4
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "density",
    "ja": {
      "term": "密度",
      "reading": "みつど",
      "alt": []
    },
    "en": {
      "term": "density",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "小学校"
      ],
      "us": [
        "Geometry",
        "Calculus III",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "単位体積あたりの質量や単位面積あたりの人数のように、ある量が単位の大きさあたりどれだけあるかを表す量。質量の密度は、質量を体積で割って求める。",
    "definition_en": "The amount of something per unit of volume or area, such as grams per cubic centimeter or people per square mile. For mass, density equals mass divided by volume.",
    "latex": "\\text{density} = \\frac{\\text{mass}}{\\text{volume}}",
    "spoken_en": "density equals mass over volume",
    "examples": [
      {
        "en": "Density is just mass per unit volume, so divide the mass by the volume and keep the units, like grams per cubic centimeter.",
        "ja": "密度は単位体積あたりの質量だから、質量を体積で割ろう。単位も g/cm³ のように付けておくこと。",
        "register": "spoken"
      },
      {
        "en": "A town has 12,000 people living in an area of 15 square miles. Find the population density of the town.",
        "ja": "面積 15 平方マイルの町に 12,000 人が住んでいる。この町の人口密度を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "単位は per で読む。g/cm³ は grams per cubic centimeter、人口密度は people per square mile のように言う。",
      "統計の確率密度（probability density）や密度曲線（density curve）は別の意味で、別エントリ。",
      "IM Geometry は 5.17 Volume and Density で 15 件使い、glossary にも入れる。"
    ],
    "related": [
      "density-curve",
      "probability-density-function"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!probability !exponential !kernel density !function !functions !curve !curves !plot !plots": 83
      },
      "written": {
        "!probability !exponential !kernel density !function !functions !curve !curves !plot !plots": 136
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivative-at-a-point",
    "ja": {
      "term": "微分係数",
      "reading": "びぶんけいすう",
      "alt": []
    },
    "en": {
      "term": "derivative at a point",
      "alt": [
        "the value of the derivative at x = a"
      ],
      "uk": null,
      "variants": [
        {
          "term": "f prime of a",
          "register": "spoken",
          "note": "記号 f′(a) の読み。話し言葉で最も多いが、ほとんどが Khan Academy。書き言葉には出てこない。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax）には「微分係数」に当たる独立した名詞が出てこず、the derivative at a point（CED topic 2.3 の言い方）／ the derivative at x = a（OpenStax Calculus・Precalculus の言い方。x = a における導関数の値）と言う。記号 f′(a) は f prime of a と読む。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数 f の x=a における変化率。グラフの点 (a, f(a)) での接線の傾きに等しい。",
    "definition_en": "How fast f is changing at x = a; it equals the slope of the line that just touches the graph at (a, f(a)).",
    "latex": "f'(a) = \\lim_{h \\to 0} \\frac{f(a+h) - f(a)}{h}",
    "spoken_en": "f prime of a equals the limit as h goes to zero of f of a plus h minus f of a, all over h",
    "examples": [
      {
        "en": "f prime of 2 tells you how steep the tangent line is at x = 2.",
        "ja": "f′(2) は x = 2 での接線の傾きを表す。",
        "register": "spoken"
      },
      {
        "en": "The derivative at a point is one number, the slope of the tangent line there; the derivative as a function hands you that number for every x.",
        "ja": "微分係数は 1 つの数で、その点での接線の傾き。導関数は、その数をどの x についても与える関数。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "evaluate the derivative at x = 2",
        "ja": "x=2 における微分係数を求める"
      },
      {
        "en": "the slope of the tangent line at x = 2",
        "ja": "x=2 での接線の傾き"
      }
    ],
    "pitfalls": [
      "differential coefficient は用例コーパス（米国の授業の書き起こしと OpenStax）にも CED にも出てこない。"
    ],
    "related": [
      "derivative",
      "instantaneous-rate-of-change",
      "slope-of-the-tangent-line",
      "sign-chart"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 2.3（derivative at a point）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "the derivative at x = …"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "the derivative at x = …"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "f prime of a": 19,
        "derivative at a point": 3
      },
      "written": {
        "evaluate the derivative at x = 2": 1,
        "derivative at a point": 6
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivative-of-a-parametric-curve",
    "ja": {
      "term": "媒介変数表示された関数の微分",
      "reading": "ばいかいへんすうひょうじされたかんすうのびぶん"
    },
    "en": {
      "term": "derivatives of parametric functions",
      "alt": [
        "derivatives of parametric equations",
        "derivative of a parametric curve"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「媒介変数表示された関数の微分」は学習指導要領解説に無い、本プロジェクトの訳語（解説の数学III は、座標が x = f(t), y = g(t) で表される点の速度を (f′(t), g′(t)) とし、媒介変数表示された曲線の長さも扱う）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "媒介変数表示された曲線で、dy/dx を (dy/dt)/(dx/dt) として求めること。",
    "definition_en": "Finding dy/dx for a curve given by parametric equations, as (dy/dt)/(dx/dt).",
    "latex": "\\frac{dy}{dx} = \\frac{dy/dt}{dx/dt}",
    "spoken_en": "d y d x equals d y d t over d x d t",
    "examples": [
      {
        "en": "For a parametric curve, dy/dx is dy/dt divided by dx/dt.",
        "ja": "媒介変数表示された曲線では、dy/dx は dy/dt を dx/dt で割ったもの。",
        "register": "spoken"
      },
      {
        "en": "Find dy/dx for x = t², y = t³ − t at t = 1.",
        "ja": "x = t², y = t³ − t のとき、t = 1 における dy/dx を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "parametric-equations",
      "parameter",
      "chain-rule",
      "derivatives-in-polar-form"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 9.1・9.2（learning objective: Calculate derivatives of parametric functions。9.2 の題 Second Derivatives of Parametric Equations は二階微分）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「媒介変数表示された関数の微分」は出てこない。数学III (2) 微分法: 平面上を運動する点の座標 x = f(t), y = g(t) と速度 (f′(t), g′(t))。(3) 積分法: 媒介変数表示された曲線の長さ"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "derivatives of parametric functions": 1
      },
      "written": {
        "!second derivatives of parametric equations": 4,
        "derivative of a parametric curve": 1
      },
      "sources": [
        "openstax-calculus",
        "yt:patrickjmt"
      ],
      "counted": "2026-10-09"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは CED の呼び方 derivatives of parametric functions（topic 9.1・topic 9.2）。register は主張しない。",
        "raised": "2026-10-09"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivative-of-a-sum",
    "ja": {
      "term": "和の微分",
      "reading": "わのびぶん",
      "alt": []
    },
    "en": {
      "term": "sum rule",
      "alt": [
        "derivative of a sum"
      ],
      "variants": [
        {
          "term": "sum and difference rule",
          "register": "both",
          "note": "差も含めた呼び方で、件数は少ない。話し言葉はすべて NancyPi、書き言葉は大半が OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "和の導関数は導関数の和になる、{f(x) + g(x)}′ = f′(x) + g′(x) という性質。",
    "definition_en": "The rule (f + g)′ = f′ + g′: to differentiate a sum, differentiate each term and add the results.",
    "latex": "(f + g)' = f' + g'",
    "spoken_en": "the derivative of f plus g is f prime plus g prime",
    "examples": [
      {
        "en": "By the sum rule, just take the derivative of each term.",
        "ja": "和の微分の性質より、項ごとに微分すればよい。",
        "register": "spoken"
      },
      {
        "en": "By the sum rule and the power rule, the derivative of x³ + x² is 3x² + 2x.",
        "ja": "和の微分と xⁿ の微分の公式より、x³ + x² の導関数は 3x² + 2x である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "場合の数の sum rule（和の法則）と同じ名前。"
    ],
    "related": [
      "power-rule",
      "differentiation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "sum rule": 24,
        "derivative of a sum": 5,
        "sum and difference rule": 4
      },
      "written": {
        "sum rule": 29,
        "sum and difference rule": 4
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivative-of-a-vector-function",
    "ja": {
      "term": "ベクトル値関数の微分",
      "reading": "べくとるちかんすうのびぶん",
      "alt": []
    },
    "en": {
      "term": "derivative of a vector-valued function",
      "alt": [],
      "variants": [
        {
          "term": "derivative of a vector function",
          "register": "spoken",
          "note": "Professor Leonard の言い方（話し言葉のほとんどが Professor Leonard）。話し言葉では derivative of a vector-valued function より多く、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトル値関数 r(t) を成分ごとに微分したもの r′(t)。曲線の接ベクトルになる。",
    "definition_en": "The vector r′(t) obtained by differentiating a vector-valued function r(t) component by component; it is tangent to the curve.",
    "latex": "\\mathbf{r}'(t) = \\langle x'(t), y'(t), z'(t) \\rangle",
    "spoken_en": "r prime of t equals the vector x prime of t, y prime of t, z prime of t",
    "examples": [
      {
        "en": "To differentiate a vector function, just differentiate each component.",
        "ja": "ベクトル値関数の微分は、成分ごとに微分するだけ。",
        "register": "spoken"
      },
      {
        "en": "Find the derivative of the vector-valued function r(t) = ⟨t², e^t, sin 2t⟩.",
        "ja": "ベクトル値関数 r(t) = ⟨t², e^t, sin 2t⟩ の導関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector-valued-function",
      "velocity-vector",
      "unit-tangent-vector",
      "derivative-of-a-parametric-curve"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "derivative of a vector-valued function": 1,
        "derivative of a vector function": 11
      },
      "written": {
        "derivative of a vector-valued function": 18
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivative-of-an-inverse-function",
    "ja": {
      "term": "逆関数の微分",
      "reading": "ぎゃくかんすうのびぶん",
      "alt": []
    },
    "en": {
      "term": "derivative of the inverse",
      "alt": [],
      "variants": [
        {
          "term": "derivative of an inverse function",
          "register": "written",
          "note": "AP の CED（topic 3.3）と OpenStax Calculus の書き方。件数は少ない。話し言葉にはほとんど出てこず、書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "inverse function theorem",
          "register": "written",
          "note": "OpenStax Calculus Volume 1（Derivatives of Inverse Functions）が、この公式を述べる定理の名前として書く。書き言葉はすべて OpenStax Calculus で、話し言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "逆関数の導関数。dx/dy = 1/(dy/dx) の関係から求める。",
    "definition_en": "The derivative of an inverse function, found from the relation dx/dy = 1/(dy/dx).",
    "latex": "(f^{-1})'(a) = \\frac{1}{f'(f^{-1}(a))}",
    "spoken_en": "the derivative of f inverse at a is one over f prime of f inverse of a",
    "examples": [
      {
        "en": "The derivative of the inverse at 3 is one over f prime at the point where f equals 3.",
        "ja": "逆関数の x = 3 での微分係数は、f の値が 3 になる点での f′ の逆数。",
        "register": "spoken"
      },
      {
        "en": "Find (f⁻¹)′(5) if f(x) = x³ + x + 3.",
        "ja": "f(x) = x³ + x + 3 のとき、(f⁻¹)′(5) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 1 は逆関数の導関数の公式 (f⁻¹)′(x) = 1 / f′(f⁻¹(x)) を Inverse Function Theorem と呼ぶ（Derivatives of Inverse Functions、14 件）。"
    ],
    "related": [
      "find-the-inverse",
      "chain-rule",
      "derivative"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 3.3（derivative of an inverse function）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "derivative of the inverse": 22,
        "derivative of an inverse function": 2
      },
      "written": {
        "derivative of the inverse": 5,
        "derivative of an inverse function": 2,
        "inverse function theorem": 16
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivative-of-the-exponential-function",
    "ja": {
      "term": "指数関数の導関数",
      "reading": "しすうかんすうのどうかんすう",
      "alt": [
        "eˣ の導関数"
      ]
    },
    "en": {
      "term": "derivative of e to the x",
      "alt": [],
      "variants": [
        {
          "term": "derivative of the exponential function",
          "register": "written",
          "note": "書き言葉の言い方だが、件数は少ない。話し言葉・書き言葉ともほとんど出てこず、書き言葉は OpenStax Calculus だけ。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学III）は「指数関数の導関数」を三角関数・対数関数の導関数と並べて挙げる（(eˣ)′ = eˣ と (aˣ)′ = aˣ log a をまとめた言い方）。英語の授業では derivative of e to the x のように関数ごとに言う（用例コーパスの話し言葉では derivative of the exponential function よりずっと多い）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "(eˣ)′ = eˣ、(aˣ)′ = aˣ log a などの、指数関数の微分の公式。",
    "definition_en": "The formulas for differentiating exponential functions: (eˣ)′ = eˣ and (aˣ)′ = aˣ ln a.",
    "latex": "\\frac{d}{dx}e^x = e^x",
    "spoken_en": "the derivative of e to the x is e to the x",
    "examples": [
      {
        "en": "The derivative of e to the x is just e to the x.",
        "ja": "eˣ の導関数は eˣ のまま。",
        "register": "spoken"
      },
      {
        "en": "Find the derivative of f(x) = 2ˣ.",
        "ja": "f(x) = 2ˣ の導関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus は aˣ ln a と書く（OpenStax Algebra and Trigonometry は底を省いた log x を常用対数とし、学習指導要領解説には ln が出てこない）。"
    ],
    "related": [
      "base-of-the-natural-logarithm",
      "derivative-of-the-logarithm",
      "chain-rule"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 微分法: 三角関数・指数関数・対数関数の導関数（ln は出てこない）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "底を省いた log x は常用対数"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "derivative of e to the x": 27,
        "derivative of the exponential function": 2
      },
      "written": {
        "derivative of the exponential function": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivative-of-the-logarithm",
    "ja": {
      "term": "対数関数の導関数",
      "reading": "たいすうかんすうのどうかんすう",
      "alt": [
        "ln x の導関数"
      ]
    },
    "en": {
      "term": "derivative of ln x",
      "alt": [],
      "variants": [
        {
          "term": "derivative of the natural log",
          "register": "spoken",
          "note": "Khan Academy に多い言い方。書き言葉には出てこない。"
        },
        {
          "term": "derivative of ln x",
          "register": "spoken",
          "note": "話し言葉では半分以上が Professor Leonard で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "(log x)′ = 1/x、(log_a x)′ = 1/(x log a) などの、対数関数の微分の公式。",
    "definition_en": "The formulas for differentiating logarithms: (ln x)′ = 1/x and (log_a x)′ = 1/(x ln a).",
    "latex": "\\frac{d}{dx}\\ln x = \\frac{1}{x}",
    "spoken_en": "the derivative of ln x is one over x",
    "examples": [
      {
        "en": "The derivative of ln x is one over x.",
        "ja": "log x の導関数は 1/x。",
        "register": "spoken"
      },
      {
        "en": "Differentiate y = ln(x² + 1).",
        "ja": "y = log(x² + 1) を微分せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax は自然対数を ln x と書き（学習指導要領解説には ln が出てこない）、ell en x や natural log of x と読む。"
    ],
    "related": [
      "base-of-the-natural-logarithm",
      "logarithmic-differentiation",
      "derivative-of-the-exponential-function"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "ln が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "derivative of the natural log": 26,
        "derivative of ln x": 14
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivative",
    "ja": {
      "term": "導関数",
      "reading": "どうかんすう",
      "alt": []
    },
    "en": {
      "term": "derivative",
      "alt": [],
      "variants": [
        {
          "term": "derivative function",
          "register": "both",
          "note": "関数であることを強調するときの言い方で、件数は少ない。話し言葉は半分以上が Khan Academy、書き言葉はほとんどが OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "高等学校学習指導要領解説（数学II）は「微分係数や導関数の意味」「x = a における微分係数」と、ある点での値（微分係数、エントリ derivative-at-a-point）と関数（導関数）を別の語で書き、日本語版 Wikipedia「微分」も、a に微分係数 f′(a) を対応させる関数 f′ を導関数と呼んで分ける。英語の derivative はどちらにも使い、AP の CED（topic 2.2）は the derivative of a function at a point とも the derivative of f is the function とも書く。OpenStax Calculus Volume 1 は 3.1 で値 f′(a) も derivative と呼び、関数であることをはっきり言うときは 3.2 の derivative function を使う。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "各 x に、その点での微分係数 f′(x) を対応させる関数。",
    "definition_en": "The function f′ whose value at each x is the slope of the graph of f there, that is, how fast f is changing at x.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The derivative of x cubed is three x squared.",
        "ja": "x³ の導関数は 3x²。",
        "register": "spoken"
      },
      {
        "en": "Find the derivative of f(x) = x³ − 2x.",
        "ja": "f(x) = x³ − 2x の導関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ある点での値（微分係数）を言うときは the derivative at x = a（f prime of a）と言う（エントリ derivative-at-a-point）。",
      "操作としての「微分」は differentiation、動詞は differentiate（take the derivative）。"
    ],
    "related": [
      "derivative-at-a-point",
      "differentiation",
      "differentiate",
      "slope-of-the-tangent-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "微分",
        "en": "Derivative"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "3.1 Defining the Derivative（the derivative of the function f(x) at a）・3.2 The Derivative as a Function（derivative function）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 2（derivative）・Topic 2.2（derivative of a function at a point）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II (5) 微分・積分の考え: 微分係数や導関数の意味、x = a における微分係数（mapping_note）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「微分」",
        "url": "https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86",
        "note": "冒頭は微分係数・微分商・導関数に英語の derivative を 1 つ当てる。節「直観的な説明」は、各点での微分係数を値にとる関数を導関数とする（mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "derivative": 7857,
        "derivative function": 11
      },
      "written": {
        "derivative": 1449,
        "derivative function": 13
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivatives-in-polar-form",
    "ja": {
      "term": "極曲線の微分",
      "reading": "きょくきょくせんのびぶん",
      "alt": []
    },
    "en": {
      "term": "differentiating in polar form",
      "alt": [
        "derivatives in polar form"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「極曲線の微分」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "極方程式 r = f(θ) の曲線の接線の傾き dy/dx を、x = r cos θ、y = r sin θ を θ で微分して求めること。",
    "definition_en": "Finding the slope dy/dx of a polar curve r = f(θ) by differentiating x = r cos θ and y = r sin θ with respect to θ.",
    "latex": "\\frac{dy}{dx} = \\frac{dy/d\\theta}{dx/d\\theta} = \\frac{r'\\sin\\theta + r\\cos\\theta}{r'\\cos\\theta - r\\sin\\theta}",
    "spoken_en": "d y d x equals d y d theta over d x d theta",
    "examples": [
      {
        "en": "Treat θ like the parameter: dy/dx is dy/dθ over dx/dθ.",
        "ja": "θ を媒介変数とみなせばいい。dy/dx は dy/dθ を dx/dθ で割ったもの。",
        "register": "spoken"
      },
      {
        "en": "Find the slope of the tangent line to r = 1 + cos θ at θ = π/2.",
        "ja": "r = 1 + cos θ の θ = π/2 における接線の傾きを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "polar-coordinates",
      "derivative-of-a-parametric-curve",
      "polar-equation"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 9.7（differentiating in polar form）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CED の呼び方 differentiating in polar form（topic 9.7）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivatives-of-inverse-trig-functions",
    "ja": {
      "term": "逆三角関数の導関数",
      "reading": "ぎゃくさんかくかんすうのどうかんすう",
      "alt": []
    },
    "en": {
      "term": "derivatives of inverse trigonometric functions",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "(arcsin x)′ = 1/√(1 − x²)、(arctan x)′ = 1/(1 + x²) などの公式。",
    "definition_en": "The derivative rule for each inverse trigonometric function, for example (arcsin x)′ = 1/√(1 − x²) and (arctan x)′ = 1/(1 + x²).",
    "latex": "\\frac{d}{dx}\\arctan x = \\frac{1}{1 + x^2}",
    "spoken_en": "the derivative of arctan x is one over one plus x squared",
    "examples": [
      {
        "en": "The derivative of arctan x is one over one plus x squared.",
        "ja": "arctan x の導関数は 1/(1 + x²)。",
        "register": "spoken"
      },
      {
        "en": "Find the derivative of y = arcsin(2x).",
        "ja": "y = arcsin(2x) の導関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説には逆三角関数が出てこない。arcsin x は inverse sine とも言い、sin⁻¹ x とも書く（1/sin x ではない）。"
    ],
    "related": [
      "derivatives-of-trigonometric-functions",
      "derivative-of-an-inverse-function",
      "inverse-trigonometric-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 3.4（derivatives of inverse trigonometric functions）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "逆三角関数が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "derivatives of inverse trigonometric functions": 6
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは CED の呼び方 derivatives of inverse trigonometric functions（topic 3.4）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "derivatives-of-trigonometric-functions",
    "ja": {
      "term": "三角関数の導関数",
      "reading": "さんかくかんすうのどうかんすう",
      "alt": []
    },
    "en": {
      "term": "derivatives of trigonometric functions",
      "alt": [
        "derivatives of trig functions"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "(sin x)′ = cos x、(cos x)′ = −sin x、(tan x)′ = 1/cos²x などの公式。",
    "definition_en": "The formulas for the derivatives of the trigonometric functions, such as (sin x)′ = cos x and (cos x)′ = −sin x.",
    "latex": "\\frac{d}{dx}\\sin x = \\cos x",
    "spoken_en": "the derivative of sine x is cosine x",
    "examples": [
      {
        "en": "The derivative of sine is cosine, but the derivative of cosine is negative sine.",
        "ja": "sin の導関数は cos、cos の導関数は −sin。",
        "register": "spoken"
      },
      {
        "en": "Find the derivative of y = sin x cos x.",
        "ja": "y = sin x cos x の導関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "(tan x)′ = 1/cos²x は、OpenStax Calculus では sec²x（secant squared x）と書く（学習指導要領解説には sec が出てこない）。"
    ],
    "related": [
      "limit-of-sine-x-over-x",
      "chain-rule",
      "derivative"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 微分法: 三角関数の導関数（sec は出てこない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "derivatives of trigonometric functions": 2,
        "derivatives of trig functions": 3
      },
      "written": {
        "derivatives of trigonometric functions": 5
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 derivatives of trigonometric functions（Derivatives of Trigonometric Functions）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "descartes-rule-of-signs",
    "ja": {
      "term": "デカルトの符号法則",
      "reading": "でかるとのふごうほうそく",
      "alt": []
    },
    "en": {
      "term": "Descartes' rule of signs",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "多項式 f(x) の係数を次数の高い順に並べたとき、符号が変わる回数から正の実数解の個数を知る法則。正の実数解の個数はその回数か、それより偶数だけ少なく、f(−x) で同じように数えると負の実数解の個数が分かる。",
    "definition_en": "A rule that uses the number of sign changes in the coefficients of f(x) to find the possible numbers of positive real zeros: that number of changes, or fewer by an even number. Counting the sign changes in f(−x) does the same for negative real zeros.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Plus to minus, then minus to plus: that's two sign changes, so Descartes' rule of signs says two or zero positive real zeros.",
        "ja": "プラスからマイナス、マイナスからプラスで、符号の変化は 2 回。だからデカルトの符号法則で、正の実数解は 2 個か 0 個。",
        "register": "spoken"
      },
      {
        "en": "Use Descartes' rule of signs to determine the possible numbers of positive and negative real zeros of f(x) = x³ − 2x² − x + 2.",
        "ja": "デカルトの符号法則を用いて、f(x) = x³ − 2x² − x + 2 の正の実数解と負の実数解の個数として考えられるものを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校では扱わない。OpenStax Algebra and Trigonometry が扱う。",
      "Descartes は「デイカート」に近く読み、語末の s は発音しない（Merriam-Webster: dā-ˈkärt）。",
      "分かるのは個数の候補だけで、解の値は分からない。係数が 0 の項は飛ばして符号の変化を数える。"
    ],
    "related": [
      "zeros-of-a-polynomial",
      "rational-root-theorem",
      "polynomial-function",
      "sign",
      "fundamental-theorem-of-algebra"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "デカルトの符号法則",
        "en": "Descartes' rule of signs"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「Descartes」",
        "url": "https://www.merriam-webster.com/dictionary/Descartes",
        "note": "発音 dā-ˈkärt（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "Descartes' rule of signs": 13
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "descriptive-statistics",
    "ja": {
      "term": "記述統計",
      "reading": "きじゅつとうけい",
      "alt": [
        "記述統計学"
      ]
    },
    "en": {
      "term": "descriptive statistics",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "データを表・グラフ・代表値・散らばりの指標でまとめ、その特徴を述べる統計の方法。",
    "definition_en": "The part of statistics that summarizes data with tables, graphs, measures of center and measures of spread.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Mean, median, a box plot: that's all descriptive statistics.",
        "ja": "平均、中央値、箱ひげ図。どれも記述統計だ。",
        "register": "spoken"
      },
      {
        "en": "Use descriptive statistics to summarize the data set.",
        "ja": "記述統計を用いてデータをまとめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics（章の名前 Descriptive Statistics）。"
    ],
    "related": [
      "statistics",
      "inferential-statistics",
      "measure-of-center",
      "spread"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "descriptive statistics": 3
      },
      "written": {
        "descriptive statistics": 20
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "determinant",
    "ja": {
      "term": "行列式",
      "reading": "ぎょうれつしき",
      "alt": []
    },
    "en": {
      "term": "determinant",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列に対して定まる 1 つの数。2 次では ad − bc。0 でないことと逆行列をもつことが同じになる。",
    "definition_en": "A number assigned to a square matrix, ad − bc for a 2 × 2 matrix; it is nonzero exactly when the matrix has an inverse.",
    "latex": "\\det\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = ad - bc",
    "spoken_en": "the determinant of a b c d is a d minus b c",
    "examples": [
      {
        "en": "If the determinant is zero, the matrix squishes space down and there's no inverse.",
        "ja": "行列式が 0 なら、行列は空間を押しつぶしていて逆行列はない。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the determinant of the 3 × 3 matrix by cofactor expansion along the first row.",
        "ja": "3 × 3 行列の行列式を、第 1 行で余因子展開して求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "det A または |A| と書き、determinant of A と読む。",
      "行列式の絶対値は、行列が表す変換で面積（体積）が何倍になるかを表す。"
    ],
    "related": [
      "matrix",
      "inverse-matrix",
      "cramers-rule",
      "square-matrix"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "行列式",
        "en": "Determinant"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "determinant": 787
      },
      "written": {
        "determinant": 282
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "determine-the-coefficients",
    "ja": {
      "term": "係数を決定する",
      "reading": "けいすうをけっていする",
      "alt": []
    },
    "en": {
      "term": "find a quadratic function",
      "alt": [
        "find the coefficients",
        "determine the coefficients"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「係数を決定する」「二次関数の決定」は係数を求めることとして言うが、参照の英語は関数そのものを求めると言う: OpenStax Intermediate Algebra は Find a Quadratic Function from its Graph（頂点と 1 点から）。3 点を通る場合は a, b, c の連立方程式を解く。find the coefficients は用例コーパス・参照では二項係数・部分分数の係数などの意味で使われ、二次関数の決定の意味では出てこない。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "与えられた条件（グラフが通る点、頂点など）から、式の中の未知の係数の値を求める。放物線が通る 3 点から a, b, c の連立方程式を立てるのが典型。",
    "definition_en": "To find the unknown coefficients in an expression from given conditions, such as points on its graph or its vertex; for a parabola through three points, this means solving a system of three equations in a, b, and c.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We've got three points, so plug each one in and we get three equations to find the coefficients a, b, and c.",
        "ja": "3 点がわかっているから、それぞれ代入すると係数 a, b, c を決める 3 つの式ができる。",
        "register": "spoken"
      },
      {
        "en": "The graph of y = ax² + bx + c passes through (0, 3), (1, 2), and (2, 5). Find a quadratic function that fits these points, that is, the values of a, b, and c.",
        "ja": "放物線 y = ax² + bx + c が 3 点 (0, 3), (1, 2), (2, 5) を通るとき、この二次関数を求めよ（係数 a, b, c を決定せよ）。",
        "register": "written"
      },
      {
        "en": "The parabola y = a(x − 1)² − 2 passes through (3, 6). Find the value of a.",
        "ja": "放物線 y = a(x − 1)² − 2 が点 (3, 6) を通る。a の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「二次関数の決定」（学習指導要領解説・共通テストには出てこない言い方）の内容はこのエントリで扱う。英語の問題文では求めるもの（the quadratic function や the values of a, b, and c）を目的語にして find で書けばよい。",
      "条件に頂点があれば vertex form y = a(x − p)² + q、通る 3 点なら standard form y = ax² + bx + c とおくと式が少なくて済む。"
    ],
    "related": [
      "vertex-form",
      "standard-form",
      "system-of-linear-equations",
      "compare-coefficients",
      "vertex"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Graph Quadratic Functions Using Transformations: Find a Quadratic Function from its Graph（4 件）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 二次関数: 「二次関数の決定」「2次関数の決定」は 0 件"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "「二次関数の決定」「2次関数の決定」は 0 件"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "find a quadratic function | find the quadratic function | quadratic function from its graph": 1
      },
      "written": {
        "determine the coefficients": 2,
        "find a quadratic function | find the quadratic function | quadratic function from its graph": 4
      },
      "sources": [
        "mit-notes",
        "openstax-calculus",
        "openstax-intalg",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 find a quadratic function | find the quadratic function | quadratic function from its graph（本文 4 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "determine-where-the-function-is-increasing-and-decreasing",
    "ja": {
      "term": "増減を調べる",
      "reading": "ぞうげんをしらべる",
      "alt": []
    },
    "en": {
      "term": "determine intervals on which … is increasing or decreasing",
      "alt": [
        "determine where … is increasing",
        "where … is increasing or decreasing",
        "intervals on which … is increasing"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "「増減を調べる」は増減表を書いて関数の様子を調べることを含むが、英語の問題文は Determine the intervals on which f is increasing or decreasing. のように、増加・減少する区間を答えさせる形になる。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "f′(x) の符号を調べて、関数が増加する区間と減少する区間を求める。",
    "definition_en": "To find, from the sign of f′(x), the intervals on which a function increases and those on which it decreases.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To figure out where it's increasing or decreasing, check the sign of f prime on each interval.",
        "ja": "増減を調べるには、各区間で f′ の符号を確かめる。",
        "register": "spoken"
      },
      {
        "en": "Determine the intervals on which f(x) = x⁴ − 2x² is increasing or decreasing.",
        "ja": "f(x) = x⁴ − 2x² の増減を調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "答えは区間で書く（increasing on (−1, 0) and (1, ∞)）。端点を含めるかどうかは教科書による。"
    ],
    "related": [
      "increasing-and-decreasing",
      "make-a-sign-chart",
      "first-derivative-test"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.3（determining intervals on which a function is increasing or decreasing）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "determine where … is increasing": 7,
        "where … is increasing or decreasing": 9,
        "intervals on which … is increasing": 4
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満）。見出しは CED の呼び方 determine intervals on which … is increasing or decreasing（topic 5.3）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "deviation",
    "ja": {
      "term": "偏差",
      "reading": "へんさ",
      "alt": [
        "平均からの偏差"
      ]
    },
    "en": {
      "term": "deviation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "各データの値から平均値を引いた差 x − x̄。",
    "definition_en": "The difference between a data value and the mean, x − x̄.",
    "latex": "x_i - \\bar{x}",
    "spoken_en": "x sub i minus x bar",
    "examples": [
      {
        "en": "The deviations always add up to zero, so you have to square them first.",
        "ja": "偏差は足すと必ず 0 になるので、先に 2 乗する。",
        "register": "spoken"
      },
      {
        "en": "Find the deviation of each data value from the mean.",
        "ja": "各データの平均値からの偏差を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "deviation from the mean",
        "ja": "平均からの偏差"
      },
      {
        "en": "squared deviation",
        "ja": "偏差の 2 乗"
      }
    ],
    "pitfalls": [
      "偏差値（T-score。エントリ standard-score）とは別。"
    ],
    "related": [
      "variance",
      "standard-deviation",
      "standard-score"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "偏差",
        "en": "Deviation (statistics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!standard !absolute !quartile deviation": 36
      },
      "written": {
        "!standard !absolute !quartile deviation": 78
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "diagonal",
    "ja": {
      "term": "対角線",
      "reading": "たいかくせん",
      "alt": []
    },
    "en": {
      "term": "diagonal",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2",
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "多角形で、となり合わない 2 つの頂点を結ぶ線分。四角形には 2 本ある。",
    "definition_en": "A segment that joins two vertices of a polygon that are not next to each other. A quadrilateral has two diagonals.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Draw a diagonal from A to C, and now the quadrilateral is split into two triangles.",
        "ja": "A から C へ対角線を引くと、四角形が 2 つの三角形に分かれるね。",
        "register": "spoken"
      },
      {
        "en": "In parallelogram ABCD, the diagonals AC and BD intersect at point O. Prove that AO = CO.",
        "ja": "平行四辺形 ABCD の対角線 AC と BD の交点を O とする。AO = CO であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "行列の対角成分（the diagonal of a matrix）も同じ diagonal で、別の意味（エントリ matrix）。",
      "形容詞の diagonal は「斜めの」（a diagonal line は斜めの線）。図形の対角線は名詞で a diagonal ／ the diagonals と言う。",
      "「対角線」と「対角」は字が似ているが別の語。対角は向かい合う角で、英語は opposite angle（エントリ opposite-angle）。"
    ],
    "related": [
      "polygon",
      "quadrilateral",
      "vertex",
      "opposite-angle",
      "matrix"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "対角線",
        "en": "Diagonal"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "diagonal of the !matrix | diagonal of a !matrix | diagonals of the !matrix | diagonals of a !matrix | length of the diagonal | diagonals bisect | diagonals intersect": 14
      },
      "written": {
        "diagonal of the !matrix | diagonal of a !matrix | diagonals of the !matrix | diagonals of a !matrix | length of the diagonal | diagonals bisect | diagonals intersect": 35
      },
      "sources": [
        "khan-middle",
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "diagonalizable",
    "ja": {
      "term": "対角化可能",
      "reading": "たいかくかかのう",
      "alt": []
    },
    "en": {
      "term": "diagonalizable",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正則行列 P で P⁻¹AP を対角行列にできること。n × n 行列では 1 次独立な固有ベクトルが n 個あることと同じ。",
    "definition_en": "Able to be made diagonal as P⁻¹AP for some invertible P; for an n × n matrix, having n linearly independent eigenvectors.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Not every matrix is diagonalizable; you need enough independent eigenvectors.",
        "ja": "どの行列も対角化可能とは限らない。1 次独立な固有ベクトルが足りないといけない。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the matrix is diagonalizable.",
        "ja": "行列が対角化可能か判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて MIT OCW 18.06。"
    ],
    "related": [
      "diagonalization",
      "eigenvector",
      "algebraic-multiplicity",
      "geometric-multiplicity"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.3 Diagonalization and Eigenvalues・5.5 Similarity and Diagonalization"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "diagonalizable": 24
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "diagonalization",
    "ja": {
      "term": "対角化",
      "reading": "たいかくか",
      "alt": []
    },
    "en": {
      "term": "diagonalization",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（diagonalization は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列 A を、正則行列 P を使って P⁻¹AP が対角行列になるように変形すること。P の列に固有ベクトルを並べる。",
    "definition_en": "Finding an invertible matrix P such that P⁻¹AP is diagonal; the columns of P are eigenvectors of A.",
    "latex": "P^{-1}AP = D",
    "spoken_en": null,
    "examples": [
      {
        "en": "Once you diagonalize A, computing A to the hundredth power is easy.",
        "ja": "A を対角化すれば、A の 100 乗も簡単に計算できる。",
        "register": "spoken"
      },
      {
        "en": "Find the diagonalization of A, if possible.",
        "ja": "可能ならば A を対角化せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語は動詞 diagonalize をよく使う。名詞の diagonalization は少なく、見出しの名詞だけを候補にした（対角化する は動詞）。"
    ],
    "related": [
      "diagonalizable",
      "eigenvalue",
      "eigenvector",
      "similar-matrices"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.3 Diagonalization and Eigenvalues"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "diagonalization": 8
      },
      "sources": [
        "mit-18.03",
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 8 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 diagonalization（3.3 Diagonalization and Eigenvalues・5.5 Similarity and Diagonalization・8.2 Orthogonal Diagonalization）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "diameter",
    "ja": {
      "term": "直径",
      "reading": "ちょっけい",
      "alt": []
    },
    "en": {
      "term": "diameter",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円の中心を通り、両端が円周上にある線分（中心を通る弦）。その長さも直径といい、半径の 2 倍になる。",
    "definition_en": "A segment through the center of a circle with both endpoints on the circle; also its length, which is twice the radius.",
    "latex": "d = 2r",
    "spoken_en": "d equals two r",
    "examples": [
      {
        "en": "The diameter goes all the way across the circle through the center, so it's twice the radius.",
        "ja": "直径は中心を通って円の端から端まで行く線だから、半径の 2 倍だね。",
        "register": "spoken"
      },
      {
        "en": "A circle has a diameter of 18 cm. Find its radius.",
        "ja": "直径 18 cm の円の半径を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の diameter も、中心を通る線分（draw a diameter）とその長さ（the diameter is 10 cm）の両方を指す。",
      "問題文が radius（半径）を与えているか diameter（直径）を与えているかで、公式に入れる値が変わる（C = πd = 2πr、エントリ circumference）。",
      "発音は di-AM-e-ter で、2 つ目の音節に強勢がある（Merriam-Webster: dī-ˈa-mə-tər）。"
    ],
    "related": [
      "radius",
      "circle",
      "center",
      "chord",
      "circumference"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「diameter」",
        "url": "https://www.merriam-webster.com/dictionary/diameter",
        "note": "発音 dī-ˈa-mə-tər（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "diameter": 149
      },
      "written": {
        "diameter": 152
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "die",
    "ja": {
      "term": "さいころ",
      "reading": "さいころ",
      "alt": []
    },
    "en": {
      "term": "die",
      "alt": [
        "number cube"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「さいころ」は 1 個でも 2 個でも同じ形だが、英語の die は 1 個のさいころで、複数形は dice（2 個なら two dice）。英語版 Wikipedia「Dice」は、dice を 1 個にも使うことがあると書く。IM（Grade 6〜8 で 28 件・9〜12 で 47 件）と OpenStax Algebra and Trigonometry・Precalculus（各 10 件）は、1〜6 の目がある立方体のさいころを number cube とも言う。",
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "立方体の 6 つの面に 1 から 6 までの目が 1 つずつついた道具。公正なさいころでは、どの目が出ることも同様に確からしい。",
    "definition_en": "A small cube whose six faces show the numbers 1 through 6, rolled to get a random result; the plural is dice. On a fair die, each number is equally likely to come up.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "One die gives you 6 outcomes, but two dice give you 6 times 6, or 36.",
        "ja": "さいころ 1 個なら結果は 6 通りだけど、2 個なら 6 × 6 で 36 通り。",
        "register": "spoken"
      },
      {
        "en": "A fair die is rolled once. Find the probability of rolling a number greater than 4.",
        "ja": "1 個のさいころを 1 回投げる。4 より大きい目が出る確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "die は動詞「死ぬ」と同じ綴りの語。さいころの die は数えられる名詞で、a die ／ the die のように冠詞が付く。",
      "「さいころを投げる」は roll a die（エントリ toss）。目は number で言い、「3 の目が出る」は roll a 3、「偶数の目が出る」は roll an even number。"
    ],
    "related": [
      "toss",
      "coin",
      "equally-likely",
      "probability",
      "number-of-possible-outcomes"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Dice",
        "note": "冒頭: A die (plural: dice, sometimes also used as singular)"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "number cube（Grade 6〜8 で 28 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "number cube（10 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "a die | the die | two dice | a fair die | roll dice | the dice": 77,
        "number cube": 2
      },
      "written": {
        "a die | the die | two dice | a fair die | roll dice | the dice": 52,
        "number cube": 11
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "difference-of-cubes",
    "ja": {
      "term": "立方の差",
      "reading": "りっぽうのさ",
      "alt": []
    },
    "en": {
      "term": "difference of cubes",
      "alt": [
        "difference of two cubes"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "a³ − b³ の形の式。(a − b)(a² + ab + b²) と因数分解できる。",
    "definition_en": "An expression of the form a³ − b³, which factors as (a − b)(a² + ab + b²).",
    "latex": "a^3 - b^3 = (a - b)(a^2 + ab + b^2)",
    "spoken_en": "a cubed minus b cubed equals the quantity a minus b, times the quantity a squared plus a b plus b squared",
    "examples": [
      {
        "en": "x cubed minus 64 is a difference of cubes, because 64 is 4 cubed.",
        "ja": "x³ − 64 は立方の差。64 は 4 の 3 乗だからね。",
        "register": "spoken"
      },
      {
        "en": "Factor the difference of cubes 8y³ − 125.",
        "ja": "立方の差 8y³ − 125 を因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a² − b² = (a + b)(a − b)（エントリ difference-of-squares）と取り違えやすい。a³ − b³ の 2 つ目の因数は a + b ではなく a² + ab + b²。",
      "2 つ目の因数 a² + ab + b² は、実数の範囲ではそれ以上因数分解できない。",
      "立方の和（エントリ sum-of-cubes）とは符号の並びが逆になる。"
    ],
    "related": [
      "sum-of-cubes",
      "difference-of-squares",
      "factoring",
      "cube",
      "polynomial"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "difference of cubes": 6,
        "difference of two cubes": 2
      },
      "written": {
        "difference of cubes": 43,
        "difference of two cubes": 2
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "difference-of-squares",
    "ja": {
      "term": "和と差の積",
      "reading": "わとさのせき",
      "alt": [
        "二乗の差"
      ]
    },
    "en": {
      "term": "difference of squares",
      "alt": [
        "difference of two squares"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「和と差の積」は左辺の積 (a + b)(a − b) の名前だが、英語の difference of squares は右辺 a² − b² の名前で、a² − b² = (a + b)(a − b) と因数分解するときに言う。OpenStax Elementary Algebra は展開の向きを Product of Conjugates Pattern（共役な 2 式の積）と呼ぶ。IM Algebra 1 7.8 は (a + b)(a − b) を a product of a sum and a difference、a² − b² を a difference of two squares と書く。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 つの数や式の和と差を掛けると、2 乗の差になるという公式。(a + b)(a − b) = a² − b² である。",
    "definition_en": "The pattern a² − b² = (a + b)(a − b): a difference of two squares factors into a sum times a difference. Multiplying (a + b)(a − b) gives back a² − b² because the middle terms cancel.",
    "latex": "(a + b)(a - b) = a^2 - b^2",
    "spoken_en": "the quantity a plus b times the quantity a minus b equals a squared minus b squared",
    "examples": [
      {
        "en": "x² − 49 is a difference of squares, so it factors as (x + 7)(x − 7).",
        "ja": "x² − 49 は x² − 7² の形だから、(x + 7)(x − 7) と因数分解できる。",
        "register": "spoken"
      },
      {
        "en": "Factor 9x² − 16 using the difference of squares pattern.",
        "ja": "和と差の積の公式を利用して、9x² − 16 を因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 乗の和 a² + b² は、実数の範囲では因数分解できない。x² + 49 を (x + 7)(x − 7) としない。",
      "9x² − 16 の 9x² は (3x)²。係数も 2 乗になっているかを確かめて (3x + 4)(3x − 4) とする。",
      "因数分解のあとに、まだ difference of squares の形が残っていないか確かめる。x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2)。"
    ],
    "related": [
      "special-products",
      "square-of-a-binomial",
      "factoring",
      "factor",
      "square"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 7.8: a product of a sum and a difference ／ a difference of two squares"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "difference of two squares": 16,
        "difference of squares": 65
      },
      "written": {
        "difference of two squares": 6,
        "difference of squares": 97
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "difference-quotient",
    "ja": {
      "term": "差分商",
      "reading": "さぶんしょう",
      "alt": []
    },
    "en": {
      "term": "difference quotient",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「差分商」は学習指導要領解説に無い（日本語版 Wikipedia「微分」にはある）。日本の高校では (f(a + h) − f(a))/h を「平均変化率」として扱い、別の名前をつけない。米国の Precalculus・Calculus では difference quotient と名前で呼ぶ。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "(f(x + h) − f(x))/h の形の式。x から x + h までの平均変化率で、h → 0 の極限が導関数。",
    "definition_en": "The expression (f(x + h) − f(x))/h: the average rate of change from x to x + h, whose limit as h → 0 is the derivative.",
    "latex": "\\frac{f(x + h) - f(x)}{h}",
    "spoken_en": "f of x plus h minus f of x, over h",
    "examples": [
      {
        "en": "Simplify the difference quotient first, and then let h go to zero.",
        "ja": "まず差分商を簡単にしてから、h を 0 に近づける。",
        "register": "spoken"
      },
      {
        "en": "For f(x) = x² − 3x, find and simplify the difference quotient (f(x + h) − f(x))/h.",
        "ja": "f(x) = x² − 3x について、(f(x + h) − f(x))/h を計算して簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "average-rate-of-change",
      "limit-definition-of-the-derivative",
      "derivative"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "差分商",
        "en": "Difference quotient"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "difference quotient": 34
      },
      "written": {
        "difference quotient": 23
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "difference",
    "ja": {
      "term": "差",
      "reading": "さ",
      "alt": []
    },
    "en": {
      "term": "difference",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "引き算の答え。たとえば 7 と 3 の差は 4 である。",
    "definition_en": "The result of subtracting one number or expression from another. For example, the difference of 7 and 3 is 7 − 3 = 4.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The difference between 7 and 3 is 4, because 7 minus 3 is 4.",
        "ja": "7 と 3 の差は 4。7 − 3 = 4 だからね。",
        "register": "spoken"
      },
      {
        "en": "The difference of two numbers is 5, and their sum is 17. Find the two numbers.",
        "ja": "2 つの数の差は 5、和は 17 である。この 2 つの数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a − b は the difference of a and b。7 と 3 なら the difference between 7 and 3 とも言う。",
      "等差数列の公差は common difference、階差数列は sequence of differences（エントリ common-difference・sequence-of-differences）。",
      "日常語の「違い」（What's the difference?）と同じ語。"
    ],
    "related": [
      "subtraction",
      "subtract",
      "sum",
      "common-difference",
      "sequence-of-differences"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the difference of | the difference between": 381
      },
      "written": {
        "the difference of | the difference between": 525
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "differentiability",
    "ja": {
      "term": "微分可能性",
      "reading": "びぶんかのうせい",
      "alt": []
    },
    "en": {
      "term": "differentiability",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数がその点（または区間）で微分可能であること。",
    "definition_en": "The property of being differentiable.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Differentiability implies continuity, but not the other way around.",
        "ja": "微分可能なら連続だが、逆は成り立たない。",
        "register": "spoken"
      },
      {
        "en": "Discuss the continuity and differentiability of f(x) = |x − 1| at x = 1.",
        "ja": "f(x) = |x − 1| の x = 1 における連続性と微分可能性を調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "differentiable",
      "continuous"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 2.4（differentiability）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "differentiability": 41
      },
      "written": {
        "differentiability": 24
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "differentiable",
    "ja": {
      "term": "微分可能",
      "reading": "びぶんかのう",
      "alt": []
    },
    "en": {
      "term": "differentiable",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "その点で微分係数（導関数の値を与える極限）が存在すること。",
    "definition_en": "Having a derivative at a point, or at every point of an interval.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's continuous at zero, but it's not differentiable there because of the corner.",
        "ja": "0 で連続だが、尖っているのでそこでは微分可能でない。",
        "register": "spoken"
      },
      {
        "en": "Show that f is differentiable at x = 1.",
        "ja": "f が x = 1 で微分可能であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "微分可能なら連続だが、連続でも微分可能とは限らない（|x| の x = 0）。英語では differentiability implies continuity と言う（OpenStax Calculus。AP の CED は topic 2.4 で微分可能と連続の関係として扱う）。"
    ],
    "related": [
      "derivative",
      "differentiation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 2.4（differentiable）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "differentiable": 298
      },
      "written": {
        "differentiable": 264
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "differential-equation",
    "ja": {
      "term": "微分方程式",
      "reading": "びぶんほうていしき",
      "alt": []
    },
    "en": {
      "term": "differential equation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II"
      ]
    },
    "definition_ja": "未知の関数とその導関数を含む方程式。条件を満たす関数を求めることを、微分方程式を解くという。",
    "definition_en": "An equation involving an unknown function and its derivatives; solving it means finding the functions that satisfy it.",
    "latex": "\\frac{dy}{dt} = ky",
    "spoken_en": "d y d t equals k y",
    "examples": [
      {
        "en": "Every solution of this differential equation is some constant times e to the k t.",
        "ja": "この微分方程式の解は、どれも定数 × e^{kt} の形になる。",
        "register": "spoken"
      },
      {
        "en": "Solve the differential equation dy/dt = 3y with y(0) = 2.",
        "ja": "微分方程式 dy/dt = 3y を、y(0) = 2 のもとで解け。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "solve the differential equation",
        "ja": "微分方程式を解く"
      }
    ],
    "pitfalls": [
      "高等学校学習指導要領（平成30年告示）の数学III には「微分方程式」の語が無く（理数数学II は dy/dx = ky 程度の簡単な微分方程式の意味と解法を扱う）。AP の CED は Unit 7 で slope fields（topic 7.3）や separation of variables（変数分離、topic 7.6）を扱い、AB の範囲に入る（Euler 法の 7.5 とロジスティックの 7.9 は BC のみ）。",
      "初期条件は initial condition、それを付けた問題は initial value problem（IVP）と言う。",
      "関数を代入して微分方程式を満たすことを確かめる問題を、CED は topic 7.2 で verifying solutions として扱う。"
    ],
    "related": [
      "separable-differential-equation",
      "logistic-differential-equation",
      "eulers-method",
      "slope-field",
      "separation-of-variables",
      "particular-solution",
      "initial-condition",
      "exponential-model"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "微分方程式",
        "en": "Differential equation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 7（Topic 7.3 slope fields、7.6 separation of variables。7.5・7.9 は BC のみ）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学II 内容の取扱い: dy/dx = ky 程度の簡単な微分方程式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "differential equation": 938
      },
      "written": {
        "differential equation": 619
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "differential",
    "ja": {
      "term": "微分 dy",
      "reading": "びぶんでぃーわい",
      "alt": []
    },
    "en": {
      "term": "differential",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "dy = f′(x) dx で定める量 dy（と dx）。Δy の近似に使う。",
    "definition_en": "The quantity dy = f′(x) dx (together with dx), used to approximate Δy.",
    "latex": "dy = f'(x)\\,dx",
    "spoken_en": "d y equals f prime of x d x",
    "examples": [
      {
        "en": "The differential dy is f prime of x times dx.",
        "ja": "微分 dy は f′(x) に dx を掛けたもの。",
        "register": "spoken"
      },
      {
        "en": "Find the differential dy for y = x² and evaluate it when x = 3 and dx = 0.1.",
        "ja": "y = x² の微分 dy を求め、x = 3、dx = 0.1 のときの値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "linear-approximation",
      "small-change",
      "leibniz-notation",
      "total-differential"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "differential !equation !equations": 273
      },
      "written": {
        "differential !equation !equations": 94
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "differentiate-both-sides",
    "ja": {
      "term": "両辺を微分する",
      "reading": "りょうへんをびぶんする",
      "alt": [
        "両辺を x で微分する"
      ]
    },
    "en": {
      "term": "differentiate both sides",
      "alt": [],
      "variants": [
        {
          "term": "take the derivative of both sides",
          "register": "spoken",
          "note": "Khan Academy に多い言い方。話し言葉では differentiate both sides より多く、書き言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "等式の両辺を同じ変数で微分する。",
    "definition_en": "To differentiate each side of an equation with respect to the same variable.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Take the derivative of both sides, and remember the chain rule on every y.",
        "ja": "両辺を微分する。y の項には合成関数の微分を忘れずに。",
        "register": "spoken"
      },
      {
        "en": "Differentiate both sides of x² + y² = 25 with respect to x.",
        "ja": "x² + y² = 25 の両辺を x で微分せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「x で微分する」は with respect to x を付けて言う。"
    ],
    "related": [
      "implicit-differentiation",
      "solve-for-y-prime",
      "differentiate"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "take the derivative of both sides": 45,
        "differentiate both sides": 12
      },
      "written": {
        "differentiate both sides": 16,
        "take the derivative of both sides": 2
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-6.042",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "differentiate-twice",
    "ja": {
      "term": "2 回微分する",
      "reading": "にかいびぶんする",
      "alt": []
    },
    "en": {
      "term": "take the second derivative",
      "alt": [],
      "variants": [
        {
          "term": "differentiate twice",
          "register": "spoken",
          "note": "少数の言い方。話し言葉はすべて MIT OCW で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数を 2 回続けて微分し、第 2 次導関数を求める。",
    "definition_en": "To differentiate a function two times in a row, getting the second derivative.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Take the second derivative to check the concavity.",
        "ja": "凹凸を調べるために 2 回微分する。",
        "register": "spoken"
      },
      {
        "en": "Differentiate y = sin 2x twice and show that y″ + 4y = 0.",
        "ja": "y = sin 2x を 2 回微分し、y″ + 4y = 0 を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "second-derivative",
      "higher-order-derivative"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "take the second derivative": 31,
        "differentiate twice": 3
      },
      "written": {
        "take the second derivative": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "differentiate",
    "ja": {
      "term": "微分する",
      "reading": "びぶんする",
      "alt": [
        "導関数を求める"
      ]
    },
    "en": {
      "term": "differentiate",
      "alt": [],
      "variants": [
        {
          "term": "take the derivative",
          "register": "both",
          "note": "話し言葉で最も多い。Khan Academy が半分を占める。書き言葉では少ない。"
        },
        {
          "term": "differentiate",
          "register": "both",
          "note": "書き言葉で最も多い。話し言葉は MIT OCW が中心。話し言葉・書き言葉ともよく使う。"
        },
        {
          "term": "find the derivative",
          "register": "both",
          "note": "問題文に多い言い方。話し言葉では take the derivative より少ない。書き言葉では differentiate と同じくらい使う（大半が OpenStax Calculus）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数の導関数を求める。",
    "definition_en": "To find the derivative of a function.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Take the derivative of x to the fifth: you get five x to the fourth.",
        "ja": "x⁵ を微分すると 5x⁴ になる。",
        "register": "spoken"
      },
      {
        "en": "Differentiate f(x) = (2x + 1)³.",
        "ja": "f(x) = (2x + 1)³ を微分せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "3 つとも同じ意味で使う。「x で微分する」は differentiate with respect to x（take the derivative with respect to x）。"
    ],
    "related": [
      "derivative",
      "differentiation",
      "find-the-derivative-at"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 3（differentiate）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "take the derivative": 825,
        "differentiate": 452,
        "find the derivative": 173
      },
      "written": {
        "find the derivative": 140,
        "differentiate": 197,
        "take the derivative": 28
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "differentiation",
    "ja": {
      "term": "微分",
      "reading": "びぶん",
      "alt": []
    },
    "en": {
      "term": "differentiation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「微分」は、微分係数や導関数を求めることを微分（differentiation）すると言い、求めた微分係数・導関数そのものも、本来の使い方ではないが微分と呼ばれることが多いと書く。英語では、OpenStax Calculus Volume 1（3.1）は derivative（値の微分係数も関数の導関数も）を求める過程を differentiation と呼び、求めた結果は derivative（エントリ derivative）と言う。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数の導関数を求めること。",
    "definition_en": "The process of finding a derivative.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Differentiation is linear, so you can go term by term.",
        "ja": "微分は線形だから、項ごとに微分してよい。",
        "register": "spoken"
      },
      {
        "en": "The rules of differentiation give f′(x) = 6x + 5.",
        "ja": "微分の公式から f′(x) = 6x + 5 となる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では、操作は differentiation、結果（導関数も微分係数も）は derivative、動詞は differentiate か take the derivative と使い分ける（「x² の微分は 2x」は The derivative of x² is 2x.）。"
    ],
    "related": [
      "derivative",
      "differentiate",
      "differentiable"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "微分",
        "en": "Derivative"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "3.1 Defining the Derivative（derivative を求める過程を differentiation と呼ぶ。用語集は the process of taking a derivative）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 2（differentiation）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「微分」",
        "url": "https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86",
        "note": "冒頭: 微分係数・導関数を求めることを微分と言い、求めた結果も、用語の濫用としてしばしば微分と呼ぶと書く（mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "differentiation": 226
      },
      "written": {
        "differentiation": 149
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "digit",
    "ja": {
      "term": "桁",
      "reading": "けた",
      "alt": []
    },
    "en": {
      "term": "digit",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "number-theory"
    ],
    "level": {
      "jp": [
        "中2",
        "数A"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Discrete Math"
      ]
    },
    "definition_ja": "位取り記数法で数を書いたとき、各位に並ぶ 1 つ 1 つの数字。「3 桁の数」は数字が 3 つ並ぶ数を言う。",
    "definition_en": "One of the symbols that fill the places when a number is written in a place-value system; a three-digit number is written with three of them.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "So we want a three-digit number whose digits add up to 12.",
        "ja": "つまり、各位の数の和が 12 になる 3 桁の数を探したい。",
        "register": "spoken"
      },
      {
        "en": "How many four-digit positive integers have no repeated digits?",
        "ja": "各位の数字がすべて異なる 4 桁の正の整数はいくつあるか。",
        "register": "written"
      },
      {
        "en": "Round the answer to two decimal places, and keep at least four significant digits along the way.",
        "ja": "答えは小数第 2 位までに丸め、途中の計算は有効数字 4 桁以上で行え。",
        "register": "written"
      },
      {
        "en": "Show that the sum of a two-digit number and the number formed by reversing its digits is a multiple of 11.",
        "ja": "2 桁の自然数と、その十の位の数字と一の位の数字を入れかえた数の和は、11 の倍数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "digit は「数字」そのもの（0〜9 の 1 つ）。「桁」の位置のほうは place（一の位 = ones place）で、「桁数」は number of digits。",
      "「3 桁の数」は a three-digit number。形容詞として使うときは digit を複数形にしない（three digits long とは言うが a three-digits number とは言わない）。",
      "十の位の数字が a、一の位の数字が b の 2 桁の数は 10a + b と表す（a two-digit number with tens digit a and ones digit b）。"
    ],
    "related": [
      "number-of-digits",
      "place",
      "leading-digit",
      "numeral-system"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "digit": 361
      },
      "written": {
        "digit": 360
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "dihedral-angle",
    "ja": {
      "term": "二面角",
      "reading": "にめんかく",
      "alt": []
    },
    "en": {
      "term": "dihedral angle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "dihedral angle という名前は、米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（4 つとも 0 件）。2 平面のなす角そのものは、OpenStax Calculus Volume 3（2.5 Equations of Lines and Planes in Space）が angle between two planes と呼び、法線ベクトルのなす角で求める。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "1 本の直線を共有する 2 つの半平面がつくる角。共有する直線上の 1 点から、それぞれの半平面上にその直線と垂直な半直線を引き、その 2 本がなす角で測る。",
    "definition_en": "The angle formed by two half-planes that share an edge, measured by the angle between two rays, one in each half-plane, drawn perpendicular to the edge from the same point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To measure the dihedral angle, pick a point on the edge and draw a line perpendicular to the edge in each face.",
        "ja": "二面角を測るには、交線上に 1 点をとり、それぞれの面の上に交線と垂直な直線を引く。",
        "register": "spoken"
      },
      {
        "en": "In a regular tetrahedron ABCD, let θ be the dihedral angle between faces ABC and ABD. Find cos θ.",
        "ja": "正四面体 ABCD において、面 ABC と面 ABD のなす二面角を θ とする。cos θ を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "交線上の点から引く 2 本の直線は、どちらも交線に垂直でなければならない。任意の 2 直線のなす角は二面角ではない。",
      "座標空間では、2 平面の法線ベクトルのなす角から求めることもできる。その場合、二面角は法線ベクトルのなす角か、その補角になる。"
    ],
    "related": [
      "angle",
      "normal-vector",
      "tetrahedron",
      "line-perpendicular-to-a-plane",
      "equation-of-a-plane"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二面角",
        "en": "Dihedral angle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.5 Equations of Lines and Planes in Space（Finding the Angle between Two Planes）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 dihedral angle（英語版「Dihedral angle」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "dilation",
    "ja": {
      "term": "拡大・縮小",
      "reading": "かくだい・しゅくしょう",
      "alt": [
        "相似拡大",
        "中心相似"
      ]
    },
    "en": {
      "term": "dilation",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説（第 3 学年）は、拡大・縮小を一つの図形を操作して新たな図形を作ることと書き、相似の位置・相似の中心を扱うが、変換の名前は出てこない（縮図・拡大図は小学校第 6 学年）。日本語版 Wikipedia「図形の相似」は、中心のある変換を相似拡大・中心相似と呼ぶ。英語の dilation は中心と scale factor で決まる変換。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "中心 O と正の数 k を決め、各点 P を半直線 OP 上で OP′ = k·OP となる点 P′ に移す変換。k > 1 なら拡大、0 < k < 1 なら縮小で、移した図形はもとの図形と相似になる。",
    "definition_en": "A transformation with a center O and a positive scale factor k that sends each point P to the point P′ on ray OP with OP′ = k · OP. The image is similar to the original: larger if k > 1 and smaller if 0 < k < 1.",
    "latex": "OP' = k \\cdot OP",
    "spoken_en": "O P prime equals k times O P",
    "examples": [
      {
        "en": "Dilate the triangle by a scale factor of 2, centered at the origin. Every coordinate just gets doubled.",
        "ja": "原点を中心に三角形を 2 倍に拡大しよう。座標がすべて 2 倍になるだけだ。",
        "register": "spoken"
      },
      {
        "en": "Triangle ABC has vertices A(1, 2), B(3, 1), and C(2, 4). Draw the image of △ABC under a dilation centered at the origin with scale factor 3.",
        "ja": "3 点 A(1, 2), B(3, 1), C(2, 4) を頂点とする △ABC を、原点を中心に 3 倍に拡大した図形をかけ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "dilate … by a scale factor of k",
        "ja": "k 倍に拡大（縮小）する"
      }
    ],
    "pitfalls": [
      "動詞は dilate（dilate the figure by a scale factor of 1/2）。縮小も dilation で、scale factor が 1 より小さいことで言い分ける。",
      "dilation は物理の time dilation（時間の遅れ）など、別の意味でも使う。関数のグラフの拡大・縮小は、OpenStax Algebra and Trigonometry・Precalculus では vertical stretch ／ vertical compression と言う（エントリ stretch-vertically）。",
      "用例コーパスでは、話し言葉はほとんどが Khan Academy の中学の講義で、書き言葉にはほとんど出てこない。IM では 295 件で、Grade 8 と Geometry の glossary の見出し。CK-12 Geometry は 7.16 Dilation in the Coordinate Plane ほかで 27 件。"
    ],
    "related": [
      "scale-factor",
      "similarity-transformation",
      "rigid-motion",
      "similar-triangles",
      "stretch-vertically"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "glossary の見出し dilation（Grade 8・Geometry）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形: 相似の位置・相似の中心（縮図や拡大図は小学校第 6 学年）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "7.16 Dilation in the Coordinate Plane ほか（27 件）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「図形の相似」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%B3%E5%BD%A2%E3%81%AE%E7%9B%B8%E4%BC%BC",
        "note": "相似拡大・中心相似"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "Unit 2 glossary: dilation（positive scale factor）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "dilation": 40
      },
      "written": {
        "dilation": 1
      },
      "sources": [
        "khan-middle",
        "mit-18.01",
        "mit-notes"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "dimension",
    "ja": {
      "term": "次元",
      "reading": "じげん",
      "alt": []
    },
    "en": {
      "term": "dimension",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "ベクトル空間の基底に含まれるベクトルの個数。どの基底をとっても同じ数になる。",
    "definition_en": "The number of vectors in a basis of a vector space; every basis has the same number.",
    "latex": "\\dim V",
    "spoken_en": null,
    "examples": [
      {
        "en": "The null space here is a plane, so its dimension is two.",
        "ja": "ここの零空間は平面なので、次元は 2 だ。",
        "register": "spoken"
      },
      {
        "en": "Find the dimension of the null space of A.",
        "ja": "A の零空間の次元を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "行列の大きさ（m × n）も英語で dimensions と言うが、ベクトル空間の次元とは別。"
    ],
    "related": [
      "basis",
      "rank",
      "rank-nullity-theorem",
      "vector-space"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "次元",
        "en": "Dimension"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.2 Independence and Dimension"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "dimension of the … space | dimension of a … space | dimension of the null | dimension of the column | dimension of the row | dimension of the subspace | dimension of that space": 60
      },
      "sources": [
        "mit-18.06",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "diophantine-equation",
    "ja": {
      "term": "不定方程式",
      "reading": "ふていほうていしき",
      "alt": [
        "ディオファントス方程式"
      ]
    },
    "en": {
      "term": "Diophantine equation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Diophantine equation は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "整数係数の方程式で、整数（または自然数）の解だけを考えるもの。日本語版 Wikipedia「ディオファントス方程式」は整係数多変数の不定方程式と説明する。",
    "definition_en": "An equation with integer coefficients for which only integer (or positive integer) solutions are sought.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is a Diophantine equation, so we only care about solutions where x and y are integers.",
        "ja": "これは不定方程式だから、x と y が整数になる解だけを考える。",
        "register": "spoken"
      },
      {
        "en": "Find all pairs of positive integers (x, y) that satisfy the Diophantine equation xy − 2x − 3y = 0.",
        "ja": "不定方程式 xy − 2x − 3y = 0 を満たす正の整数の組 (x, y) をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「不定」を indefinite と直訳しない。英語はディオファントス（Diophantus）の名前で Diophantine equation と言い、D は大文字。",
      "xy − 2x − 3y = 0 のような式は (x − 3)(y − 2) = 6 と積の形にして、6 の約数の組を調べる。"
    ],
    "related": [
      "linear-diophantine-equation",
      "integer-solution",
      "prime-factorization",
      "multiple"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2 Introduction to Number Theory"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "ディオファントス方程式",
        "en": "Diophantine equation"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成28年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "note": "第4問: 不定方程式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Levin の呼び方 Diophantine equation（6.2 Introduction to Number Theory）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "direct-proof",
    "ja": {
      "term": "直接証明",
      "reading": "ちょくせつしょうめい",
      "alt": []
    },
    "en": {
      "term": "direct proof",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（direct proof は CED・OpenStax・IM・CK-12 とも 0 件）。見出しの「直接証明」は学習指導要領解説に無い（日本語版 Wikipedia「証明 (数学)」にはある）。数I に置くのは対偶を利用した証明・背理法と対比する語として。",
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "仮定から出発し、定義や既に知っている定理を順に使って結論を導く証明。背理法や対偶を使った証明と対比して使う。",
    "definition_en": "A proof that starts from the hypothesis and reaches the conclusion through a chain of definitions and known results, without assuming that the conclusion is false.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's try a direct proof first: start with what we're given and work toward what we want.",
        "ja": "まず直接証明を試そう。与えられたことから出発して、示したいことに向かって進める。",
        "register": "spoken"
      },
      {
        "en": "Give a direct proof that the sum of two odd integers is even.",
        "ja": "2 つの奇数の和が偶数であることを直接証明で示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは MIT OCW と MIT の講義ノートにしか出てこない。",
      "direct proof は、背理法（proof by contradiction）や対偶を使った証明（proof by contrapositive）と区別するための名前。"
    ],
    "related": [
      "proof",
      "proof-by-contradiction",
      "proof-by-contrapositive",
      "contrapositive"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.4 Proofs"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「証明 (数学)」",
        "url": "https://ja.wikipedia.org/wiki/%E8%A8%BC%E6%98%8E%20%28%E6%95%B0%E5%AD%A6%29",
        "note": "P をそのまま証明することを直接証明という"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「直接証明」は 0 件（対偶を利用した証明・背理法による証明はある）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "direct proof": 2
      },
      "written": {
        "direct proof": 2
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは Levin の呼び方 direct proof（1.4 Proofs・1.5 Proofs about Discrete Structures・1.6 Chapter Summary）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "direct-proportion",
    "ja": {
      "term": "比例",
      "reading": "ひれい",
      "alt": [
        "比例関係",
        "〜に比例する"
      ]
    },
    "en": {
      "term": "proportional relationship",
      "alt": [],
      "variants": [
        {
          "term": "varies directly",
          "register": "both",
          "note": "書き言葉で最も多い（OpenStax Elementary Algebra・Algebra and Trigonometry）。y varies directly with x の形。"
        },
        {
          "term": "direct variation",
          "register": "both",
          "note": "関係の名前として Algebra の教科書（OpenStax）が使う言い方。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説（第 1 学年）の「比例」は、a を比例定数として y = ax で表される関係。米国の 7 年生の教材（IM Grade 7）はこの関係を proportional relationship と呼び、glossary の見出しにしている。Algebra の教科書（OpenStax）は direct variation（y varies directly with x）と呼ぶ。「y は x に比例する」は y is proportional to x と言う（AP Calculus の CED topic 7.8 もこの形）。",
    "domains": [
      "functions",
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの量 x、y の関係が、0 でない定数 a を使って y = ax の式で表されること。x が 2 倍、3 倍になると、y も 2 倍、3 倍になる。",
    "definition_en": "A relationship between two quantities that can be written as y = ax, where a is a nonzero constant. When x doubles or triples, y doubles or triples too.",
    "latex": "y = ax",
    "spoken_en": "y equals a x",
    "examples": [
      {
        "en": "Careful, y = 2x + 1 isn't a proportional relationship: when x is 0, y is 1, not 0.",
        "ja": "気をつけて。y = 2x + 1 は比例じゃない。x が 0 のとき、y は 0 じゃなくて 1 だからね。",
        "register": "spoken"
      },
      {
        "en": "The distance d, in miles, is proportional to the time t, in hours. A car travels 150 miles in 3 hours. Write an equation relating d and t.",
        "ja": "進んだ道のり d マイルは、走った時間 t 時間に比例する。車が 3 時間で 150 マイル進むとき、d を t の式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "y = ax + b（b ≠ 0）は比例ではない。グラフが原点を通らないので、proportional relationship とも direct variation とも言わない。",
      "名詞の proportion は「比例式」（a : b = c : d、エントリ proportion）の意味で使う。「比例する」は形容詞の proportional で y is proportional to x と言い、関係の名前としては proportional relationship（IM Grade 7）や direct variation（Algebra の教科書）と言う。",
      "direct variation の direct は、反比例の inverse variation（エントリ inverse-proportion）と区別するための語。"
    ],
    "related": [
      "constant-of-proportionality",
      "inverse-proportion",
      "proportion",
      "origin",
      "graph"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "比例",
        "en": "Proportionality (mathematics)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Grade 7 glossary: proportional relationship"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 C 関数: 比例は a を比例定数として y＝ax で表される関係"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 7.8（is proportional to）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "varies directly": 35,
        "direct variation": 24,
        "proportional relationship": 77
      },
      "written": {
        "proportional relationship": 3,
        "varies directly": 87,
        "direct variation": 42
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "directed-graph",
    "ja": {
      "term": "有向グラフ",
      "reading": "ゆうこうぐらふ",
      "alt": []
    },
    "en": {
      "term": "directed graph",
      "alt": [],
      "variants": [
        {
          "term": "directed graph",
          "register": "both",
          "note": "話し言葉はすべて MIT OCW、書き言葉はすべて MIT の講義ノート。"
        },
        {
          "term": "digraph",
          "register": "written",
          "note": "略した言い方。書き言葉はすべて MIT の講義ノートで、話し言葉（MIT OCW）では少ない。書き言葉では directed graph と並ぶ。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "各辺に向きが付いたグラフのこと。辺は頂点の順序対 (u, v) で表し、u から v への矢印でかく。",
    "definition_en": "A graph in which every edge has a direction. Each edge is an ordered pair (u, v) of vertices and is drawn as an arrow from u to v.",
    "latex": "\\sum_{v} \\deg^{+}(v) = \\sum_{v} \\deg^{-}(v) = |E|",
    "spoken_en": "the sum of the out-degrees equals the sum of the in-degrees equals the number of edges",
    "examples": [
      {
        "en": "In a directed graph, the edge from u to v is not the same as the edge from v to u.",
        "ja": "有向グラフでは、u から v への辺と v から u への辺は別の辺だ。",
        "register": "spoken"
      },
      {
        "en": "Draw the directed graph with vertices 1, 2, 3, 4 and an edge from a to b whenever a < b. Find the in-degree and out-degree of each vertex.",
        "ja": "頂点 1, 2, 3, 4 をもち、a < b のとき a から b への辺をもつ有向グラフをかけ。各頂点の入次数と出次数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "書き言葉では digraph と略すことがある（用例コーパスではすべて MIT で、話し言葉より書き言葉に多い）。directed graph は話し言葉（MIT OCW）・書き言葉（MIT の講義ノート）とも digraph より多い。",
      "有向グラフの次数は、入ってくる辺の数 in-degree（入次数）と出ていく辺の数 out-degree（出次数）に分けて言う。",
      "集合 A 上の関係は、A を頂点とする有向グラフでかける。Levin の本では 2.6 Relations and Graphs に 4 件、Nicholson の線形代数の本では 2.3 Matrix Multiplication に 5 件。"
    ],
    "related": [
      "vertex-graph",
      "path-in-a-graph",
      "degree-vertex",
      "edge-of-a-graph",
      "graph-network",
      "adjacency-matrix",
      "relation"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "directed graph": 24,
        "digraph": 4
      },
      "written": {
        "directed graph": 50,
        "digraph": 17
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "directed-segment",
    "ja": {
      "term": "有向線分",
      "reading": "ゆうこうせんぶん",
      "alt": []
    },
    "en": {
      "term": "directed line segment",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "向きを考えた線分。始点から終点へ矢印をつけて表す。",
    "definition_en": "A line segment with a direction, drawn as an arrow from its initial point to its terminal point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We draw a vector as a directed line segment, an arrow from one point to another.",
        "ja": "ベクトルは有向線分、つまりある点から別の点への矢印でかく。",
        "register": "spoken"
      },
      {
        "en": "Draw the directed line segment from P(1, 2) to Q(4, 0).",
        "ja": "P(1, 2) から Q(4, 0) への有向線分をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector",
      "initial-point",
      "terminal-point"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "directed line segment": 21
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "direction-angle",
    "ja": {
      "term": "方向角",
      "reading": "ほうこうかく",
      "alt": []
    },
    "en": {
      "term": "direction angle",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "ベクトルが x 軸の正の向き（空間では各座標軸の正の向き）となす角。",
    "definition_en": "The angle a vector makes with the positive x-axis (in space, with each positive coordinate axis).",
    "latex": "\\tan\\theta = \\frac{b}{a}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The direction angle is measured counterclockwise from the positive x-axis.",
        "ja": "方向角は x 軸の正の向きから反時計回りに測る。",
        "register": "spoken"
      },
      {
        "en": "Find the magnitude and direction angle of the vector v = ⟨−3, 3⟩.",
        "ja": "ベクトル v = (−3, 3) の大きさと方向角を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "空間ベクトルの方向角の余弦は direction cosines（方向余弦）。用例コーパスの書き言葉はほとんどが OpenStax Calculus Volume 3。"
    ],
    "related": [
      "vector",
      "component-form"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "direction angle": 16
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "direction-vector",
    "ja": {
      "term": "方向ベクトル",
      "reading": "ほうこうべくとる",
      "alt": []
    },
    "en": {
      "term": "direction vector",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "直線の向きを表す、直線に平行な 0 でないベクトル。",
    "definition_en": "A nonzero vector parallel to a line, giving its direction.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Any nonzero multiple of the direction vector works just as well.",
        "ja": "方向ベクトルの 0 でない実数倍でも同じように使える。",
        "register": "spoken"
      },
      {
        "en": "Find a direction vector for the line through (1, 0, 3) and (4, 2, −1).",
        "ja": "2 点 (1, 0, 3), (4, 2, −1) を通る直線の方向ベクトルを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector-equation-of-a-line",
      "normal-vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "direction vector": 5
      },
      "written": {
        "direction vector": 29
      },
      "sources": [
        "mit-18.02",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "directional-derivative",
    "ja": {
      "term": "方向微分",
      "reading": "ほうこうびぶん",
      "alt": [
        "方向微分係数"
      ]
    },
    "en": {
      "term": "directional derivative",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "多変数関数の、ある点での単位ベクトル u の向きの変化率。D_u f = ∇f · u。",
    "definition_en": "The rate of change of a function of several variables at a point in the direction of a unit vector u, D_u f = ∇f · u.",
    "latex": "D_{\\mathbf{u}} f = \\nabla f \\cdot \\mathbf{u}",
    "spoken_en": "D sub u of f equals del f dot u",
    "examples": [
      {
        "en": "The directional derivative is biggest in the direction of the gradient.",
        "ja": "方向微分は、勾配の向きで最大になる。",
        "register": "spoken"
      },
      {
        "en": "Find the directional derivative of f(x, y) = x²y at (1, 2) in the direction of ⟨3, 4⟩.",
        "ja": "f(x, y) = x²y の点 (1, 2) における ⟨3, 4⟩ 方向の方向微分を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "u は単位ベクトルにしてから内積をとる。"
    ],
    "related": [
      "gradient",
      "partial-derivative",
      "level-curve"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "方向微分",
        "en": "Directional derivative"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "directional derivative": 34
      },
      "written": {
        "directional derivative": 59
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "directrix",
    "ja": {
      "term": "準線",
      "reading": "じゅんせん",
      "alt": []
    },
    "en": {
      "term": "directrix",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "放物線を「焦点と準線から等距離にある点の軌跡」として定めるときの定直線。",
    "definition_en": "The fixed line that, together with the focus, defines a parabola as the set of points equidistant from both.",
    "latex": "y^2 = 4px: \\; x = -p",
    "spoken_en": null,
    "examples": [
      {
        "en": "Every point on the parabola is the same distance from the focus and the directrix.",
        "ja": "放物線上のどの点も、焦点と準線から同じ距離にある。",
        "register": "spoken"
      },
      {
        "en": "Find the focus and the directrix of the parabola y² = 12x.",
        "ja": "放物線 y² = 12x の焦点と準線を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形は directrices。話し言葉にはほとんど出てこず、書き言葉（OpenStax Algebra and Trigonometry）の語。"
    ],
    "related": [
      "focus",
      "conic-section",
      "eccentricity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "directrix": 4
      },
      "written": {
        "directrix": 182
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "discontinuous",
    "ja": {
      "term": "不連続",
      "reading": "ふれんぞく",
      "alt": []
    },
    "en": {
      "term": "discontinuous",
      "alt": [],
      "variants": [
        {
          "term": "discontinuous",
          "register": "both",
          "note": "書き言葉で最も多い。話し言葉では Khan Academy が半分近くを占め、書き言葉はほとんどが OpenStax Precalculus と OpenStax Calculus。"
        },
        {
          "term": "not continuous",
          "register": "both",
          "note": "話し言葉ではこちらが多い（Khan Academy・Professor Leonard）。書き言葉でも使う（OpenStax Calculus・OpenStax Precalculus）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "その点で連続でないこと。",
    "definition_en": "Not continuous at a point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "There's a jump there, so it's not continuous at x = 1.",
        "ja": "そこで跳んでいるので、x = 1 で連続でない。",
        "register": "spoken"
      },
      {
        "en": "f is discontinuous at x = 3 because f(3) is undefined.",
        "ja": "f(3) が定義されないので、f は x = 3 で不連続である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "不連続の種類は、英語では removable（除去可能）・jump（跳躍）・infinite（無限）と名前で呼び分ける。"
    ],
    "related": [
      "continuous",
      "continuous-function",
      "limit"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.13（discontinuous）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "discontinuous": 47,
        "not continuous": 68
      },
      "written": {
        "discontinuous": 77,
        "not continuous": 42
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "discrete-random-variable",
    "ja": {
      "term": "離散型確率変数",
      "reading": "りさんがたかくりつへんすう",
      "alt": []
    },
    "en": {
      "term": "discrete random variable",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "とる値がとびとびで、数え上げられる確率変数。",
    "definition_en": "A random variable whose possible values are separate and can be listed.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The number of heads is a discrete random variable, but the time you wait isn't.",
        "ja": "表の出る回数は離散型確率変数だが、待ち時間はそうではない。",
        "register": "spoken"
      },
      {
        "en": "Is X a discrete or a continuous random variable? Explain.",
        "ja": "X は離散型確率変数か連続型確率変数か。説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "continuous-random-variable",
      "probability-distribution",
      "random-variable"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "4 章 Discrete Random Variables"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Random_variable",
        "note": "節 Discrete random variable"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.9（discrete random variable）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "discrete random variable": 6
      },
      "written": {
        "discrete random variable": 41
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "discriminant",
    "ja": {
      "term": "判別式",
      "reading": "はんべつしき",
      "alt": []
    },
    "en": {
      "term": "discriminant",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "二次方程式 ax²+bx+c=0 の b²−4ac のこと。符号で実数解の個数が決まる。",
    "definition_en": "The quantity b²−4ac attached to ax²+bx+c=0; its sign tells you how many real solutions there are.",
    "latex": "b^2 - 4ac",
    "spoken_en": "b squared minus four a c",
    "examples": [
      {
        "en": "The discriminant is negative, so there are no real solutions.",
        "ja": "判別式が負なので実数解はありません。",
        "register": "spoken"
      },
      {
        "en": "Since b² − 4ac = 0, the equation has a double root.",
        "ja": "b²−4ac=0 より、この方程式は重解をもつ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the discriminant is positive/zero/negative",
        "ja": "判別式が正／0／負である"
      },
      {
        "en": "check the discriminant",
        "ja": "判別式を調べる"
      }
    ],
    "pitfalls": [
      "共通テスト・センター試験（数学II）の問題文は「判別式を D とすると」と置き（平成 28・31 年度 第4問、令和 5 年度 第4問）、日本語版 Wikipedia「判別式」も D で表記する。OpenStax Algebra and Trigonometry は二次方程式の判別式に文字を置かず b² − 4ac のまま the discriminant と呼び、OpenStax Calculus Volume 3 は円錐曲線や 2 変数関数の判別式を D と置く。"
    ],
    "related": [
      "quadratic-formula"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "判別式",
        "en": "Discriminant"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "平成 28・31 年度 数学II 第4問、令和 5 年度 数学II 第4問: 判別式を D とする"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "円錐曲線の判別式 D = 4AC − B²、2 変数関数の判別式 D"
      }
    ],
    "confidence": "verified",
    "respelling": "dih-SKRIM-uh-nunt",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "discriminant": 32
      },
      "written": {
        "discriminant": 101
      },
      "sources": [
        "khan-algebra",
        "mit-18.02",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "disk-method",
    "ja": {
      "term": "円板法",
      "reading": "えんばんほう",
      "alt": []
    },
    "en": {
      "term": "disk method",
      "alt": [],
      "variants": [
        {
          "term": "disk method",
          "register": "both",
          "note": "話し言葉は Khan Academy が中心、書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "disc method",
          "register": "spoken",
          "note": "AP の CED の綴り（topic 8.9・8.10）。話し言葉では disk method より少なく、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学III）には、回転体の体積の求め方の名前が出てこない。OpenStax Calculus は disk method ／ washer method ／ shell method と名前で呼び分ける。見出しの「円板法」は学習指導要領解説に無い（日本語版 Wikipedia「回転体」にはある）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "回転体を回転軸に垂直な薄い円板に分け、その体積を積分して回転体の体積を求める方法。",
    "definition_en": "A way to get the volume of a solid of revolution: cut it into thin disks perpendicular to the axis of rotation and integrate their volumes.",
    "latex": "V = \\pi \\int_a^b \\bigl(R(x)\\bigr)^2\\,dx",
    "spoken_en": "V equals pi times the integral from a to b of R of x, squared, dx",
    "examples": [
      {
        "en": "There's no hole in the middle, so the disk method works here.",
        "ja": "真ん中に穴がないから、ここは円板法でいける。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は disc method と綴る（topic 8.9・8.10）。OpenStax Calculus Volume 1 は disk method。"
    ],
    "related": [
      "washer-method",
      "shell-method",
      "disk",
      "volume-of-a-solid-of-revolution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.9–8.10（disc method）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「回転体」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%9E%E8%BB%A2%E4%BD%93",
        "note": "円板法"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III: 回転体の体積の求め方の名前が出てこない・「円板法」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "disc method": 5,
        "disk method": 20
      },
      "written": {
        "disk method": 25
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "disk",
    "ja": {
      "term": "円板",
      "reading": "えんばん",
      "alt": []
    },
    "en": {
      "term": "disk",
      "alt": [],
      "uk": "disc",
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Geometry",
        "AP Calculus AB",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "円とその内部をあわせた図形。回転体の体積では、薄い円板の体積を足し合わせて考える。",
    "definition_en": "A circle together with its interior; in volume problems, a thin disk is one slice of a solid of revolution.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each slice is a thin disk with radius f of x, so its volume is pi f of x squared times dx.",
        "ja": "切り口は半径 f(x) の薄い円板だから、体積は π f(x)² dx。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英国の伝統的な綴りは disc。OpenStax Calculus は disk と綴るが、AP の CED は disc method（topic 8.9・8.10）と書く。",
      "circle は円周（線）だけを指す。中身まで含むなら disk。"
    ],
    "related": [
      "disk-method",
      "volume-of-a-solid-of-revolution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円板",
        "en": "Disk (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences",
        "note": "記事「American and British English spelling differences」の disc ／ disk の行（伝統的に disc は英国、disk は米国の綴り）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.9–8.10（disc method）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "disk": 186
      },
      "written": {
        "disk": 212
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "displacement",
    "ja": {
      "term": "変位",
      "reading": "へんい",
      "alt": []
    },
    "en": {
      "term": "displacement",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "動く点の、始めの位置から終わりの位置までの位置の変化。速度の定積分 ∫ₐᵇ v(t)dt で求まり、終わりの位置は x(b) = x(a) + ∫ₐᵇ v(t)dt。",
    "definition_en": "The change in position of a moving object, final minus initial, given by ∫ₐᵇ v(t) dt; the final position is x(b) = x(a) + ∫ₐᵇ v(t) dt.",
    "latex": "\\int_a^b v(t)\\,dt",
    "spoken_en": "the integral from a to b of v of t dt",
    "examples": [
      {
        "en": "Displacement can be negative. It just means you ended up to the left of where you started.",
        "ja": "変位は負にもなる。出発点より左で止まったというだけ。",
        "register": "spoken"
      },
      {
        "en": "If you know where the particle starts, integrating the velocity tells you where it ends up.",
        "ja": "点が最初どこにあるか分かっていれば、速度を積分すると最後にどこにいるかが分かる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（数学III）は「位置の変化」と言い、「変位」は物理の項目にだけ出てくる。",
      "実際に動いた道のりは total distance traveled で、速度の絶対値を積分する。",
      "OpenStax Calculus は数直線上を動く点を a particle moving along a line と言い、位置を s(t)、速度を v(t) と書く。AP の CED は rectilinear motion と言う（topic 4.2・8.2）。",
      "velocity は向き（符号）を含み、speed はその絶対値。"
    ],
    "related": [
      "total-distance-traveled",
      "net-change",
      "definite-integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.2（位置・速度・加速度と積分。displacement と total distance）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 積分法: 時刻 t₁ から t₂ までの点 P の位置の変化"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "displacement": 70
      },
      "written": {
        "displacement": 96
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "distance-formula",
    "ja": {
      "term": "距離の公式",
      "reading": "きょりのこうしき",
      "alt": [
        "2 点間の距離"
      ]
    },
    "en": {
      "term": "distance formula",
      "alt": [
        "distance between two points"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3",
        "数II"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "座標平面上の 2 点 (x₁, y₁)、(x₂, y₂) の間の距離 d = √((x₂ − x₁)² + (y₂ − y₁)²) を求める公式。2 点を結ぶ線分を斜辺とする直角三角形に三平方の定理を使って導く。",
    "definition_en": "The formula d = √((x₂ − x₁)² + (y₂ − y₁)²) giving the distance between two points of the coordinate plane. It is the Pythagorean theorem applied to the right triangle whose hypotenuse joins the two points.",
    "latex": "d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}",
    "spoken_en": "d equals the square root of x sub two minus x sub one, squared, plus y sub two minus y sub one, squared",
    "examples": [
      {
        "en": "Plug the points into the distance formula: 3 squared plus 4 squared is 25, and the square root of 25 is 5.",
        "ja": "2 点を距離の公式に代入しよう。3 の 2 乗 + 4 の 2 乗 = 25 で、25 の平方根は 5 だ。",
        "register": "spoken"
      },
      {
        "en": "Use the distance formula to find the distance between the points A(−2, 1) and B(4, 9).",
        "ja": "距離の公式を用いて、2 点 A(−2, 1)、B(4, 9) の間の距離を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "中学校学習指導要領解説は、三平方の定理の活用として座標平面における 2 点間の距離を求めることを挙げ、高等学校学習指導要領解説（数学II）は座標を用いて二点間の距離を表すことを扱う。OpenStax Algebra and Trigonometry（2.1）・Intermediate Algebra（11.1）は Distance Formula という名前の公式として扱う。",
      "x どうし・y どうしを引く。引く順番は逆でもよい（2 乗すると符号が消える）が、x と y を混ぜない。",
      "中点の公式（midpoint formula）と取り違えない（エントリ midpoint-formula）。"
    ],
    "related": [
      "pythagorean-theorem",
      "coordinate-plane",
      "midpoint-formula",
      "distance",
      "distance-from-a-point-to-a-line"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 三平方の定理: 座標平面における 2 点間の距離"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 図形と方程式: 座標を用いて二点間の距離を表す"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "distance formula": 13,
        "distance between two points": 3
      },
      "written": {
        "distance between two points": 10,
        "distance formula": 71
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "distance-from-a-point-to-a-line",
    "ja": {
      "term": "点と直線の距離",
      "reading": "てんとちょくせんのきょり",
      "alt": []
    },
    "en": {
      "term": "distance from a point to a line",
      "alt": [
        "distance between a point and a line"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "点から直線に下ろした垂線の長さ。点 (x₁, y₁) と直線 ax + by + c = 0 の距離は |ax₁ + by₁ + c|/√(a² + b²)。",
    "definition_en": "The length of the perpendicular from a point to a line; for (x₁, y₁) and ax + by + c = 0 it is |ax₁ + by₁ + c|/√(a² + b²).",
    "latex": "d = \\frac{|ax_1 + by_1 + c|}{\\sqrt{a^2 + b^2}}",
    "spoken_en": "d equals the absolute value of a x one plus b y one plus c, over the square root of a squared plus b squared",
    "examples": [
      {
        "en": "The distance from a point to a line is measured along the perpendicular.",
        "ja": "点と直線の距離は垂線に沿って測る。",
        "register": "spoken"
      },
      {
        "en": "Find the distance from the point (2, 3) to the line 3x + 4y − 5 = 0.",
        "ja": "点 (2, 3) と直線 3x + 4y − 5 = 0 の距離を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "点 (x₁, y₁) と直線 ax + by + c = 0 の距離の公式は、CED・OpenStax・IM・CK-12 に出てこない。IM Geometry（7.3）・CK-12 Geometry（4.21）は点と直線の最短距離が垂線に沿うことだけを使い、OpenStax Calculus Volume 3（2.5）は空間の点と直線の距離を d = ‖PM⃗ × v‖/‖v‖ で求める。"
    ],
    "related": [
      "perpendicular-lines",
      "equation-of-a-line",
      "tangent-line",
      "distance",
      "foot-of-the-perpendicular",
      "distance-formula",
      "position-of-a-line-relative-to-a-circle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "点と直線の距離",
        "en": "Distance from a point to a line"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.5 Equations of Lines and Planes in Space（Distance from a Point to a Line）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "7.3 Tangent Lines（the shortest distance from a point to a line）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.21（点と直線の最短距離）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "distance from a point to a line": 3,
        "distance between a point and a line": 3
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 distance from a point to a line（本文 3 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "distance",
    "ja": {
      "term": "距離",
      "reading": "きょり",
      "alt": [
        "道のり"
      ]
    },
    "en": {
      "term": "distance",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 点を結ぶ線分の長さ。座標平面では √((x₂ − x₁)² + (y₂ − y₁)²)。点と図形の距離は、最も近い点までの長さ。",
    "definition_en": "The length of the segment joining two points, √((x₂ − x₁)² + (y₂ − y₁)²) on the coordinate plane; from a point to a figure, the length to its closest point.",
    "latex": "\\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}",
    "spoken_en": "the square root of x two minus x one squared plus y two minus y one squared",
    "examples": [
      {
        "en": "The distance is never negative, so take the absolute value.",
        "ja": "距離は負にならないので、絶対値をとる。",
        "register": "spoken"
      },
      {
        "en": "Find the distance between the points A(−1, 3) and B(5, −5).",
        "ja": "2 点 A(−1, 3), B(5, −5) 間の距離を求めよ。",
        "register": "written"
      },
      {
        "en": "A car travels at 45 miles per hour for 3 hours. Find the distance it travels.",
        "ja": "時速 45 マイルで 3 時間走る車が進む道のりを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "distance between",
        "ja": "〜の間の距離"
      }
    ],
    "pitfalls": [
      "2 点間の距離の式を、OpenStax と CK-12 Geometry は distance formula と名前で呼ぶ。",
      "運動の「道のり」も英語は distance（total distance traveled）。「変位」は displacement。"
    ],
    "related": [
      "distance-from-a-point-to-a-line",
      "midpoint",
      "total-distance-traveled",
      "displacement"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "距離",
        "en": "Distance"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.37 Distance Formula and the Pythagorean Theorem（distance formula）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "distance": 1583
      },
      "written": {
        "distance": 1355
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "distinguish",
    "ja": {
      "term": "区別する",
      "reading": "くべつする",
      "alt": []
    },
    "en": {
      "term": "distinguish",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "英語の問題文では「区別する／区別しない」を動詞ではなく形容詞で言うことがある。OpenStax Algebra and Trigonometry（13.5 Counting Principles）は distinct objects ／ non-distinct objects（indistinguishable とも）と書き、Levin（3.5 Counting Multisets）は identical objects・indistinguishable と書く。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "2 つ以上のものを、互いに違うものとして見分けて扱う。場合の数では、同じ種類のものを別々に数えるかどうかで答えが変わる。",
    "definition_en": "To treat two or more objects as different from each other; in counting, whether identical-looking objects are distinguished changes the count.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If the red balls are identical, we can't distinguish them, so swapping two of them doesn't give a new arrangement.",
        "ja": "赤玉を区別しないなら、2 つを入れ替えても新しい並べ方にはならない。",
        "register": "spoken"
      },
      {
        "en": "Do we distinguish the two boxes or not? That changes the answer.",
        "ja": "2 つの箱を区別するかしないかで答えが変わるよ。",
        "register": "spoken"
      },
      {
        "en": "In how many ways can 5 identical balls be placed into 3 distinct boxes?",
        "ja": "区別のつかない 5 個の玉を、区別のある 3 つの箱に入れる方法は何通りあるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "identical は「同一の」ではなく「見た目が同じで区別しない」の意味で使う（identical balls ＝ 区別しない玉）。区別する側は distinct か different。",
      "distinguish A from B ／ distinguish between A and B は「A と B を見分ける」の一般的な意味でも使う。"
    ],
    "related": [
      "permutation-of-a-multiset",
      "combination-with-repetition",
      "partitioning-into-groups",
      "counting"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles（distinct ／ non-distinct objects）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.5 Counting Multisets（identical objects、indistinguishable）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "distinguish": 27
      },
      "written": {
        "distinguish": 40
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "distribute",
    "ja": {
      "term": "分配する",
      "reading": "ぶんぱいする",
      "alt": []
    },
    "en": {
      "term": "distribute",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "分配法則を使って、かっこの外の数や式をかっこの中のそれぞれの項に掛け、かっこを外す。",
    "definition_en": "To use the distributive property by multiplying the number or expression outside the parentheses by each term inside, so that the parentheses go away.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Distribute the 3 to both terms: 3 times x is 3x, and 3 times 4 is 12.",
        "ja": "3 を両方の項に分配しよう。3 × x で 3x、3 × 4 で 12。",
        "register": "spoken"
      },
      {
        "en": "Distribute and simplify: 4(2x − 3) − 2(x + 5).",
        "ja": "分配法則を使ってかっこを外し、4(2x − 3) − 2(x + 5) を整理せよ。",
        "register": "written"
      },
      {
        "en": "Now distribute the negative, so every sign inside flips: −(x − 5) becomes −x + 5.",
        "ja": "今度はマイナスをかっこの中に分配するから、中の符号が全部変わる。−(x − 5) は −x + 5 になる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "distribute の目的語は、かっこの外で掛ける数。distribute the 3 は「3 をかっこの中の各項に掛ける」、distribute the negative は「−1 を分配して、かっこの中の符号をすべて変える」。",
      "かっこの中のすべての項に掛ける。3(x + 4) を 3x + 4 とするのは、2 つ目の項への掛け忘れ。",
      "名詞の distribution は、統計では「分布」（エントリ distribution）の意味で使う。分配法則は distributive property（エントリ distributive-property）。"
    ],
    "related": [
      "distributive-property",
      "expand",
      "parentheses",
      "combine-like-terms",
      "simplify"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A 数と式: 一次式の加法と減法（2(3x + 4) − 3(x − 5)）。第 2 学年: 整式の加法と減法（level.jp）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "distribute": 694
      },
      "written": {
        "distribute": 354
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "distribution",
    "ja": {
      "term": "分布",
      "reading": "ぶんぷ",
      "alt": []
    },
    "en": {
      "term": "distribution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "AP Statistics",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データの値がどのように散らばっているかの様子。ヒストグラムなどで形・中心・散らばりを見る。",
    "definition_en": "The way the values of the data are spread out, seen in a graph such as a histogram through its shape, center, and spread.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When you describe a distribution, talk about its shape, center, and spread, and point out any outliers.",
        "ja": "分布を説明するときは、形・中心・散らばりと外れ値について述べる。",
        "register": "spoken"
      },
      {
        "en": "Describe the distribution of the test scores.",
        "ja": "テストの得点の分布を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "確率分布（probability distribution）も英語は distribution で、統計では両方の意味で使う。"
    ],
    "related": [
      "histogram",
      "be-skewed",
      "spread",
      "measure-of-center"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "distribution": 1041
      },
      "written": {
        "distribution": 1546
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "distributive-property",
    "ja": {
      "term": "分配法則",
      "reading": "ぶんぱいほうそく",
      "alt": []
    },
    "en": {
      "term": "distributive property",
      "alt": [
        "distributive law"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic",
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "かっこの外の数を、かっこの中のそれぞれの項に掛けてから足しても結果が変わらないという法則。a(b + c) = ab + ac。",
    "definition_en": "The rule that multiplying a number by a sum gives the same result as multiplying the number by each term and then adding: a(b + c) = ab + ac.",
    "latex": "a(b + c) = ab + ac",
    "spoken_en": "a times the quantity b plus c equals a b plus a c",
    "examples": [
      {
        "en": "Distribute the 3: the distributive property says 3(x + 4) is 3x + 12.",
        "ja": "3 を分配しよう。分配法則から 3(x + 4) は 3x + 12 だ。",
        "register": "spoken"
      },
      {
        "en": "Use the distributive property to simplify −2(x − 5).",
        "ja": "分配法則を使って、−2(x − 5) を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "授業では、分配法則を使う操作を動詞で distribute the 3（3 をかっこの中の各項に掛ける）のように言う。",
      "負の数を分配するときは符号に注意。−(x − 5) は −x + 5 で、−x − 5 ではない。かっこの中のすべての項に −1 を掛ける。",
      "かっこの中が積のときは分配しない。2(3x) は 6x で、3 と x の両方に 2 を掛けて 12x とするのは誤り。"
    ],
    "related": [
      "commutative-property",
      "associative-property",
      "parentheses",
      "term",
      "like-terms"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "分配法則",
        "en": "Distributive property"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "distributive law": 16,
        "distributive property": 69
      },
      "written": {
        "distributive property": 177,
        "distributive law": 2
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "diverge-to-negative-infinity",
    "ja": {
      "term": "負の無限大に発散する",
      "reading": "ふのむげんだいにはっさんする",
      "alt": []
    },
    "en": {
      "term": "decrease without bound",
      "alt": [
        "diverge to negative infinity"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "英語の教科書（OpenStax）では decreases without bound と書くことが多く、diverges to negative infinity は少ない。式では lim f(x) = −∞ と書き、口頭では goes to negative infinity（minus infinity）とも言う。",
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "変数を近づけたとき、関数や数列の値が負で、その絶対値が限りなく大きくなる。",
    "definition_en": "Of a function or sequence: to become less than any given number as the variable approaches its target.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "As x approaches zero from the left, one over x goes to negative infinity.",
        "ja": "x を左から 0 に近づけると、1/x は負の無限大に発散する。",
        "register": "spoken"
      },
      {
        "en": "As x → 0⁺, ln x decreases without bound.",
        "ja": "x → +0 のとき、log x は負の無限大に発散する。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は自然対数を ln x、底を省いた log x を常用対数（底 10）と書く（学習指導要領解説には ln が出てこない）。"
    ],
    "related": [
      "diverge-to-positive-infinity",
      "divergence",
      "infinity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "ln が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "decrease without bound": 33,
        "diverge to negative infinity": 3
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "diverge-to-positive-infinity",
    "ja": {
      "term": "正の無限大に発散する",
      "reading": "せいのむげんだいにはっさんする",
      "alt": [
        "無限大に発散する"
      ]
    },
    "en": {
      "term": "increase without bound",
      "alt": [
        "diverge to positive infinity"
      ],
      "variants": [
        {
          "term": "diverge to infinity",
          "register": "both",
          "note": "件数は少ない。話し言葉にはほとんど出てこず、書き言葉はすべて OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「正の無限大に発散する」は、英語の教科書（OpenStax）では increases without bound と書くことが多く、diverges to infinity は少ない。式では lim f(x) = ∞ と書き、口頭では goes to infinity とも言う。lim f(x) = ∞ と書いても、極限が実数として存在するわけではない。",
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "変数を近づけたとき、関数や数列の値が限りなく大きくなる。",
    "definition_en": "Of a function or sequence: to become larger than any given number as the variable approaches its target.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "As x goes to zero from the right, one over x just goes to infinity.",
        "ja": "x を右から 0 に近づけると、1/x は正の無限大に発散する。",
        "register": "spoken"
      },
      {
        "en": "As n → ∞, the sequence aₙ = n² increases without bound.",
        "ja": "n → ∞ のとき、数列 aₙ = n² は正の無限大に発散する。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "diverge-to-negative-infinity",
      "divergence",
      "infinity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "diverge to positive infinity": 2,
        "diverge to infinity": 2
      },
      "written": {
        "increase without bound": 42,
        "diverge to infinity": 6
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "diverge",
    "ja": {
      "term": "発散する",
      "reading": "はっさんする",
      "alt": []
    },
    "en": {
      "term": "diverge",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "数列・級数・関数が収束しない。正または負の無限大に発散する場合と、振動する場合がある。",
    "definition_en": "Not to converge: to grow without bound in either direction, or to oscillate without settling.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The terms don't go to zero, so the series diverges.",
        "ja": "項が 0 に近づかないから、この級数は発散する。",
        "register": "spoken"
      },
      {
        "en": "Show that the sequence aₙ = (−1)ⁿ diverges.",
        "ja": "数列 aₙ = (−1)ⁿ が発散することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "diverge to infinity",
        "ja": "正の無限大に発散する"
      }
    ],
    "pitfalls": [
      "振動する数列も diverge。OpenStax Calculus Volume 2（5.1）は収束しない数列を divergent sequence と呼び、振動する (−1)ⁿ もその例に挙げる。名詞は divergence（divergence を参照）。"
    ],
    "related": [
      "converge",
      "divergence",
      "diverge-to-positive-infinity",
      "oscillate",
      "nth-term-test",
      "limit-of-a-sequence",
      "infinite-series"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences（divergent sequence、(−1)ⁿ）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 10、Topic 10.1–10.9（converge ／ diverge）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "diverge": 310
      },
      "written": {
        "diverge": 292
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "divergence-theorem",
    "ja": {
      "term": "発散定理",
      "reading": "はっさんていり",
      "alt": [
        "ガウスの発散定理"
      ]
    },
    "en": {
      "term": "divergence theorem",
      "alt": [],
      "variants": [
        {
          "term": "Gauss's theorem",
          "register": "spoken",
          "note": "MIT OCW（18.02）の言い方。話し言葉はすべて MIT OCW で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "閉曲面 S を通る外向きの流束を、S で囲まれた立体 E 上の発散の三重積分で表す定理。",
    "definition_en": "The theorem that the outward flux through a closed surface S equals the triple integral of the divergence over the solid E it encloses.",
    "latex": "\\iint_S \\mathbf{F} \\cdot d\\mathbf{S} = \\iiint_E \\nabla \\cdot \\mathbf{F}\\,dV",
    "spoken_en": null,
    "examples": [
      {
        "en": "For a closed surface, the divergence theorem is usually the easy way to get the flux.",
        "ja": "閉曲面なら、流束は発散定理で求めるのがたいてい楽。",
        "register": "spoken"
      },
      {
        "en": "Use the divergence theorem to find the outward flux of F = ⟨x, y, z⟩ across the sphere x² + y² + z² = 9.",
        "ja": "発散定理を用いて、球面 x² + y² + z² = 9 を通る F = ⟨x, y, z⟩ の外向きの流束を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "divergence-vector",
      "surface-integral",
      "triple-integral",
      "stokes-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "発散定理",
        "en": "Divergence theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "divergence theorem": 31,
        "Gauss's theorem": 10
      },
      "written": {
        "divergence theorem": 98
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "divergence-vector",
    "ja": {
      "term": "発散（ベクトル解析）",
      "reading": "はっさん・べくとるかいせき",
      "alt": []
    },
    "en": {
      "term": "divergence",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトル場 F の各点での湧き出しの強さを表すスカラー ∇ · F。",
    "definition_en": "The scalar ∇ · F that measures how much a vector field F flows outward from each point.",
    "latex": "\\operatorname{div}\\mathbf{F} = \\nabla \\cdot \\mathbf{F} = \\frac{\\partial P}{\\partial x} + \\frac{\\partial Q}{\\partial y} + \\frac{\\partial R}{\\partial z}",
    "spoken_en": "div F equals del dot F",
    "examples": [
      {
        "en": "Positive divergence means more is flowing out than in.",
        "ja": "発散が正なら、流れ込むより流れ出るほうが多い。",
        "register": "spoken"
      },
      {
        "en": "Compute the divergence of F(x, y, z) = ⟨x², yz, xz²⟩.",
        "ja": "F(x, y, z) = ⟨x², yz, xz²⟩ の発散を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数列・級数の発散（divergence）と同じ語。",
      "書くときは div F とも ∇ · F とも書く（div F は書き言葉によく出てくる）。"
    ],
    "related": [
      "curl-vector",
      "vector-field",
      "divergence-theorem",
      "divergence"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the divergence of": 22
      },
      "written": {
        "the divergence of": 49
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "divergence",
    "ja": {
      "term": "発散",
      "reading": "はっさん",
      "alt": []
    },
    "en": {
      "term": "divergence",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "数列や関数が収束しないこと。正の無限大・負の無限大に発散する場合と、振動する場合がある。",
    "definition_en": "Failing to converge: going to infinity or negative infinity, or oscillating.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "No limit means divergence: it either blows up or bounces around.",
        "ja": "極限がなければ発散。限りなく大きくなるか、振動するかのどちらか。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the sequence aₙ = (−1)ⁿ converges or diverges.",
        "ja": "数列 aₙ = (−1)ⁿ が収束するか発散するかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の divergence はベクトル場の発散（div）も指す（多変数の微積分）。",
      "日本語版 Wikipedia「極限」が振動も発散の一種とするのと同じく、英語でも振動する数列は diverge する（OpenStax Calculus Volume 2 は (−1)ⁿ の数列が oscillates するので diverges と書く）。収束しないもの全体が divergence。"
    ],
    "related": [
      "convergence",
      "oscillate",
      "diverge-to-positive-infinity",
      "diverge-to-negative-infinity",
      "diverge",
      "divergence-vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 10（divergence）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「極限」",
        "url": "https://ja.wikipedia.org/wiki/%E6%A5%B5%E9%99%90",
        "note": "振動も発散の一種"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "divergence": 173
      },
      "written": {
        "divergence": 267
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "divide",
    "ja": {
      "term": "割る",
      "reading": "わる",
      "alt": []
    },
    "en": {
      "term": "divide",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "割り算をする。a を b で割るとは、b に何を掛けると a になるかを求めることで、a ÷ b と書く（b は 0 でない）。",
    "definition_en": "To find how many times one number fits into another, or to split an amount into equal parts. Dividing a by b is written a ÷ b or a/b, and b cannot be zero.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We've got 2x = 14, so divide both sides by 2, and x is 7.",
        "ja": "2x = 14 だから、両辺を 2 で割って x = 7。",
        "register": "spoken"
      },
      {
        "en": "Divide 3/4 by 2/5, and write your answer as a fraction in simplest form.",
        "ja": "3/4 を 2/5 で割り、答えを既約分数で表せ。",
        "register": "written"
      },
      {
        "en": "Show that 4 divides the sum of any two consecutive odd numbers.",
        "ja": "連続する 2 つの奇数の和は 4 で割り切れることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「A を B で割る」は divide A by B（A ÷ B）。「両辺を 2 で割る」は divide both sides by 2。",
      "divide into は「〜に分ける」の意味で使う（divide the class into 4 groups でクラスを 4 つのグループに分ける）。",
      "数論の B divides A は「B が A を割り切る」（A は B で割り切れる。記号では B | A）。3 divides 12 と 12 divided by 3（12 ÷ 3）では、3 と 12 の並びが逆になる。"
    ],
    "related": [
      "division",
      "multiply",
      "reciprocal",
      "remainder",
      "divisibility"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "divide": 6121
      },
      "written": {
        "divide": 1821
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "dividend",
    "ja": {
      "term": "被除数",
      "reading": "ひじょすう",
      "alt": [
        "割られる数"
      ]
    },
    "en": {
      "term": "dividend",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "小学校",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "割り算で割られる数。a ÷ b の a のこと。",
    "definition_en": "The number being divided in a division problem. In a ÷ b, the dividend is a.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In 24 divided by 6, 24 is the dividend and 6 is the divisor.",
        "ja": "24 ÷ 6 では、24 が割られる数で、6 が割る数だよ。",
        "register": "spoken"
      },
      {
        "en": "In 84 ÷ 7 = 12, identify the dividend, the divisor, and the quotient.",
        "ja": "84 ÷ 7 = 12 で、被除数・除数・商はそれぞれどれか答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "株の「配当」も dividend で、同じ語。",
      "割る数は divisor（エントリ divisor-in-division）。divisor は「約数」の意味にもなる（エントリ divisor）。",
      "a ÷ b を分数 a/b で書くと、被除数 a は分子（numerator）になる。"
    ],
    "related": [
      "divisor-in-division",
      "quotient",
      "remainder",
      "division",
      "divide"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "総説の小学校の内容の表: 第 4 学年「被除数，除数，商及び余りの間の関係」（level.jp 小学校）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "dividend": 23
      },
      "written": {
        "dividend": 95
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "divisibility-rules",
    "ja": {
      "term": "倍数の判定法",
      "reading": "ばいすうのはんていほう",
      "alt": [
        "倍数判定法"
      ]
    },
    "en": {
      "term": "divisibility test",
      "alt": [
        "divisibility rule"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math",
        "Pre-Algebra"
      ]
    },
    "definition_ja": "実際に割り算をせずに、各位の数字の和や下の何桁かだけを見て、ある整数が 3・4・9 などの倍数かどうかを見分ける方法。",
    "definition_en": "A shortcut for deciding whether an integer is divisible by a number such as 3, 4, or 9 by looking at its digit sum or its last few digits, without dividing.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "By the divisibility rule for 3, add up the digits: 1 + 2 + 3 is 6, so 123 is a multiple of 3.",
        "ja": "3 の倍数の判定法で、各位の数字を足すと 1 + 2 + 3 = 6 だから、123 は 3 の倍数。",
        "register": "spoken"
      },
      {
        "en": "Use the divisibility test for 9 to decide whether 4,356 is divisible by 9.",
        "ja": "9 の倍数の判定法を用いて、4356 が 9 の倍数かどうか調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は「〜の倍数」の側から言うが、英語は「割り切れる」の側から言うことができる。「3 の倍数である」は is a multiple of 3 とも is divisible by 3 とも言える。",
      "4 の倍数は下 2 桁、8 の倍数は下 3 桁で判定する。英語では the last two digits ／ the last three digits。",
      "教科書（OpenStax Prealgebra・Elementary Algebra）では divisibility test が多く、divisibility rule はほとんど使わない。"
    ],
    "related": [
      "multiple",
      "digit",
      "classification-by-remainder",
      "modulus",
      "divisibility"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2 Introduction to Number Theory"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "2.4 Find Multiples and Factors（Use Common Divisibility Tests）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Introduction to Whole Numbers（Identify Multiples and Apply Divisibility Tests）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「倍数判定法」",
        "url": "https://ja.wikipedia.org/wiki/%E5%80%8D%E6%95%B0%E5%88%A4%E5%AE%9A%E6%B3%95",
        "note": "整除性の判定法、倍数の判定法ともいう"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "divisibility rule": 3
      },
      "written": {
        "divisibility test": 20,
        "divisibility rule": 1
      },
      "sources": [
        "khan-middle",
        "openstax-elemalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "divisibility",
    "ja": {
      "term": "整除性",
      "reading": "せいじょせい",
      "alt": [
        "整除",
        "割り切れること",
        "割り切れる"
      ]
    },
    "en": {
      "term": "divisibility",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "number-theory"
    ],
    "level": {
      "jp": [
        "中2",
        "数A",
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Discrete Math"
      ]
    },
    "definition_ja": "整数 a が整数 b で割り切れるかどうか、つまり a = bk となる整数 k があるかどうかという性質のこと。b が a を割り切ることを b | a と書く。",
    "definition_en": "The property of one integer dividing another evenly: b divides a when a = bk for some integer k. This is written b | a.",
    "latex": "b \\mid a \\iff a = bk \\text{ for some integer } k",
    "spoken_en": "b divides a if and only if a equals b k for some integer k",
    "examples": [
      {
        "en": "Let's check divisibility by 3: add up the digits and see if the sum is a multiple of 3.",
        "ja": "3 で割り切れるかどうかを調べよう。各位の数字を足して、その和が 3 の倍数かどうかを見る。",
        "register": "spoken"
      },
      {
        "en": "Use the definition of divisibility to prove that if 6 | n, then 3 | n.",
        "ja": "n が 6 の倍数ならば 3 の倍数でもあることを、倍数の定義を用いて証明せよ。",
        "register": "written"
      },
      {
        "en": "Explain why the sum of three consecutive integers is always divisible by 3.",
        "ja": "連続する 3 つの整数の和がいつも 3 で割り切れる理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "b | a は「b が a を割り切る」（b divides a）で、縦棒の左が約数。分数 b/a や a ÷ b と読み違えない。",
      "学習指導要領解説（中学校・高等学校）は約数・倍数の言い方で扱い、「整除性」「整除」は出てこない。センター試験の問題文は「37a が 4 で割り切れる」のように「割り切れる」と言う。",
      "OpenStax Prealgebra（2.4）・Elementary Algebra（1.1）の divisibility は、ほとんどが divisibility tests（倍数の判定法。エントリ divisibility-rules）の形で出てくる。整除性そのもの（b が a を割り切る関係）を divisibility と呼ぶのは、Levin（6.2 の the divisibility relation）や MIT の講義ノート（6.042）。",
      "「a は 3 で割り切れる」は a is divisible by 3。中 2 の証明では、3 の倍数になること（3 × 整数の形に書けること）を示して割り切れることを言う。"
    ],
    "related": [
      "divisibility-rules",
      "remainder",
      "euclidean-algorithm",
      "multiple",
      "greatest-common-divisor"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「整除性」「整除」は出てこない（約数・倍数の言い方）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "「整除性」「整除」は出てこない（約数・倍数の言い方）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成29年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20180601/https://www.dnc.ac.jp/albums/abm.php?f=abm00009412.pdf&n=%E6%95%B0%E5%AD%A6%E2%85%A0%E3%83%BB%E6%95%B0%E5%AD%A6%EF%BC%A1.pdf",
        "note": "第 4 問: 37a が 4 で割り切れるのは"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "2.4 Find Multiples and Factors（Use Common Divisibility Tests。pitfalls[2]）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.1 Introduction to Whole Numbers（Apply Divisibility Tests。pitfalls[2]）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2 Introduction to Number Theory（the divisibility relation。pitfalls[2]）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "divisibility": 19
      },
      "written": {
        "divisibility": 33
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "division-algorithm",
    "ja": {
      "term": "除法の原理",
      "reading": "じょほうのげんり",
      "alt": []
    },
    "en": {
      "term": "division algorithm",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "number-theory"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Discrete Math",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "整式 A と 0 でない整式 B に対して、A = BQ + R（R = 0 か、R の次数 < B の次数）を満たす商 Q と余り R がただ 1 組あるという事実。整数でも同じで、余りは 0 ≦ R < |B|。",
    "definition_en": "The fact that for polynomials A and B ≠ 0 there is exactly one pair Q, R with A = BQ + R and R = 0 or deg R < deg B; for integers the remainder satisfies 0 ≤ R < |B|.",
    "latex": "A = BQ + R",
    "spoken_en": "A equals B Q plus R",
    "examples": [
      {
        "en": "By the division algorithm, f of x equals x minus 2 times q of x, plus some constant r.",
        "ja": "除法の原理から、f(x) = (x − 2)q(x) + r（r は定数）と書ける。",
        "register": "spoken"
      },
      {
        "en": "Use the division algorithm to write x³ + 1 in the form (x − 1)Q(x) + R.",
        "ja": "除法の原理を使って、x³ + 1 を (x − 1)Q(x) + R の形に表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは書き言葉（大半が OpenStax Algebra and Trigonometry）にだけ出てきて、話し言葉には出てこない。名前は algorithm（手順）だが、OpenStax Algebra and Trigonometry（Dividing Polynomials）と Levin（6.2）は商と余りがただ 1 組決まるという事実を the division algorithm と呼び、Nicholson（Appendix D Polynomials）は割り算の手順そのものをこう呼ぶ。"
    ],
    "related": [
      "polynomial-division",
      "remainder",
      "remainder-theorem",
      "euclidean-algorithm"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Dividing Polynomials（the Division Algorithm）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「除法の原理」",
        "url": "https://ja.wikipedia.org/wiki/%E9%99%A4%E6%B3%95%E3%81%AE%E5%8E%9F%E7%90%86",
        "note": "記事名（英: division theorem）: n = am + b（0 ≤ b < m）の a, b がただ 1 組"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「剰余の定理」",
        "url": "https://ja.wikipedia.org/wiki/%E5%89%B0%E4%BD%99%E3%81%AE%E5%AE%9A%E7%90%86",
        "note": "多項式における除法の原理（deg r < deg d）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式（因数定理の項）: A＝BQ＋R（R の次数＜B の次数）という関係式"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2 Introduction to Number Theory（the division algorithm）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "Appendix D Polynomials（the division algorithm）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "division algorithm": 18
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "division",
    "ja": {
      "term": "除法",
      "reading": "じょほう",
      "alt": []
    },
    "en": {
      "term": "division",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある数を別の数で割る計算（割り算）。結果を商といい、0 で割ることはできない。",
    "definition_en": "The operation of dividing one number by another; the answer is called the quotient. Division by zero is undefined.",
    "latex": "a \\div b = \\frac{a}{b} \\quad (b \\neq 0)",
    "spoken_en": "a divided by b equals a over b, where b is not zero",
    "examples": [
      {
        "en": "Division by zero is undefined, so 7 divided by 0 just doesn't have an answer.",
        "ja": "0 で割る割り算は定義されないから、7 ÷ 0 には答えがないんだ。",
        "register": "spoken"
      },
      {
        "en": "Rewrite the division as a multiplication, and then find the value of (−2/3) ÷ (4/9).",
        "ja": "除法を乗法に直して、(−2/3) ÷ (4/9) を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "÷ は divided by と読む（12 divided by 4）。演算の名前は division、動詞は divide（エントリ divide）、答えの「商」は quotient。",
      "0 で割ること（division by zero）は定義されない（undefined）。0 を 0 でない数で割るのはふつうに計算でき、0 ÷ 5 = 0。",
      "a ÷ b は分数 a/b と同じ値。a/b は a over b と読む。"
    ],
    "related": [
      "divide",
      "multiplication",
      "reciprocal",
      "fraction",
      "remainder"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "除法",
        "en": "Division (mathematics)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "division": 468
      },
      "written": {
        "division": 590
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "divisor-in-division",
    "ja": {
      "term": "除数",
      "reading": "じょすう",
      "alt": [
        "割る数"
      ]
    },
    "en": {
      "term": "divisor",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "割り算で、割るほうの数。a ÷ b の b のこと。",
    "definition_en": "The number you divide by. In a ÷ b, the divisor is b.",
    "latex": "\\text{dividend} \\div \\text{divisor} = \\text{quotient}",
    "spoken_en": "the dividend divided by the divisor equals the quotient",
    "examples": [
      {
        "en": "In 24 divided by 6, 6 is the divisor, the number you're dividing by.",
        "ja": "24 ÷ 6 では、6 が除数、つまり割る数だよ。",
        "register": "spoken"
      },
      {
        "en": "In 84 ÷ 7 = 12, name the dividend, the divisor, and the quotient.",
        "ja": "84 ÷ 7 = 12 で、被除数・除数・商をそれぞれ答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "divisor は「約数」の意味でも使う（エントリ divisor）。6 is a divisor of 24 は「6 は 24 の約数」で、割り算の除数の話ではない。",
      "割られる数（被除数）は dividend（エントリ dividend）、答えの商は quotient（エントリ quotient）。dividend ÷ divisor = quotient の順に並ぶ。",
      "分数 a/b で割る数に当たる b は、denominator（分母、エントリ denominator）と呼ぶ。"
    ],
    "related": [
      "division",
      "dividend",
      "quotient",
      "divisor",
      "remainder"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the divisor is | divide by the divisor | dividend and divisor | dividend by the divisor | divisor and the dividend": 4
      },
      "written": {
        "the divisor is | divide by the divisor | dividend and divisor | dividend by the divisor | divisor and the dividend": 17
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "divisor",
    "ja": {
      "term": "約数",
      "reading": "やくすう",
      "alt": []
    },
    "en": {
      "term": "factor",
      "alt": [],
      "variants": [
        {
          "term": "divisor",
          "register": "spoken",
          "note": "話し言葉では factor に次いで使う。数論や割り算の文脈の言い方。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "米国の小中学校の教材（IM・OpenStax Prealgebra）は「約数」を factor と言い、the factors of 12 のように使う。divisor も同じ意味だが、数論や割り算の文脈で使われる。factor は多項式の「因数」の意味にもなるので、日本語の「約数」と「因数」の両方に当たる。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "ある整数を割り切ることのできる整数。たとえば 12 の約数は 1, 2, 3, 4, 6, 12 である。",
    "definition_en": "A whole number that divides another number evenly, with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "What are the factors of 18? Think of pairs that multiply to 18: 1 and 18, 2 and 9, 3 and 6.",
        "ja": "18 の約数は何かな。かけて 18 になる組を考えよう。1 と 18、2 と 9、3 と 6。",
        "register": "spoken"
      },
      {
        "en": "List all the factors of 36.",
        "ja": "36 の約数をすべて書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "factor は多項式の「因数」（x − 2 is a factor of x² − 4、エントリ factor）や、動詞「因数分解する」（factor x² − 4）にもなる。約数の意味かどうかは文脈で決まる。",
      "divisor は割り算の「割る数」（a ÷ b の b、エントリ divisor-in-division）の意味にもなる。",
      "「a は b の約数」は a is a factor of b ／ a divides b。逆向きの「b は a の倍数」は b is a multiple of a（エントリ multiple）。"
    ],
    "related": [
      "multiple",
      "number-of-divisors",
      "greatest-common-divisor",
      "prime-factorization",
      "divisor-in-division",
      "factor"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "約数",
        "en": "Divisor"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 6)",
        "url": "https://im.kendallhunt.com/MS/students/1/index.html",
        "note": "Grade 6: the factors of 21 and 6（greatest common factor の単元）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "is a factor of | are factors of | factors of": 346,
        "divisor": 141
      },
      "written": {
        "is a factor of | are factors of | factors of": 454,
        "divisor": 140
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "domain-and-range",
    "ja": {
      "term": "変域",
      "reading": "へんいき",
      "alt": [
        "定義域と値域"
      ]
    },
    "en": {
      "term": "domain and range",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「変域」1 語に当たる語は用例コーパスと参照に出てこない。x の変域は domain（定義域）、y の変域は range（値域）と、x と y で言い分ける。見出しの domain and range は、2 つをまとめて言うときの形。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "関数で、変数がとる値の範囲。x の変域を定義域、y の変域を値域という。",
    "definition_en": "The sets of values that the variables of a function can take. The possible x-values form the domain, and the resulting y-values form the range.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's find the domain and range: this graph runs from x = 1 to x = 4, and from y = 2 to y = 8.",
        "ja": "変域を求めよう。このグラフは x = 1 から x = 4 まで、y = 2 から y = 8 までだね。",
        "register": "spoken"
      },
      {
        "en": "For y = 2x, the values of x go from −1 to 3. Write the domain and range as inequalities.",
        "ja": "y = 2x で、x の値は −1 から 3 までとする。x の変域と y の変域を不等式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の −1 ≦ x ≦ 3（共通テスト・センター試験の問題文も ≦ を使う）は、英語では ≤ を使って −1 ≤ x ≤ 3 と書く。区間記法 [−1, 3] で書くこともある（エントリ interval）。",
      "range は統計では「範囲」（最大値 − 最小値）の意味でも使う。関数の値域とは別の意味。",
      "domain を「領域」と訳さない。不等式が表す平面の「領域」は region と言う。"
    ],
    "related": [
      "function",
      "find-the-domain",
      "interval",
      "inequality",
      "variable"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "問題文の不等号 ≦"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "domain and range": 14
      },
      "written": {
        "domain and range": 143
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "domain",
    "ja": {
      "term": "定義域",
      "reading": "ていぎいき",
      "alt": []
    },
    "en": {
      "term": "domain",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "関数で、x がとることのできる値全体の範囲。中学校学習指導要領解説は「x の変域」と言う。",
    "definition_en": "The set of all input values (x-values) that a function can take.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The domain is all the x-values you're allowed to plug in.",
        "ja": "定義域というのは、代入していい x の値全部のこと。",
        "register": "spoken"
      },
      {
        "en": "If the domain of y = −2x + 5 is −1 ≤ x ≤ 3, find all possible values of y.",
        "ja": "関数 y = −2x + 5 の定義域が −1 ≦ x ≦ 3 のとき、y のとりうる値の範囲を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "中学の「x の変域」も英語では domain（エントリ domain-and-range）。",
      "定義域は domain、値域は range。domain は入力（x）の側、range は出力（y）の側で、取り違えやすい。",
      "定義域を求めるは find the domain（エントリ find-the-domain）。"
    ],
    "related": [
      "domain-and-range",
      "find-the-domain",
      "function",
      "restricted-domain",
      "linear-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "定義域",
        "en": "Domain of a function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "変域（第 1 学年 比例・反比例ほか）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the domain !and | domain of": 298
      },
      "written": {
        "the domain !and | domain of": 884
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "dot-plot",
    "ja": {
      "term": "ドットプロット",
      "reading": "どっとぷろっと",
      "alt": []
    },
    "en": {
      "term": "dot plot",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数直線の上に、データの値ごとに点（●）を 1 つずつ積み上げて表した図。どの値がいくつあるかと、データの散らばり方がひと目で分かる。",
    "definition_en": "A graph that shows data by stacking one dot for each data value above a number line. It shows how often each value occurs and how the data are spread out.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each dot on the dot plot is one student, so if you read 4 books, add a dot above the 4.",
        "ja": "ドットプロットの点 1 つが生徒 1 人。本を 4 冊読んだなら、4 の上に点を 1 つ足してね。",
        "register": "spoken"
      },
      {
        "en": "The dot plot shows the number of books each student read last month. How many students read more than 3 books?",
        "ja": "ドットプロットは、各生徒が先月読んだ本の冊数を表している。4 冊以上読んだ生徒は何人か。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "IM（6〜8 年生の教材）は、dot plot を line plot と呼ぶこともあると書く。line plot は折れ線グラフ（line graph、エントリ line-graph）とは別の図。",
      "dot plot は 1 つの変数の値を数直線に並べる図。2 つの変数を点で表す散布図（scatter plot、エントリ scatter-plot）とは違う。"
    ],
    "related": [
      "histogram",
      "line-graph",
      "scatter-plot",
      "stem-and-leaf-plot",
      "number-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ドットプロット",
        "en": "Dot plot (statistics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 8.1 Got Data?（dot plot を line plot とも呼ぶ）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "dot plot": 43
      },
      "written": {
        "dot plot": 6
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "dot-product",
    "ja": {
      "term": "内積",
      "reading": "ないせき",
      "alt": [
        "ドット積"
      ]
    },
    "en": {
      "term": "dot product",
      "alt": [
        "inner product",
        "scalar product"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "高校の「内積」は英語の dot product（OpenStax は scalar product とも書く）。大学の線形代数では一般の内積を inner product と呼び、dot product はその特別な場合。用例コーパスの inner product は、話し言葉のほぼすべてが MIT 18.06。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "definition_ja": "2 つのベクトル a, b について |a||b| cos θ で定める実数 a·b。成分では a₁b₁ + a₂b₂。",
    "definition_en": "The number a·b = |a||b| cos θ for vectors a and b; in components, a₁b₁ + a₂b₂.",
    "latex": "\\vec{a}\\cdot\\vec{b} = |\\vec{a}||\\vec{b}|\\cos\\theta = a_1 b_1 + a_2 b_2",
    "spoken_en": "a dot b equals the magnitude of a times the magnitude of b times cosine theta",
    "examples": [
      {
        "en": "If the dot product is zero, the vectors are perpendicular.",
        "ja": "内積が 0 なら、2 つのベクトルは垂直。",
        "register": "spoken"
      },
      {
        "en": "Find the dot product of a = (3, −2) and b = (4, 1), and the angle between them.",
        "ja": "a = (3, −2), b = (4, 1) の内積と、なす角を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a·b は a dot b と読む。",
      "線形代数では inner product（内積）とも言う（用例コーパスの話し言葉ではほとんどが MIT OCW）。OpenStax Calculus は scalar product とも書く（37 件）。"
    ],
    "related": [
      "vector",
      "cauchy-schwarz-inequality",
      "orthogonal-projection",
      "vector-projection",
      "angle-between-vectors",
      "cross-product"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ドット積",
        "en": "Dot product"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "dot product": 169,
        "inner product": 52,
        "scalar product": 2
      },
      "written": {
        "inner product": 2,
        "dot product": 119,
        "scalar product": 37
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "double-angle-formulas",
    "ja": {
      "term": "2 倍角の公式",
      "reading": "にばいかくのこうしき",
      "alt": []
    },
    "en": {
      "term": "double-angle formulas",
      "alt": [
        "double-angle identities"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "sin 2α、cos 2α、tan 2α を α の三角関数で表す公式。加法定理で β = α とおいて得られる。",
    "definition_en": "The formulas for sin 2α, cos 2α, and tan 2α in terms of trigonometric functions of α, obtained from the addition formulas with β = α.",
    "latex": "\\sin 2\\alpha = 2\\sin\\alpha\\cos\\alpha,\\quad \\cos 2\\alpha = \\cos^2\\alpha - \\sin^2\\alpha",
    "spoken_en": null,
    "examples": [
      {
        "en": "Cosine of 2θ has three versions — pick the one that has only cosine in it.",
        "ja": "cos 2θ には 3 通りの形がある。cos だけのものを選ぼう。",
        "register": "spoken"
      },
      {
        "en": "Use a double-angle formula to solve cos 2θ + cos θ = 0 for 0 ≤ θ < 2π.",
        "ja": "2 倍角の公式を用いて、0 ≦ θ < 2π のとき cos 2θ + cos θ = 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "use the double-angle formula",
        "ja": "2 倍角の公式を使う"
      }
    ],
    "pitfalls": [
      "sin・cos・tan のどれの式かは、the double-angle formula for cosine のように for で言う（OpenStax Algebra and Trigonometry、講義でも同じ形）。"
    ],
    "related": [
      "angle-addition-formulas",
      "half-angle-formulas",
      "triple-angle-formulas"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "double-angle formulas": 27,
        "double-angle identities": 3
      },
      "written": {
        "double-angle formulas": 44,
        "double-angle identities": 7
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "double-count",
    "ja": {
      "term": "重複して数える",
      "reading": "ちょうふくしてかぞえる",
      "alt": []
    },
    "en": {
      "term": "double count",
      "alt": [
        "overcount"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Geometry"
      ]
    },
    "definition_ja": "同じ場合を 2 回以上数えてしまい、個数を実際より多く数えること。場合分けの重なりや、並べ方の入れ替えを別々に数えたときに起こる。",
    "definition_en": "To count the same case more than once, so the total comes out too large; it happens when cases overlap or when different orders of the same selection are counted separately.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Careful — if you just add the multiples of 2 and the multiples of 3, you double count the multiples of 6.",
        "ja": "2 の倍数と 3 の倍数をただ足すと、6 の倍数を重複して数えてしまうから気をつけて。",
        "register": "spoken"
      },
      {
        "en": "We overcounted by a factor of 2, because AB and BA are the same pair, so divide by 2.",
        "ja": "AB と BA は同じ組なのに 2 回数えたので、2 で割る。",
        "register": "spoken"
      },
      {
        "en": "To avoid double counting, subtract the number of outcomes in A ∩ B once.",
        "ja": "重複して数えないように、A ∩ B の場合の数を 1 回引く。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語は double count（名詞形 double counting）か overcount（多く数えすぎる）という動詞で言う。「ダブり」のような重なりそのものは overlap。",
      "書き言葉では、IM Geometry（8.6 The Addition Rule）や Levin（3.3・3.8）のように double counting や is counted twice と書く。",
      "組合せ論の double counting は「同じものを 2 通りの方法で数えて等式を示す証明法」の名前でもある（英語版 Wikipedia「Double counting (proof technique)」）。数えすぎのミスとは別の意味。"
    ],
    "related": [
      "inclusion-exclusion-principle",
      "undercount",
      "union-of-events",
      "venn-diagram",
      "counting"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.3（counted twice）・3.8 Advanced Counting Using PIE（double counting、overcount）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 8.6 The Addition Rule（double counting、counted twice）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Double_counting_%28proof_technique%29",
        "note": "記事「Double counting (proof technique)」の冒頭（double counting・counting in two ways。数学カテゴリから 2 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "double count": 21
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "double-integral",
    "ja": {
      "term": "二重積分",
      "reading": "にじゅうせきぶん",
      "alt": []
    },
    "en": {
      "term": "double integral",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "2 変数関数を平面上の領域で積分したもの。f ≥ 0 なら曲面の下の立体の体積を表す。",
    "definition_en": "The integral of a function of two variables over a region of the plane; for f ≥ 0 it gives the volume under the surface.",
    "latex": "\\iint_R f(x, y)\\,dA",
    "spoken_en": "the double integral over R of f of x comma y d A",
    "examples": [
      {
        "en": "Set up the double integral as an iterated integral first.",
        "ja": "まず二重積分を累次積分の形にしよう。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the double integral ∬_R (x + 2y) dA, where R = [0, 1] × [0, 2].",
        "ja": "R = [0, 1] × [0, 2] のとき、二重積分 ∬_R (x + 2y) dA を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "極座標では dA = r dr dθ（r を忘れない）。"
    ],
    "related": [
      "iterated-integral",
      "fubinis-theorem",
      "changing-the-order-of-integration",
      "triple-integral",
      "change-of-variables"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "double integral": 65
      },
      "written": {
        "double integral": 225
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "double-root",
    "ja": {
      "term": "重解",
      "reading": "じゅうかい",
      "alt": []
    },
    "en": {
      "term": "double root",
      "alt": [],
      "variants": [
        {
          "term": "repeated root",
          "register": "spoken",
          "note": "話し言葉で double root に次いで使う。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語では「解」ではなく root（根）や solution・zero の語で言う。OpenStax Elementary Algebra は解が重なるときを double root と呼び、Algebra and Trigonometry は判別式の表で double solution、多項式の零点では double zero と書く。repeated root は Levin（漸化式の特性方程式）ほかにある。用例コーパスの話し言葉の double root・repeated root はほぼ MIT OCW。多項式の根が何回重なっているかは multiplicity（重複度、エントリ multiplicity）で言い、x = 3 is a root of multiplicity 2 のように数で表す。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3",
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "二次方程式の 2 つの解が重なって 1 つになったもの。判別式が 0 のときで、x² − 6x + 9 = 0 は (x − 3)² = 0 となり、x = 3 が重解。",
    "definition_en": "A solution of a quadratic equation where the two solutions are the same number, so it counts twice. This happens when the discriminant is 0, as in (x − 3)² = 0.",
    "latex": "x^2 - 6x + 9 = (x - 3)^2 = 0 \\Rightarrow x = 3",
    "spoken_en": "x squared minus six x plus nine equals the quantity x minus three, squared, which equals zero, so x equals three",
    "examples": [
      {
        "en": "See how the graph just touches the x-axis at 3 instead of crossing it? That's a double root.",
        "ja": "グラフが x 軸を横切らずに、x = 3 で接しているだけでしょ？ だから x = 3 は重解。",
        "register": "spoken"
      },
      {
        "en": "Show that x² + 10x + 25 = 0 has a double root, and find the root.",
        "ja": "方程式 x² + 10x + 25 = 0 が重解をもつことを示し、その重解を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "解の個数の数え方に注意。日本語版 Wikipedia「判別式」は、判別式が 0 のとき重複する実数解を 1 個もつと書く。重なりを込めて 2 つと数えるときは、Nicholson のように counting multiplicities と断る（エントリ fundamental-theorem-of-algebra）。",
      "グラフでは、重解の x 座標のところで放物線が x 軸に接する（横切らない）。",
      "判別式 b² − 4ac が 0 なら重解、正なら異なる 2 つの実数解、負なら実数解はない（エントリ discriminant）。"
    ],
    "related": [
      "discriminant",
      "quadratic-equation",
      "zeros-of-a-polynomial",
      "fundamental-theorem-of-algebra",
      "solving-by-factoring"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction (4th ed.)",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "漸化式の特性方程式の repeated root"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「判別式」",
        "url": "https://ja.wikipedia.org/wiki/%E5%88%A4%E5%88%A5%E5%BC%8F",
        "note": "判別式が 0 のとき 1 個の重複する実数解をもつ"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "counting multiplicities（重複を込めて数える）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学I: 二次方程式 - 判別式（level.jp 数I）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "double root": 29,
        "repeated root": 14
      },
      "written": {
        "double root": 7,
        "repeated root": 7
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "draw-a-graph",
    "ja": {
      "term": "グラフをかく",
      "reading": "ぐらふをかく",
      "alt": []
    },
    "en": {
      "term": "graph",
      "alt": [
        "draw the graph"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "「グラフをかく」は、英語では動詞の graph 1 語で言う（Graph y = 2x − 1.）。draw the graph とも言い、概形だけをかくときは sketch the graph と言う。名詞の graph と同じ語で、問題文の文頭の Graph は命令文の動詞。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "式や関数を満たす点を座標平面上にとり、それらを結んでグラフをかくこと。",
    "definition_en": "To draw the graph of an equation or function on the coordinate plane, usually by plotting points and connecting them.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To graph y = 2x − 1, make a quick table, plot a few points, and connect them with a straight line.",
        "ja": "y = 2x − 1 のグラフをかくには、簡単な表を作って点をいくつかとり、直線で結べばいい。",
        "register": "spoken"
      },
      {
        "en": "Graph y = −3x and y = 12/x on the same coordinate plane.",
        "ja": "y = −3x と y = 12/x のグラフを、同じ座標平面上にかけ。",
        "register": "written"
      },
      {
        "en": "When you graph y = 12/x, don't connect the two pieces across the y-axis.",
        "ja": "y = 12/x のグラフをかくときは、y 軸をまたいで 2 つの部分をつながないこと。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「かく」を write と訳さない。グラフは動詞の graph か、draw the graph で言う。",
      "sketch the graph（エントリ sketch-the-graph）は、増減や漸近線など形の特徴が分かるように概形をかくこと。"
    ],
    "related": [
      "graph",
      "plot-a-point",
      "table-of-values",
      "sketch-the-graph",
      "coordinate-plane"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "draw the graph": 25,
        "graph the | graph each | graph this | graph these | graph it": 460
      },
      "written": {
        "graph the | graph each | graph this | graph these | graph it": 964,
        "draw the graph": 50
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "draw",
    "ja": {
      "term": "取り出す",
      "reading": "とりだす",
      "alt": []
    },
    "en": {
      "term": "draw",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語の「取り出す」は中から外へ出す一般の動詞だが、英語の確率の問題では、袋から玉やカードを取り出すことを draw と言う（draw a marble from the bag、draw a card）。「無作為に取り出す」は draw at random。取り出したものを戻すかどうかは with replacement ／ without replacement で言い分ける（エントリ sampling-with-replacement、sampling-without-replacement）。",
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "袋や箱の中から、玉やカードなどを中を見ずに取り出す。",
    "definition_en": "To take an object such as a marble or a card out of a bag, box, or deck without looking, so that the result is random.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Draw a marble, write down its color, and put it back before you draw again.",
        "ja": "玉を 1 個取り出して色を書きとめたら、次を取り出す前に袋に戻して。",
        "register": "spoken"
      },
      {
        "en": "A bag contains 4 red marbles and 6 blue marbles. Two marbles are drawn one at a time without replacement. Find the probability that both marbles are red.",
        "ja": "袋の中に赤玉 4 個と青玉 6 個が入っている。この袋から玉を 1 個ずつ 2 回取り出す。ただし、取り出した玉は袋に戻さない。2 個とも赤玉である確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "draw には「（図やグラフを）かく」の意味もある（エントリ draw-a-graph）。draw a card は「カードを 1 枚引く」、draw a line は「直線をひく」。",
      "過去形は drew、過去分詞は drawn（Two cards are drawn from the deck.）。",
      "取り出したものを戻すかどうかは、draw の後に with replacement ／ without replacement を付けて言う（draw two cards without replacement）。戻すかどうかで確率が変わる。"
    ],
    "related": [
      "sampling-with-replacement",
      "sampling-without-replacement",
      "select-at-random",
      "probability",
      "draw-a-graph"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "draw a card | draw a marble | draw a ball | drawn at random | drawn from | draw two | draw one | draw a chip | draw a name": 50
      },
      "written": {
        "draw a card | draw a marble | draw a ball | drawn at random | drawn from | draw two | draw one | draw a chip | draw a name": 57
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "dynamical-system",
    "ja": {
      "term": "動的システム",
      "reading": "どうてきしすてむ",
      "alt": [
        "力学系",
        "離散力学系"
      ]
    },
    "en": {
      "term": "dynamical system",
      "alt": [
        "discrete dynamical system"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（dynamical system ／ discrete dynamical system は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "状態ベクトルが行列を掛けるたびに x_{k+1} = Ax_k のように移り変わる仕組み。固有値と固有ベクトルで長い時間の後の様子が分かる。",
    "definition_en": "A system whose state vector changes by a matrix at each step, x_{k+1} = Ax_k; eigenvalues and eigenvectors describe its long-run behavior.",
    "latex": "\\vec{x}_{k+1} = A\\vec{x}_k",
    "spoken_en": null,
    "examples": [
      {
        "en": "The biggest eigenvalue tells you whether the population grows or dies out in the long run.",
        "ja": "いちばん大きい固有値で、個体数が長い目で見て増えるか減るかが分かる。",
        "register": "spoken"
      },
      {
        "en": "Describe the long-term behavior of the dynamical system x_{k+1} = Ax_k.",
        "ja": "動的システム x_{k+1} = Ax_k の長期的なふるまいを説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「力学系」は微分方程式で表す連続の系も含む。線形代数の教科書（Nicholson 3.3）は行列による離散の系を扱う。"
    ],
    "related": [
      "eigenvalue",
      "eigenvector",
      "markov-chain",
      "diagonalization"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "力学系",
        "en": "Dynamical system"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.3 Diagonalization and Eigenvalues"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "dynamical system": 1
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは Nicholson の呼び方 dynamical system（3.3 Diagonalization and Eigenvalues・3.4 An Application to Linear Recurrences・5.5 Similarity and Diagonalization）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "eccentricity",
    "ja": {
      "term": "離心率",
      "reading": "りしんりつ",
      "alt": []
    },
    "en": {
      "term": "eccentricity",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "二次曲線上の点の、焦点までの距離と準線までの距離の比 e。楕円は 0 < e < 1、放物線は e = 1、双曲線は e > 1。",
    "definition_en": "The ratio e of a conic's distance to the focus to its distance to the directrix: 0 < e < 1 for an ellipse, e = 1 for a parabola, e > 1 for a hyperbola.",
    "latex": "e = \\frac{c}{a}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The closer the eccentricity is to zero, the rounder the ellipse.",
        "ja": "離心率が 0 に近いほど、楕円は円に近い。",
        "register": "spoken"
      },
      {
        "en": "Find the eccentricity of the ellipse x²/25 + y²/16 = 1.",
        "ja": "楕円 x²/25 + y²/16 = 1 の離心率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "find the eccentricity",
        "ja": "離心率を求める"
      }
    ],
    "pitfalls": [
      "話し言葉にはほとんど出てこず、書き言葉（OpenStax Calculus・Algebra and Trigonometry）の語。"
    ],
    "related": [
      "conic-section",
      "focus",
      "directrix",
      "ellipse",
      "polar-equation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "離心率",
        "en": "Eccentricity (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "eccentricity": 6
      },
      "written": {
        "eccentricity": 81
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "edge-of-a-graph",
    "ja": {
      "term": "辺（グラフ）",
      "reading": "へん",
      "alt": []
    },
    "en": {
      "term": "edge",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフで 2 つの頂点を結ぶ線。",
    "definition_en": "A line joining two vertices of a graph.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Every edge has two ends, so the degrees add up to twice the number of edges.",
        "ja": "どの辺にも端が 2 つあるので、次数の和は辺の数の 2 倍になる。",
        "register": "spoken"
      },
      {
        "en": "How many edges does a complete graph on 6 vertices have?",
        "ja": "頂点が 6 個の完全グラフの辺の数はいくつか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "立方体などの立体の辺も英語で edge（エントリ edge）。"
    ],
    "related": [
      "vertex-graph",
      "graph-network",
      "path-in-a-graph",
      "adjacency-matrix",
      "edge",
      "eulers-formula-for-polyhedra"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.1 Problems and Definitions"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "edges of the graph | edge of the graph | edges in the graph | an edge between | edge from | number of edges | vertices and edges | edges and vertices | edge connecting | edges connecting": 60
      },
      "written": {
        "edges of the graph | edge of the graph | edges in the graph | an edge between | edge from | number of edges | vertices and edges | edges and vertices | edge connecting | edges connecting": 62
      },
      "sources": [
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "edge",
    "ja": {
      "term": "稜",
      "reading": "りょう",
      "alt": [
        "辺"
      ]
    },
    "en": {
      "term": "edge",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "多面体で、となり合う 2 つの面が交わってできる線分。立方体には 12 本ある。",
    "definition_en": "A line segment where two faces of a solid meet. A cube has 12 edges.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Count the edges on this cube: four on top, four on the bottom, and four going up the sides. That's 12.",
        "ja": "この立方体の辺を数えてみよう。上に 4 本、下に 4 本、縦に 4 本で、12 本。",
        "register": "spoken"
      },
      {
        "en": "Each edge of a cube is 5 cm long. Find the sum of the lengths of all its edges.",
        "ja": "立方体の 1 辺の長さが 5 cm のとき、すべての辺の長さの和を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では、立体の辺は edge と言い（立方体の辺は 12 edges。面・頂点と並べて faces, edges, and vertices）、多角形の辺の side（エントリ side）とは呼び名が違う。ただし立方体の辺の長さは edge length のほかに side length とも言う（IM Grade 6 は a cube with side length s、OpenStax Prealgebra 9.6 は cube with sides of length s と書く）。",
      "グラフ理論の edge（エントリ edge-of-a-graph）も同じ語だが、頂点と頂点を結ぶ線のことで、別の概念。",
      "日常語の edge は「ふち・端」の意味（the edge of the table）。"
    ],
    "related": [
      "face",
      "vertex",
      "side",
      "edge-of-a-graph",
      "eulers-formula-for-polyhedra"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 6)",
        "url": "https://im.kendallhunt.com/MS/students/1/index.html",
        "note": "Grade 6 4.15 Volume of Prisms・4.14 Fractional Lengths in Triangles and Prisms・1.18 Surface Area of a Cube（a cube with side length s）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "9.6 Solve Geometry Applications: Volume and Surface Area（cube with sides of length s。pitfalls[0]）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "edge of the cube | edge of a cube | edge length | edge of the prism | edge of a polyhedron | faces, edges | faces and edges | edges and vertices | edges, and vertices": 2
      },
      "written": {
        "edge of the cube | edge of a cube | edge length | edge of the prism | edge of a polyhedron | faces, edges | faces and edges | edges and vertices | edges, and vertices": 5
      },
      "sources": [
        "mit-notes",
        "openstax-calculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは IM の呼び方 edge of the cube | edge of a cube | edge length | edge of the prism | edge of a polyhedron | faces, edges | faces and edges | edges and vertices | edges, and vertices（Grade 6 4.15 Volume of Prisms・Grade 6 4.14 Fractional Lengths in Triangles and Prisms・Grade 6 1.18 Surface Area of a Cube・計 94 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "eigenspace",
    "ja": {
      "term": "固有空間",
      "reading": "こゆうくうかん",
      "alt": []
    },
    "en": {
      "term": "eigenspace",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（eigenspace は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "固有値 λ に対応する固有ベクトル全体に 0 ベクトルを加えた部分空間。A − λI の零空間。",
    "definition_en": "The subspace made of all eigenvectors for an eigenvalue λ together with the zero vector; the null space of A − λI.",
    "latex": "E_\\lambda = \\operatorname{Nul}(A - \\lambda I)",
    "spoken_en": null,
    "examples": [
      {
        "en": "The eigenspace for lambda equals two is a whole plane here.",
        "ja": "ここでは λ = 2 の固有空間は平面全体だ。",
        "register": "spoken"
      },
      {
        "en": "Find a basis for each eigenspace of the matrix.",
        "ja": "行列の各固有空間の基底を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスには出てこない。"
    ],
    "related": [
      "eigenvalue",
      "eigenvector",
      "null-space",
      "geometric-multiplicity"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.5 Similarity and Diagonalization"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 eigenspace（9.3 Invariant Subspaces and Direct Sums・5.5 Similarity and Diagonalization・11.1 Block Triangular Form）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "eigenvalue",
    "ja": {
      "term": "固有値",
      "reading": "こゆうち",
      "alt": []
    },
    "en": {
      "term": "eigenvalue",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列 A について、0 でないベクトル v で Av = λv となるものがあるときの数 λ。",
    "definition_en": "A number λ for which some nonzero vector v satisfies Av = λv for a square matrix A.",
    "latex": "A\\vec{v} = \\lambda\\vec{v}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The eigenvalues are the roots of the characteristic polynomial.",
        "ja": "固有値は特性多項式の根だ。",
        "register": "spoken"
      },
      {
        "en": "Find the eigenvalues of the matrix and a corresponding eigenvector for each.",
        "ja": "行列の固有値と、それぞれに対応する固有ベクトルを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "eigen はドイツ語由来で「アイゲン」と読む。",
      "用例コーパスの書き言葉はすべて MIT の講義ノート。"
    ],
    "related": [
      "eigenvector",
      "characteristic-polynomial",
      "eigenspace",
      "diagonalization"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "固有値と固有ベクトル",
        "en": "Eigenvalues and eigenvectors"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.3 Diagonalization and Eigenvalues"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "eigenvalue": 869
      },
      "written": {
        "eigenvalue": 64
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "eigenvector",
    "ja": {
      "term": "固有ベクトル",
      "reading": "こゆうべくとる",
      "alt": []
    },
    "en": {
      "term": "eigenvector",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列 A について、Av = λv を満たす 0 でないベクトル v。A を掛けても向きが変わらない（実数倍になる）。",
    "definition_en": "A nonzero vector v with Av = λv for a square matrix A; multiplying by A only scales it.",
    "latex": "A\\vec{v} = \\lambda\\vec{v}",
    "spoken_en": null,
    "examples": [
      {
        "en": "An eigenvector stays on its own line when you apply the transformation.",
        "ja": "変換をしても、固有ベクトルは自分の直線の上にとどまる。",
        "register": "spoken"
      },
      {
        "en": "Find an eigenvector corresponding to the eigenvalue λ = 3.",
        "ja": "固有値 λ = 3 に対応する固有ベクトルを 1 つ求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "0 ベクトルは固有ベクトルに含めない。",
      "用例コーパスの書き言葉はすべて MIT の講義ノート。"
    ],
    "related": [
      "eigenvalue",
      "eigenspace",
      "diagonalization"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "固有値と固有ベクトル",
        "en": "Eigenvalues and eigenvectors"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.3 Diagonalization and Eigenvalues"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "eigenvector": 607
      },
      "written": {
        "eigenvector": 30
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "element",
    "ja": {
      "term": "要素",
      "reading": "ようそ",
      "alt": [
        "元"
      ]
    },
    "en": {
      "term": "element",
      "alt": [
        "member"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Discrete Math",
        "Calculus I"
      ]
    },
    "definition_ja": "集合を作っている一つ一つのもの。a が集合 A の要素であることを a ∈ A と書き、a は A に属するという。",
    "definition_en": "One of the objects that make up a set; a ∈ A says that a is an element of the set A.",
    "latex": "a \\in A",
    "spoken_en": "a is an element of A (or: a is in A)",
    "examples": [
      {
        "en": "Is 5 in this set? Yeah, 5 is an element of A, but 4 isn't.",
        "ja": "5 はこの集合に入っている？ そう、5 は A の要素だけど 4 は違う。",
        "register": "spoken"
      },
      {
        "en": "Determine whether −3 is an element of the set {x | x² + x − 6 = 0}.",
        "ja": "−3 が集合 {x | x² + x − 6 = 0} の要素かどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "∈ は is an element of と読むほか、話し言葉では is in とも言う。「A に属する」は belongs to A。",
      "element は行列の「成分」の意味でも使われる。行列の成分の見出しは entry。",
      "話し言葉では element の形・member の形ともほとんど出てこず、どちらが多いとは言えない。"
    ],
    "related": [
      "set",
      "entry",
      "solution-set",
      "all"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "元 (数学)",
        "en": "Element of a set"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.1 Review of Functions（x is an element of A。level.us Calculus I）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "member of the set | members of the set": 5,
        "is an element of | are elements of | element of the set | elements of the set | elements of a set": 4
      },
      "written": {
        "is an element of | are elements of | element of the set | elements of the set | elements of a set": 13
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "elementary-event",
    "ja": {
      "term": "根元事象",
      "reading": "こんげんじしょう",
      "alt": []
    },
    "en": {
      "term": "outcome",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）には根元事象に当たる名前は出てこず、起こりうる 1 つ 1 つの結果を outcome と呼ぶ（「同様に確からしい」は each outcome is equally likely）。英語版 Wikipedia には記事「Elementary event」がある（elementary event・atomic event・sample point を太字で定義）。日本語版 Wikipedia「標本空間」は単純事象（simple event）とも書く。",
    "domains": [
      "combinatorics",
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "試行の結果として起こりうることのうち、それ以上細かく分けられない 1 つ 1 つの事象。全事象の要素 1 個だけからなる事象。",
    "definition_en": "A single possible result of an experiment that cannot be broken down further; as an event, it is the set containing just that one result.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When you roll two dice there are 36 outcomes, and every outcome is equally likely.",
        "ja": "2 個のさいころを投げると根元事象は 36 個あり、どれも同様に確からしい。",
        "register": "spoken"
      },
      {
        "en": "List all the outcomes in the sample space for tossing a coin three times.",
        "ja": "硬貨を 3 回投げる試行について、根元事象をすべて書き出せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "厳密には、outcome は全事象の要素 ω で、根元事象はそれだけを含む集合 {ω}。AP Statistics の CED（topic 2.3）と OpenStax Introductory Statistics（3.1 Terminology）は、outcome を 1 回の試行の結果、event を outcome の集まりとして分けて定義する。",
      "outcome は「結果」一般にも使う語。確率の文脈で「根元事象の個数」は the number of outcomes と言う。"
    ],
    "related": [
      "sample-space",
      "trial",
      "probability",
      "element",
      "event",
      "equally-likely"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（outcome、event）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "根元事象",
        "en": "Elementary event"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「標本空間」",
        "url": "https://ja.wikipedia.org/wiki/%E6%A8%99%E6%9C%AC%E7%A9%BA%E9%96%93",
        "note": "単純事象（simple event）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.3（outcome、event）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "outcome": 310
      },
      "written": {
        "outcome": 485
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "elementary-matrix",
    "ja": {
      "term": "基本行列",
      "reading": "きほんぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "elementary matrix",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（elementary matrix は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "単位行列に行基本変形を 1 回行ってできる行列。左から掛けるとその行基本変形を行うことになる。",
    "definition_en": "A matrix obtained from the identity matrix by one elementary row operation; multiplying by it on the left performs that operation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each row operation is the same as multiplying by an elementary matrix on the left.",
        "ja": "行基本変形はどれも、基本行列を左から掛けることと同じだ。",
        "register": "spoken"
      },
      {
        "en": "Write A⁻¹ as a product of elementary matrices.",
        "ja": "A⁻¹ を基本行列の積で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉（MIT OCW）に少し出てくるだけ。"
    ],
    "related": [
      "row-operations",
      "invertible-matrix",
      "identity-matrix",
      "lu-decomposition"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "行列の基本変形",
        "en": "Elementary matrix"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "2.5 Elementary Matrices"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "elementary matrix": 7
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 elementary matrix（2.5 Elementary Matrices・3.2 Determinants and Matrix Inverses・2.7 LU-Factorization）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "elementary-symmetric-polynomial",
    "ja": {
      "term": "基本対称式",
      "reading": "きほんたいしょうしき",
      "alt": []
    },
    "en": {
      "term": "elementary symmetric polynomial",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（elementary symmetric polynomial は CED・OpenStax・IM・CK-12 とも 0 件）。用例コーパスの話・書のどちらにも出てこない。見出しは英語版 Wikipedia の記事名「Elementary symmetric polynomial」（冒頭で太字で定義。2 変数では x + y と xy）。授業の英語では名前を出さず、the sum x + y and the product xy と中身を言えば足りる。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": []
    },
    "definition_ja": "対称式を組み立てる基本になる式。2 文字なら x + y と xy、3 文字なら x + y + z、xy + yz + zx、xyz。",
    "definition_en": "The basic symmetric polynomials from which every symmetric polynomial can be built: x + y and xy for two variables, and x + y + z, xy + yz + zx, and xyz for three.",
    "latex": "x + y + z,\\quad xy + yz + zx,\\quad xyz",
    "spoken_en": null,
    "examples": [
      {
        "en": "For two variables, the elementary symmetric polynomials are just the sum x + y and the product xy.",
        "ja": "2 文字の基本対称式は、和 x + y と積 xy だけだ。",
        "register": "spoken"
      },
      {
        "en": "Express x² + y² in terms of the elementary symmetric polynomials x + y and xy.",
        "ja": "x² + y² を基本対称式 x + y, xy で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "解と係数の関係とつながる: 2 次方程式の 2 つの解 α, β の基本対称式 α + β, αβ は係数から求まる。英語の名前は Vieta's formulas（エントリ relationship-between-roots-and-coefficients）。"
    ],
    "related": [
      "symmetric-expression",
      "relationship-between-roots-and-coefficients",
      "polynomial"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Elementary_symmetric_polynomial",
        "note": "記事名 Elementary symmetric polynomial（en.term と同じ。冒頭で太字で定義。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 複素数と方程式: 二つの解を用いて基本対称式の値を計算（1 件）"
      }
    ],
    "confidence": "verified",
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 elementary symmetric polynomial（英語版「Elementary symmetric polynomial」、en.term のリダイレクト先）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "eliminate",
    "ja": {
      "term": "消去する",
      "reading": "しょうきょする",
      "alt": []
    },
    "en": {
      "term": "eliminate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "連立方程式で、式を足したり引いたり、一方の式を他方に代入したりして、ある文字をふくまない式をつくること。",
    "definition_en": "To get rid of a variable in a system of equations by adding or subtracting the equations or by substituting, leaving an equation with one fewer variable.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If I multiply the second equation by 2, I can eliminate x when I subtract.",
        "ja": "2 つ目の式を 2 倍すれば、引いたときに x を消去できる。",
        "register": "spoken"
      },
      {
        "en": "We know y = 2x + 1, so substitute that in and you've eliminated y.",
        "ja": "y = 2x + 1 とわかっているから、それを代入すれば y を消去できる。",
        "register": "spoken"
      },
      {
        "en": "Eliminate y from the system x + y = 5, 2x − 3y = 0, and find the value of x.",
        "ja": "連立方程式 x + y = 5、2x − 3y = 0 から y を消去して、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "eliminate は解く人を主語にして、消す文字を目的語にとる（eliminate x、eliminate the y's）。項どうしが主語になって「消える」ときは cancel out（エントリ cancel-out）。",
      "加減法の英語 elimination method（エントリ elimination）はこの動詞の名詞形を使う。ただし eliminate は、代入法（エントリ substitution-method）で文字を消すときにも使う。"
    ],
    "related": [
      "elimination",
      "cancel-out",
      "substitution-method",
      "substitute",
      "system-of-linear-equations"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "eliminate": 145
      },
      "written": {
        "eliminate": 234
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "elimination",
    "ja": {
      "term": "加減法",
      "reading": "かげんほう",
      "alt": [
        "消去法"
      ]
    },
    "en": {
      "term": "elimination",
      "alt": [
        "addition method"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説の「加減法」は式を足す・引くという操作から付いた名前で、英語は文字を消去することから elimination method（method of elimination）と呼ぶ。OpenStax Algebra and Trigonometry は addition method とも呼ぶ。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "連立方程式の 2 つの式を、必要なら何倍かしてから足したり引いたりして、1 つの文字を消去して解く方法。",
    "definition_en": "A method for solving a system of equations: add or subtract the equations, after multiplying one or both by a number if needed, so that one variable is eliminated.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The y's already have opposite coefficients, so the elimination method is the way to go: just add the equations.",
        "ja": "y の係数がもう符号違いだから、ここは加減法がいい。2 つの式を足すだけ。",
        "register": "spoken"
      },
      {
        "en": "Solve the system 2x + 3y = 7, 4x − 3y = 5 by the elimination method.",
        "ja": "連立方程式 2x + 3y = 7、4x − 3y = 5 を加減法で解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "代入法は substitution method（エントリ substitution-method）。加減法と代入法のどちらでも文字を「消去する」（eliminate、エントリ eliminate）が、elimination method と呼ぶのは加減法のほう。",
      "行列を使うガウスの消去法（Gaussian elimination、エントリ gaussian-elimination）とは別の語。"
    ],
    "related": [
      "substitution-method",
      "eliminate",
      "cancel-out",
      "system-of-linear-equations",
      "gaussian-elimination"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 連立二元一次方程式: 加減法・代入法"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "addition method（11 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "by elimination | using elimination | use elimination | elimination method | method of elimination": 50
      },
      "written": {
        "addition method": 12,
        "by elimination | using elimination | use elimination | elimination method | method of elimination": 105
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "ellipse",
    "ja": {
      "term": "楕円",
      "reading": "だえん",
      "alt": []
    },
    "en": {
      "term": "ellipse",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "2 つの焦点からの距離の和が一定である点の軌跡。",
    "definition_en": "The set of points whose distances to two fixed points, the foci, add up to a constant.",
    "latex": "\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1",
    "spoken_en": "x squared over a squared plus y squared over b squared equals one",
    "examples": [
      {
        "en": "If a equals b, the ellipse is just a circle.",
        "ja": "a = b なら、楕円はただの円。",
        "register": "spoken"
      },
      {
        "en": "Find the foci of the ellipse x²/25 + y²/9 = 1.",
        "ja": "楕円 x²/25 + y²/9 = 1 の焦点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "focus",
      "major-axis",
      "minor-axis",
      "conic-section",
      "eccentricity"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "楕円",
        "en": "Ellipse"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "ellipse": 104
      },
      "written": {
        "ellipse": 541
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "empirical-rule",
    "ja": {
      "term": "経験則",
      "reading": "けいけんそく",
      "alt": [
        "68-95-99.7 則"
      ]
    },
    "en": {
      "term": "empirical rule",
      "alt": [
        "68-95-99.7 rule"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の数B では正規分布表を使って確率を求め、この目安に名前を付けない。日本語の「経験則」は一般には rule of thumb（大ざっぱな目安）の意味で、統計のこの規則を指すとは限らない。",
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "分布が釣鐘型（正規分布に近い形）のとき、データのおよそ 68% が平均から標準偏差 1 つ分以内に、95% が 2 つ分以内に、99.7% が 3 つ分以内に入るという目安。",
    "definition_en": "For a bell-shaped, approximately normal distribution, about 68% of the data fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The heights are roughly bell-shaped, so by the empirical rule about 95% of them are within two standard deviations of the mean.",
        "ja": "身長の分布はおおよそ釣鐘型だから、経験則により、およそ 95% が平均から標準偏差 2 つ分以内に入る。",
        "register": "spoken"
      },
      {
        "en": "Test scores are approximately normal with a mean of 70 and a standard deviation of 8. Use the empirical rule to estimate the percentage of scores between 62 and 78.",
        "ja": "テストの得点は平均 70、標準偏差 8 の正規分布で近似できる。経験則を用いて、得点が 62 と 78 の間にある割合を推定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "使えるのは分布が釣鐘型（正規分布に近い）ときだけ。OpenStax Introductory Statistics もこの条件を注意している。左右に大きく偏った分布には当てはめない。",
      "68% は平均の両側を合わせた割合。平均より上の片側だけなら 34%、平均 + 標準偏差 1 つ分より上は 16% になる。"
    ],
    "related": [
      "normal-distribution",
      "standard-deviation",
      "standard-normal-distribution",
      "z-score",
      "standard-normal-table"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "empirical rule": 9
      },
      "written": {
        "empirical rule": 13,
        "68-95-99.7 rule": 1
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "empty-set",
    "ja": {
      "term": "空集合",
      "reading": "くうしゅうごう",
      "alt": []
    },
    "en": {
      "term": "empty set",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "要素を 1 つも持たない集合。記号 ∅ で表し、どの集合の部分集合でもある。",
    "definition_en": "The set that has no elements, written ∅; it is a subset of every set.",
    "latex": "\\varnothing",
    "spoken_en": "the empty set",
    "examples": [
      {
        "en": "There's no real x with x² = −1, so the solution set is the empty set.",
        "ja": "x² = −1 を満たす実数 x はないので、解の集合は空集合だ。",
        "register": "spoken"
      },
      {
        "en": "Show that A ∩ B is the empty set.",
        "ja": "A ∩ B が空集合であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "{0} は 0 という要素を 1 つ持つ集合で、空集合ではない。{∅} も空集合ではない。",
      "英語版 Wikipedia「Empty set」は null set とも呼ばれると書くが、参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）と用例コーパスには null set は出てこない。empty set 自体も用例コーパスには少なく、話し言葉はすべて MIT OCW、書き言葉はほとんどが MIT の講義ノート。Levin は 17 件使う。"
    ],
    "related": [
      "set",
      "subset",
      "solution-set",
      "element"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "空集合",
        "en": "Empty set"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "0.2.2 Sets: The empty set, written ∅（17 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "empty set": 9
      },
      "written": {
        "empty set": 7
      },
      "sources": [
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 9 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 empty set（本文 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "end-behavior",
    "ja": {
      "term": "関数の終端挙動",
      "reading": "かんすうのしゅうたんきょどう",
      "alt": [
        "終端挙動"
      ]
    },
    "en": {
      "term": "end behavior",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「関数の終端挙動」は学習指導要領解説に無い、本プロジェクトの訳語。日本の数III では x → ±∞ のときの極限（エントリ limit-at-infinity）として扱い、まとめた名前はない。",
    "domains": [
      "functions",
      "algebra"
    ],
    "level": {
      "jp": [
        "数III",
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "x を限りなく大きくしたとき（x → ∞）と限りなく小さくしたとき（x → −∞）に、関数の値がどうなるかという性質。多項式関数では最高次の項で決まる。",
    "definition_en": "What happens to the values of a function as x goes to positive infinity and to negative infinity. For a polynomial, the leading term determines it.",
    "latex": "\\text{as } x \\to \\infty,\\ f(x) \\to -\\infty",
    "spoken_en": "as x approaches infinity, f of x approaches negative infinity",
    "examples": [
      {
        "en": "The leading term is negative 2x cubed, so the end behavior is up on the left and down on the right.",
        "ja": "最高次の項が −2x³ だから、終端挙動は左で上がって右で下がる形だね。",
        "register": "spoken"
      },
      {
        "en": "Describe the end behavior of f(x) = −2x³ + 5x − 1.",
        "ja": "f(x) = −2x³ + 5x − 1 の x → ∞、x → −∞ のときの様子を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "多項式関数は最高次の項（leading term、エントリ leading-term）だけを見ればよい。y = −2x³ なら x → ∞ で y → −∞、x → −∞ で y → ∞。",
      "OpenStax Algebra and Trigonometry は、答えを As x → ∞, f(x) → −∞ の形で書かせる。日本の数III の lim を使う書き方とは形が違う。",
      "behavior は米国の綴り（英国は behaviour）。"
    ],
    "related": [
      "leading-term",
      "limit-at-infinity",
      "polynomial",
      "as-x-approaches-infinity",
      "leading-coefficient"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences#our_or",
        "note": "記事「American and British English spelling differences」の節 -our, -or（英国の behaviour）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "end behavior": 57
      },
      "written": {
        "end behavior": 128
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "endpoint",
    "ja": {
      "term": "端点",
      "reading": "たんてん",
      "alt": [
        "区間の端"
      ]
    },
    "en": {
      "term": "endpoint",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "geometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Geometry",
        "AP Calculus AB"
      ]
    },
    "definition_ja": "線分や半直線の端の点。区間では両端の数（点）で、区間 a ≦ x ≦ b の端点は a と b。",
    "definition_en": "A point at either end of a segment or the starting point of a ray; for an interval, either of the two numbers that bound it, so the endpoints of [a, b] are a and b.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the max on a closed interval, check the vertex and both endpoints.",
        "ja": "閉区間での最大値は、頂点と両端の値を比べて求める。",
        "register": "spoken"
      },
      {
        "en": "Is the endpoint included? If it is, use a bracket; if not, use a parenthesis.",
        "ja": "端点は含まれる？ 含まれるなら角かっこ、含まれないなら丸かっこを使う。",
        "register": "spoken"
      },
      {
        "en": "On the interval [0, 5], f(x) = x² − 4x + 1 has its minimum at the vertex x = 2 and its maximum at the endpoint x = 5.",
        "ja": "0 ≦ x ≦ 5 において、f(x) = x² − 4x + 1 は頂点 x = 2 で最小、区間の端 x = 5 で最大となる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語は線分・半直線・区間のどの端も endpoint 1 語で言い（IM・OpenStax は end point と 2 語に綴ることもある）、「両端」は the endpoints (of the interval)。高等学校学習指導要領解説（数学I 図形と計量）と共通テストは線分・半直線の端を「端点」と書き、区間の端は日本語版 Wikipedia「区間 (数学)」が端点 (endpoints) と呼ぶ。",
      "端点を含むかどうかは区間記法のかっこで表す。[ ] は含む、( ) は含まない。∞ は数ではないので端点として含められず、必ず丸かっこ（英語版 Wikipedia「Interval (mathematics)」は infinite endpoints の節で扱う）。"
    ],
    "related": [
      "interval",
      "closed-interval",
      "open-interval",
      "vertex",
      "maximum",
      "interval-notation",
      "segment"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「区間 (数学)」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8C%BA%E9%96%93_(%E6%95%B0%E5%AD%A6)",
        "note": "用語と表記: 端点 (endpoints)。区間の最小値と最大値を示す 2 つの値"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.1 Geometry Terms: An endpoint is a point at the end of a line segment（endpoint 110 件）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 原点を一方の端点とする半直線"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "平成 30 年度 数学I・A: 辺 AC の端点。令和 8 年度 数学I 第2問: 点 B を端点とする半直線"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.5（endpoints: 閉区間の最大・最小は critical points か endpoints で）・10.13（endpoints of the interval: 収束区間の両端）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Interval_(mathematics)",
        "note": "記事「Interval (mathematics)」: 区間の endpoints（infinite endpoints の節）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/1/index.html",
        "note": "end point と 2 語に綴る（4 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "endpoint | end point": 316
      },
      "written": {
        "endpoint | end point": 319
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "entry",
    "ja": {
      "term": "成分",
      "reading": "せいぶん",
      "alt": [
        "要素"
      ]
    },
    "en": {
      "term": "entry",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「成分」はベクトルの成分にも行列の成分にも使うが、英語はベクトルでは component、行列では entry（または element）と言い分ける。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列の中に並んでいるそれぞれの数。第 i 行第 j 列の成分を (i, j) 成分という。",
    "definition_en": "Each number in a matrix; the one in row i and column j is the (i, j) entry.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The entry in row 2, column 3 is negative 4.",
        "ja": "2 行 3 列の成分は −4。",
        "register": "spoken"
      },
      {
        "en": "Find the entry in row 1, column 2 of the product AB.",
        "ja": "積 AB の 1 行 2 列の成分を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「要素」は集合の要素（element）の意味でも使う。行列の成分は entry、集合の要素は element で、見出しを分けた。"
    ],
    "related": [
      "matrix",
      "components",
      "element"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "entry in row | entries of the matrix": 4
      },
      "written": {
        "entry in row | entries of the matrix": 26
      },
      "sources": [
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "enumeration",
    "ja": {
      "term": "数え上げ",
      "reading": "かぞえあげ",
      "alt": []
    },
    "en": {
      "term": "list all",
      "alt": [],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "用例コーパスでは名詞の enumeration は話し言葉に出てこず、書き言葉にもわずか。list all the possibilities や make a list of all possible outcomes（OpenStax Algebra and Trigonometry 13.7、IM Geometry 8.3）のように動詞句で言う。英語版 Wikipedia には記事「Enumerative combinatorics」（数え上げの組合せ論）がある。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "中2",
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "樹形図や辞書式の順などを使って、起こりうる場合を漏れなく重複なく書き出して数えること。",
    "definition_en": "Counting by writing out every possible case in an organized way, such as with a tree diagram or in dictionary order, so that none is missed or repeated.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The numbers are small, so let's just list all the outcomes and count them.",
        "ja": "数が小さいので、全部書き出して数え上げよう。",
        "register": "spoken"
      },
      {
        "en": "List all the possible outcomes when three coins are tossed.",
        "ja": "3 枚の硬貨を投げるときに起こりうる結果をすべて書き出せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "辞書式に並べるは list in lexicographic (dictionary) order、樹形図は tree diagram。"
    ],
    "related": [
      "counting",
      "tree-diagram",
      "lexicographic-order",
      "permutation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.7 Probability（a list of all possible outcomes）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Enumeration",
        "note": "記事「Enumeration」の冒頭（集まりの要素の完全な並べ上げ）と節 Combinatorics（数学カテゴリから 3 段）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Enumerative_combinatorics",
        "note": "記事「Enumerative combinatorics」の冒頭（数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 8.3 Sample Spaces（list all the possible outcomes）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 D データの活用: 起こり得る場合を順序よく整理し，落ちや重なりがないように数え上げる"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "list all the possible | list all possible | list all the outcomes | list all outcomes | list all the possibilities | list all your options | list all the ways | list all the subsets | list all the permutations | list all the arrangements | list all the combinations": 1
      },
      "written": {
        "list all the possible | list all possible | list all the outcomes | list all outcomes | list all the possibilities | list all your options | list all the ways | list all the subsets | list all the permutations | list all the arrangements | list all the combinations": 9
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-introstats",
        "yt:nancypi"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 list all the possible | list all possible | list all the outcomes | list all outcomes | list all the possibilities | list all your options | list all the ways | list all the subsets | list all the permutations | list all the arrangements | list all the combinations（本文 9 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "epsilon-delta-definition",
    "ja": {
      "term": "ε-δ 論法",
      "reading": "いぷしろんでるたろんぽう",
      "alt": [
        "イプシロン・デルタ論法",
        "極限の厳密な定義"
      ]
    },
    "en": {
      "term": "epsilon-delta definition",
      "alt": [],
      "variants": [
        {
          "term": "precise definition of a limit",
          "register": "written",
          "note": "OpenStax Calculus の節の名前（The Precise Definition of a Limit）。話し言葉には出てこず、書き言葉（OpenStax Calculus）にもほとんど出てこない。"
        },
        {
          "term": "formal definition of a limit",
          "register": "both",
          "note": "少数の言い方。話し言葉は Khan Academy だけ、書き言葉はほとんどが OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "「任意の ε > 0 に対して、0 < |x − a| < δ ならば |f(x) − L| < ε となる δ > 0 がある」という形の、極限の厳密な定義。",
    "definition_en": "The precise definition of a limit: for every ε > 0 there is a δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The epsilon-delta definition makes 'gets close' precise.",
        "ja": "ε-δ 論法は「近づく」を厳密に言い直したもの。",
        "register": "spoken"
      },
      {
        "en": "Use the epsilon-delta definition to prove that lim_{x→2} (3x − 1) = 5.",
        "ja": "ε-δ 論法を用いて lim_{x→2} (3x − 1) = 5 を証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（数学III）には出てこない。AP の CED も topic 1.2 で、ε-δ による定義は AP の試験では問わないと断っている。"
    ],
    "related": [
      "limit",
      "limit-laws"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.2（epsilon-delta definition、試験の対象外）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 極限: ε-δ は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "epsilon-delta definition": 5,
        "formal definition of a limit": 2
      },
      "written": {
        "precise definition of a limit": 2,
        "formal definition of a limit": 5,
        "epsilon-delta definition": 15
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equal-angles",
    "ja": {
      "term": "等しい角",
      "reading": "ひとしいかく",
      "alt": []
    },
    "en": {
      "term": "congruent angles",
      "alt": [
        "equal angles"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "CK-12 Geometry は congruent angles と呼ぶ（1.11 Congruent Angles and Angle Bisectors ほか、31 件。equal angles は 3 件）。用例コーパスには congruent angles が話し言葉に少しあるだけ（大半が The Organic Chemistry Tutor）。大きさの等しい角は congruent angles とも equal angles とも言う。CK-12 Geometry（1.11）は、角そのものの関係を ∠A ≅ ∠B（the angles are congruent）、角の大きさ（measure）の関係を m∠A = m∠B と書き分ける。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2",
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "大きさ（角度）が同じである 2 つ以上の角。",
    "definition_en": "Two or more angles that have the same measure.",
    "latex": "m\\angle A = m\\angle B",
    "spoken_en": "the measure of angle A equals the measure of angle B",
    "examples": [
      {
        "en": "These are congruent angles, so if this one is 40 degrees, that one is 40 too.",
        "ja": "これらは等しい角だから、こっちが 40° ならあっちも 40° だ。",
        "register": "spoken"
      },
      {
        "en": "The base angles of an isosceles triangle are equal angles.",
        "ja": "二等辺三角形の底角は等しい角である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（中学校、第 2 学年）は ∠AOC = ∠BOD のように角の大きさの等しさを = で書く。∠A ≅ ∠B と m∠A = m∠B を書き分けるのは CK-12 Geometry の書き方で、m は measure（大きさ）の m。"
    ],
    "related": [
      "congruent",
      "angle",
      "isosceles-triangle",
      "supplementary-angle",
      "isosceles-triangle-theorem"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.11 Congruent Angles and Angle Bisectors・4.12 Congruent Triangles・5.20 Regular and Irregular Polygons"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 B 図形: ∠AOC＝∠BOD（角の大きさの等しさ）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "congruent angles": 11
      },
      "written": {
        "equal angles": 2
      },
      "sources": [
        "khan-middle",
        "mit-notes",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equal-sign",
    "ja": {
      "term": "等号",
      "reading": "とうごう",
      "alt": []
    },
    "en": {
      "term": "equal sign",
      "alt": [
        "equals sign"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "記号 =。その左右の数や式が等しいことを表す。",
    "definition_en": "The symbol =, which shows that the expressions on its two sides have the same value.",
    "latex": "a = b",
    "spoken_en": "a equals b",
    "examples": [
      {
        "en": "The equal sign means both sides have the same value, not 'write your answer here.'",
        "ja": "等号は、両側が同じ値だという意味。「ここに答えを書く」という意味じゃないよ。",
        "register": "spoken"
      },
      {
        "en": "Fill in the blank so that the equal sign makes a true statement: 8 + 5 = ___ + 6.",
        "ja": "等号の左右が等しくなるように、□ に当てはまる数を求めよ。8 + 5 = □ + 6",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "= は equals または is equal to と読む。3 + 4 = 7 は three plus four equals seven。",
      "「等号が成り立つ」（不等式 A ≧ B で A = B となる）は equality holds と言う（エントリ equality-holds）。"
    ],
    "related": [
      "equation",
      "both-sides",
      "equality-holds",
      "inequality-sign",
      "left-hand-side"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "等号",
        "en": "Equals sign"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "equal sign": 85
      },
      "written": {
        "equal sign": 99
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equal-vectors",
    "ja": {
      "term": "ベクトルの相等",
      "reading": "べくとるのそうとう",
      "alt": []
    },
    "en": {
      "term": "equivalent vectors",
      "alt": [
        "equal vectors"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「ベクトルの相等」は a = b となる条件のこと。OpenStax Calculus は大きさと向きが同じベクトルを equivalent vectors と呼ぶ（11 件）。成分では対応する成分がすべて等しいこと。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "2 つのベクトルの大きさと向きがともに等しいこと。位置が違っても等しいベクトルとみなす。",
    "definition_en": "Two vectors are equal when they have the same magnitude and the same direction, wherever they are placed.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These two arrows are in different places, but they're the same vector.",
        "ja": "この 2 本の矢印は場所が違うが、同じベクトル。",
        "register": "spoken"
      },
      {
        "en": "Show that the vectors from A to B and from C to D are equivalent.",
        "ja": "A から B へのベクトルと C から D へのベクトルが等しいことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector",
      "component-form"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "equal vectors": 1
      },
      "written": {
        "equivalent vectors": 11
      },
      "sources": [
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equal",
    "ja": {
      "term": "等しい",
      "reading": "ひとしい",
      "alt": []
    },
    "en": {
      "term": "equal",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの数や量が同じであること。記号 = で表す。",
    "definition_en": "Having the same value, size, or amount. The symbol = is written between two quantities that are equal.",
    "latex": "A = B",
    "spoken_en": "A is equal to B",
    "examples": [
      {
        "en": "AB and AC are equal, so this is an isosceles triangle.",
        "ja": "AB と AC が等しいから、これは二等辺三角形だ。",
        "register": "spoken"
      },
      {
        "en": "In △ABC, AB = AC. Prove that the measures of ∠B and ∠C are equal.",
        "ja": "△ABC で AB = AC のとき、∠B = ∠C であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "A = B は A equals B か A is equal to B と読む。形容詞で言うときは to が要る（A is equal B とは言わない）。",
      "CK-12 Geometry は、線分そのものが重なることを congruent（≅）、その長さが同じことを equal（=）と分けて書く（AB ≅ CD のとき AB = CD）。共通テスト・センター試験の問題文は、線分の長さや角にも = を使う（AB = AC、∠ABC = B）（エントリ congruent、equal-angles）。",
      "記号 = そのものは equal sign（エントリ equal-sign）。「等号が成り立つ」は equality holds（エントリ equality-holds）。"
    ],
    "related": [
      "equal-sign",
      "congruent",
      "equal-angles",
      "equation",
      "equality-holds"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.4 Midpoints and Segment Bisector（congruent segments と長さ AB = BC を分けて書く）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成27年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20160601/https://www.dnc.ac.jp/albums/abm.php?f=abm00004808.pdf&n=2701-0402+suugaku1A.pdf",
        "note": "AB = AC = 5 と長さに = を使う"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和3年度 大学入学共通テスト 第1日程 数学Ⅰ（問題）",
        "url": "https://web.archive.org/web/20220601/https://www.dnc.ac.jp/albums/abm.php?d=88&f=abm00001271.pdf&n=R3_%E6%95%B0%E5%AD%A6%E2%85%A0%EF%BC%88%EF%BC%91%E6%9C%8816%E6%97%A5%E3%83%BB17%E6%97%A5%EF%BC%89%E3%81%AE%E5%95%8F%E9%A1%8C.pdf",
        "note": "∠ABC = B と角に = を使う"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "equal to | are equal | is equal | equal in | be equal": 16166
      },
      "written": {
        "equal to | are equal | is equal | equal in | be equal": 1124
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equality-holds",
    "ja": {
      "term": "等号成立",
      "reading": "とうごうせいりつ",
      "alt": [
        "等号が成り立つ"
      ]
    },
    "en": {
      "term": "equality holds",
      "alt": [],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスで equality holds は話し言葉・書き言葉ともほとんど出てこない）。Nicholson は Cauchy inequality の等号の場合を if equality holds（5.3）、equality occurs if and only if …（10.1）と書き、OpenStax Calculus Volume 1（5.2 The Definite Integral）は where equality holds only at the endpoints と書く。「等号成立条件」に当たる condition for equality・equality condition は、用例コーパスにも参照にも出てこない。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Calculus I",
        "Linear Algebra"
      ]
    },
    "definition_ja": "不等式 A ≧ B で A = B となること。",
    "definition_en": "In an inequality A ≥ B, the case A = B.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "And the two sides are equal exactly when a equals b.",
        "ja": "そして両辺が等しくなるのは、ちょうど a = b のとき。",
        "register": "spoken"
      },
      {
        "en": "Use the AM-GM inequality to find the minimum value of x + 4/x for x > 0 and the value of x at which equality holds.",
        "ja": "相加平均と相乗平均の関係を用いて、x > 0 のとき x + 4/x の最小値と、等号が成り立つときの x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "相加平均と相乗平均の関係で最小値を求めるとき、等号が成り立つ x が範囲内にあることを確かめる。"
    ],
    "related": [
      "am-gm-inequality",
      "prove",
      "equal",
      "hold",
      "cauchy-schwarz-inequality"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.3 Orthogonality（if equality holds）・10.1 Inner Products and Norms（equality occurs if and only if）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "5.2 The Definite Integral（where equality holds only at the endpoints）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "equality holds": 3
      },
      "written": {
        "equality holds": 4
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 3 件 ／ 書 4 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "equality-of-complex-numbers",
    "ja": {
      "term": "複素数の相等",
      "reading": "ふくそすうのそうとう",
      "alt": []
    },
    "en": {
      "term": "equality of complex numbers",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。Nicholson（Appendix A）は、2 つの複素数は実部どうし・虚部どうしが等しいときに等しいと定め、実部と虚部を等しいとおく手順を equating real and imaginary parts と書く。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "a, b, c, d が実数のとき、a + bi = c + di は a = c かつ b = d と同じであること。",
    "definition_en": "For real a, b, c, d, a + bi = c + di exactly when a = c and b = d.",
    "latex": "a + bi = c + di \\iff a = c,\\ b = d",
    "spoken_en": "a plus b i equals c plus d i if and only if a equals c and b equals d",
    "examples": [
      {
        "en": "Set the real parts equal and the imaginary parts equal, and you get two equations.",
        "ja": "実部どうし、虚部どうしを等しいとおくと、式が 2 つできる。",
        "register": "spoken"
      },
      {
        "en": "If x and y are real and (x + 2i)(1 − i) = 5 + yi, find x and y.",
        "ja": "x, y が実数で (x + 2i)(1 − i) = 5 + yi のとき、x と y を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a, b, c, d が実数であることが前提。その条件を書き落とさない（where a, b, c, d are real）。"
    ],
    "related": [
      "complex-number",
      "real-part",
      "imaginary-part"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "Appendix A Complex Numbers（equal if and only if they have the same real parts and the same imaginary parts、equating real and imaginary parts）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「複素数」",
        "url": "https://ja.wikipedia.org/wiki/%E8%A4%87%E7%B4%A0%E6%95%B0",
        "note": "節「相等関係」"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "equally-likely",
    "ja": {
      "term": "同様に確からしい",
      "reading": "どうようにたしからしい",
      "alt": []
    },
    "en": {
      "term": "equally likely",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "ある試行で、どの結果が起こることも同じ程度に期待できること。",
    "definition_en": "Having the same chance of happening. When all the outcomes of an experiment are equally likely, each outcome has the same probability.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's a fair die, so all six outcomes are equally likely.",
        "ja": "公正なさいころだから、6 つの目の出方はどれも同様に確からしい。",
        "register": "spoken"
      },
      {
        "en": "A bag contains 3 red marbles and 5 blue marbles, and each marble is equally likely to be drawn. Find the probability of drawing a red marble.",
        "ja": "袋の中に赤玉 3 個と青玉 5 個が入っていて、どの玉が取り出されることも同様に確からしい。赤玉を取り出す確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は「同様に確からしい」という決まった句だが、英語は形容詞 likely に副詞 equally を付けた形で、equally likely outcomes ／ Each outcome is equally likely. のように使う。",
      "結果が 2 通りだからといって、同様に確からしいとは限らない。画びょうを投げて針が上を向くか下を向くかは equally likely ではない。",
      "fair（公正な）は、硬貨やさいころのどの面も equally likely に出ることを言う形容詞（a fair coin、a fair die）。"
    ],
    "related": [
      "probability",
      "number-of-possible-outcomes",
      "elementary-event",
      "experimental-probability",
      "coin"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "equally likely": 83
      },
      "written": {
        "equally likely": 58
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equation-in-quadratic-form",
    "ja": {
      "term": "2 次方程式の形の方程式",
      "reading": "にじほうていしきのかたちのほうていしき",
      "alt": []
    },
    "en": {
      "term": "equation in quadratic form",
      "alt": [
        "quadratic in form"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「2 次方程式の形の方程式」は学習指導要領解説に無い、本プロジェクトの訳語。日本では「x² = t とおく」のように、おき換えの手順として扱い、この種類の方程式に名前を付けない（x⁴ + ax² + b = 0 の形は「複 2 次式」と呼ぶことがある）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "式の一部を新しい文字におき換えると 2 次方程式になる方程式。x⁴ − 5x² + 4 = 0 は u = x² とおくと u² − 5u + 4 = 0 になる。",
    "definition_en": "An equation that becomes a quadratic equation when you substitute a new variable for part of it. For example, x⁴ − 5x² + 4 = 0 becomes u² − 5u + 4 = 0 with u = x².",
    "latex": "x^4 - 5x^2 + 4 = 0 \\;\\Rightarrow\\; u^2 - 5u + 4 = 0 \\quad (u = x^2)",
    "spoken_en": "x to the fourth minus five x squared plus four equals zero becomes u squared minus five u plus four equals zero, where u equals x squared",
    "examples": [
      {
        "en": "See how x to the fourth is just x squared, squared? That's an equation in quadratic form, so let u equal x squared.",
        "ja": "x⁴ は x² の 2 乗になっているよね。これは 2 次方程式の形の方程式だから、u = x² とおこう。",
        "register": "spoken"
      },
      {
        "en": "The equation x⁴ − 5x² + 4 = 0 is an equation in quadratic form. Let u = x² and find all real values of x.",
        "ja": "方程式 x⁴ − 5x² + 4 = 0 は 2 次方程式の形の方程式である。u = x² とおいて、実数 x の値をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "u の値を求めたら終わりではない。u = x² に戻して x を求める（u = 4 なら x = ±2、u = 1 なら x = ±1）。u が負になったら、その u から実数の x は出ない。",
      "見分け方は、最高次の項の指数が中の項の指数の 2 倍になっているか（x⁴ と x²、x⁶ と x³）。OpenStax Algebra and Trigonometry はこの見方で解く手順を示す。",
      "(x + 1)² − 3(x + 1) + 2 = 0 のように、かたまりを u = x + 1 とおく形も同じ種類（エントリ substitute-new-variable）。"
    ],
    "related": [
      "quadratic-equation",
      "substitute-new-variable",
      "solving-by-factoring",
      "factoring",
      "quadratic-formula"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "equation in quadratic form": 30,
        "quadratic in form": 5
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg",
        "openstax-precalculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equation-of-a-circle",
    "ja": {
      "term": "円の方程式",
      "reading": "えんのほうていしき",
      "alt": []
    },
    "en": {
      "term": "equation of a circle",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "中心 (a, b)、半径 r の円を表す方程式 (x − a)² + (y − b)² = r²。",
    "definition_en": "The equation (x − a)² + (y − b)² = r² of the circle with center (a, b) and radius r.",
    "latex": "(x - a)^2 + (y - b)^2 = r^2",
    "spoken_en": "x minus a squared plus y minus b squared equals r squared",
    "examples": [
      {
        "en": "Read off the center and the radius: center (2, −1), radius 3.",
        "ja": "中心と半径を読み取ろう。中心 (2, −1)、半径 3。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of the circle whose diameter has endpoints (1, 2) and (5, 6).",
        "ja": "2 点 (1, 2), (5, 6) を直径の両端とする円の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax（Intermediate Algebra 11.1・Algebra and Trigonometry 11.3）・IM Geometry（6.4）・CK-12 Geometry（6.21）は中心を (h, k) として (x − h)² + (y − k)² = r² と書き、OpenStax はこれを standard form of the equation of a circle と呼ぶ。",
      "用例コーパスでは話し言葉には少ない。冠詞の違う equation of the circle も書き言葉ではよく使い、話し言葉では少ない。"
    ],
    "related": [
      "general-form-of-a-circle",
      "tangent-line",
      "find-the-equation",
      "position-of-a-line-relative-to-a-circle",
      "relative-positions-of-two-circles",
      "vector-equation-of-a-circle",
      "circle-through-the-intersections-of-two-circles"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.3（standard form of the equation of a circle）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "11.1 Distance and Midpoint Formulas; Circles（standard form of the equation of a circle）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "6.4 Distances and Circles"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.21 Circles in the Coordinate Plane"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "equation of a circle": 9
      },
      "written": {
        "equation of a circle": 37
      },
      "sources": [
        "khan-algebra",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equation-of-a-line",
    "ja": {
      "term": "直線の式",
      "reading": "ちょくせんのしき",
      "alt": [
        "直線の方程式"
      ]
    },
    "en": {
      "term": "equation of a line",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "geometry"
    ],
    "level": {
      "jp": [
        "中2",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "座標平面上の直線を表す x, y の 1 次方程式。傾きと通る点、または通る 2 点から求める。",
    "definition_en": "A first-degree equation in x and y whose graph is a given line, found from its slope and a point, or from two points.",
    "latex": "y - y_1 = m(x - x_1)",
    "spoken_en": "y minus y one equals m times x minus x one",
    "examples": [
      {
        "en": "Once you have the slope and a point, you can write the equation of the line.",
        "ja": "傾きと 1 点がわかれば直線の式が書ける。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of the line passing through (1, −2) and (3, 4).",
        "ja": "2 点 (1, −2), (3, 4) を通る直線の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "冠詞だけの違いの equation of the line もよく使う（話し言葉では見出しの a と同じくらい、書き言葉では a のほうが多い）。",
      "OpenStax（Algebra and Trigonometry・Elementary Algebra ほか）と IM は、直線の式の形に名前をつけて呼ぶ: slope-intercept form（y = mx + b。OpenStax Algebra and Trigonometry 54 件・IM 8 件）、point-slope form（y − y₁ = m(x − x₁)。12 件・21 件）、standard form（Ax + By = C）。"
    ],
    "related": [
      "parallel-lines",
      "find-the-equation",
      "distance-from-a-point-to-a-line",
      "perpendicular-lines"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 1)",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 ほか（slope-intercept form 8 件・point-slope form 21 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "slope-intercept form ほか"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "equation of a line": 22
      },
      "written": {
        "equation of a line": 180
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equation-of-a-plane",
    "ja": {
      "term": "平面の方程式",
      "reading": "へいめんのほうていしき",
      "alt": []
    },
    "en": {
      "term": "equation of a plane",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "空間の平面上の点 (x, y, z) が満たす方程式。法線ベクトル (a, b, c) と平面上の 1 点から ax + by + cz + d = 0 の形に書ける。",
    "definition_en": "The equation satisfied by the points (x, y, z) of a plane in space, ax + by + cz + d = 0, where (a, b, c) is a normal vector.",
    "latex": "a(x - x_0) + b(y - y_0) + c(z - z_0) = 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "The coefficients in the equation of the plane give you a normal vector right away.",
        "ja": "平面の方程式の係数から、法線ベクトルがすぐわかる。",
        "register": "spoken"
      },
      {
        "en": "Find an equation of the plane through (1, 0, 2) with normal vector (3, −1, 4).",
        "ja": "点 (1, 0, 2) を通り、法線ベクトルが (3, −1, 4) の平面の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは、書き言葉を中心に the equation of the plane のように the をつけることも多い。",
      "数C では法線ベクトルとの内積で導く。英語でも normal vector を使う。"
    ],
    "related": [
      "coordinates-in-space",
      "tangent-plane",
      "normal-vector",
      "coplanarity-condition"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "equation of a plane": 1
      },
      "written": {
        "equation of a plane": 17
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equation-of-a-sphere",
    "ja": {
      "term": "球面の方程式",
      "reading": "きゅうめんのほうていしき",
      "alt": []
    },
    "en": {
      "term": "equation of a sphere",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "中心 (a, b, c)、半径 r の球面上の点 (x, y, z) が満たす方程式 (x − a)² + (y − b)² + (z − c)² = r²。",
    "definition_en": "The equation (x − a)² + (y − b)² + (z − c)² = r² satisfied by the points (x, y, z) on the sphere with center (a, b, c) and radius r.",
    "latex": "(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2",
    "spoken_en": null,
    "examples": [
      {
        "en": "It's just the circle equation with a z term added.",
        "ja": "円の方程式に z の項を足しただけ。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of the sphere with center (1, −2, 3) and radius 4.",
        "ja": "中心 (1, −2, 3)、半径 4 の球面の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「球面」（表面）と「球」（中身を含む）の区別は、英語では sphere（表面）と ball（中身を含む）。"
    ],
    "related": [
      "equation-of-a-circle",
      "coordinates-in-space",
      "equation-of-a-plane"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "equation of a sphere": 8
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは OpenStax の呼び方 equation of a sphere（本文 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "equation",
    "ja": {
      "term": "方程式",
      "reading": "ほうていしき",
      "alt": [
        "等式"
      ]
    },
    "en": {
      "term": "equation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語の equation は等号でつないだ式すべてを指し、2 + 3 = 5 のような文字のない等式も、x + 3 = 5 のような方程式も equation と言う。日本語の「等式」と「方程式」（文字の値によって成り立ったり成り立たなかったりする等式）を言い分ける 1 語はない。どんな値でも成り立つ等式（恒等式）は identity（エントリ identity）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "等号 = を使って、2 つの式が等しいことを表した式（等式）。とくに、文字の値によって成り立ったり成り立たなかったりする等式を方程式という。",
    "definition_en": "A mathematical statement, written with an equal sign, that two expressions are equal. An equation with a variable, like x + 3 = 5, can be true for some values of the variable and false for others.",
    "latex": "x + 3 = 5",
    "spoken_en": "x plus three equals five",
    "examples": [
      {
        "en": "2x + 3 = 11 is an equation because it has an equal sign; 2x + 3 by itself is just an expression.",
        "ja": "2x + 3 = 11 は等号があるから方程式。2x + 3 だけなら、ただの式だよ。",
        "register": "spoken"
      },
      {
        "en": "Solve the equation 4x − 7 = 13.",
        "ja": "方程式 4x − 7 = 13 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "等号を含まない 2x + 3 は equation ではなく expression（エントリ expression）。expression は solve するものではなく、simplify（整理する）や evaluate（値を求める）の対象になる。",
      "「方程式を解く」は solve the equation、「方程式をつくる」は write an equation（エントリ write-an-equation）。"
    ],
    "related": [
      "expression",
      "identity",
      "solve",
      "linear-equation",
      "write-an-equation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "方程式",
        "en": "Equation"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "equation": 5806
      },
      "written": {
        "equation": 11037
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equiangular-triangle",
    "ja": {
      "term": "等角三角形",
      "reading": "とうかくさんかくけい",
      "alt": []
    },
    "en": {
      "term": "equiangular triangle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「等角三角形」は学習指導要領解説に無い、本プロジェクトの訳語。日本では 3 つの角が等しい三角形は正三角形（equilateral triangle）と呼び、角で呼ぶ名前を使わない（平面では同じ三角形）。CK-12 Geometry は equiangular triangle と呼ぶ（4.17 Triangle Angle Sum Theorem ほか、12 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "3 つの角の大きさがすべて等しい三角形。どの角も 60° で、正三角形と同じ図形になる。",
    "definition_en": "A triangle whose three angles are all congruent. Each angle measures 60°, so it is the same shape as an equilateral triangle.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If a triangle is equiangular, each angle has to be 180 divided by 3, so 60 degrees.",
        "ja": "三角形の 3 つの角が等しければ、どの角も 180 ÷ 3 で 60°。",
        "register": "spoken"
      },
      {
        "en": "Prove that every equiangular triangle is equilateral.",
        "ja": "3 つの角が等しい三角形は正三角形であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "三角形では equiangular と equilateral は同じ三角形を指すが、四角形では違う。長方形は equiangular だが、equilateral（4 辺が等しい）とは限らない。",
      "用例コーパスに equiangular triangle は出てこない。IM でも equiangular は Geometry に 1 度出てくるだけ。"
    ],
    "related": [
      "triangle",
      "scalene-triangle",
      "isosceles-triangle",
      "equal-angles"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.17 Triangle Angle Sum Theorem・4.1 Classify Triangles・4.2 Classify Triangles by Angle Measurement"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 equiangular triangle（CK-12 Geometry 4.17 Triangle Angle Sum Theorem・CK-12 Geometry 4.1 Classify Triangles・CK-12 Geometry 4.2 Classify Triangles by Angle Measurement・計 12 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "equilateral-triangle",
    "ja": {
      "term": "正三角形",
      "reading": "せいさんかくけい",
      "alt": []
    },
    "en": {
      "term": "equilateral triangle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "3 辺の長さがすべて等しい三角形。3 つの角も等しく、どれも 60° である。",
    "definition_en": "A triangle whose three sides are all the same length. Its three angles are also equal, and each one measures 60°.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In an equilateral triangle, all three sides are the same length, so every angle has to be 60 degrees.",
        "ja": "正三角形は 3 辺が全部同じ長さだから、角はどれも 60° になる。",
        "register": "spoken"
      },
      {
        "en": "Prove that the three angles of an equilateral triangle are all equal.",
        "ja": "正三角形の 3 つの角はすべて等しいことを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "equilateral は「辺が等しい」、equiangular は「角が等しい」。三角形では同じ三角形を指す（エントリ equiangular-triangle）。",
      "日本語の「正」に当たる英語は形によって変わる。正三角形は equilateral triangle、正方形は square（エントリ square-shape）、正多角形は regular polygon（エントリ regular-polygon）。",
      "正三角形は、どの 2 辺も等しいので、二等辺三角形の特別な場合でもある（エントリ isosceles-triangle）。"
    ],
    "related": [
      "equiangular-triangle",
      "isosceles-triangle",
      "regular-polygon",
      "triangle",
      "square-shape"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正三角形",
        "en": "Equilateral triangle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "equilateral triangle": 24
      },
      "written": {
        "equilateral triangle": 34
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equivalence-class",
    "ja": {
      "term": "同値類",
      "reading": "どうちるい",
      "alt": []
    },
    "en": {
      "term": "equivalence class",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete",
      "set-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "同値関係 ~ のもとで、元 a と同値な元をすべて集めた集合のこと。[a] と書き、異なる同値類どうしは共通の元をもたない。",
    "definition_en": "The set of all elements that are equivalent to a given element a under an equivalence relation, written [a]. Two different equivalence classes never share an element.",
    "latex": "[a] = \\{x \\in A \\mid x \\sim a\\}",
    "spoken_en": "the equivalence class of a equals the set of all x in A such that x is related to a",
    "examples": [
      {
        "en": "Under congruence mod 3, the equivalence class of 1 is every integer that leaves a remainder of 1 when you divide by 3.",
        "ja": "3 を法とする合同では、1 の同値類は 3 で割って 1 余る整数全体だ。",
        "register": "spoken"
      },
      {
        "en": "Define a ~ b on ℤ if a ≡ b (mod 4). Find the equivalence class [3] and the number of distinct equivalence classes.",
        "ja": "ℤ 上の関係を a ≡ b (mod 4) のとき a ~ b と定める。同値類 [3] と、異なる同値類の個数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "同値類全体は集合の分割（partition）になる。同値類全体の集合は商集合（quotient set）で、A/~ と書く。",
      "mod n の同値類は日本語で「剰余類」とも呼び、英語では residue class や congruence class とも言う。",
      "用例コーパスでは MIT だけに出てきて、話し言葉（MIT OCW）のほうが書き言葉（MIT の講義ノート）より多い。Levin の本では 19 件（2.6 Relations and Graphs ほか）。"
    ],
    "related": [
      "equivalence-relation",
      "relation",
      "congruence-modulo-n"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "同値類",
        "en": "Equivalence class"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "equivalence class": 18
      },
      "written": {
        "equivalence class": 5
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equivalence-relation",
    "ja": {
      "term": "同値関係",
      "reading": "どうちかんけい",
      "alt": []
    },
    "en": {
      "term": "equivalence relation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete",
      "set-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "反射律・対称律・推移律の 3 つをみたす二項関係のこと。集合を、互いに同値な元どうしの組（同値類）に分ける。",
    "definition_en": "A relation on a set that is reflexive, symmetric, and transitive. It splits the set into groups of elements that are equivalent to each other.",
    "latex": "a \\sim a, \\quad a \\sim b \\Rightarrow b \\sim a, \\quad (a \\sim b \\land b \\sim c) \\Rightarrow a \\sim c",
    "spoken_en": "a is related to a; if a is related to b, then b is related to a; and if a is related to b and b is related to c, then a is related to c",
    "examples": [
      {
        "en": "Congruence mod 5 is an equivalence relation, so it breaks the integers into five classes.",
        "ja": "5 を法とする合同は同値関係なので、整数全体を 5 つの類に分ける。",
        "register": "spoken"
      },
      {
        "en": "Let R be the relation on ℤ defined by a R b if a − b is even. Determine whether R is an equivalence relation, and justify your answer.",
        "ja": "整数全体の集合 ℤ 上の関係 R を、a − b が偶数のとき a R b と定める。R が同値関係であるかどうかを調べ、理由を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "3 つの性質は形容詞で言う（R is reflexive / symmetric / transitive）。米国の Geometry で習う reflexive property などは、等号や合同についての同じ性質の名前。",
      "日本の高校で習う「命題 p と q は同値」は equivalent（論理の同値）で、同値関係（equivalence relation）とは別の話。",
      "用例コーパスでは MIT だけに出てきて、話し言葉（MIT OCW）と書き言葉（MIT の講義ノート）が同じくらい。Levin の本では 37 件（2.6 Relations and Graphs ほか）。"
    ],
    "related": [
      "relation",
      "equivalence-class",
      "congruence-modulo-n",
      "reflexive-property",
      "symmetric-property",
      "transitive-property"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "同値関係",
        "en": "Equivalence relation"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "equivalence relation": 12
      },
      "written": {
        "equivalence relation": 13
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "equivalent",
    "ja": {
      "term": "同値",
      "reading": "どうち",
      "alt": []
    },
    "en": {
      "term": "equivalent",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "2 つの命題・条件について、一方が成り立てば他方も成り立ち、その逆も言えること。p ⇔ q と書く。",
    "definition_en": "Describes two statements or conditions that imply each other, so that one is true exactly when the other is true; written p ⇔ q.",
    "latex": "p \\iff q",
    "spoken_en": "p is equivalent to q",
    "examples": [
      {
        "en": "These two conditions are equivalent, so you can use whichever one is easier to check.",
        "ja": "この 2 つの条件は同値だから、確かめやすい方を使えばいい。",
        "register": "spoken"
      },
      {
        "en": "Show that the conditions x² − 5x + 6 = 0 and \"x = 2 or x = 3\" are equivalent.",
        "ja": "条件 x² − 5x + 6 = 0 と「x = 2 または x = 3」が同値であることを示せ。",
        "register": "written"
      },
      {
        "en": "Multiplying both sides by a nonzero number gives an equivalent equation.",
        "ja": "両辺に 0 でない数を掛けても、同値な方程式が得られる。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "logically equivalent",
        "ja": "論理的に同値"
      }
    ],
    "pitfalls": [
      "論理の同値をはっきり言うときは logically equivalent（用例コーパスでは書き言葉に出てくる）。見出しの equivalent を含むので別の言い方としては数えない。",
      "英語の equivalent は equivalent fractions（等しい分数）、equivalent expressions など論理以外にも広く使うので、件数にはその意味も含まれる。",
      "同値変形は、方程式を equivalent equation に変形していくこと（中学校学習指導要領解説（第 1 学年 方程式）は「同値な方程式」「同値変形」と書く）。両辺を 2 乗するなどの変形は同値でなくなることがある。"
    ],
    "related": [
      "necessary-and-sufficient-condition",
      "contrapositive",
      "proposition",
      "condition",
      "biconditional"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.3 Rules of Logic: logically equivalent（44 件）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 方程式: もとの方程式と同値な方程式、同値変形（3 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "equivalent": 543
      },
      "written": {
        "equivalent": 513
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "error-bound",
    "ja": {
      "term": "誤差限界",
      "reading": "ごさげんかい",
      "alt": [
        "誤差の範囲",
        "許容誤差"
      ]
    },
    "en": {
      "term": "margin of error",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "error bound",
          "register": "written",
          "note": "OpenStax Introductory Statistics の言い方（EBM ／ EBP）。Lagrange error bound を除いて数えた。"
        }
      ],
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "信頼区間で、点推定値の両側にとる幅。信頼区間は 点推定値 ± 誤差限界。",
    "definition_en": "The amount added to and subtracted from the point estimate to make a confidence interval: point estimate ± margin of error.",
    "latex": "\\bar{x} \\pm E",
    "spoken_en": null,
    "examples": [
      {
        "en": "The poll says 52% with a margin of error of 3 points, so anywhere from 49 to 55.",
        "ja": "世論調査は 52%、誤差の範囲は 3 ポイントなので、49% から 55% のどこか。",
        "register": "spoken"
      },
      {
        "en": "Find the error bound for the population mean and construct a 90% confidence interval.",
        "ja": "母平均の誤差限界を求め、90% 信頼区間を作れ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP Statistics の CED（2026 年版）と Khan Academy は margin of error、OpenStax Introductory Statistics は error bound（EBM は平均、EBP は比率）と書く。",
      "微積分の Lagrange error bound（テイラー多項式の誤差の上限）は別の概念。"
    ],
    "related": [
      "confidence-interval",
      "point-estimate",
      "standard-error",
      "critical-value"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "誤差の範囲",
        "en": "Margin of error"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.3・3.4"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "margin of error": 54
      },
      "written": {
        "!lagrange !legrange error bound": 132,
        "margin of error": 25
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "error",
    "ja": {
      "term": "誤差",
      "reading": "ごさ",
      "alt": []
    },
    "en": {
      "term": "error",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語の error は符号の扱いが資料で分かれる。IM Grade 7（4.13 Measurement Error）は measurement error を測定値と実際の値の差の大きさ（正の数）とし、OpenStax Calculus Volume 2（3.6）は absolute error を |A − B| と定義する。OpenStax Calculus Volume 1（4.2）は、測定値 a + dx の dx を符号つきで measurement error と呼ぶ。日本語の誤差 = 近似値 − 真の値 は符号をもつ（pitfalls）。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "近似値（測定値）と真の値との差。誤差 = 近似値 − 真の値 で求める。",
    "definition_en": "The difference between an approximate or measured value and the true value: error = approximate value − true value.",
    "latex": "\\text{error} = \\text{approximate value} - \\text{true value}",
    "spoken_en": "error equals approximate value minus true value",
    "examples": [
      {
        "en": "The true length is 12.6 centimeters and your measurement is 12.8, so the error is 0.2 centimeters.",
        "ja": "本当の長さは 12.6 cm で、測定値は 12.8 cm だから、誤差は 0.2 cm。",
        "register": "spoken"
      },
      {
        "en": "The number 3.786 is rounded to the nearest tenth, 3.8. Find the error in this approximation.",
        "ja": "3.786 の小数第 2 位を四捨五入して 3.8 とした。この近似値の誤差を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の error は「まちがい」の意味でもよく使うので、Find the error. だけだと「まちがいを見つけよ」ともとれる。誤差の意味では absolute error（OpenStax Calculus Volume 1）や rounding error（OpenStax Algebra and Trigonometry）のように前に語を付けて言う。",
      "統計の margin of error（誤差限界、エントリ error-bound）や標本誤差（sampling error、エントリ sampling-error）は別の用語。",
      "誤差 = 近似値 − 真の値 なので、近似値が真の値より小さいと誤差は負の数になる（真の値 3.14159… に対して近似値 3.14 なら、誤差は約 −0.0016）。"
    ],
    "related": [
      "approximate-value",
      "measurement",
      "significant-figures",
      "round",
      "error-bound"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "誤差",
        "en": "Errors and residuals"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Grade 7 4.13 Measurement Error: 測定値と実際の値の差（正の数）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "absolute error | measurement error | error in the measurement | error in measurement | amount of error | round-off error | rounding error": 1
      },
      "written": {
        "absolute error | measurement error | error in the measurement | error in measurement | amount of error | round-off error | rounding error": 34
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "estimation",
    "ja": {
      "term": "推定",
      "reading": "すいてい",
      "alt": [
        "推定する"
      ]
    },
    "en": {
      "term": "estimate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B",
        "中3"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Algebra 1",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "標本から得たデータをもとに、母平均や母比率など母集団の値を見積もること。また、見積もった値。",
    "definition_en": "Using sample data to find an approximate value of a population quantity such as the mean or a proportion; also the value found.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We use the sample proportion to estimate the population proportion.",
        "ja": "標本比率を使って母比率を推定する。",
        "register": "spoken"
      },
      {
        "en": "Use the sample data to estimate the population mean with 95% confidence.",
        "ja": "標本のデータを用いて、信頼度 95% で母平均を推定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP Statistics の CED と OpenStax Introductory Statistics は、推定を動詞 estimate と名詞 estimate（point estimate・interval estimate）で書き、名詞 estimation はどちらも 1 件。名詞の estimation は、用例コーパスでは多くが微積分の「見積もり」（積分の値の評価など）の意味だった。",
      "推定の方法は、1 つの値で推定する点推定（point estimate）と、区間で推定する区間推定（interval estimate、confidence interval）に分かれる。",
      "√10 が 3 と 4 の間にあることを調べるような、値の範囲をはさむ「評価する」も英語は estimate（エントリ bound-estimate）で、統計の推定とは別の意味。"
    ],
    "related": [
      "point-estimate",
      "interval-estimation",
      "confidence-interval",
      "bound-estimate"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.10（estimation）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "estimate the population | estimate of the population | estimate a population | estimate of a population": 21
      },
      "written": {
        "estimate the population | estimate of the population | estimate a population | estimate of a population": 26
      },
      "sources": [
        "khan-ap-stats",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "euclidean-algorithm",
    "ja": {
      "term": "ユークリッドの互除法",
      "reading": "ゆーくりっどのごじょほう",
      "alt": [
        "互除法"
      ]
    },
    "en": {
      "term": "Euclidean algorithm",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Euclidean algorithm は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "2 つの自然数の最大公約数を、割り算を繰り返して求める方法。a を b で割った余りを r とすると a と b の最大公約数は b と r の最大公約数に等しいことを使い、余りが 0 になるまで続ける。",
    "definition_en": "A method for finding the greatest common divisor of two positive integers by repeated division: replace the pair (a, b) with (b, r), where r is the remainder of a divided by b, until the remainder is 0.",
    "latex": "a = bq + r \\;\\Rightarrow\\; \\gcd(a, b) = \\gcd(b, r)",
    "spoken_en": "if a equals b q plus r, then the gcd of a and b equals the gcd of b and r",
    "examples": [
      {
        "en": "Let's run the Euclidean algorithm: 252 divided by 105 is 2 with a remainder of 42, so now we look at 105 and 42.",
        "ja": "互除法を使おう。252 ÷ 105 は 2 余り 42 だから、次は 105 と 42 を考える。",
        "register": "spoken"
      },
      {
        "en": "Use the Euclidean algorithm to find gcd(1071, 462).",
        "ja": "互除法を用いて、1071 と 462 の最大公約数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「互除法」だけでは英語の名前にならない。Euclidean algorithm と言う。",
      "Levin（6.2）は最大公約数を gcd(a, b) と書く。",
      "互除法の式を下から逆にたどると、ax + by = gcd(a, b) の整数解が 1 つ見つかる（一次不定方程式を解くときに使う。エントリ extended-euclidean-algorithm）。"
    ],
    "related": [
      "greatest-common-divisor",
      "division-algorithm",
      "remainder",
      "linear-diophantine-equation",
      "extended-euclidean-algorithm",
      "bezouts-identity"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ユークリッドの互除法",
        "en": "Euclidean algorithm"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2 Introduction to Number Theory"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 数学と人間の活動: ユークリッドの互除法"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "Euclidean algorithm": 1
      },
      "sources": [
        "mit-6.042"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Levin の呼び方 Euclidean algorithm（6.2 Introduction to Number Theory）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "eulerian-path",
    "ja": {
      "term": "一筆書き",
      "reading": "ひとふでがき",
      "alt": []
    },
    "en": {
      "term": "Euler trail",
      "alt": [
        "Euler circuit",
        "Euler path",
        "Euler walk",
        "Euler tour"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「一筆書き」は始点に戻らない場合も戻る場合も含む。見出しは Levin の呼び方で、すべての辺をちょうど 1 回ずつ通る道が Euler trail（Euler path・Euler walk とも。本文 51 件）、そのうち始点に戻るものが Euler circuit（22 件）。用例コーパスはほとんどが MIT OCW 6.042 の Euler tour（講義と講義ノート）で、6.042 は始点に戻るものをこう呼ぶ。講義ノートには Euler walk も少し出てくる。",
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフのすべての辺をちょうど 1 回ずつ通る道。始点と終点が一致するものと、しないものがある。",
    "definition_en": "A walk through a graph that uses every edge exactly once (an Euler trail); one that also ends where it started is an Euler circuit.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "You can draw it in one stroke only if zero or two vertices have odd degree.",
        "ja": "一筆書きできるのは、次数が奇数の頂点が 0 個か 2 個のときだけ。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the graph has an Euler trail. If it does, is the trail an Euler circuit?",
        "ja": "グラフが一筆書きできるか判定せよ。できるなら、始点に戻る一筆書きか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ケーニヒスベルクの橋の問題（Seven Bridges of Königsberg）が起源。",
      "教科書によっては Euler path を始点に戻らないものに限り、Euler circuit と対にする。Levin の Euler trail は始点に戻るものも含む。",
      "MIT OCW 6.042 の Euler tour は始点に戻るもの（Euler circuit）の意味。"
    ],
    "related": [
      "vertex-graph",
      "tree"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.4 Euler Trails and Circuits"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "一筆書き",
        "en": "Seven Bridges of Königsberg"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "Euler tour": 23
      },
      "written": {
        "Euler walk": 5,
        "Euler tour": 14
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "eulers-formula-for-polyhedra",
    "ja": {
      "term": "オイラーの多面体定理",
      "reading": "おいらーのためんたいていり",
      "alt": [
        "オイラーの多面体公式"
      ]
    },
    "en": {
      "term": "Euler's formula",
      "alt": [
        "Euler's polyhedron formula"
      ],
      "variants": [
        {
          "term": "Euler's theorem",
          "register": "written",
          "note": "CK-12 Geometry（9.1 Polyhedrons）の呼び方（F + V = E + 2 の形）。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "discrete"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math",
        "Geometry"
      ]
    },
    "definition_ja": "凸多面体の頂点の数を V、辺の数を E、面の数を F とすると V − E + F = 2 が成り立つという定理。連結な平面グラフでも同じ式が成り立つ。",
    "definition_en": "If a convex polyhedron has V vertices, E edges, and F faces, then V − E + F = 2; a connected planar graph satisfies the same equation.",
    "latex": "V - E + F = 2",
    "spoken_en": "V minus E plus F equals 2",
    "examples": [
      {
        "en": "A cube has 8 vertices, 12 edges, and 6 faces: 6 plus 8 is 12 plus 2, just as Euler's formula says.",
        "ja": "立方体は頂点が 8、辺が 12、面が 6 で、6 + 8 = 12 + 2。オイラーの多面体定理のとおりになる。",
        "register": "spoken"
      },
      {
        "en": "A convex polyhedron has 12 faces and 30 edges. Use Euler's formula to find the number of vertices.",
        "ja": "面の数が 12、辺の数が 30 の凸多面体がある。オイラーの多面体定理を用いて頂点の数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "Euler's formula for planar graphs",
        "ja": "平面グラフのオイラーの多面体定理"
      }
    ],
    "pitfalls": [
      "e^(iθ) = cos θ + i sin θ（オイラーの公式、eulers-formula）も英語で同じ Euler's formula と呼ぶ（Nicholson Appendix A・OpenStax Calculus Volume 2・3）。",
      "CK-12 Geometry（9.1 Polyhedrons）は F + V = E + 2 の形で Euler's Theorem と呼ぶが、Euler's theorem は整数論の定理（a と n が互いに素のとき a^φ(n) ≡ 1 (mod n)。英語版 Wikipedia「Euler's theorem」）の名前でもある。英語版 Wikipedia「Euler characteristic」（節 Polyhedra）は多面体の定理を Euler's polyhedron formula と呼び（太字）、節 Proof of Euler's formula では Euler's formula とも書く。",
      "Levin（2.3 Planar Graphs）は平面グラフの定理を Euler's formula（定理の題は Euler's formula for planar graphs）と呼び、多面体にも当てる。",
      "頂点・辺・面の数は number of vertices ／ edges ／ faces。vertex の複数形は vertices。"
    ],
    "related": [
      "eulers-formula",
      "tetrahedron",
      "vertex",
      "edge-of-a-graph",
      "vertex-graph",
      "polyhedron",
      "face",
      "edge",
      "planar-graph",
      "regular-polyhedron"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.3 Planar Graphs（Euler's formula for planar graphs、v − e + f = 2）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "9.1 Polyhedrons（Euler's Theorem: F + V = E + 2）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Euler_characteristic",
        "note": "記事「Euler characteristic」の節 Polyhedra（Euler's polyhedron formula を太字。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Euler%27s_theorem",
        "note": "記事「Euler's theorem」（整数論の定理）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "Appendix A Complex Numbers（Euler's formula e^(iθ) = cos θ + i sin θ）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "6.4 Working with Taylor Series（Euler's formula）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "7.1 Second-Order Linear Equations（Euler's formula）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「多面体」",
        "url": "https://ja.wikipedia.org/wiki/%E5%A4%9A%E9%9D%A2%E4%BD%93",
        "note": "節 オイラーの多面体定理（オイラーの多面体公式）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "five.eight.three Euler's formula | five.eight.three (Euler's formula | as Euler's formula claims | planar graph like Euler's formula | five.eight.five with Euler's formula | by Euler's formula, and substitute | use Euler's formula to show | in Euler's formula, we find | substitute into Euler's formula | in harmony with Euler's formula | relates to Euler's formula | use Euler's formula to confirm | Euler's formula for planar graphs": 3
      },
      "written": {
        "five.eight.three Euler's formula | five.eight.three (Euler's formula | as Euler's formula claims | planar graph like Euler's formula | five.eight.five with Euler's formula | by Euler's formula, and substitute | use Euler's formula to show | in Euler's formula, we find | substitute into Euler's formula | in harmony with Euler's formula | relates to Euler's formula | use Euler's formula to confirm | Euler's formula for planar graphs": 10
      },
      "sources": [
        "mit-notes",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "eulers-formula",
    "ja": {
      "term": "オイラーの公式",
      "reading": "おいらーのこうしき",
      "alt": []
    },
    "en": {
      "term": "Euler's formula",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers",
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "e^{iθ} = cos θ + i sin θ という等式。極形式を指数の形で書ける。",
    "definition_en": "The identity e^{iθ} = cos θ + i sin θ, which lets the polar form be written as an exponential.",
    "latex": "e^{i\\theta} = \\cos\\theta + i\\sin\\theta",
    "spoken_en": "e to the i theta equals cosine theta plus i sine theta",
    "examples": [
      {
        "en": "Euler's formula turns multiplication of complex numbers into adding exponents.",
        "ja": "オイラーの公式で、複素数の掛け算が指数の足し算になる。",
        "register": "spoken"
      },
      {
        "en": "Use Euler's formula to show that e^{iπ} + 1 = 0.",
        "ja": "オイラーの公式を用いて e^{iπ} + 1 = 0 を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校の数C では扱わない。用例コーパスでは 3Blue1Brown・MIT OCW・MIT の講義ノートが中心で、OpenStax にはほとんど出てこない。",
      "多面体の V − E + F = 2 も Euler's formula と呼ぶ。"
    ],
    "related": [
      "polar-form",
      "de-moivres-theorem",
      "taylor-series",
      "eulers-formula-for-polyhedra"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "オイラーの公式",
        "en": "Euler's formula"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "Euler's formula": 48
      },
      "written": {
        "Euler's formula": 19
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "eulers-method",
    "ja": {
      "term": "オイラー法",
      "reading": "おいらーほう",
      "alt": []
    },
    "en": {
      "term": "Euler's method",
      "alt": [],
      "variants": [
        {
          "term": "Euler's method",
          "register": "both",
          "note": "話し言葉は MIT OCW（18.03）が中心。書き言葉は大半が OpenStax Calculus。"
        },
        {
          "term": "Euler method",
          "register": "spoken",
          "note": "所有格の 's を付けない言い方。MIT OCW だけ。話し言葉では Euler's method より少なく、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "微分方程式の解を、接線に沿って小さな幅 h ずつ進んで近似する方法。yₙ₊₁ = yₙ + h f(xₙ, yₙ)。",
    "definition_en": "A way to approximate a solution of a differential equation by stepping along tangent lines: yₙ₊₁ = yₙ + h·f(xₙ, yₙ).",
    "latex": "y_{n+1} = y_n + h\\,f(x_n, y_n)",
    "spoken_en": "y sub n plus one equals y sub n plus h times f of x sub n comma y sub n",
    "tts_text": "Oiler's method",
    "respelling": "OY-lerz method",
    "examples": [
      {
        "en": "Use Euler's method with a step size of point one to estimate y of one.",
        "ja": "刻み幅 0.1 のオイラー法で y(1) を見積もろう。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Euler は OY-ler と読む（ユーラーではない）。",
      "刻み幅は step size と言う。"
    ],
    "related": [
      "differential-equation",
      "logistic-differential-equation",
      "slope-field",
      "initial-condition"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "オイラー法",
        "en": "Euler method"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Euler's method": 26,
        "Euler method": 6
      },
      "written": {
        "Euler's method": 37
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "evaluate-the-integral",
    "ja": {
      "term": "積分を計算する",
      "reading": "せきぶんをけいさんする",
      "alt": [
        "定積分を計算する"
      ]
    },
    "en": {
      "term": "evaluate the integral",
      "alt": [],
      "variants": [
        {
          "term": "compute the integral",
          "register": "spoken",
          "note": "話し言葉ではこれが最も多いが、大学の講義（MIT OCW、特に 18.02）に集中している。書き言葉では evaluate the integral よりずっと少ない。"
        },
        {
          "term": "evaluate the integral",
          "register": "both",
          "note": "書き言葉ではほぼこれ（大半が OpenStax Calculus）。話し言葉では Khan Academy と MIT OCW。話し言葉では compute the integral と同じくらい使う。"
        },
        {
          "term": "evaluate the definite integral",
          "register": "both",
          "note": "話し言葉ではすべて Khan Academy、書き言葉ではすべて OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "積分（不定積分・定積分）を実際に計算すること。",
    "definition_en": "To work out an integral, definite or indefinite.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's evaluate the definite integral from one to four: find an antiderivative, then plug in.",
        "ja": "1 から 4 までの定積分を計算しよう。原始関数を求めて、代入する。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the integral ∫₁⁴ √x dx.",
        "ja": "定積分 ∫₁⁴ √x dx を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus の問題文の指示語は Evaluate the integral が多く、Find は find the antiderivative などに使う。",
      "積分の値を「求めよ」は、書き言葉では evaluate the integral が compute the integral よりずっと多い。"
    ],
    "related": [
      "integrate",
      "definite-integral",
      "indefinite-integral",
      "fundamental-theorem-of-calculus"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "compute the integral": 29,
        "evaluate the integral": 20,
        "evaluate the definite integral": 18
      },
      "written": {
        "evaluate the integral": 115,
        "compute the integral": 14,
        "evaluate the definite integral": 14
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "evaluate",
    "ja": {
      "term": "値を求める",
      "reading": "あたいをもとめる",
      "alt": [
        "式を評価する"
      ]
    },
    "en": {
      "term": "evaluate",
      "alt": [
        "find the value of"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語の「値を求める」は、方程式の解・図形の長さ・確率など何の値にも使う。evaluate は、式や関数に値を代入して計算する（evaluate the expression when x = 2、evaluate f at 3）ときと、極限・定積分などを計算するときの動詞で、方程式を解くのは solve、長さや確率を求めるのは find と言い分ける。用例コーパスでは話し言葉・書き言葉とも evaluate が find the value of よりずっと多い。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "式や関数の文字に与えられた数を代入して、その値を計算する。極限や定積分の値を計算するときにも使う。",
    "definition_en": "To work out the numerical value of an expression or function by substituting given numbers for its variables; also used for computing a limit or a definite integral.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To evaluate this at x = 2, just plug in 2 everywhere you see an x.",
        "ja": "x = 2 のときの値を求めるには、x のところ全部に 2 を代入すればいい。",
        "register": "spoken"
      },
      {
        "en": "Evaluate 3x² − 5x + 1 when x = −2.",
        "ja": "x = −2 のとき、式 3x² − 5x + 1 の値を求めよ。",
        "register": "written"
      },
      {
        "en": "Evaluate the integral ∫₀¹ x² dx.",
        "ja": "定積分 ∫₀¹ x² dx を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "evaluate the expression",
        "ja": "式の値を求める"
      },
      {
        "en": "evaluate the integral",
        "ja": "積分を計算する"
      }
    ],
    "pitfalls": [
      "ja.alt の「式を評価する」は直訳に近い。学習指導要領解説（中学校・高等学校）は「式の値を求める」と書く。",
      "「f(3) を求めよ」は evaluate f at 3 または evaluate f(3)。前置詞は at を使う。",
      "方程式の解を求めるのは solve で、evaluate は使わない。"
    ],
    "related": [
      "substitute",
      "polynomial",
      "find-the-limit",
      "definite-integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A 数と式: 文字に数を代入して式の値を求める"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "evaluate": 1735,
        "find the value of": 117
      },
      "written": {
        "evaluate": 2327,
        "find the value of": 169
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "even-function",
    "ja": {
      "term": "偶関数",
      "reading": "ぐうかんすう",
      "alt": []
    },
    "en": {
      "term": "even function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry",
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "f(−x) = f(x) がすべての x で成り立つ関数。グラフは y 軸に関して対称。",
    "definition_en": "A function with f(−x) = f(x) for every x; reflecting its graph across the y-axis leaves it unchanged.",
    "latex": "f(-x) = f(x)",
    "spoken_en": "f of negative x equals f of x",
    "examples": [
      {
        "en": "Cosine is an even function, so cos(−θ) is just cos θ.",
        "ja": "cos は偶関数なので、cos(−θ) は cos θ。",
        "register": "spoken"
      },
      {
        "en": "Determine whether f(x) = x⁴ − 3x² + 1 is even, odd, or neither.",
        "ja": "f(x) = x⁴ − 3x² + 1 が偶関数か奇関数か、どちらでもないかを判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は、関数が even, odd, or neither（どちらでもない）のどれかを判定させる問題を扱う（13 件）。IM Algebra 2（5.6 Symmetry in Equations）も同じ 3 択で聞く。"
    ],
    "related": [
      "odd-function",
      "symmetric",
      "integrals-of-even-and-odd-functions"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "偶関数と奇関数",
        "en": "Even and odd functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 5.6 Symmetry in Equations（even, odd, or neither）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "even function": 69
      },
      "written": {
        "even function": 55
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "even-number",
    "ja": {
      "term": "偶数",
      "reading": "ぐうすう",
      "alt": []
    },
    "en": {
      "term": "even number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 で割り切れる整数。n を整数として 2n と表せる数で、0 も偶数である。",
    "definition_en": "An integer that can be divided by 2 with no remainder, such as −4, 0, 2, or 10. Every even number can be written as 2n, where n is an integer.",
    "latex": "2n",
    "spoken_en": "two n",
    "examples": [
      {
        "en": "Remember, zero is an even number too, since 2 times 0 is 0.",
        "ja": "0 も偶数だよ。2 × 0 = 0 だからね。",
        "register": "spoken"
      },
      {
        "en": "Let m and n be integers. Show that the sum of the even numbers 2m and 2n is also an even number.",
        "ja": "m, n を整数とする。偶数 2m と 2n の和も偶数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「12 は偶数だ」は 12 is even. と形容詞 even だけでも言える。",
      "0 も偶数（0 = 2 × 0）。負の数も −4 = 2 × (−2) のように 2n の形になれば偶数。",
      "偶関数（even function、エントリ even-function）の even と同じ語。偶数か奇数かという性質は parity（エントリ parity）。"
    ],
    "related": [
      "odd-number",
      "parity",
      "integer",
      "multiple",
      "even-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "偶数",
        "en": "Even number"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "even number": 94
      },
      "written": {
        "even number": 56
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "event",
    "ja": {
      "term": "事象",
      "reading": "じしょう",
      "alt": []
    },
    "en": {
      "term": "event",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "試行の結果のうち、ある条件に当てはまるものの集まり。全事象（起こりうる結果全体の集合）の部分集合である。",
    "definition_en": "A set of outcomes of an experiment, such as rolling an even number on a die. An event is a subset of the sample space.",
    "latex": "A \\subseteq S",
    "spoken_en": "A is a subset of S",
    "examples": [
      {
        "en": "Rolling an even number is an event, and it's made up of three outcomes: 2, 4, and 6.",
        "ja": "偶数の目が出るというのは 1 つの事象で、2、4、6 の 3 つの結果でできている。",
        "register": "spoken"
      },
      {
        "en": "A card is drawn from a standard 52-card deck. Let A be the event that the card is a heart. Find P(A).",
        "ja": "ジョーカーを除く 52 枚のトランプから 1 枚を取り出す。取り出したカードがハートである事象を A とするとき、P(A) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日常語の event は「行事・出来事・イベント」。確率の event は結果の集まり（集合）で、1 つ 1 つの結果（outcome、エントリ elementary-event）とは区別する。",
      "「全事象」は event を使わず sample space（エントリ sample-space）と言う。「A の余事象」は the complement of A（エントリ complementary-event）。",
      "「事象 A が起こる」は event A occurs（エントリ occur）。"
    ],
    "related": [
      "sample-space",
      "elementary-event",
      "complementary-event",
      "probability",
      "mutually-exclusive-events"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "事象 (確率論)",
        "en": "Event (probability theory)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "event": 395
      },
      "written": {
        "event": 807
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "excenter",
    "ja": {
      "term": "傍心",
      "reading": "ぼうしん",
      "alt": []
    },
    "en": {
      "term": "excenter",
      "alt": [],
      "uk": "excentre"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（excenter は CED・OpenStax・IM・CK-12 のどれにも、用例コーパスにも出てこない）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "definition_ja": "三角形の 1 つの内角の二等分線と、他の 2 つの頂点における外角の二等分線が交わる点。1 辺と他の 2 辺の延長に接する円（傍接円）の中心で、1 つの三角形に 3 つある。",
    "definition_en": "The point where the bisector of one interior angle of a triangle meets the exterior angle bisectors at the other two vertices. It is the center of an excircle, a circle tangent to one side and to the extensions of the other two, and every triangle has three of them.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Every triangle has three excenters, one opposite each vertex, and they all lie outside the triangle.",
        "ja": "三角形には傍心が 3 つあり、それぞれ 1 つの頂点に対応していて、どれも三角形の外部にある。",
        "register": "spoken"
      },
      {
        "en": "Show that the bisector of ∠A and the exterior angle bisectors at B and C meet at one point. This point is the excenter opposite A.",
        "ja": "∠A の二等分線と、∠B, ∠C の外角の二等分線は 1 点で交わることを示せ。この点が ∠A 内の傍心である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語版 Wikipedia「Incircle and excircles」は、傍接円を excircle、その半径を exradius と書く。内心・内接円（incenter・incircle）と対になる言い方。",
      "英国式のつづりは excentre。"
    ],
    "related": [
      "exterior-angle-bisector",
      "incenter",
      "inscribed-circle",
      "the-five-centers-of-a-triangle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences#re_er",
        "note": "記事「American and British English spelling differences」の節 -re, -er（英国の綴りは centre、米国は center）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Incircle_and_excircles",
        "note": "記事「Incircle and excircles」の冒頭と節 Excircles and excenters（excenter・excircle を太字で定義）、節 Exradii。数学カテゴリから 5 段"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元 3 の前の修正」の、残った ③ を人間が決めた行）。en.term は excenter のまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "exist",
    "ja": {
      "term": "存在する",
      "reading": "そんざいする",
      "alt": []
    },
    "en": {
      "term": "exist",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "domains": [
      "algebra",
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "条件を満たすものが少なくとも 1 つある。極限や逆関数などの値が定まる、の意味でも使う。",
    "definition_en": "To be at least one thing that satisfies a given condition; also used to say that a value such as a limit or an inverse is defined.",
    "latex": "\\exists x \\in \\mathbb{R},\\ x^2 = 2",
    "spoken_en": "there exists a real number x such that x squared equals 2",
    "examples": [
      {
        "en": "The limit doesn't exist here, because the left-hand and right-hand limits don't match.",
        "ja": "左側極限と右側極限が一致しないので、ここでは極限は存在しない。",
        "register": "spoken"
      },
      {
        "en": "Show that there exists a real number x such that x³ + x = 1.",
        "ja": "x³ + x = 1 となる実数 x が存在することを示せ。",
        "register": "written"
      },
      {
        "en": "Determine whether the inverse of the matrix exists.",
        "ja": "この行列の逆行列が存在するかどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the limit does not exist",
        "ja": "極限は存在しない"
      }
    ],
    "pitfalls": [
      "記号 ∃ は there exists と読む。「ただ 1 つ存在する」は there exists a unique … ／ there is exactly one …。",
      "「存在する」の否定を言うときは doesn't exist（話し言葉）／ does not exist（答案）。there is no x such that … も同じ意味。",
      "日本語の「〜となる x が存在する」（∃）は、英語では there exists x such that … ／ there is … と、存在するものを後ろに置く形になる。exist は the limit exists ／ the inverse exists のように、値や物が「ある・定まる」の意味でも使う。"
    ],
    "related": [
      "at-least-one",
      "all",
      "the-limit-does-not-exist",
      "intermediate-value-theorem",
      "counterexample"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "exist": 718
      },
      "written": {
        "exist": 678
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "existence-proof",
    "ja": {
      "term": "存在証明",
      "reading": "そんざいしょうめい",
      "alt": []
    },
    "en": {
      "term": "existence proof",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "英語版 Wikipedia の Constructive proof（existence proof のリダイレクト先）は構成的証明で、別の概念。",
    "domains": [
      "proof"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "ある条件を満たすものが少なくとも 1 つ存在することを示す証明のこと。実際に例を作って示す方法（構成的）と、例を作らずに存在だけを示す方法（非構成的）がある。",
    "definition_en": "A proof that at least one object with a given property exists. It can be constructive, by giving an example, or it can show that such an object must exist without actually finding it.",
    "latex": "\\exists x\\ P(x)",
    "spoken_en": "there exists x such that P of x",
    "examples": [
      {
        "en": "For an existence proof, one example is enough. Just show me a number that works.",
        "ja": "存在証明なら例は 1 つで十分。条件を満たす数を 1 つ見せてくれればいい。",
        "register": "spoken"
      },
      {
        "en": "Give an existence proof that the equation x³ + x − 1 = 0 has a real solution between 0 and 1.",
        "ja": "方程式 x³ + x − 1 = 0 が 0 と 1 の間に実数解をもつことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "存在を示すには例が 1 つあれば足りるが、「すべての…」を示すときは例をいくつ挙げても証明にならない。",
      "数III の中間値の定理を使って「解が存在することを示せ」と答えるのは、解を具体的に求めない（非構成的な）存在証明にあたる。",
      "用例コーパスにも参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）にも existence proof は出てこない。"
    ],
    "related": [
      "uniqueness-proof",
      "exist",
      "existential-quantifier",
      "intermediate-value-theorem"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。そのまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "existential-quantifier",
    "ja": {
      "term": "存在量化子",
      "reading": "そんざいりょうかし",
      "alt": [
        "存在記号"
      ]
    },
    "en": {
      "term": "existential quantifier",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "notation"
    ],
    "level": {
      "jp": [
        "数I",
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "「ある x について」「x が存在する」を表す量化子で、記号 ∃ で書く。∃x P(x) は、P(x) を真にする x が少なくとも 1 つあることを表す。",
    "definition_en": "The quantifier \"there exists,\" written ∃. The statement ∃x P(x) says that at least one x in the domain makes P(x) true.",
    "latex": "\\exists x \\in \\mathbb{R},\\ x^2 = 2",
    "spoken_en": "there exists x in the real numbers such that x squared equals two",
    "examples": [
      {
        "en": "The backward E is the existential quantifier, so read this line as \"there exists an x such that…\"",
        "ja": "裏返しの E は存在記号だから、この行は「…となる x が存在する」と読もう。",
        "register": "spoken"
      },
      {
        "en": "Write the statement \"Some real number x satisfies x² = 2\" using the existential quantifier, and decide whether it is true.",
        "ja": "命題「x² = 2 を満たす実数 x が存在する」を存在記号を用いて書き、その真偽を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "∃ は「少なくとも 1 つある」の意味で、ちょうど 1 つを言うときは ∃! と書く。",
      "¬∃x P(x) は ∀x ¬P(x) と同じ。数I の「ある…」の否定が「すべての…でない」になるのと同じ規則。",
      "用例コーパスで existential quantifier は書き言葉にわずかに出てくるだけで、話し言葉には出てこない。Levin の本では 4 件。"
    ],
    "related": [
      "quantifier",
      "universal-quantifier",
      "exist",
      "predicate",
      "existence-proof"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "存在記号",
        "en": "Existential quantification"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "existential quantifier": 2
      },
      "sources": [
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは OpenStax の呼び方 existential quantifier（本文 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "expand",
    "ja": {
      "term": "展開する",
      "reading": "てんかいする",
      "alt": []
    },
    "en": {
      "term": "expand",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "かっこのある式の積を、かっこを外して単項式の和の形に書き直す。",
    "definition_en": "To rewrite a product of expressions as a sum of terms by multiplying out the parentheses; for example, (x + 2)(x + 3) expands to x² + 5x + 6.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To expand (x + 2)(x + 3), multiply every term in the first by every term in the second.",
        "ja": "(x + 2)(x + 3) を展開するには、前のかっこのすべての項に後ろのかっこのすべての項を掛ける。",
        "register": "spoken"
      },
      {
        "en": "Expand (2x − 1)(x + 4).",
        "ja": "(2x − 1)(x + 4) を展開せよ。",
        "register": "written"
      },
      {
        "en": "Careful: when you expand (x + 3) squared, you get x squared plus 6x plus 9, not just x squared plus 9.",
        "ja": "気をつけて。(x + 3)² を展開すると x² + 6x + 9 で、x² + 9 じゃないよ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "名詞は expansion（エントリ expansion）。同じ操作を multiply out とも言う。",
      "展開の逆は因数分解（factor）。問題文の Expand と Factor は向きが逆の指示なので、取り違えない。因数分解した積の形は factored form（エントリ factored-form）。",
      "(x + 3)² は x² + 9 ではない。(a + b)² = a² + 2ab + b² の 2ab を落とさない。"
    ],
    "related": [
      "expansion",
      "distribute",
      "distributive-property",
      "factored-form",
      "polynomial"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "expand": 262
      },
      "written": {
        "expand": 316
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "expanding-and-condensing-logs",
    "ja": {
      "term": "対数の展開と圧縮",
      "reading": "たいすうのてんかいとあっしゅく",
      "alt": []
    },
    "en": {
      "term": "expand the logarithm",
      "alt": [],
      "uk": null,
      "register": "written",
      "variants": [
        {
          "term": "condense the logarithm",
          "register": "written",
          "note": "いくつかの対数を 1 つにまとめる向き。書き言葉はすべて OpenStax（Intermediate Algebra・Algebra and Trigonometry）で、話し言葉には出てこない。"
        },
        {
          "term": "expand the logarithm",
          "register": "written",
          "note": "1 つの対数を和・差に分ける向き。書き言葉はすべて OpenStax（Intermediate Algebra・Algebra and Trigonometry）で、話し言葉にはほとんど出てこない。"
        }
      ]
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": "見出しの「対数の展開と圧縮」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書では「対数の性質を使って式を変形する」とだけ言う。米国の授業では、対数の性質で 1 つの log を和・差に分けることを expand、和・差を 1 つの log にまとめることを condense と呼ぶ。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "対数の性質を使って、log(MN) を log M + log N のように分ける（展開）、または逆に 1 つの対数にまとめる（圧縮）こと。",
    "definition_en": "Using the properties of logarithms to split one logarithm into a sum or difference (expand), or to combine several into one (condense).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To condense, go backwards: the coefficient becomes an exponent, and the plus becomes a product.",
        "ja": "まとめるときは逆にたどる。係数は指数に、足し算は掛け算になる。",
        "register": "spoken"
      },
      {
        "en": "Expand the logarithm log₂(8x³/√y). Then condense 2 ln x − ln(x + 1) into a single logarithm.",
        "ja": "log₂(8x³/√y) を対数の和・差に分けよ。また 2 ln x − ln(x + 1) を 1 つの対数にまとめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉（OpenStax Intermediate Algebra・Algebra and Trigonometry）では condense the logarithm と expand the logarithm がどちらもよく使われ、両方が並ぶ。話し言葉にはほとんど出てこない（MIT OCW の expand the logarithm だけ）。"
    ],
    "related": [
      "properties-of-logarithms",
      "logarithm"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "expand the logarithm": 1
      },
      "written": {
        "expand the logarithm": 10,
        "condense the logarithm": 12
      },
      "sources": [
        "mit-18.01",
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "expansion",
    "ja": {
      "term": "展開",
      "reading": "てんかい",
      "alt": [
        "展開式"
      ]
    },
    "en": {
      "term": "expansion",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "名詞の「展開」（展開式）は expansion。式を展開する操作は、IM Algebra 1 6.8 Equivalent Quadratic Expressions が multiply out（expand）と書き、OpenStax Elementary Algebra の問題文は Multiply: … の形。用例コーパスの話し言葉でも multiply out を使う。二項定理の展開は binomial expansion（OpenStax Algebra and Trigonometry 13.6）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3",
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "積の形の式を、分配法則で単項式の和に書き直すこと。また、書き直した式（展開式）。",
    "definition_en": "Rewriting a product as a sum of terms by distributing, or the resulting sum.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In the expansion of (x + 2)^5, the coefficient of x³ is 40.",
        "ja": "(x + 2)⁵ の展開式で、x³ の係数は 40。",
        "register": "spoken"
      },
      {
        "en": "Find the coefficient of x²y³ in the expansion of (2x − y)^5.",
        "ja": "(2x − y)⁵ の展開式における x²y³ の係数を求めよ。",
        "register": "written"
      },
      {
        "en": "Expand and simplify (x + 3)(x − 2) − (x − 1)².",
        "ja": "(x + 3)(x − 2) − (x − 1)² を展開して整理せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "expansion は series expansion（級数展開。OpenStax Calculus）や decimal expansion（小数展開。IM Grade 8）にも使う。",
      "「展開せよ」は Expand …（または Multiply out …）。"
    ],
    "related": [
      "expand",
      "binomial-theorem",
      "factoring",
      "distribute"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 6.8 Equivalent Quadratic Expressions: multiply out（expand）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "Properties of Power Series・Working with Taylor Series（series expansion）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "expansion of (": 15
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "expected-count",
    "ja": {
      "term": "期待度数",
      "reading": "きたいどすう",
      "alt": []
    },
    "en": {
      "term": "expected frequency",
      "alt": [
        "expected count"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "帰無仮説が正しいとしたときに、各カテゴリ（分割表の各セル）に入ると見込まれる度数。",
    "definition_en": "The count that each category (or cell of a two-way table) would be expected to have if the null hypothesis were true.",
    "latex": "E = \\frac{(\\text{row total})(\\text{column total})}{\\text{grand total}}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Check that every expected count is at least five before you use the chi-square test.",
        "ja": "カイ二乗検定を使う前に、期待度数がすべて 5 以上か確かめる。",
        "register": "spoken"
      },
      {
        "en": "Compute the expected frequency for each cell of the table.",
        "ja": "表の各セルの期待度数を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は expected frequency、AP Statistics の CED（2026 年版）は expected count と書く。",
      "期待値（expected value）とは別の語だが、OpenStax は式の説明で E = expected values とも書く。"
    ],
    "related": [
      "observed-count",
      "chi-square-test",
      "expected-value"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.14・3.15"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "expected frequency": 17
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "expected-value",
    "ja": {
      "term": "期待値",
      "reading": "きたいち",
      "alt": []
    },
    "en": {
      "term": "expected value",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "expectation",
          "register": "written",
          "note": "話し言葉では expected value より少なく（大半が MIT OCW）、書き言葉は多くが MIT の講義ノート。書き言葉では expected value と並ぶ。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "数A",
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "確率変数のとる値に、その値をとる確率を掛けて足し合わせた値。確率変数の平均。E(X) と書く。",
    "definition_en": "The sum of each value of a random variable times its probability: the mean of the random variable, written E(X).",
    "latex": "E(X) = \\sum_{i} x_i p_i",
    "spoken_en": "the expected value of X is the sum of x i times p i",
    "examples": [
      {
        "en": "The expected value is what you'd average out to if you played this game over and over.",
        "ja": "期待値は、このゲームを何度も繰り返したときの平均になる値。",
        "register": "spoken"
      },
      {
        "en": "Find the expected value of the number of heads in three tosses of a coin.",
        "ja": "硬貨を 3 回投げるときの表の出る回数の期待値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "compute the expected value",
        "ja": "期待値を計算する"
      }
    ],
    "pitfalls": [
      "E(X) は E of X と読む。AP Statistics の CED（topic 2.9）は expected value (or mean) と呼び、E(X) と μ_X で表す。"
    ],
    "related": [
      "random-variable",
      "probability-distribution",
      "arithmetic-mean"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "期待値",
        "en": "Expected value"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.9（expected value (or mean)、E(X)、μ_X）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "expectation": 81,
        "expected value": 364
      },
      "written": {
        "expected value": 128,
        "expectation": 124
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "experiment",
    "ja": {
      "term": "実験",
      "reading": "じっけん",
      "alt": []
    },
    "en": {
      "term": "experiment",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "研究者が被験者（実験の単位）に処理を割り当てて、その効果を調べる研究。処理を無作為に割り当てれば、因果関係を示せる。",
    "definition_en": "A study in which researchers assign treatments to the experimental units and measure the effect. When the treatments are assigned at random, an experiment can show cause and effect.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We're the ones deciding who gets the new drug, so this is an experiment, not an observational study.",
        "ja": "誰に新薬を与えるかをこちらで決めているから、これは観察研究ではなく実験だ。",
        "register": "spoken"
      },
      {
        "en": "In an experiment, tomato plants were randomly assigned to receive either fertilizer A or fertilizer B. Identify the treatments and the response variable.",
        "ja": "ある実験で、トマトの苗を無作為に分け、肥料 A か肥料 B のどちらかを与えた。処理と応答変数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "確率の「試行」（コインを投げるなど、エントリ trial）も英語で experiment と呼ぶことがある。統計の研究としての「実験」か、確率の「試行」かを場面で区別する。",
      "実験と観察研究（エントリ observational-study）の違いは、処理を割り当てるかどうか。因果関係を言えるのは、処理を無作為に割り当てた（random assignment、エントリ randomization）実験。"
    ],
    "related": [
      "observational-study",
      "randomization",
      "treatment",
      "control-group",
      "completely-randomized-design",
      "trial"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "observational study or an experiment | observational studies and experiments | designed experiment | randomized experiment | controlled experiment | well-designed experiment | randomized comparative experiment | conduct an experiment | design an experiment": 2
      },
      "written": {
        "observational study or an experiment | observational studies and experiments | designed experiment | randomized experiment | controlled experiment | well-designed experiment | randomized comparative experiment | conduct an experiment | design an experiment": 11
      },
      "sources": [
        "khan-ap-stats",
        "mit-notes",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "experimental-probability",
    "ja": {
      "term": "統計的確率",
      "reading": "とうけいてきかくりつ",
      "alt": []
    },
    "en": {
      "term": "empirical probability",
      "alt": [
        "experimental probability"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "OpenStax Introductory Statistics は empirical probability と呼ぶ。experimental probability は参照（CED・OpenStax・IM・CK-12）には出てこず、用例コーパスでは Khan Academy の中学の講義に少し出てくるだけ。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "実験や観察を何回も繰り返したとき、ある事柄が起こった相対度数を確率とみなしたもの。",
    "definition_en": "The relative frequency with which an event occurred in many repeated trials, taken as its probability.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We flipped the coin 200 times and got 94 heads, so the empirical probability is 0.47.",
        "ja": "硬貨を 200 回投げて表が 94 回出たので、統計的確率は 0.47。",
        "register": "spoken"
      },
      {
        "en": "Use the results of the experiment to estimate how likely a dropped thumbtack is to land point up.",
        "ja": "実験の結果を使って、落とした画びょうが上向きになる確率を推定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "理論から求める確率は theoretical probability（OpenStax Introductory Statistics）。"
    ],
    "related": [
      "relative-frequency"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "experimental probability": 7
      },
      "written": {
        "empirical probability": 5
      },
      "sources": [
        "khan-middle",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 empirical probability（本文 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "explain",
    "ja": {
      "term": "説明する",
      "reading": "せつめいする",
      "alt": []
    },
    "en": {
      "term": "explain",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "答えの出し方や、あることがなぜ正しいのかを、言葉や式を使って相手に分かるように述べる。",
    "definition_en": "To tell clearly, using words and math, how you got an answer or why something is true.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Don't just give me the number. Explain how you got it.",
        "ja": "数だけ言うんじゃなくて、どうやって出したか説明して。",
        "register": "spoken"
      },
      {
        "en": "If x is an integer, is 2x + 3 always odd? Explain your reasoning.",
        "ja": "x が整数のとき、2x + 3 はいつも奇数になるか。理由を説明せよ。",
        "register": "written"
      },
      {
        "en": "Explain why the sum of two even numbers is always even.",
        "ja": "2 つの偶数の和がいつも偶数になる理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "問題文の「理由を説明せよ」は Explain your reasoning. や Explain why …。根拠をはっきり示すことを求めるときは justify（根拠を示す）も使う（Justify your answer.）。",
      "「私に説明して」は Explain it to me.。explain は説明を受ける人を直接の目的語にとらないので、Explain me … とは言わない。",
      "「証明する」「〜を示せ」は prove ／ show that（エントリ prove）で、explain とは別の動詞。"
    ],
    "related": [
      "prove",
      "proof",
      "deductive-reasoning",
      "counterexample",
      "even-number"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "explain": 381
      },
      "written": {
        "explain": 868
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "explanatory-variable",
    "ja": {
      "term": "説明変数",
      "reading": "せつめいへんすう",
      "alt": []
    },
    "en": {
      "term": "explanatory variable",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "2 つの変量の関係を調べるとき、もう一方の変量を説明・予測するのに使う変量。散布図の横軸にとる。",
    "definition_en": "In a relationship between two variables, the variable used to explain or predict the other; it goes on the horizontal axis of a scatter plot.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Hours studied is the explanatory variable, and the test score is the response.",
        "ja": "勉強時間が説明変数で、テストの得点が応答変数。",
        "register": "spoken"
      },
      {
        "en": "Identify the explanatory variable and the response variable.",
        "ja": "説明変数と応答変数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数学の independent variable（独立変数）に当たる役割。"
    ],
    "related": [
      "response-variable",
      "scatter-plot",
      "regression-line"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "explanatory variable": 7
      },
      "written": {
        "explanatory variable": 23
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "exponent",
    "ja": {
      "term": "指数",
      "reading": "しすう",
      "alt": []
    },
    "en": {
      "term": "exponent",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "累乗 aⁿ で、a の右肩に小さく書く数 n。a を何個掛け合わせるかを表す。",
    "definition_en": "The small raised number written up and to the right of the base in a power. It tells how many times the base is used as a factor.",
    "latex": "a^{n}",
    "spoken_en": "a to the n",
    "examples": [
      {
        "en": "The exponent tells you how many 3s to multiply, so 3 to the fourth is 3 times 3 times 3 times 3, or 81.",
        "ja": "指数は 3 を何個掛けるかを表す。だから 3⁴ は 3 × 3 × 3 × 3 で 81。",
        "register": "spoken"
      },
      {
        "en": "Name the base and the exponent of (−4)³, and then evaluate it.",
        "ja": "(−4)³ の底と指数を答え、その値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "x⁵ は x to the fifth、x to the fifth power、x to the power of 5 と読む。2 乗は squared、3 乗は cubed とも読む。",
      "exponent は右肩の数 n だけを指す。aⁿ 全体は power（エントリ power）、下の a は base（エントリ base-of-a-power）。",
      "「指数関数」の「指数」は形容詞 exponential で言う（exponential function、エントリ exponential-function）。"
    ],
    "related": [
      "power",
      "base-of-a-power",
      "laws-of-exponents",
      "negative-exponent",
      "zero-exponent"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "exponent": 1269
      },
      "written": {
        "exponent": 1266
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "exponential-decay",
    "ja": {
      "term": "指数関数的減少",
      "reading": "しすうかんすうてきげんしょう",
      "alt": [
        "指数関数的減衰"
      ]
    },
    "en": {
      "term": "exponential decay",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1",
        "Integrated Math 3",
        "Calculus I",
        "Algebra 1"
      ]
    },
    "definition_ja": "一定の時間ごとに一定の倍率で減ること。y = a·bᵗ（0 < b < 1）や y = Ae^(−kt)（k > 0）で表す。",
    "definition_en": "Decrease by the same factor over each equal time interval, modeled by y = a·bᵗ (0 < b < 1) or y = Ae^(−kt) (k > 0).",
    "latex": "y = A e^{-kt}\\quad(k > 0)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Radioactive decay is the classic example of exponential decay — think half-life.",
        "ja": "放射性崩壊は指数関数的減少の代表例。半減期を考えよう。",
        "register": "spoken"
      },
      {
        "en": "A substance has a half-life of 8 days. How much of 100 g remains after 20 days?",
        "ja": "半減期 8 日の物質 100 g は、20 日後に何 g 残るか。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "exponential decay model",
        "ja": "指数関数的減少のモデル"
      }
    ],
    "pitfalls": [],
    "related": [
      "exponential-growth",
      "exponential-model",
      "newtons-law-of-cooling"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "exponential decay": 28
      },
      "written": {
        "exponential decay": 52
      },
      "sources": [
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "exponential-distribution",
    "ja": {
      "term": "指数分布",
      "reading": "しすうぶんぷ",
      "alt": []
    },
    "en": {
      "term": "exponential distribution",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "次の出来事が起こるまでの待ち時間などが従う連続型の確率分布。確率密度は λe^{−λx}（x ≥ 0）。",
    "definition_en": "A continuous distribution for waiting times until the next event; its density is λe^{−λx} for x ≥ 0.",
    "latex": "f(x) = \\lambda e^{-\\lambda x},\\ x \\ge 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "The exponential distribution is memoryless: having waited ten minutes doesn't make the bus come sooner.",
        "ja": "指数分布には記憶がない。10 分待ったからといって、バスが早く来るわけではない。",
        "register": "spoken"
      },
      {
        "en": "The lifetime of a battery follows an exponential distribution with mean 3 years. Find the probability that it lasts more than 5 years.",
        "ja": "電池の寿命は平均 3 年の指数分布に従う。5 年より長くもつ確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics。"
    ],
    "related": [
      "poisson-distribution",
      "uniform-distribution",
      "probability-density-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "指数分布",
        "en": "Exponential distribution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "exponential distribution": 3
      },
      "written": {
        "exponential distribution": 50
      },
      "sources": [
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "exponential-equation",
    "ja": {
      "term": "指数方程式",
      "reading": "しすうほうていしき",
      "alt": []
    },
    "en": {
      "term": "exponential equation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "未知数を指数に含む方程式。底をそろえるか、両辺の対数をとって解く。",
    "definition_en": "An equation with the unknown in an exponent, solved by writing both sides with the same base or by taking logs of both sides.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you can write both sides as powers of 2, just set the exponents equal.",
        "ja": "両辺を 2 の累乗で書ければ、指数どうしを等しいとおけばいい。",
        "register": "spoken"
      },
      {
        "en": "Solve the exponential equation 9^x − 4·3^x + 3 = 0.",
        "ja": "方程式 9ˣ − 4·3ˣ + 3 = 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは書き言葉の大半が OpenStax Algebra and Trigonometry で、残りはほとんど OpenStax Intermediate Algebra。話し言葉にはほとんど出てこない。"
    ],
    "related": [
      "exponential-inequality",
      "write-with-the-same-base",
      "take-the-log-of-both-sides",
      "one-to-one-property",
      "exponential-function",
      "logarithmic-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "exponential equation": 3
      },
      "written": {
        "exponential equation": 94
      },
      "sources": [
        "khan-algebra",
        "mit-notes",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "exponential-function",
    "ja": {
      "term": "指数関数",
      "reading": "しすうかんすう",
      "alt": []
    },
    "en": {
      "term": "exponential function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1",
        "Integrated Math 3",
        "Algebra 1"
      ]
    },
    "definition_ja": "a > 0, a ≠ 1 の定数 a について y = aˣ で表される関数。a > 1 なら増加、0 < a < 1 なら減少する。",
    "definition_en": "A function y = aˣ with a constant base a > 0, a ≠ 1; it is increasing when a > 1 and decreasing when 0 < a < 1.",
    "latex": "y = a^x\\quad(a > 0,\\ a \\ne 1)",
    "spoken_en": "y equals a to the x",
    "examples": [
      {
        "en": "The graph of the exponential function y = a to the x always passes through (0, 1).",
        "ja": "指数関数 y = aˣ のグラフは必ず (0, 1) を通る。",
        "register": "spoken"
      },
      {
        "en": "Sketch the graph of the exponential function y = 2^(x−1) + 3 and state its asymptote.",
        "ja": "指数関数 y = 2^(x−1) + 3 のグラフをかき、漸近線を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry（6.1）は y = abˣ（a は initial value、b は x が 1 増えるごとに掛かる倍率 growth factor）の形も exponential function と呼ぶ。"
    ],
    "related": [
      "exponential-equation",
      "logarithmic-function",
      "asymptote",
      "exponential-growth",
      "decreasing"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "指数関数",
        "en": "Natural exponential function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.1 Exponential Functions（f(x) = abˣ、growth factor）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "exponential function": 153
      },
      "written": {
        "exponential function": 339
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "exponential-growth",
    "ja": {
      "term": "指数関数的増加",
      "reading": "しすうかんすうてきぞうか",
      "alt": [
        "指数関数的成長"
      ]
    },
    "en": {
      "term": "exponential growth",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1",
        "Integrated Math 3",
        "Calculus I",
        "Algebra 1"
      ]
    },
    "definition_ja": "一定の時間ごとに一定の倍率で増えること。y = a·bᵗ（b > 1）や y = Ae^(kt)（k > 0）で表す。",
    "definition_en": "Growth by the same factor over each equal time interval, modeled by y = a·bᵗ (b > 1) or y = Ae^(kt) (k > 0).",
    "latex": "y = A e^{kt}\\quad(k > 0)",
    "spoken_en": "y equals A e to the k t",
    "examples": [
      {
        "en": "Exponential growth starts slow and then takes off.",
        "ja": "指数関数的増加は、最初はゆっくりで、あるところから急に増える。",
        "register": "spoken"
      },
      {
        "en": "A town's population shows exponential growth of 3% per year. Write a model for the population.",
        "ja": "ある町の人口は毎年 3% の割合で指数関数的に増加する。人口を表す式をつくれ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "exponential-decay",
      "grow-exponentially",
      "exponential-model",
      "logistic-growth"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "指数関数的成長",
        "en": "Exponential growth"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "exponential growth": 18
      },
      "written": {
        "exponential growth": 102
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "exponential-inequality",
    "ja": {
      "term": "指数不等式",
      "reading": "しすうふとうしき",
      "alt": []
    },
    "en": {
      "term": "exponential inequality",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。参照は指数を含む不等式を名前で呼ばず、式を変形して解く問題も無い。IM Algebra 1（5.4 ほか）・IM Algebra 2（4.2・4.5・4.16）は、指数関数のモデルで「何回で〜より小さくなるか」のように境目を問い、表やグラフで答えさせる。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "definition_ja": "未知数を指数に含む不等式。底が 1 より大きいか小さいかで、指数の大小と不等号の向きの関係が変わる。",
    "definition_en": "An inequality with the unknown in an exponent; whether the base is greater or less than 1 decides whether the inequality sign keeps or flips when comparing exponents.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The base is one half, which is less than 1, so the inequality flips when you compare the exponents.",
        "ja": "底が 1/2 で 1 より小さいので、指数を比べると不等号の向きが変わる。",
        "register": "spoken"
      },
      {
        "en": "Solve (1/3)^(2x − 1) > 1/27.",
        "ja": "不等式 (1/3)^(2x − 1) > 1/27 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "exponential-equation",
      "logarithmic-inequality",
      "compare"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 1 5.4 ほか・Algebra 2 4.2・4.5・4.16（指数関数のモデルで境目を問う文章題）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学II: 指数方程式と指数不等式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "exponential-model",
    "ja": {
      "term": "指数モデル",
      "reading": "しすうもでる",
      "alt": [
        "指数増加・減衰のモデル"
      ]
    },
    "en": {
      "term": "exponential model",
      "alt": [],
      "variants": [
        {
          "term": "exponential model",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry の言い方（書き言葉の大半）。データに指数関数を当てはめる節の名前にもなっている。話し言葉は少なく、すべて Khan Academy。"
        },
        {
          "term": "exponential growth and decay",
          "register": "written",
          "note": "OpenStax Calculus の節の名前（Exponential Growth and Decay）で、AP の CED（topic 7.8）も使う。話し言葉には出てこず、書き言葉（OpenStax Algebra and Trigonometry・Calculus・Intermediate Algebra）にだけ出てくる。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「指数モデル」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "変化率が量に比例する（dy/dt = ky）現象を、y = Ce^{kt} で表すモデル。k > 0 で増加、k < 0 で減衰。",
    "definition_en": "A model where the amount changes at a rate proportional to itself, dy/dt = ky, so y = Ce^{kt}: growth when k > 0, decay when k < 0.",
    "latex": "\\frac{dy}{dt} = ky \\;\\Rightarrow\\; y = Ce^{kt}",
    "spoken_en": "d y d t equals k y, so y equals C e to the k t",
    "examples": [
      {
        "en": "If the rate is proportional to the amount, you're looking at an exponential model.",
        "ja": "変化率が量に比例するなら、それは指数モデル。",
        "register": "spoken"
      },
      {
        "en": "A culture grows according to an exponential model with k = 0.3 per hour. How long does it take to double?",
        "ja": "ある培養は k = 0.3（1 時間あたり）の指数モデルに従って増える。2 倍になるまでの時間を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉ではどの言い方も少ない。",
      "半減期は half-life、2 倍になる時間は doubling time。"
    ],
    "related": [
      "differential-equation",
      "separable-differential-equation",
      "logistic-differential-equation",
      "initial-condition"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 7.8（exponential models ／ exponential growth and decay）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「指数モデル」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "exponential model": 5
      },
      "written": {
        "exponential model": 41,
        "exponential growth and decay": 18
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "exponential-regression",
    "ja": {
      "term": "指数回帰",
      "reading": "しすうかいき",
      "alt": []
    },
    "en": {
      "term": "exponential regression",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "データに最もよく合う指数関数 y = abˣ を求めること。回帰直線を、直線の代わりに指数関数で当てはめたもの。",
    "definition_en": "Finding the exponential function y = ab^x that best fits a set of data. It works like finding a regression line, but the model is an exponential curve instead of a line.",
    "latex": "y = ab^x",
    "spoken_en": "y equals a times b to the x",
    "examples": [
      {
        "en": "These points curve up way too fast for a line, so let's run an exponential regression on the calculator.",
        "ja": "この点は直線にしては急に上がりすぎるから、電卓で指数回帰をしてみよう。",
        "register": "spoken"
      },
      {
        "en": "Use exponential regression to find a function of the form y = ab^x that models the data in the table. Round a and b to the nearest hundredth.",
        "ja": "指数回帰を用いて、表のデータに合う y = abˣ の形の関数を求めよ。a、b は小数第 2 位まで求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校では扱わない。グラフ電卓の ExpReg（Exponential Regression）機能で a と b を求める。",
      "回帰直線（エントリ regression-line）は直線 y = a + bx を当てはめる。指数回帰の b は傾きではなく、x が 1 増えるごとにかかる倍率（b > 1 なら増加、0 < b < 1 なら減少）。"
    ],
    "related": [
      "regression-line",
      "exponential-model",
      "exponential-function",
      "scatter-plot",
      "quadratic-regression"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "exponential regression": 14
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "expression",
    "ja": {
      "term": "式",
      "reading": "しき",
      "alt": []
    },
    "en": {
      "term": "expression",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「式」は等式や不等式も指す（中学校学習指導要領解説（第 1 学年）は、数量の関係を表す式を等式または不等式に表すとし、日本語版 Wikipedia「等式」は等式を相等関係を表す数式とする）が、英語の expression は等号を含まないもの（3x + 2）だけ。等号を含むもの（3x + 2 = 8）は equation、不等号を含むものは inequality と言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数や文字を、演算の記号（+、−、×、÷）やかっこで結んで、1 つの値を表したもの。英語の expression は等号を含まないものを指す。",
    "definition_en": "A combination of numbers, variables, and operation symbols, such as 3x + 2, that stands for a value. Unlike an equation, it has no equals sign.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is an expression, not an equation, so there's nothing to solve; we just simplify it.",
        "ja": "これは方程式じゃなくて式だから、解くものはない。簡単にするだけだよ。",
        "register": "spoken"
      },
      {
        "en": "Simplify the expression 4(x + 3) − 2x.",
        "ja": "式 4(x + 3) − 2x を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "expression は solve しない。式は simplify（簡単にする）か evaluate（値を求める。エントリ evaluate）で、solve するのは equation。",
      "「式を立てる」は、等号を含むなら write an equation、含まないなら write an expression と言い分ける。"
    ],
    "related": [
      "algebraic-expression",
      "term",
      "value-of-the-expression",
      "evaluate",
      "polynomial"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「等式」",
        "url": "https://ja.wikipedia.org/wiki/%E7%AD%89%E5%BC%8F",
        "note": "等式は二つの対象の相等関係を表す数式（mapping_note）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A 数と式: 数量の関係を表す式を等式または不等式に表す（mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "expression": 2164
      },
      "written": {
        "expression": 2838
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "extended-euclidean-algorithm",
    "ja": {
      "term": "拡張ユークリッド互除法",
      "reading": "かくちょうゆーくりっどごじょほう",
      "alt": []
    },
    "en": {
      "term": "extended Euclidean algorithm",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "ユークリッドの互除法の計算を使って、gcd(a, b) だけでなく ax + by = gcd(a, b) を満たす整数 x, y も求める手順のこと。互除法の各段の式を逆にたどって x, y を得る。",
    "definition_en": "A version of the Euclidean algorithm that finds not only gcd(a, b) but also integers x and y with ax + by = gcd(a, b). The values of x and y come from working back through the division steps.",
    "latex": "9 \\cdot 7 - 2 \\cdot 31 = 1",
    "spoken_en": "nine times seven minus two times thirty-one equals one",
    "examples": [
      {
        "en": "Once you get a remainder of 1, work backward through the steps. That's the extended Euclidean algorithm.",
        "ja": "余りが 1 になったら、式を逆にたどっていこう。これが拡張ユークリッド互除法だ。",
        "register": "spoken"
      },
      {
        "en": "Use the extended Euclidean algorithm to find integers x and y such that 31x + 7y = 1.",
        "ja": "互除法を用いて、31x + 7y = 1 を満たす整数 x, y の組を 1 つ求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数A の一次不定方程式では、この手順を名前を付けずに「互除法の式を逆にたどる」形で使う。",
      "求まる (x, y) は 1 組だけ。d = gcd(a, b) とすると、ほかの整数解は x を b/d ずつ、y を a/d ずつ逆向きにずらして得られる。",
      "用例コーパスにも参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）にも出てこない。"
    ],
    "related": [
      "euclidean-algorithm",
      "bezouts-identity",
      "linear-diophantine-equation",
      "modular-inverse",
      "greatest-common-divisor"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。そのまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "exterior-angle-bisector",
    "ja": {
      "term": "外角の二等分線",
      "reading": "がいかくのにとうぶんせん",
      "alt": []
    },
    "en": {
      "term": "exterior angle bisector",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（exterior angle bisector は 0 件）。英語版 Wikipedia「Angle bisector theorem」の節 Exterior angle bisectors が、外角の二等分線が対辺を外分する定理を書く。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "definition_ja": "三角形の 1 つの頂点で、外角を 2 等分する直線。∠A の外角の二等分線が直線 BC と交わる点は、辺 BC を AB : AC に外分する。",
    "definition_en": "The line that cuts an exterior angle of a triangle into two equal angles. The exterior angle bisector at A meets line BC at a point that divides BC externally in the ratio AB : AC.",
    "latex": "\\frac{BE}{EC} = \\frac{AB}{AC}",
    "spoken_en": "BE over EC equals AB over AC",
    "examples": [
      {
        "en": "The exterior angle bisector at A hits line BC outside the triangle, and it splits BC externally in the ratio AB to AC.",
        "ja": "∠A の外角の二等分線は三角形の外で直線 BC と交わり、BC を AB : AC に外分する。",
        "register": "spoken"
      },
      {
        "en": "In triangle ABC, AB = 6 and AC = 4. The exterior angle bisector at A meets line BC at E. Find BE : EC.",
        "ja": "△ABC において AB = 6, AC = 4 とする。∠A の外角の二等分線と直線 BC の交点を E とするとき、BE : EC を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "交点は辺 BC 上ではなく BC の延長上にある（外分点）。内角の二等分線（内分）と比は同じ AB : AC。",
      "AB = AC のときは外角の二等分線が BC に平行になり、交点ができない。",
      "内角の二等分線と外角の二等分線は直交する。"
    ],
    "related": [
      "angle-bisector-theorem",
      "point-of-external-division",
      "excenter",
      "triangle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Angle_bisector_theorem",
        "note": "記事「Angle bisector theorem」の節 Exterior angle bisectors（数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Bisection",
        "note": "記事「Bisection」（内角と外角の二等分線は直交する）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 図形の性質: 外角の場合も含めた角の二等分線と辺の比の関係"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元 3 の前の修正」の、残った ③ を人間が決めた行）。en.term は exterior angle bisector のまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "exterior-angle-theorem",
    "ja": {
      "term": "外角定理",
      "reading": "がいかくていり",
      "alt": [
        "三角形の外角の性質"
      ]
    },
    "en": {
      "term": "exterior angle theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "三角形の 1 つの外角は、それと隣り合わない 2 つの内角の和に等しい、という定理。",
    "definition_en": "The theorem that an exterior angle of a triangle equals the sum of the two interior angles that are not adjacent to it.",
    "latex": "m\\angle ACD = m\\angle A + m\\angle B",
    "spoken_en": "the measure of angle ACD equals the measure of angle A plus the measure of angle B",
    "examples": [
      {
        "en": "The exterior angle equals the two remote interior angles added together, so x is 50 plus 65, which is 115.",
        "ja": "外角は隣り合わない 2 つの内角の和に等しいから、x = 50 + 65 = 115。",
        "register": "spoken"
      },
      {
        "en": "In △ABC, side BC is extended through C to point D. If m∠A = 48° and m∠ACD = 121°, find m∠B.",
        "ja": "△ABC で、辺 BC を C の側に延長した直線上に点 D をとる。∠A = 48°、∠ACD = 121° のとき、∠B を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "外角と隣り合わない 2 つの内角は、英語で remote interior angles（nonadjacent interior angles）と呼ぶ。",
      "英語の exterior angle theorem は、外角が隣り合わない内角のどちらよりも大きいという不等式（ユークリッド原論 I.16）を指すこともある。",
      "用例コーパスに exterior angle theorem は出てこない。IM では exterior angle が 5 件（Grade 8）あるが、定理の名前としては使われていない。CK-12 Geometry は 9 件（4.18 Exterior Angles and Theorems・3.6 Alternate Exterior Angles ほか）。"
    ],
    "related": [
      "triangle",
      "angle",
      "exterior-angle-bisector",
      "supplementary-angle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "外角定理",
        "en": "Exterior angle theorem"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.18 Exterior Angles and Theorems・3.6 Alternate Exterior Angles・5.28 Exterior Angles in Convex Polygons"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 exterior angle theorem（CK-12 Geometry 4.18 Exterior Angles and Theorems・CK-12 Geometry 3.6 Alternate Exterior Angles・CK-12 Geometry 5.28 Exterior Angles in Convex Polygons・計 9 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "exterior-angle",
    "ja": {
      "term": "外角",
      "reading": "がいかく",
      "alt": []
    },
    "en": {
      "term": "exterior angle",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "多角形の 1 つの辺と、となりの辺を延長した直線とがつくる、多角形の外側の角。1 つの頂点で、内角と外角の和は 180° になる。",
    "definition_en": "The angle formed outside a polygon by one side and the extension of the side next to it. At each vertex, the interior angle and the exterior angle add up to 180°.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Extend this side out past the vertex, and the angle you get outside the triangle is the exterior angle.",
        "ja": "この辺を頂点の先まで延ばすと、三角形の外側にできる角が外角。",
        "register": "spoken"
      },
      {
        "en": "In △ABC, m∠A = 45° and m∠B = 70°. Find the measure of the exterior angle at vertex C.",
        "ja": "△ABC で ∠A = 45°、∠B = 70° のとき、頂点 C における外角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "三角形の外角が、となり合わない 2 つの内角の和に等しいことは exterior angle theorem（エントリ exterior-angle-theorem）。",
      "exterior は「外側の」の意味。平行線の alternate exterior angles（エントリ alternate-exterior-angles）の exterior は平行な 2 直線の外側のことで、多角形の外角とは指す角が違う。",
      "外角をすべて足したものは exterior angle sum（エントリ sum-of-the-exterior-angles）。"
    ],
    "related": [
      "interior-angle",
      "exterior-angle-theorem",
      "sum-of-the-exterior-angles",
      "polygon",
      "triangle"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.18 Exterior Angles and Theorems・5.28 Exterior Angles in Convex Polygons"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "exterior angle": 44
      },
      "sources": [
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "extraneous-solution",
    "ja": {
      "term": "無縁解",
      "reading": "むえんかい",
      "alt": []
    },
    "en": {
      "term": "extraneous solution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1"
      ]
    },
    "definition_ja": "方程式を解く途中で両辺を 2 乗したり分母を払ったりしたときに出てくるが、もとの方程式を満たさない値。解の候補に入れてはいけない。",
    "definition_en": "A value that comes out of the solving process, for example after squaring both sides or clearing denominators, but does not satisfy the original equation. It must be thrown out.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Squaring both sides can give you an extraneous solution, so always plug your answers back into the original equation.",
        "ja": "両辺を 2 乗すると無縁解が出ることがあるから、答えは必ずもとの方程式に代入して確かめよう。",
        "register": "spoken"
      },
      {
        "en": "Solve √(2x + 3) = x and identify any extraneous solution.",
        "ja": "方程式 √(2x + 3) = x を解け。無縁解があればそれも示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "√(x + 3) = x − 3 の両辺を 2 乗すると x = 1、6 が出るが、x = 1 は左辺が 2、右辺が −2 になるので無縁解。答えは x = 6 だけ。",
      "日本では「解の吟味」（エントリ checking-whether-the-solution-makes-sense）で除くと言う。英語では、見つけた値が extraneous solution かどうかを check して除く。",
      "無縁解があっても、残りの値が解になることはある。方程式にまったく解がない場合（エントリ no-solution）とは区別する。"
    ],
    "related": [
      "radical-equation",
      "checking-whether-the-solution-makes-sense",
      "clear-the-denominators",
      "logarithmic-equation",
      "no-solution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "extraneous solution": 21
      },
      "written": {
        "extraneous solution": 50
      },
      "sources": [
        "khan-algebra",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "extrapolation",
    "ja": {
      "term": "外挿",
      "reading": "がいそう",
      "alt": [
        "補外"
      ]
    },
    "en": {
      "term": "extrapolation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics",
        "Precalculus"
      ]
    },
    "definition_ja": "データの範囲の外側の値を、モデルを延長して推定すること。範囲から離れるほど信頼できなくなる。",
    "definition_en": "Estimating a value outside the range of the data by extending a model; the farther from the data, the less reliable it is.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Using the line to predict way beyond your data is extrapolation, and it can go badly wrong.",
        "ja": "データのずっと外側を直線で予測するのが外挿で、大きく外れることがある。",
        "register": "spoken"
      },
      {
        "en": "Explain why using the regression line to predict the value at x = 100 is extrapolation.",
        "ja": "回帰直線を使って x = 100 での値を予測することが外挿である理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "interpolation",
      "regression-line",
      "predict"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "外挿",
        "en": "Extrapolation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 5.3（5.3.A.3 extrapolation）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "2.4 Fitting Linear Models to Data（Recognizing Interpolation or Extrapolation）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「外挿」",
        "url": "https://ja.wikipedia.org/wiki/%E5%A4%96%E6%8C%BF",
        "note": "外挿（英: extrapolation）・補外"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "extrapolation": 3
      },
      "written": {
        "extrapolation": 22
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "extreme-value-theorem",
    "ja": {
      "term": "最大値・最小値の定理",
      "reading": "さいだいち・さいしょうちのていり",
      "alt": [
        "極値定理"
      ]
    },
    "en": {
      "term": "extreme value theorem",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "ja.alt の「極値定理」は学習指導要領解説に無い（日本語版 Wikipedia「最大値最小値定理」は extreme value theorem の訳として挙げる）。",
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "閉区間 [a, b] で連続な関数は、その区間で最大値と最小値をとるという定理。",
    "definition_en": "A function continuous on [a, b], a closed interval, reaches both a largest and a smallest value somewhere in [a, b].",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The extreme value theorem only works on a closed interval.",
        "ja": "最大値・最小値の定理は閉区間でしか使えない。",
        "register": "spoken"
      },
      {
        "en": "By the extreme value theorem, f(x) = x³ − 3x attains a maximum and a minimum on [0, 2].",
        "ja": "最大値・最小値の定理より、f(x) = x³ − 3x は [0, 2] で最大値と最小値をとる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の名前の extreme（極端な）は極値（local extremum）ではなく、最大・最小（absolute extrema）のこと。"
    ],
    "related": [
      "continuous-function",
      "intermediate-value-theorem",
      "local-extremum"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "最大値最小値定理",
        "en": "Extreme value theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.2（extreme value theorem）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「最大値最小値定理」",
        "url": "https://ja.wikipedia.org/wiki/%E6%9C%80%E5%A4%A7%E5%80%A4%E6%9C%80%E5%B0%8F%E5%80%A4%E5%AE%9A%E7%90%86",
        "note": "英: extreme value theorem の訳語として「極値定理」"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「極値定理」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "extreme value theorem": 21
      },
      "written": {
        "extreme value theorem": 13
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "f-distribution",
    "ja": {
      "term": "F 分布",
      "reading": "えふぶんぷ",
      "alt": []
    },
    "en": {
      "term": "F distribution",
      "alt": [
        "F-distribution"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "2 つの分散の比が従う連続型の確率分布。分子と分母の 2 つの自由度で形が決まり、分散分析や 2 つの分散の比較に使う。",
    "definition_en": "The continuous distribution of a ratio of two variances; its shape depends on two degrees of freedom, and it is used in ANOVA and to compare two variances.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The F distribution is skewed right and never goes below zero.",
        "ja": "F 分布は右に歪んでいて、0 より小さくならない。",
        "register": "spoken"
      },
      {
        "en": "Find the p-value using the F distribution with 2 and 12 degrees of freedom.",
        "ja": "自由度 2 と 12 の F 分布を用いて p 値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics（F distribution がほとんどで、F-distribution は少ない）。"
    ],
    "related": [
      "anova",
      "degrees-of-freedom",
      "chi-square-test"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "F distribution": 27,
        "F-distribution": 2
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "face",
    "ja": {
      "term": "面",
      "reading": "めん",
      "alt": []
    },
    "en": {
      "term": "face",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "多面体を囲んでいる、平らな多角形の部分。立方体には 6 つの面がある。",
    "definition_en": "One of the flat polygon surfaces that form the boundary of a polyhedron. A cube has 6 faces.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A cube has six faces, and every face is a square.",
        "ja": "立方体には面が 6 つあって、どの面も正方形だよ。",
        "register": "spoken"
      },
      {
        "en": "Complete the table by counting the faces, edges, and vertices of each solid.",
        "ja": "それぞれの立体の面・辺・頂点の数を数えて、表を完成させよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "限りなく広がる平らな「面」（平面）は plane（エントリ plane）。face は立体の表面のうちの 1 枚の多角形。",
      "角柱・角錐の底面以外の面は lateral face（側面、エントリ lateral-face）。円柱・円錐の曲がった側面は多角形ではないので、face とは区別して lateral surface と言う。",
      "オイラーの多面体定理 V − E + F = 2 の F は faces の頭文字（エントリ eulers-formula-for-polyhedra）。"
    ],
    "related": [
      "edge",
      "vertex",
      "lateral-face",
      "polyhedron",
      "eulers-formula-for-polyhedra"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "face of the | face of a | faces, edges | faces and edges | lateral face | number of faces": 19
      },
      "written": {
        "face of the | face of a | faces, edges | faces and edges | lateral face | number of faces": 26
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "factor-theorem",
    "ja": {
      "term": "因数定理",
      "reading": "いんすうていり",
      "alt": []
    },
    "en": {
      "term": "factor theorem",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus"
      ]
    },
    "definition_ja": "整式 P(x) が x − a を因数にもつことと、P(a) = 0 であることが同じだという定理。",
    "definition_en": "The theorem that a polynomial P(x) has x − a as a factor exactly when P(a) = 0.",
    "latex": "P(a) = 0 \\iff (x - a) \\mid P(x)",
    "spoken_en": "P of a equals zero if and only if x minus a is a factor of P of x",
    "examples": [
      {
        "en": "P of 1 is zero, so by the factor theorem x − 1 is a factor.",
        "ja": "P(1) = 0 だから、因数定理より x − 1 は因数。",
        "register": "spoken"
      },
      {
        "en": "Use the factor theorem to factor x³ − 6x² + 11x − 6 completely.",
        "ja": "因数定理を用いて x³ − 6x² + 11x − 6 を因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "by the factor theorem",
        "ja": "因数定理より"
      }
    ],
    "pitfalls": [
      "OpenStax Intermediate Algebra は剰余の定理と組にして教える（Use the Remainder and Factor Theorems）。OpenStax Algebra and Trigonometry の Zero Factor Theorem（zero-product property）は別の性質。"
    ],
    "related": [
      "remainder-theorem",
      "synthetic-division",
      "rational-root-theorem",
      "polynomial-equation-of-higher-degree"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "因数定理",
        "en": "Factor theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Use the Remainder and Factor Theorems"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "Zeros of Polynomial Functions（Factor Theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "factor theorem": 3
      },
      "written": {
        "factor theorem": 31
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "factor",
    "ja": {
      "term": "因数",
      "reading": "いんすう",
      "alt": []
    },
    "en": {
      "term": "factor",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "積をつくっている 1 つ 1 つの数や式。x² − 4 = (x + 2)(x − 2) では、x + 2 と x − 2 が x² − 4 の因数である。",
    "definition_en": "One of the numbers or expressions that are multiplied together to make a product. Since x² − 4 = (x + 2)(x − 2), both x + 2 and x − 2 are factors of x² − 4.",
    "latex": "x^2 - 4 = (x + 2)(x - 2)",
    "spoken_en": "x squared minus four equals the quantity x plus two times the quantity x minus two",
    "examples": [
      {
        "en": "So x + 2 is one factor and x − 2 is the other factor; multiply them back together and you get x² − 4.",
        "ja": "つまり x + 2 が 1 つの因数で、x − 2 がもう 1 つの因数。掛け合わせると x² − 4 に戻る。",
        "register": "spoken"
      },
      {
        "en": "Show that x + 3 is a factor of x² + 7x + 12.",
        "ja": "x + 3 が x² + 7x + 12 の因数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の factor は、整数の約数（エントリ divisor）の意味にもなる。3 is a factor of 12 は「3 は 12 の約数」、x + 2 is a factor of x² − 4 は「x + 2 は x² − 4 の因数」で、数の話か式の話かで見分ける。",
      "動詞の factor は「因数分解する」（エントリ factoring）。問題文の Factor x² − 4. は「x² − 4 を因数分解せよ」で、「因数を答えよ」ではない。",
      "どの項にも共通して含まれる因数は common factor（共通因数、エントリ common-factor）。"
    ],
    "related": [
      "factoring",
      "divisor",
      "common-factor",
      "prime-factor",
      "product"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "linear factor | quadratic factor | each factor | two factors | the factors are | irreducible factor": 88
      },
      "written": {
        "linear factor | quadratic factor | each factor | two factors | the factors are | irreducible factor": 256
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "factored-form",
    "ja": {
      "term": "因数分解形",
      "reading": "いんすうぶんかいけい",
      "alt": [
        "分解形"
      ]
    },
    "en": {
      "term": "factored form",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "intercept form",
          "register": "both",
          "note": "x 切片が読める形という名前。slope-intercept form（傾き・切片形）と区別して、前に slope が付かない形で数えた。用例コーパスにわずかで、参照（OpenStax・IM・CK-12）には出てこない。factored form よりずっと少ない。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「因数分解形」は学習指導要領解説に無い（日本語版 Wikipedia「二次関数」にはある）。日本語版 Wikipedia「二次関数」は f(x) = a(x − s)(x − t) の形を因数分解形（単に分解形とも）と呼び、高等学校学習指導要領解説（数学I）はこの形に名前を付けず、二次方程式の解を二次関数のグラフと x 軸との共有点の x 座標として扱う。英語ではこの形に factored form ／ intercept form という名前があり、話し言葉・書き言葉とも factored form が多い（書き言葉は OpenStax Algebra and Trigonometry が中心）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "二次関数を y = a(x − α)(x − β) と一次式の積で表した形。グラフと x 軸の交点 x = α, β が読み取れる。",
    "definition_en": "A quadratic function written as a product of linear factors, y = a(x − r)(x − s), which shows the x-intercepts r and s.",
    "latex": "y = a(x - \\alpha)(x - \\beta)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Factored form is nice because you can see the zeros right away.",
        "ja": "因数分解形の良いところは、零点がすぐに見えることだ。",
        "register": "spoken"
      },
      {
        "en": "Put it in intercept form and the x-intercepts are right there: 2 and −5.",
        "ja": "因数分解形にすれば、x 切片は 2 と −5 だとすぐわかる。",
        "register": "spoken"
      },
      {
        "en": "Write f(x) = x² + 3x − 10 in factored form and find the x-intercepts of its graph.",
        "ja": "f(x) = x² + 3x − 10 を因数分解した形で表し、グラフと x 軸の交点の x 座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "判別式が負の二次関数はグラフが x 軸と交わらず、実数の範囲では factored form に書けない。",
      "頂点の x 座標は 2 つの x 切片のちょうど真ん中、x = (α + β)/2 になる。"
    ],
    "related": [
      "x-intercept",
      "vertex-form",
      "standard-form",
      "zeros-of-a-polynomial",
      "discriminant"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「二次関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BA%8C%E6%AC%A1%E9%96%A2%E6%95%B0",
        "note": "因数分解形 f(x) = a(x − s)(x − t)（factored form）、単に分解形とも"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 二次関数: 二次方程式の解を二次関数のグラフと x 軸との共有点の x 座標として扱う（因数分解形の名前は無い）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "factored form": 20,
        "!slope intercept form": 1
      },
      "written": {
        "factored form": 48,
        "!slope intercept form": 2
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "factorial",
    "ja": {
      "term": "階乗",
      "reading": "かいじょう",
      "alt": []
    },
    "en": {
      "term": "factorial",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "1 から n までの自然数をすべて掛けた積 n!。0! = 1 と約束する。",
    "definition_en": "The product of all positive integers from 1 to n, written n!; by convention 0! = 1.",
    "latex": "n! = n(n-1)(n-2)\\cdots 2 \\cdot 1",
    "spoken_en": "n factorial equals n times n minus one times n minus two, all the way down to one",
    "examples": [
      {
        "en": "Five factorial is 5 times 4 times 3 times 2 times 1, which is 120.",
        "ja": "5 の階乗は 5 × 4 × 3 × 2 × 1 で 120 だ。",
        "register": "spoken"
      },
      {
        "en": "Simplify 10!/8!.",
        "ja": "10!/8! を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "n! は n factorial と読む（日本語の「n の階乗」と同じく n が先）。the factorial of n は OpenStax Algebra and Trigonometry（13.1）の定義の文などに出てくるが、講義では five factorial のように数を先に言う。",
      "0! は 0 ではなく 1。₅C₀ や ₙP₀ の計算で使う。",
      "! は驚きの記号ではない。数式を読み上げるとき「five!」と強く言うのではなく、five factorial と読む。"
    ],
    "related": [
      "permutation",
      "combination",
      "circular-permutation",
      "binomial-coefficient"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "階乗",
        "en": "Factorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.1 Sequences and Their Notations（the factorial of n）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "factorial": 571
      },
      "written": {
        "factorial": 52
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "factoring-by-grouping",
    "ja": {
      "term": "グループ分けによる因数分解",
      "reading": "ぐるーぷわけによるいんすうぶんかい",
      "alt": []
    },
    "en": {
      "term": "factoring by grouping",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「グループ分けによる因数分解」は学習指導要領解説に無い、本プロジェクトの訳語。学習指導要領解説にも日本語版 Wikipedia「因数分解」にも、この方法の名前は出てこない（Wikipedia の因数分解の節は共通因数でくくる・たすきがけ・因数定理などで、項を組にしてくくる方法の節はない）。OpenStax Elementary Algebra（Greatest Common Factor and Factor by Grouping）は Factor by Grouping という名前で扱い、ax² + bx + c を因数分解する ac method（grouping method とも）もこの方法を使う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "4 つ以上の項を 2 つずつ組にし、それぞれの組から共通因数をくくり出したあと、どの組にも現れる式でさらにくくる因数分解の方法。",
    "definition_en": "A way to factor a polynomial with four or more terms: group the terms in pairs, factor out the common factor from each pair, and then factor out the expression the pairs share.",
    "latex": "ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)",
    "spoken_en": "a x plus a y plus b x plus b y equals a times the quantity x plus y, plus b times the quantity x plus y, which equals the quantity a plus b times the quantity x plus y",
    "examples": [
      {
        "en": "For x cubed plus 3x squared plus 2x plus 6, try factoring by grouping: x squared comes out of the first pair, 2 comes out of the second, and both leave x plus 3.",
        "ja": "x³ + 3x² + 2x + 6 は、組にしてくくってみよう。前の 2 項から x²、後ろの 2 項から 2 をくくり出すと、どちらにも x + 3 が残る。",
        "register": "spoken"
      },
      {
        "en": "Use factoring by grouping to factor xy + 5x − 2y − 10.",
        "ja": "項を 2 つずつ組にして、xy + 5x − 2y − 10 を因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 つ目の組から負の数をくくり出すときは符号に注意する。xy + 5x − 2y − 10 = x(y + 5) − 2(y + 5) = (x − 2)(y + 5) で、−2 でくくると −10 は +5 になる。",
      "ax² + bx + c を因数分解する ac method（たすき掛け（エントリ cross-method）とは別の手順）は、bx を 2 つの項に分けて 4 項にし、この方法でくくる。2x² + 7x + 3 = 2x² + 6x + x + 3 = 2x(x + 3) + (x + 3) = (2x + 1)(x + 3)。",
      "組の作り方によっては共通の式が出てこない。そのときは項の順番を入れ替えて組み直す。1 つの文字について整理する方法（エントリ arrange-by-the-variable-of-lowest-degree）とは別の手順。"
    ],
    "related": [
      "factoring",
      "common-factor",
      "cross-method",
      "distributive-property",
      "arrange-by-the-variable-of-lowest-degree"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 数と式: 「グループ分けによる因数分解」は 0 件（見出しは本プロジェクトの訳語）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「因数分解」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%A0%E6%95%B0%E5%88%86%E8%A7%A3",
        "note": "共通因数でくくる（全ての項に共通している因数でくくり出す）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「グループ分けによる因数分解」は 0 件"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "factoring by grouping": 39
      },
      "written": {
        "factoring by grouping": 38
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "factoring",
    "ja": {
      "term": "因数分解",
      "reading": "いんすうぶんかい",
      "alt": [
        "因数分解する"
      ]
    },
    "en": {
      "term": "factoring",
      "alt": [
        "factorization"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「因数分解」は名詞で、「因数分解する」と動詞にして使う。OpenStax Elementary Algebra（第 7 章 Factoring）の問題文は動詞 factor で書き（Factor completely: …、65 件）、名詞は factoring または factorization。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3",
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "多項式を、いくつかの式（因数）の積の形に表すこと。展開の逆の操作である。",
    "definition_en": "Writing a polynomial as a product of simpler expressions, called its factors. Factoring undoes multiplying out, or expanding.",
    "latex": "x^2 + 5x + 6 = (x + 2)(x + 3)",
    "spoken_en": "x squared plus five x plus six equals the quantity x plus two times the quantity x plus three",
    "examples": [
      {
        "en": "Factoring is like multiplying in reverse: we start with x² + 5x + 6 and work back to (x + 2)(x + 3).",
        "ja": "因数分解は掛け算を逆にたどるようなもの。x² + 5x + 6 から始めて、(x + 2)(x + 3) に戻す。",
        "register": "spoken"
      },
      {
        "en": "Solve x² − 7x + 12 = 0 by factoring.",
        "ja": "因数分解を利用して、x² − 7x + 12 = 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「〜を因数分解せよ」は動詞 factor を使って Factor x² − 9. と書く。factor completely は、これ以上因数分解できないところまで因数分解すること。",
      "名詞の factor は「因数」（エントリ factor）。因数分解した積の形は factored form（エントリ factored-form）。",
      "整数を素数の積に表すのは prime factorization（素因数分解、エントリ prime-factorization）で、多項式の因数分解とは別の見出し。"
    ],
    "related": [
      "factor",
      "factored-form",
      "expand",
      "common-factor",
      "prime-factorization"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "因数分解",
        "en": "Factorization"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I・数学II: 二次・三次の乗法公式と因数分解の公式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "factor the | factor this | factor each | factor it | factor completely": 406,
        "!prime factorization": 32
      },
      "written": {
        "factor the | factor this | factor each | factor it | factor completely": 521,
        "!prime factorization": 15
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "fail-to-reject",
    "ja": {
      "term": "棄却されない",
      "reading": "ききゃくされない",
      "alt": []
    },
    "en": {
      "term": "fail to reject",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "do not reject",
          "register": "written",
          "note": "話し言葉にはほとんど出てこない。書き言葉では fail to reject よりずっと多い（すべて OpenStax Introductory Statistics）。"
        }
      ],
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本の教科書は「棄却されない」「棄却できない」と受け身・可能で書くが、英語は主語を we にして fail to reject（話し言葉）／ do not reject（書き言葉）と言う。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "仮説検定で、データが帰無仮説を否定するのに十分でなく、帰無仮説を棄却しないこと。",
    "definition_en": "In a hypothesis test, not rejecting the null hypothesis because the data do not give enough evidence against it.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The p-value is 0.2, which is bigger than alpha, so we fail to reject the null.",
        "ja": "p 値は 0.2 で α より大きいので、帰無仮説は棄却されない。",
        "register": "spoken"
      },
      {
        "en": "At the 5% level, do not reject H₀; there is not sufficient evidence to conclude that the mean has changed.",
        "ja": "有意水準 5% で H₀ は棄却されない。平均が変わったと結論する十分な証拠はない。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "帰無仮説が正しいと証明されたわけではないので、accept the null hypothesis とは言わない（用例コーパスでも accept the null はほとんど出てこない）。"
    ],
    "related": [
      "reject",
      "null-hypothesis",
      "hypothesis-testing"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "fail to reject": 41,
        "do not reject": 2
      },
      "written": {
        "do not reject": 62,
        "fail to reject": 4
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "false",
    "ja": {
      "term": "偽",
      "reading": "ぎ",
      "alt": []
    },
    "en": {
      "term": "false",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "命題が正しくないこと。真理値表では F と書く。",
    "definition_en": "Describes a statement that is not correct; in a truth table it is marked F.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "That's false. One counterexample is enough: try x = −2.",
        "ja": "それは偽だ。反例は 1 つあれば十分で、x = −2 を試してごらん。",
        "register": "spoken"
      },
      {
        "en": "The equation does not hold when x = 0, so the statement is false.",
        "ja": "x = 0 のときこの等式は成り立たないので、命題は偽である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「成り立たない」は does not hold または fails（The identity fails at x = 0.）。命題そのものについては is false と言う。",
      "偽であることを示すには反例（counterexample）を 1 つ挙げればよい。"
    ],
    "related": [
      "true",
      "counterexample",
      "truth-value",
      "proposition"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "false": 228
      },
      "written": {
        "false": 293
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "feasible-region",
    "ja": {
      "term": "実行可能領域",
      "reading": "じっこうかのうりょういき",
      "alt": []
    },
    "en": {
      "term": "feasible region",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数II",
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "線形計画法で、すべての制約条件（不等式）を満たす点全体の領域。",
    "definition_en": "In linear programming, the region of all points satisfying every constraint.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The feasible region is where all the shaded parts overlap.",
        "ja": "実行可能領域は、塗った部分がすべて重なるところ。",
        "register": "spoken"
      },
      {
        "en": "Graph the feasible region for x ≥ 0, y ≥ 0, x + y ≤ 5, 2x + y ≤ 8, and find its vertices.",
        "ja": "x ≧ 0, y ≧ 0, x + y ≦ 5, 2x + y ≦ 8 の表す領域を図示し、その頂点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校の教科書では「連立不等式の表す領域」と言う。用例コーパスでは書き言葉（OpenStax Algebra and Trigonometry）に少し出てくるだけ。"
    ],
    "related": [
      "region",
      "linear-programming",
      "objective-function",
      "constraint"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "実行可能領域",
        "en": "Feasible region"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "feasible region": 9
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 feasible region（本文 9 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "fermats-little-theorem",
    "ja": {
      "term": "フェルマーの小定理",
      "reading": "ふぇるまーのしょうていり",
      "alt": []
    },
    "en": {
      "term": "Fermat's little theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Fermat's little theorem は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": []
    },
    "definition_ja": "p が素数で、整数 a が p の倍数でないとき、a^{p−1} ≡ 1 (mod p) が成り立つという定理。両辺に a をかけた aᵖ ≡ a (mod p) はすべての整数 a で成り立つ。",
    "definition_en": "If p is prime and a is an integer not divisible by p, then a^(p−1) ≡ 1 (mod p). Multiplying both sides by a gives a^p ≡ a (mod p), which holds for every integer a.",
    "latex": "a^{p-1} \\equiv 1 \\pmod{p}",
    "spoken_en": "a to the p minus 1 is congruent to 1 mod p",
    "examples": [
      {
        "en": "Seven is prime and it doesn't divide three, so Fermat's little theorem says three to the sixth is congruent to one mod seven.",
        "ja": "7 は素数で 3 を割り切らないから、フェルマーの小定理より 3⁶ ≡ 1 (mod 7) となる。",
        "register": "spoken"
      },
      {
        "en": "Use Fermat's little theorem to find the remainder when 3¹⁰⁰ is divided by 7.",
        "ja": "フェルマーの小定理を用いて、3¹⁰⁰ を 7 で割った余りを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "little（小）は、フェルマーの最終定理（Fermat's last theorem）と区別するための呼び名。",
      "逆は成り立たない。aⁿ⁻¹ ≡ 1 (mod n) でも n が素数とは限らない（n = 341 = 11 × 31 で 2³⁴⁰ ≡ 1 (mod 341)）。",
      "用例コーパスでは話し言葉（MIT OCW）・書き言葉（MIT の講義ノート）とも少しだけ。CED・OpenStax・IM・CK-12 では 0 件。"
    ],
    "related": [
      "remainder",
      "prime-factorization",
      "modulus",
      "modular-inverse",
      "cryptography",
      "congruence-modulo-n"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "フェルマーの小定理",
        "en": "Fermat's little theorem"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "Fermat's little theorem": 3
      },
      "written": {
        "Fermat's little theorem": 3
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは Wikipedia の記事名 Fermat's little theorem（英語版「Fermat's little theorem」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "fibonacci-sequence",
    "ja": {
      "term": "フィボナッチ数列",
      "reading": "ふぃぼなっちすうれつ",
      "alt": []
    },
    "en": {
      "term": "Fibonacci numbers",
      "alt": [],
      "variants": [
        {
          "term": "Fibonacci sequence",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry の書き方（6 件）。話し言葉にはほとんど出てこず、書き言葉では Fibonacci numbers より少ない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Discrete Math"
      ]
    },
    "definition_ja": "初めの 2 項が 1、1 で、それ以降の各項が直前の 2 項の和になる数列。",
    "definition_en": "The sequence that starts 1, 1 and in which every later term is the sum of the two before it.",
    "latex": "F_{n+2} = F_{n+1} + F_n, \\quad F_1 = F_2 = 1",
    "spoken_en": "F sub n plus two equals F sub n plus one plus F sub n",
    "examples": [
      {
        "en": "The Fibonacci numbers grow roughly like powers of the golden ratio.",
        "ja": "フィボナッチ数はおよそ黄金比の累乗のように増える。",
        "register": "spoken"
      },
      {
        "en": "Find the 10th term of the Fibonacci sequence.",
        "ja": "フィボナッチ数列の第 10 項を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "MIT の講義と講義ノートは数列全体を the Fibonacci numbers と複数形で呼ぶ。OpenStax Algebra and Trigonometry（13.1）と英語版 Wikipedia の記事名は Fibonacci sequence。1 つの項は the nth Fibonacci number。",
      "F₀ = 0, F₁ = 1 から始める本もある（Levin 4.1、英語版 Wikipedia「Fibonacci sequence」）。第何項かが 1 つずれる。"
    ],
    "related": [
      "recurrence-relation",
      "sequence",
      "characteristic-equation",
      "linear-recurrence-relation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "フィボナッチ数",
        "en": "Fibonacci number"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.1 Sequences and Their Notations（Fibonacci sequence）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.1 Describing Sequences（F₀ = 0 から始める Fibonacci numbers）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Fibonacci_sequence",
        "note": "記事名 Fibonacci sequence、F₀ = 0 から始める"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "Fibonacci numbers": 21,
        "Fibonacci sequence": 1
      },
      "written": {
        "Fibonacci numbers": 25,
        "Fibonacci sequence": 8
      },
      "sources": [
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-a-common-denominator",
    "ja": {
      "term": "通分する",
      "reading": "つうぶんする",
      "alt": [
        "通分"
      ]
    },
    "en": {
      "term": "common denominator",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "英語は動詞句で言う（用例コーパスの話し言葉では find a common denominator と get a common denominator が同じくらいで、put them over a common denominator の形は少ない）。見出しの common denominator は、どの動詞句にも入る共通部分。",
    "domains": [
      "arithmetic",
      "algebra"
    ],
    "level": {
      "jp": [
        "小学校",
        "中1",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "分母の異なる分数を、分母をそろえた形に書き直すこと。",
    "definition_en": "To rewrite fractions so that they share the same denominator.",
    "latex": "\\frac{1}{2} + \\frac{1}{3} = \\frac{3}{6} + \\frac{2}{6}",
    "spoken_en": "one half plus one third equals three sixths plus two sixths",
    "examples": [
      {
        "en": "Get a common denominator first, then add the numerators.",
        "ja": "まず通分して、それから分子を足す。",
        "register": "spoken"
      },
      {
        "en": "Putting both terms over a common denominator gives a single fraction.",
        "ja": "両方の項を通分すると 1 つの分数にまとまる。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the least common denominator (LCD)",
        "ja": "最小公倍数を分母にした形"
      }
    ],
    "pitfalls": [
      "日本語は名詞の「通分」「約分」で言うが、英語はどちらも動詞句で言う（通分は find a common denominator、約分はエントリ reduce の simplify the fraction）。"
    ],
    "related": [
      "reduce",
      "rational-expression",
      "least-common-denominator",
      "clear-the-denominators"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Add and Subtract Fractions with Different Denominators（common denominator）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Add and Subtract Fractions・Add and Subtract Rational Expressions with Unlike Denominators（common denominator）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "付録: 小学校学習指導要領 算数 第5学年〔用語・記号〕通分"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "common denominator": 265
      },
      "written": {
        "common denominator": 244
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-an-antiderivative",
    "ja": {
      "term": "原始関数を見つける",
      "reading": "げんしかんすうをみつける",
      "alt": []
    },
    "en": {
      "term": "find an antiderivative",
      "alt": [],
      "variants": [
        {
          "term": "take the antiderivative",
          "register": "spoken",
          "note": "Khan Academy（AP Calculus）の言い方で、大半が Khan Academy。ほかのソースでは find an antiderivative のほうが多い。書き言葉にはほとんど出てこない。"
        },
        {
          "term": "find an antiderivative",
          "register": "both",
          "note": "大学の講義（MIT OCW）の言い方で、書き言葉でも使う。原始関数は 1 つに決まらないので、数学的には an が正確。話し言葉は MIT OCW が中心、書き言葉はほとんどが OpenStax Calculus。"
        },
        {
          "term": "find the antiderivative",
          "register": "both",
          "note": "書き言葉ではこれが最も多いが、すべて OpenStax Calculus。話し言葉は大半が Khan Academy。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "微分すると与えられた関数になる関数を 1 つ求めること。",
    "definition_en": "To come up with a function whose derivative is the given function.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Take the antiderivative of each term, then plug in the bounds.",
        "ja": "各項の原始関数を求めて、それから上端と下端を代入する。",
        "register": "spoken"
      },
      {
        "en": "Find the antiderivative F(x) of f(x) = 3x² that satisfies F(1) = 2.",
        "ja": "F(1) = 2 を満たす f(x) = 3x² の原始関数 F(x) を求めよ。",
        "register": "written"
      },
      {
        "en": "First find an antiderivative of the integrand; then the definite integral is just F of b minus F of a.",
        "ja": "まず被積分関数の原始関数を 1 つ見つける。定積分はそれを使って F(b) − F(a) で済む。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "言い方はソースで分かれる。Khan Academy は take the antiderivative、MIT OCW は find an antiderivative、OpenStax Calculus は find the antiderivative が多い。"
    ],
    "related": [
      "antiderivative",
      "indefinite-integral"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "take the antiderivative": 107,
        "find the antiderivative": 21,
        "find an antiderivative": 29
      },
      "written": {
        "find an antiderivative": 16,
        "find the antiderivative": 29,
        "take the antiderivative": 6
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-arc-length",
    "ja": {
      "term": "曲線の長さを求める",
      "reading": "きょくせんのながさをもとめる",
      "alt": []
    },
    "en": {
      "term": "find the arc length",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線のある区間の長さを、定積分 ∫ₐᵇ √(1 + {f′(x)}²) dx で求めること。",
    "definition_en": "To compute the length of a curve over an interval with ∫ₐᵇ √(1 + (f′(x))²) dx.",
    "latex": "L = \\int_a^b \\sqrt{1 + \\bigl(f'(x)\\bigr)^2}\\,dx",
    "spoken_en": "L equals the integral from a to b of the square root of one plus f prime of x squared, dx",
    "examples": [
      {
        "en": "To find the arc length, take the derivative, square it, add one, and integrate the square root.",
        "ja": "曲線の長さは、微分して 2 乗して 1 を足し、そのルートを積分する。",
        "register": "spoken"
      },
      {
        "en": "Find the arc length of y = x^{3/2} from x = 0 to x = 4.",
        "ja": "曲線 y = x^{3/2}（0 ≦ x ≦ 4）の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数IIIの「曲線の長さ」は英語では arc length。length of the curve とも言う。"
    ],
    "related": [
      "integrand",
      "arc-length-of-a-curve"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "find the arc length": 14
      },
      "written": {
        "find the arc length": 19
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-asymptotes",
    "ja": {
      "term": "グラフの漸近線を求める",
      "reading": "ぐらふのぜんきんせんをもとめる",
      "alt": []
    },
    "en": {
      "term": "find the vertical asymptotes",
      "alt": [
        "find the asymptotes"
      ],
      "variants": [
        {
          "term": "find the horizontal asymptote",
          "register": "both",
          "note": "水平な漸近線を求めるときの言い方。話し言葉はすべて NancyPi、書き言葉はすべて OpenStax Algebra and Trigonometry で、どちらも少ない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本の「漸近線を求める」は 1 つの指示だが、英語の問題や説明では垂直な漸近線（vertical asymptotes）と水平な漸近線（horizontal asymptote）を分けて求めることが多い（用例コーパスでは find the vertical asymptotes のほうが find the asymptotes より多い）。",
    "domains": [
      "functions",
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "関数のグラフが限りなく近づく直線（漸近線）を求める。",
    "definition_en": "To find the lines that a graph approaches, such as its vertical and horizontal asymptotes.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the vertical asymptotes, set the denominator equal to zero.",
        "ja": "垂直な漸近線を求めるには、分母 = 0 とおく。",
        "register": "spoken"
      },
      {
        "en": "Find the vertical asymptotes and the horizontal asymptote of f(x) = (x + 1)/(x² − 4).",
        "ja": "f(x) = (x + 1)/(x² − 4) のグラフの漸近線を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "斜めの漸近線は slant asymptote（oblique asymptote）と言う。"
    ],
    "related": [
      "rational-function",
      "limit"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "find the asymptotes": 1,
        "find the vertical asymptotes": 10,
        "find the horizontal asymptote": 4
      },
      "written": {
        "find the vertical asymptotes": 7,
        "find the horizontal asymptote": 2,
        "find the asymptotes": 2
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-derivative-at",
    "ja": {
      "term": "微分係数を求める",
      "reading": "びぶんけいすうをもとめる",
      "alt": []
    },
    "en": {
      "term": "evaluate the derivative at",
      "alt": [
        "find the derivative at",
        "find f prime of"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数の x = a における微分係数 f′(a) の値を求める。",
    "definition_en": "To find the value f′(a) of the derivative at a given point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Evaluate the derivative at x = 2: just plug 2 into f prime.",
        "ja": "x = 2 での微分係数を求める。f′ に 2 を代入するだけ。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the derivative of f(x) = x³ − x at x = 2.",
        "ja": "f(x) = x³ − x の x = 2 における微分係数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語は「微分係数」に当たる名詞を立てないので、find f′(2) や the derivative at x = 2 と言う。"
    ],
    "related": [
      "derivative-at-a-point",
      "differentiate",
      "derivative"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "find f prime of": 8,
        "find the derivative at": 4,
        "evaluate the derivative at": 3
      },
      "written": {
        "find the derivative at": 2,
        "evaluate the derivative at": 5
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満 ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 evaluate the derivative at（本文 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-domain",
    "ja": {
      "term": "定義域を求める",
      "reading": "ていぎいきをもとめる",
      "alt": []
    },
    "en": {
      "term": "find the domain",
      "alt": [],
      "variants": [
        {
          "term": "determine the domain",
          "register": "written",
          "note": "少数の書き方。話し言葉にはほとんど出てこず、書き言葉は OpenStax Algebra and Trigonometry と OpenStax Calculus。"
        },
        {
          "term": "state the domain",
          "register": "written",
          "note": "少数の書き方。話し言葉にはほとんど出てこず、書き言葉は大半が OpenStax Algebra and Trigonometry。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I",
        "数III"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "関数の式が意味をもつ x の値の範囲を求める。",
    "definition_en": "To work out the set of x-values for which a function is defined.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the domain, look for anything that would make you divide by zero or take the square root of a negative.",
        "ja": "定義域を求めるには、0 で割るところや負の数の平方根になるところを探す。",
        "register": "spoken"
      },
      {
        "en": "Find the domain of f(x) = √(x − 2)/(x − 5).",
        "ja": "f(x) = √(x − 2)/(x − 5) の定義域を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "答えは区間の記法（interval notation）で [2, 5) ∪ (5, ∞) のように書くことがある。"
    ],
    "related": [
      "radical-function",
      "rational-function",
      "find-the-inverse"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "find the domain": 40,
        "state the domain": 3,
        "determine the domain": 1
      },
      "written": {
        "find the domain": 164,
        "determine the domain": 26,
        "state the domain": 27
      },
      "sources": [
        "khan-algebra",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-equation-of-the-tangent-line",
    "ja": {
      "term": "接線の方程式を求める",
      "reading": "せっせんのほうていしきをもとめる",
      "alt": []
    },
    "en": {
      "term": "find an equation of the tangent line",
      "alt": [],
      "variants": [
        {
          "term": "find the equation of the tangent line",
          "register": "spoken",
          "note": "話し言葉で最も多いが、Khan Academy に多く、講義によって偏る。書き言葉では find an equation of the tangent line よりずっと少ない。"
        },
        {
          "term": "find the tangent line",
          "register": "spoken",
          "note": "equation を省いた言い方。話し言葉では find the equation of the tangent line と同じくらいで、書き言葉では少ない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線上の点での接線を、y = mx + b などの式で表す。",
    "definition_en": "To write the tangent line at a point on a curve as an equation such as y = mx + b.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the equation of the tangent line, you need a point and a slope.",
        "ja": "接線の方程式を求めるには、1 点と傾きがあればよい。",
        "register": "spoken"
      },
      {
        "en": "Find an equation of the tangent line to y = √x at x = 4.",
        "ja": "x = 4 における y = √x の接線の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "問題文（OpenStax Calculus など）は an equation of the tangent line と書き、授業では the equation of the tangent line と言うことが多い。意味は同じ。"
    ],
    "related": [
      "slope-of-the-tangent-line",
      "point-of-tangency",
      "normal-line"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "find the tangent line": 10,
        "find the equation of the tangent line": 11
      },
      "written": {
        "find an equation of the tangent line": 35,
        "find the equation of the tangent line": 3,
        "find the tangent line": 4
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-equation",
    "ja": {
      "term": "式を求める",
      "reading": "しきをもとめる",
      "alt": [
        "方程式を求める"
      ]
    },
    "en": {
      "term": "find the equation",
      "alt": [],
      "variants": [
        {
          "term": "find the equation",
          "register": "both",
          "note": "話し言葉では write an equation と同じくらい、書き言葉では write an equation より多い（OpenStax Algebra and Trigonometry・Calculus・Elementary Algebra ほか）。"
        },
        {
          "term": "write an equation",
          "register": "both",
          "note": "特に文章題や条件から式を立てる問い。話し言葉は大半が Khan Academy（特に中学の講義）、書き言葉は多くが OpenStax Algebra and Trigonometry。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "geometry"
    ],
    "level": {
      "jp": [
        "中2",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "与えられた条件を満たす直線・円・関数などの式を求める。",
    "definition_en": "To determine the equation of a line, circle, function, etc. that satisfies the given conditions.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's find the equation of the circle — we need the center and the radius.",
        "ja": "円の方程式を求めよう。中心と半径がわかればいい。",
        "register": "spoken"
      },
      {
        "en": "A circle has its center at (2, −1) and passes through the origin. Find its equation.",
        "ja": "中心 (2, −1) で原点を通る円の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "冠詞だけ違う find an equation も書き言葉に多い（特に OpenStax Calculus）。"
    ],
    "related": [
      "equation-of-a-line",
      "find-the-equation-of-the-tangent-line",
      "equation-of-a-circle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "write an equation": 27,
        "find the equation": 39
      },
      "written": {
        "find the equation": 215,
        "write an equation": 75
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-inflection-points",
    "ja": {
      "term": "変曲点を求める",
      "reading": "へんきょくてんをもとめる",
      "alt": []
    },
    "en": {
      "term": "find the points of inflection",
      "alt": [
        "find the inflection points"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線の凹凸が入れ替わる点を求める。",
    "definition_en": "To find the points where a graph changes concavity.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the points of inflection, set f double prime equal to zero and check that it actually changes sign.",
        "ja": "変曲点を求めるには、f″ = 0 とおいて、そこで本当に符号が変わるかを確かめる。",
        "register": "spoken"
      },
      {
        "en": "Find the points of inflection of f(x) = x³ − 6x² + 9x.",
        "ja": "f(x) = x³ − 6x² + 9x の変曲点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "inflection-point",
      "check-the-concavity",
      "second-derivative"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.6（points of inflection）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "find the inflection points": 1
      },
      "sources": [
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-24。docs/DECISIONS.md「Phase 2 数列・級数の単元の前の修正」の、人間が見出しを決めた ③ の行（number-of-real-solutions・prime-notation・find-the-inflection-points））。register は主張しない。",
        "raised": "2026-09-24"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-inverse",
    "ja": {
      "term": "逆関数を求める",
      "reading": "ぎゃくかんすうをもとめる",
      "alt": []
    },
    "en": {
      "term": "find the inverse",
      "alt": [
        "find f inverse"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "関数 y = f(x) の逆関数 f⁻¹(x) を求める。",
    "definition_en": "To find the inverse function f⁻¹ of a function f.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the inverse, swap x and y and solve for y.",
        "ja": "逆関数を求めるには、x と y を入れ替えて y について解く。",
        "register": "spoken"
      },
      {
        "en": "Find the inverse of f(x) = 2x + 3.",
        "ja": "f(x) = 2x + 3 の逆関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "f⁻¹(x) は f inverse of x と読む。1/f(x) ではない。"
    ],
    "related": [
      "one-to-one",
      "composite-function",
      "find-the-domain",
      "inverse-function"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "find the inverse": 36
      },
      "written": {
        "find the inverse": 108
      },
      "sources": [
        "khan-algebra",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-limit",
    "ja": {
      "term": "極限を求める",
      "reading": "きょくげんをもとめる",
      "alt": []
    },
    "en": {
      "term": "find the limit",
      "alt": [],
      "variants": [
        {
          "term": "evaluate the limit",
          "register": "written",
          "note": "書き言葉（問題文）で最も多い。話し言葉では find the limit よりずっと少ない。"
        },
        {
          "term": "compute the limit",
          "register": "spoken",
          "note": "少数の言い方。話し言葉は大半が MIT OCW で、書き言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "式の極限の値を求める。",
    "definition_en": "To work out the value of a limit.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the limit, try plugging in first. If you get zero over zero, factor.",
        "ja": "極限を求めるには、まず代入してみる。0/0 になったら因数分解する。",
        "register": "spoken"
      },
      {
        "en": "Evaluate lim_{x→1} (x² − 1)/(x − 1).",
        "ja": "lim_{x→1} (x² − 1)/(x − 1) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "約分や有理化で極限を求めることを、AP の CED は topic 1.6 で algebraic manipulation と呼ぶ。"
    ],
    "related": [
      "limit",
      "take-the-limit",
      "indeterminate-form",
      "limit-laws"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.6（algebraic manipulation）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "find the limit": 100,
        "evaluate the limit": 13,
        "compute the limit": 10
      },
      "written": {
        "find the limit": 60,
        "evaluate the limit": 74,
        "compute the limit": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-local-extrema",
    "ja": {
      "term": "極値を求める",
      "reading": "きょくちをもとめる",
      "alt": []
    },
    "en": {
      "term": "find the local extrema",
      "alt": [
        "find all local extrema"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数の極大値・極小値と、それをとる x の値を求める。",
    "definition_en": "To find the local maximum and minimum values of a function and where they occur.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the local extrema, first find where f prime is zero or undefined.",
        "ja": "極値を求めるには、まず f′ が 0 か存在しない点を探す。",
        "register": "spoken"
      },
      {
        "en": "Find the local extrema of f(x) = x³ − 6x² + 9x.",
        "ja": "f(x) = x³ − 6x² + 9x の極値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "extrema は extremum の複数形。1 つだけなら a local extremum。"
    ],
    "related": [
      "local-extremum",
      "critical-point",
      "first-derivative-test"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "find all local extrema": 1,
        "find the local extrema": 3
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは OpenStax の呼び方 find the local extrema（本文 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-nth-term",
    "ja": {
      "term": "一般項を求める",
      "reading": "いっぱんこうをもとめる",
      "alt": []
    },
    "en": {
      "term": "find the general term",
      "alt": [
        "find the nth term"
      ],
      "variants": [
        {
          "term": "find the general term",
          "register": "written",
          "note": "OpenStax Intermediate Algebra と OpenStax Algebra and Trigonometry の問題の言い方。話し言葉にはほとんど出てこない（Professor Leonard だけ）。"
        },
        {
          "term": "write an explicit formula",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry（13.2・13.3）の問題文の言い方（Write an explicit formula for …）。"
        },
        {
          "term": "find an explicit formula",
          "register": "written",
          "note": "同じ問いの別の言い方。explicit formula は漸化式（recursive formula）と対になる語。話し言葉には出てこず、書き言葉では find the general term より少ない（OpenStax Calculus・Algebra and Trigonometry）。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数列の第 n 項 aₙ を n の式で表す。",
    "definition_en": "To express the nth term aₙ of a sequence as a formula in n.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's find the general term — look at how each term relates to n.",
        "ja": "一般項を求めよう。各項が n とどう関係しているか見て。",
        "register": "spoken"
      },
      {
        "en": "Find an explicit formula for the nth term of the sequence defined by a₁ = 3 and aₙ₊₁ = aₙ + 4.",
        "ja": "a₁ = 3、aₙ₊₁ = aₙ + 4 で定まる数列の一般項を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉ではどの言い方も少ない。",
      "explicit formula（n の式）と recursive formula（漸化式）を対にして覚えるとよい。"
    ],
    "related": [
      "nth-term",
      "recurrence-relation",
      "solve-the-recurrence",
      "sequence",
      "general-term"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.2 Arithmetic Sequences・13.3 Geometric Sequences（write an explicit formula）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.2 Arithmetic Sequences・12.3 Geometric Sequences and Series（Find the General Term (nth Term)）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "find the general term": 2,
        "find the nth term": 2
      },
      "written": {
        "find an explicit formula": 11,
        "find the general term": 34,
        "find the nth term": 1,
        "write an explicit formula": 21
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find-the-sum",
    "ja": {
      "term": "和を求める",
      "reading": "わをもとめる",
      "alt": []
    },
    "en": {
      "term": "find the sum",
      "alt": [
        "compute the sum",
        "evaluate the sum"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "数列の項の和や Σ で表された和の値を計算する。",
    "definition_en": "To work out the value of a sum of terms, including one written with sigma notation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Before you find the sum, count how many terms there are.",
        "ja": "和を求める前に、項がいくつあるか数えよう。",
        "register": "spoken"
      },
      {
        "en": "Find the sum Σₖ₌₁¹⁰ (2k + 1).",
        "ja": "和 Σₖ₌₁¹⁰ (2k + 1) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "summation-notation",
      "summation-formulas",
      "partial-sum",
      "sum-of-an-arithmetic-sequence",
      "sum"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "compute the sum": 5,
        "find the sum": 34
      },
      "written": {
        "find the sum": 181,
        "compute the sum": 7,
        "evaluate the sum": 6
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "find",
    "ja": {
      "term": "求めよ",
      "reading": "もとめよ",
      "alt": [
        "求める"
      ]
    },
    "en": {
      "term": "find",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "問題文の「〜を求めよ」は Find … が基本。ただし日本語の「求める」が指す操作によって英語の動詞が分かれ、方程式を解くのは solve（エントリ solve）、式の値を出すのは evaluate（エントリ evaluate）、計算するのは calculate（エントリ compute）と言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "答えとなる数・式・長さなどを、計算や考えによって出すこと。問題文では「〜を求めよ」の形で使う。",
    "definition_en": "To work out an answer, such as a number, an expression, or a length. In directions, a problem asks for it with \"Find …\".",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Now we have enough information to find the missing side.",
        "ja": "これで、残りの辺の長さを求める材料がそろったね。",
        "register": "spoken"
      },
      {
        "en": "Find the perimeter of a rectangle that is 8 cm long and 5 cm wide.",
        "ja": "縦 8 cm、横 5 cm の長方形の周の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "find のふつうの意味は「見つける」。問題文の Find … は「探せ」ではなく「求めよ」。",
      "「x の値を求めよ」は Find the value of x.。方程式から x を出すなら Solve for x.（エントリ solve-for）とも書く。"
    ],
    "related": [
      "solve",
      "evaluate",
      "compute",
      "solve-for",
      "check"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "find": 5059
      },
      "written": {
        "find": 11392
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "first-derivative-test",
    "ja": {
      "term": "第 1 次導関数判定法",
      "reading": "だいいちじどうかんすうはんていほう",
      "alt": [
        "第 1 次導関数テスト"
      ]
    },
    "en": {
      "term": "first derivative test",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学II・III）には、f′(x) の符号の変化から極大・極小を判定する方法の名前が出てこない。AP の CED（topic 5.4）と OpenStax Calculus は first derivative test と名前で呼ぶ。見出しの「第 1 次導関数判定法」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "f′(x) の符号が x = c の前後で正から負に変われば極大、負から正に変われば極小と判定する方法。",
    "definition_en": "A test that classifies a critical point c: if f′ changes from positive to negative at c, f has a local maximum there; from negative to positive, a local minimum.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "By the first derivative test, since f prime goes from positive to negative at x = 2, that's a relative max.",
        "ja": "第 1 次導関数判定法より、x = 2 で f′ が正から負に変わるので極大。",
        "register": "spoken"
      },
      {
        "en": "Use the first derivative test to classify the critical points of f(x) = x³ − 12x.",
        "ja": "第 1 次導関数判定法を使って、f(x) = x³ − 12x の臨界点が極大か極小かを判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は Unit 5 の各 topic で結論の理由（justification）を求める。増減表を書くなら、f′ の符号が変わることを文で添える。"
    ],
    "related": [
      "local-extremum",
      "critical-point",
      "sign-chart",
      "increasing-and-decreasing",
      "changes-sign"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.4（first derivative test）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II・III: f′(x) の符号の変化から極大・極小を判定する方法の名前が出てこない・「第 1 次導関数判定法」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "first derivative test": 21
      },
      "written": {
        "first derivative test": 18
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "first-order-linear-differential-equation",
    "ja": {
      "term": "1 階線形微分方程式",
      "reading": "いっかいせんけいびぶんほうていしき",
      "alt": [
        "線形一階微分方程式"
      ]
    },
    "en": {
      "term": "first-order linear differential equation",
      "alt": [
        "first-order linear equation"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "y′ + P(x)y = Q(x) の形の微分方程式。積分因子を掛けて解く。",
    "definition_en": "A differential equation of the form y′ + P(x)y = Q(x), solved by multiplying by an integrating factor.",
    "latex": "y' + P(x)\\,y = Q(x)",
    "spoken_en": "y prime plus P of x times y equals Q of x",
    "examples": [
      {
        "en": "First put it in standard form, with y′ by itself in front.",
        "ja": "まず y′ の係数を 1 にした標準形に直そう。",
        "register": "spoken"
      },
      {
        "en": "Solve the first-order linear differential equation y′ + 2y = e^{−x}.",
        "ja": "1 階線形微分方程式 y′ + 2y = e^{−x} を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "integrating-factor",
      "differential-equation",
      "separable-differential-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "first-order linear equation": 6,
        "first-order linear differential equation": 5
      },
      "written": {
        "first-order linear differential equation": 22,
        "first-order linear equation": 9
      },
      "sources": [
        "mit-18.03",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "first-term",
    "ja": {
      "term": "初項",
      "reading": "しょこう",
      "alt": []
    },
    "en": {
      "term": "first term",
      "alt": [],
      "variants": [
        {
          "term": "first term",
          "register": "written",
          "note": "数列・級数の初項。IM Algebra 2（1.3）も first term of a sequence と書く。式の最初の項にも使う。"
        },
        {
          "term": "initial term",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry（13.1 ほか）・Calculus Volume 2 と Levin（4.1〜4.4）の言い方。Levin は番号を 0 から始めて初項を a₀ とすることがある。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数列の最初の項 a₁。",
    "definition_en": "The first number a₁ of a sequence.",
    "latex": "a_1",
    "spoken_en": "a sub one",
    "examples": [
      {
        "en": "The first term is 3 and each term doubles.",
        "ja": "初項は 3 で、項ごとに 2 倍になる。",
        "register": "spoken"
      },
      {
        "en": "Find the first term and the common difference of the arithmetic sequence.",
        "ja": "この等差数列の初項と公差を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "first term は式の最初の項にも使う。"
    ],
    "related": [
      "sequence",
      "nth-term",
      "last-term",
      "common-difference",
      "common-ratio",
      "term"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.1 Describing Sequences・4.2 Arithmetic and Geometric Sequences（initial term）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.2 Infinite Series（initial term）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 1.3（first term of a sequence）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "first term of … sequence | first term of … series | where the first term is": 5,
        "initial term": 1
      },
      "written": {
        "initial term": 33,
        "first term of … sequence | first term of … series | where the first term is": 38
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "five-number-summary",
    "ja": {
      "term": "五数要約",
      "reading": "ごすうようやく",
      "alt": []
    },
    "en": {
      "term": "five-number summary",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "最小値・第 1 四分位数・中央値・第 3 四分位数・最大値の 5 つの値でデータをまとめたもの。",
    "definition_en": "The minimum, Q1, the median, Q3, and the maximum of a data set, listed together.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Once you have the five-number summary, drawing the box plot is easy.",
        "ja": "五数要約がわかれば、箱ひげ図は簡単にかける。",
        "register": "spoken"
      },
      {
        "en": "Give the five-number summary of the data.",
        "ja": "データの五数要約を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "box-plot",
      "quartile",
      "median",
      "minimum",
      "maximum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "five-number summary": 7
      },
      "written": {
        "five-number summary": 1
      },
      "sources": [
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは AP Statistics の CED の呼び方 five-number summary（topic 1.8）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "floor-function",
    "ja": {
      "term": "ガウス記号",
      "reading": "がうすきごう",
      "alt": [
        "床関数"
      ]
    },
    "en": {
      "term": "floor function",
      "alt": [
        "floor of"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Calculus I",
        "Calculus II"
      ]
    },
    "definition_ja": "実数 x に、x を超えない最大の整数を対応させる関数。日本語版 Wikipedia「床関数と天井関数」はガウス記号 [x] と書き、英語の本は ⌊x⌋ と書く（OpenStax Calculus Volume 2 は [x] を使うときに the greatest integer less than or equal to x と断る）。",
    "definition_en": "The function whose value at a real number x is the largest integer not exceeding x, written ⌊x⌋.",
    "latex": "\\lfloor x \\rfloor = n \\iff n \\le x < n + 1",
    "spoken_en": "the floor of x equals n if and only if n is less than or equal to x, which is less than n plus 1",
    "examples": [
      {
        "en": "The floor of 3.7 is 3, and the floor of negative 3.7 is negative 4.",
        "ja": "[3.7] = 3、[−3.7] = −4 だ。",
        "register": "spoken"
      },
      {
        "en": "Sketch the graph of the floor function y = ⌊x⌋ for −2 ≤ x < 3.",
        "ja": "−2 ≦ x < 3 で y = [x] のグラフをかけ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the graph of the floor function",
        "ja": "ガウス記号のグラフ（y = [x] のグラフ）"
      }
    ],
    "pitfalls": [
      "日本語の「ガウス記号」は記号 [x] の名前で、英語の見出しは関数の名前（floor function）。記号は ⌊x⌋ と書き、the floor of x と読む。",
      "日本語版 Wikipedia「床関数と天井関数」は、日本の高校数学ではガウス記号 [x] が使われることがほとんどと書く。英語で [x] を使うなら OpenStax Calculus Volume 2 のように the greatest integer less than or equal to x と断る。記号で書くなら ⌊x⌋ を使う（OpenStax Calculus Volume 1 は ⌊x⌋ にも定義を添える）。",
      "負の数に注意。⌊−3.7⌋ は −3 ではなく −4（−3.7 を超えない最大の整数）。",
      "切り上げの ⌈x⌉ は ceiling function（the ceiling of x）。床関数と天井関数は対になっている。"
    ],
    "related": [
      "integer-part",
      "fractional-part",
      "real-number",
      "number-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "床関数と天井関数",
        "en": "Floor and ceiling functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "4.4 The Mean Value Theorem の練習問題（f(x) = ⌊x⌋、floor function）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「床関数と天井関数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%BA%8A%E9%96%A2%E6%95%B0%E3%81%A8%E5%A4%A9%E4%BA%95%E9%96%A2%E6%95%B0",
        "note": "ガウス記号 [x]"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "級数の練習問題: [x] は the greatest integer less than or equal to x"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%20%28%E6%95%99%E7%A7%91%29",
        "note": "数学A 整数の性質: ガウス記号"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "floor function": 4,
        "!top floor of !a !an !the !this !that !his !her !their !its !your": 4
      },
      "written": {
        "floor function": 1
      },
      "sources": [
        "mit-6.042",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 8 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは OpenStax の呼び方 floor function（本文 1 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "flowchart-proof",
    "ja": {
      "term": "フローチャート証明",
      "reading": "ふろーちゃーとしょうめい",
      "alt": []
    },
    "en": {
      "term": "flowchart proof",
      "alt": [
        "flow proof"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「フローチャート証明」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書には、根拠を矢印でつなぐこの答案の形式はない。",
    "domains": [
      "proof",
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "主張を 1 つずつ箱に書いて理由を箱の下に添え、どの主張からどの主張が出るかを矢印でつないで示す証明の書き方。",
    "definition_en": "A proof in which each statement is written in a box with its reason underneath, and arrows show which statements lead to which.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a flowchart proof, the arrows tell you which statements you need before you can write the next one.",
        "ja": "フローチャート証明では、次の主張を書く前にどの主張が必要かを矢印が示している。",
        "register": "spoken"
      },
      {
        "en": "Complete the flowchart proof by filling in the missing statements and reasons.",
        "ja": "空欄の主張と理由を埋めて、フローチャート証明を完成させよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "flow proof とも言う（同じもの）。",
      "2 つ以上の主張から 1 つの結論が出るところ（2 辺とその間の角から合同を言うなど）は、複数の矢印を 1 つの箱に集める。",
      "用例コーパスにも参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）にも flowchart proof・flow proof・flow diagram proof はどれも出てこない。"
    ],
    "related": [
      "two-column-proof",
      "paragraph-proof",
      "proof"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。数え直しても決まらず、en.term は flowchart proof、flow proof は en.alt。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "focus",
    "ja": {
      "term": "焦点",
      "reading": "しょうてん",
      "alt": []
    },
    "en": {
      "term": "focus",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "放物線・楕円・双曲線を定める定点。楕円と双曲線には 2 つ、放物線には 1 つある。",
    "definition_en": "A fixed point used to define a conic: a parabola has one, an ellipse and a hyperbola have two.",
    "latex": "F(c, 0), \\; c = \\sqrt{a^2 - b^2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The two foci sit on the major axis, c units from the center.",
        "ja": "2 つの焦点は長軸上、中心から c の距離にある。",
        "register": "spoken"
      },
      {
        "en": "Find the foci of the hyperbola x²/16 − y²/9 = 1.",
        "ja": "双曲線 x²/16 − y²/9 = 1 の焦点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "find the foci",
        "ja": "焦点を求める"
      }
    ],
    "pitfalls": [
      "複数形は foci（フォウサイ）。",
      "focus（焦点）は話し言葉では少なく、ほとんどが書き言葉（OpenStax）の語。"
    ],
    "related": [
      "ellipse",
      "directrix",
      "conic-section",
      "major-axis",
      "eccentricity"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "焦点 (幾何学)",
        "en": "Focus (geometry)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "focus of the": 5
      },
      "written": {
        "focus of the": 27
      },
      "sources": [
        "mit-18.01",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "foil",
    "ja": {
      "term": "FOIL",
      "reading": "ふぉいる",
      "alt": []
    },
    "en": {
      "term": "FOIL",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "日本では分配法則で 1 項ずつ掛けて展開し、順序の覚え方に名前はない。英語の FOIL は二項式どうしの積を展開する順序の名前で、OpenStax Elementary Algebra が扱う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 つの二項式の積 (a + b)(c + d) を展開するときの、項を掛ける順序の覚え方。First（ac）、Outer（ad）、Inner（bc）、Last（bd）の頭文字。",
    "definition_en": "A way to remember how to multiply two binomials: multiply the First terms, then the Outer terms, then the Inner terms, then the Last terms. For (a + b)(c + d), this gives ac + ad + bc + bd.",
    "latex": "(a + b)(c + d) = ac + ad + bc + bd",
    "spoken_en": "the quantity a plus b times the quantity c plus d equals a c plus a d plus b c plus b d",
    "examples": [
      {
        "en": "Let's FOIL it out: first, outer, inner, last.",
        "ja": "FOIL で展開しよう。最初どうし、外側どうし、内側どうし、最後どうしの順に掛ける。",
        "register": "spoken"
      },
      {
        "en": "Use FOIL to multiply (x + 4)(x − 3).",
        "ja": "FOIL を使って (x + 4)(x − 3) を展開せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "FOIL が使えるのは二項式 × 二項式だけ。3 項以上の式を掛けるときは分配法則（エントリ distributive-property）で 1 項ずつ掛ける。",
      "授業では動詞としても使う（FOIL it out、FOIL the binomials）。",
      "Outer と Inner の項は同類項になることが多いので、FOIL のあとでまとめる（(x + 4)(x − 3) = x² − 3x + 4x − 12 = x² + x − 12）。"
    ],
    "related": [
      "distributive-property",
      "binomial",
      "expand",
      "like-terms",
      "special-products"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "FOIL": 71
      },
      "written": {
        "FOIL": 137
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "foot-of-the-perpendicular",
    "ja": {
      "term": "垂線の足",
      "reading": "すいせんのあし",
      "alt": []
    },
    "en": {
      "term": "foot of the perpendicular",
      "alt": [
        "foot of a perpendicular"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（foot of the perpendicular は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "点から直線（または平面）に垂線を下ろしたとき、垂線と直線（平面）が交わる点。",
    "definition_en": "The point where a line drawn from a given point perpendicular to a line or plane meets that line or plane.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Drop a perpendicular from A down to BC, and call the foot of the perpendicular H.",
        "ja": "A から BC に垂線を下ろし、垂線の足を H とする。",
        "register": "spoken"
      },
      {
        "en": "Let H be the foot of the perpendicular from the point P(1, 2, 3) to the plane x + y + z = 0. Find the coordinates of H.",
        "ja": "点 P(1, 2, 3) から平面 x + y + z = 0 に下ろした垂線の足を H とする。H の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「垂線を下ろす」は drop a perpendicular (from A to BC)。",
      "ベクトルで求めるときは、垂線の足は点の直線（平面）への正射影（orthogonal projection）になる。"
    ],
    "related": [
      "perpendicular-lines",
      "orthogonal-projection",
      "distance-from-a-point-to-a-line",
      "vector-projection",
      "tetrahedron"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "4.2 Projections and Planes"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Perpendicular",
        "note": "節 Foot of a perpendicular"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "foot of the perpendicular": 1
      },
      "sources": [
        "mit-18.01"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 foot of the perpendicular（4.2 Projections and Planes）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "fraction",
    "ja": {
      "term": "分数",
      "reading": "ぶんすう",
      "alt": []
    },
    "en": {
      "term": "fraction",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "小学校",
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "a ÷ b（b は 0 でない）の商を a/b の形で表した数。横線の上の a を分子、下の b を分母という。",
    "definition_en": "A number written as a/b, meaning a divided by b, where b is not 0. The number on top is the numerator and the number on the bottom is the denominator.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Remember, a fraction is just division, so three fourths means 3 divided by 4.",
        "ja": "分数はただの割り算だよ。4 分の 3 は 3 ÷ 4 ということ。",
        "register": "spoken"
      },
      {
        "en": "Write 0.35 as a fraction in simplest form.",
        "ja": "0.35 を既約分数で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "読む順番が日本語と逆。英語は分子から読み、分母を序数にする（3/4 は three fourths、2/5 は two fifths、1/2 は one half）。日本語の「4 分の 3」は分母から読む。",
      "帯分数は mixed number、仮分数は improper fraction、真分数は proper fraction。分数の横線は fraction bar。",
      "分子は numerator（エントリ numerator）、分母は denominator（エントリ denominator）。"
    ],
    "related": [
      "numerator",
      "denominator",
      "reduce",
      "find-a-common-denominator",
      "decimal"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "分数",
        "en": "Fraction"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "fraction": 1748
      },
      "written": {
        "fraction": 2069
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "fractional-part",
    "ja": {
      "term": "小数部分",
      "reading": "しょうすうぶぶん",
      "alt": []
    },
    "en": {
      "term": "fractional part",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（fractional part of の形で CED・OpenStax・IM・CK-12 とも 0 件。IM Grade 6 の fractional parts 2 件と CK-12 Algebra の 2 件は、1 や小数の位を分数に分けた部分の意味）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": []
    },
    "definition_ja": "実数 x から整数部分を引いた値 x − ⌊x⌋。0 以上 1 未満になる。",
    "definition_en": "The value x − ⌊x⌋ left after subtracting the integer part from a real number x; it is always at least 0 and less than 1.",
    "latex": "x - \\lfloor x \\rfloor",
    "spoken_en": "x minus the floor of x",
    "examples": [
      {
        "en": "The integer part of √5 is 2, so its fractional part is √5 minus 2.",
        "ja": "√5 の整数部分は 2 だから、小数部分は √5 − 2。",
        "register": "spoken"
      },
      {
        "en": "Let b be the fractional part of √3. Find the value of b² + 2b.",
        "ja": "√3 の小数部分を b とする。b² + 2b の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the fractional part of √5",
        "ja": "√5 の小数部分"
      }
    ],
    "pitfalls": [
      "無理数の小数部分は 0.236… と書き並べず、√5 − 2 のように「元の数 − 整数部分」の式で表す。",
      "fractional part の fractional は「分数の」ではなく「1 に満たない端の」の意味。答えが分数になるわけではない。",
      "負の数では流儀が分かれる。英語版 Wikipedia「Fractional part」（For negative numbers の節）は x − ⌊x⌋ 以外の流儀も挙げるので、負の数のときはどちらの意味かを書く（エントリ integer-part）。"
    ],
    "related": [
      "integer-part",
      "floor-function",
      "repeating-decimal",
      "nested-radical"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "小数部分",
        "en": "Fractional part"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 6)",
        "url": "https://im.kendallhunt.com/MS/students/1/index.html",
        "note": "Grade 6 4.3・7.x: fractional parts は 1 を分数に分けた部分の意味（2 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "a fractional part of a power of one-tenth（小数の位の意味、2 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!a fractional part of | fractional part is": 1
      },
      "sources": [
        "khan-middle"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 fractional part（英語版「Fractional part」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "free-variable",
    "ja": {
      "term": "自由変数",
      "reading": "じゆうへんすう",
      "alt": []
    },
    "en": {
      "term": "free variable",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "連立一次方程式を行階段形にしたとき、ピボットのない列に対応する変数。任意の値をとれるパラメータになる。",
    "definition_en": "In a row-reduced linear system, a variable whose column has no pivot; it can take any value and acts as a parameter.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "x3 is a free variable, so set it equal to t and solve for the others.",
        "ja": "x3 は自由変数なので t とおいて、ほかの変数を求める。",
        "register": "spoken"
      },
      {
        "en": "Identify the free variables and write the general solution in parametric form.",
        "ja": "自由変数を答え、一般解をパラメータ表示で書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "台帳の langlink（自由変数と束縛変数 → Free variables and bound variables）は論理学の自由変数で、別の概念。Levin の free variable も論理学の意味。",
      "Nicholson（1.2）は nonleading variable（先頭の 1 の無い列の変数）と書き、それを free variable とも呼ぶ。自由変数に置く新しい文字（s、t）は parameter と呼ぶ。"
    ],
    "related": [
      "basic-variable",
      "pivot",
      "system-of-linear-equations"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "1.2 Gaussian Elimination（nonleading variables を “free” variables とも呼び、新しい文字 s, t を parameters と呼ぶ。索引 free variables）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "論理の節の predicate（free variable は述語の自由変数。pitfalls[0]）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-07",
    "evidence": {
      "spoken": {
        "free variable": 93
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-10-07"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "frequency-polygon",
    "ja": {
      "term": "度数折れ線",
      "reading": "どすうおれせん",
      "alt": [
        "度数分布多角形"
      ]
    },
    "en": {
      "term": "frequency polygon",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ヒストグラムの各柱の上の辺の中点を結んだ折れ線。両端は度数 0 の階級の中点まで結ぶ。",
    "definition_en": "A line graph through the middle of the top of each histogram bar, brought down to zero at a class with no data at each end.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Put a dot in the middle of the top of each bar, connect the dots, and that's your frequency polygon.",
        "ja": "柱の上の辺の真ん中に点を打って結ぶと、度数折れ線になる。",
        "register": "spoken"
      },
      {
        "en": "Draw frequency polygons for the two classes on the same axes and compare them.",
        "ja": "2 つのクラスの度数折れ線を同じ座標軸にかいて比べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 つの分布を重ねて比べるときに使う（OpenStax Introductory Statistics）。"
    ],
    "related": [
      "histogram",
      "frequency-table"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "frequency polygon": 19
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "frequency-table",
    "ja": {
      "term": "度数分布表",
      "reading": "どすうぶんぷひょう",
      "alt": [
        "度数分布"
      ]
    },
    "en": {
      "term": "frequency table",
      "alt": [
        "frequency distribution"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Intro Statistics",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データをいくつかの階級に分け、各階級の度数を並べた表。表が表している度数の散らばり方を度数分布という。",
    "definition_en": "A table that lists each class of the data together with its frequency; the pattern it shows is the frequency distribution.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "First make a frequency table, then draw the histogram from it.",
        "ja": "まず度数分布表を作って、それからヒストグラムをかこう。",
        "register": "spoken"
      },
      {
        "en": "Make a frequency table for the data using classes of width 10.",
        "ja": "階級の幅を 10 として、このデータの度数分布表を作れ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「度数分布」（表が表す分布）と「度数分布表」（表そのもの）は、英語でも frequency distribution と frequency table で分けるが、用例コーパスでは話し言葉・書き言葉とも表（frequency table）のほうが多い。"
    ],
    "related": [
      "frequency",
      "class",
      "histogram",
      "relative-frequency"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "度数分布",
        "en": "Frequency distribution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "frequency table": 13,
        "frequency distribution": 5
      },
      "written": {
        "frequency table": 30,
        "frequency distribution": 10
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "frequency",
    "ja": {
      "term": "度数",
      "reading": "どすう",
      "alt": []
    },
    "en": {
      "term": "frequency",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "度数分布表で、それぞれの階級（や値）に入るデータの個数。",
    "definition_en": "The number of data values that fall in a class or take a given value.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The frequency is just how many times each value shows up.",
        "ja": "度数は、それぞれの値が何回出てくるかということ。",
        "register": "spoken"
      },
      {
        "en": "Complete the frequency column of the table.",
        "ja": "表の度数の欄を埋めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "frequency は物理の「振動数・周波数」（natural frequency など）と同じ語。",
      "相対度数は relative frequency、累積度数は cumulative frequency。"
    ],
    "related": [
      "frequency-table",
      "relative-frequency",
      "cumulative-frequency",
      "class"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "frequency table | frequency column | frequency polygon | frequency distribution": 18
      },
      "written": {
        "frequency table | frequency column | frequency polygon | frequency distribution": 74
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "fubinis-theorem",
    "ja": {
      "term": "フビニの定理",
      "reading": "ふびにのていり",
      "alt": []
    },
    "en": {
      "term": "Fubini's theorem",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "連続関数の長方形上の二重積分は、どちらの順序の累次積分で計算しても同じ値になるという定理。",
    "definition_en": "The theorem that the double integral of a continuous function over a rectangle equals the iterated integral taken in either order.",
    "latex": "\\iint_R f\\,dA = \\int_a^b \\int_c^d f\\,dy\\,dx = \\int_c^d \\int_a^b f\\,dx\\,dy",
    "spoken_en": null,
    "examples": [
      {
        "en": "By Fubini's theorem, you can integrate in whichever order is easier.",
        "ja": "フビニの定理より、楽なほうの順序で積分してよい。",
        "register": "spoken"
      },
      {
        "en": "Use Fubini's theorem to evaluate ∬_R x e^{xy} dA over R = [0, 1] × [0, 2] in the more convenient order.",
        "ja": "フビニの定理を用いて、R = [0, 1] × [0, 2] 上の ∬_R x e^{xy} dA を計算しやすい順序で求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉には出てこず、書き言葉はすべて OpenStax Calculus。"
    ],
    "related": [
      "iterated-integral",
      "double-integral",
      "changing-the-order-of-integration"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "フビニの定理",
        "en": "Fubini's theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "Fubini's theorem": 27
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "function-notation",
    "ja": {
      "term": "関数記号",
      "reading": "かんすうきごう",
      "alt": [
        "関数記法"
      ]
    },
    "en": {
      "term": "function notation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「関数記号」は学習指導要領解説に無い、本プロジェクトの訳語（日本語版 Wikipedia「関数記号」は数理論理学の別の意味）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "関数に f などの名前を付け、x に対応する値を f(x) と書き表す書き方。",
    "definition_en": "The way of naming a function, such as f, and writing its output for the input x as f(x).",
    "latex": "y = f(x)",
    "spoken_en": "y equals f of x",
    "examples": [
      {
        "en": "In function notation, instead of y = 3x + 2 we write f(x) = 3x + 2.",
        "ja": "関数記号を使うと、y = 3x + 2 の代わりに f(x) = 3x + 2 と書く。",
        "register": "spoken"
      },
      {
        "en": "Rewrite the equation y = x² − 5 using function notation, and then find f(3).",
        "ja": "y = x² − 5 を関数記号を用いて書き直し、f(3) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "f(x) は「f かける x」ではない。括弧は掛け算ではなく、f に x を入れた値を表す。",
      "英語版 Wikipedia の Function symbol（日本語版「関数記号」の対応記事）は数理論理学の用語。f(x) という書き方そのものは function notation と言う。"
    ],
    "related": [
      "value-of-the-function",
      "evaluate",
      "composite-function",
      "inverse-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Function_symbol",
        "note": "記事「Function symbol」（数理論理学の function symbol。日本語版「関数記号」の対応記事で、見出しの意味の記事ではない）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 二次関数: f(x) の書き方（「関数記号」は 0 件）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「関数記号」",
        "url": "https://ja.wikipedia.org/wiki/%E9%96%A2%E6%95%B0%E8%A8%98%E5%8F%B7",
        "note": "数理論理学の用語（見出しの意味の記事ではない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "function notation": 19
      },
      "written": {
        "function notation": 38
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "function-of-several-variables",
    "ja": {
      "term": "多変数関数",
      "reading": "たへんすうかんすう",
      "alt": []
    },
    "en": {
      "term": "function of several variables",
      "alt": [
        "multivariable function"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "2 つ以上の変数の値に対して 1 つの値を対応させる関数 f(x, y) など。",
    "definition_en": "A function, such as f(x, y), that assigns one value to each choice of two or more input variables.",
    "latex": "z = f(x, y)",
    "spoken_en": "z equals f of x comma y",
    "examples": [
      {
        "en": "With a function of two variables, the graph is a surface, not a curve.",
        "ja": "2 変数関数のグラフは、曲線ではなく曲面になる。",
        "register": "spoken"
      },
      {
        "en": "Find the domain of the function of two variables f(x, y) = √(9 − x² − y²).",
        "ja": "2 変数関数 f(x, y) = √(9 − x² − y²) の定義域を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "授業と教科書で多いのは 2 変数の場合の function of two variables（特に書き言葉に多い）。"
    ],
    "related": [
      "partial-derivative",
      "level-curve",
      "tangent-plane",
      "double-integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "function of several variables": 2,
        "multivariable function": 2
      },
      "written": {
        "function of several variables": 7,
        "multivariable function": 2
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 function of several variables（Functions of Several Variables）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "function",
    "ja": {
      "term": "関数",
      "reading": "かんすう",
      "alt": [
        "函数"
      ]
    },
    "en": {
      "term": "function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの数量 x, y があり、x の値を決めると y の値がただ 1 つに決まるとき、y は x の関数であるという。",
    "definition_en": "A rule that gives exactly one output for each input. If each value of x gives exactly one value of y, then y is a function of x.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For each x you get exactly one y, so y is a function of x.",
        "ja": "x を 1 つ決めると y が 1 つに決まるから、y は x の関数だね。",
        "register": "spoken"
      },
      {
        "en": "A notebook costs 2 dollars. The total cost y of x notebooks is a function of x. Write an equation for y in terms of x.",
        "ja": "1 冊 2 ドルのノートを x 冊買うときの代金を y ドルとすると、y は x の関数である。y を x の式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「函数」は古い書き方で、意味は同じ。",
      "「y は x の関数」は y is a function of x と of を使う。f(x) は function notation（エントリ function-notation）で、f of x と読む。「f かける x」ではない。",
      "function は「機能・働き」の意味でも使う。数学の function は「関数」。"
    ],
    "related": [
      "function-notation",
      "value-of-the-function",
      "direct-proportion",
      "inverse-proportion",
      "domain-and-range"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "関数 (数学)",
        "en": "Function (mathematics)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "function": 9753
      },
      "written": {
        "function": 11622
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "fundamental-theorem-of-algebra",
    "ja": {
      "term": "代数学の基本定理",
      "reading": "だいすうがくのきほんていり",
      "alt": []
    },
    "en": {
      "term": "fundamental theorem of algebra",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "複素数係数の n 次方程式（n ≧ 1）は、複素数の範囲に少なくとも 1 つの解をもつという定理。重複も数えると解はちょうど n 個になる。",
    "definition_en": "The theorem that every polynomial equation of degree n ≥ 1 with complex coefficients has at least one complex root, so exactly n roots counted with multiplicity.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "By the fundamental theorem of algebra, a degree-5 polynomial has exactly five roots, counting multiplicity.",
        "ja": "代数学の基本定理から、5 次の多項式には重複も数えてちょうど 5 個の解がある。",
        "register": "spoken"
      },
      {
        "en": "Explain, using the fundamental theorem of algebra, why x⁴ + 1 = 0 has four complex solutions.",
        "ja": "代数学の基本定理を用いて、x⁴ + 1 = 0 が 4 つの複素数解をもつ理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は定理として扱い、多項式の解の個数の根拠に使う。"
    ],
    "related": [
      "polynomial-equation-of-higher-degree",
      "complex-number",
      "imaginary-solution",
      "zeros-of-a-polynomial",
      "double-root",
      "descartes-rule-of-signs"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "代数学の基本定理",
        "en": "Fundamental theorem of algebra"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "5.5 Zeros of Polynomial Functions（Fundamental Theorem of Algebra）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "3.6 Zeros of Polynomial Functions"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "Appendix A Complex Numbers・3.3 Diagonalization and Eigenvalues（fundamental theorem of algebra）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "fundamental theorem of algebra": 2
      },
      "written": {
        "fundamental theorem of algebra": 10
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "fundamental-theorem-of-calculus-part-1",
    "ja": {
      "term": "微積分学の基本定理 第 1 部",
      "reading": "びせきぶんがくのきほんていり だいいちぶ",
      "alt": []
    },
    "en": {
      "term": "fundamental theorem of calculus, part 1",
      "alt": [
        "fundamental theorem of calculus part 1"
      ],
      "variants": [
        {
          "term": "fundamental theorem of calculus, part 1",
          "register": "written",
          "note": "OpenStax Calculus の書き方。話し言葉では Professor Leonard が part one と言う。書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "first fundamental theorem of calculus",
          "register": "written",
          "note": "MIT の講義ノートの呼び方。話し言葉にはほとんど出てこず（Khan Academy だけ）、書き言葉はすべて MIT の講義ノート。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学II）は d/dx ∫ₐˣ f(t)dt = f(x) を「微分と積分の関係」に着目して扱い、定理の名前や番号は出てこない。OpenStax Calculus Volume 1 はこれを Part 1 と呼ぶ。番号や名前（first ／ second fundamental theorem）の付け方は教科書による。AP の CED は番号を付けず、topic 6.4 で accumulation functions とあわせて扱う。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "f が連続なら、上端を変数とする定積分 ∫ₐˣ f(t)dt を x で微分すると f(x) になる、という定理。",
    "definition_en": "If f is continuous, then the function ∫ₐˣ f(t) dt has derivative f(x).",
    "latex": "\\frac{d}{dx}\\int_a^x f(t)\\,dt = f(x)",
    "spoken_en": "the derivative with respect to x of the integral from a to x of f of t dt equals f of x",
    "examples": [
      {
        "en": "Part 1 says that if you integrate f and then differentiate, you get f back.",
        "ja": "第 1 部は、f を積分してから微分すると f に戻る、ということ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "上端が x² のような式なら連鎖律が要る: d/dx ∫ₐ^{x²} f(t)dt = f(x²)·2x。AP の CED も topic 6.4 でこの形を扱う。"
    ],
    "related": [
      "fundamental-theorem-of-calculus",
      "fundamental-theorem-of-calculus-part-2",
      "accumulation-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.4（番号は付けない）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 微分・積分の考え: 微分と積分の関係"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "first fundamental theorem of calculus": 1,
        "fundamental theorem of calculus part 1": 4
      },
      "written": {
        "fundamental theorem of calculus, part 1": 10,
        "first fundamental theorem of calculus": 3
      },
      "sources": [
        "khan-ap-calc",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "fundamental-theorem-of-calculus-part-2",
    "ja": {
      "term": "微積分学の基本定理 第 2 部",
      "reading": "びせきぶんがくのきほんていり だいにぶ",
      "alt": []
    },
    "en": {
      "term": "fundamental theorem of calculus, part 2",
      "alt": [
        "fundamental theorem of calculus part 2"
      ],
      "variants": [
        {
          "term": "fundamental theorem of calculus, part 2",
          "register": "written",
          "note": "OpenStax Calculus の書き方。話し言葉には出てこず、書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "evaluation theorem",
          "register": "written",
          "note": "OpenStax Calculus がこの名前も挙げる。話し言葉には出てこず、書き言葉（すべて OpenStax Calculus）でも少ない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学II）には、∫ₐᵇ f(x)dx = F(b) − F(a) を定理として名前で呼ぶ扱いが出てこない（定積分は面積 S(t) や区分求積法の考えと関連付けて導入する）。OpenStax Calculus Volume 1 は定積分を Riemann sum の極限で定義したあと、この式を Part 2 として示し、Evaluation Theorem という名前も挙げる。番号と名前の付け方は教科書による。AP の CED は番号を付けず topic 6.7 で扱う。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "f が連続で F がその原始関数なら、∫ₐᵇ f(x)dx = F(b) − F(a) が成り立つ、という定理。",
    "definition_en": "If f is continuous and F is any antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a).",
    "latex": "\\int_a^b f(x)\\,dx = F(b) - F(a)",
    "spoken_en": "the integral from a to b of f of x dx equals F of b minus F of a",
    "examples": [
      {
        "en": "Thanks to Part 2, we never have to compute a limit of Riemann sums by hand again.",
        "ja": "第 2 部のおかげで、リーマン和の極限を手で計算する必要はもうない。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Khan Academy の AP Calculus の動画は、主にこの式を the second fundamental theorem of calculus と呼ぶ。番号は教材によって違うので、番号ではなく式で確かめる。"
    ],
    "related": [
      "fundamental-theorem-of-calculus",
      "fundamental-theorem-of-calculus-part-1",
      "definite-integral",
      "net-change"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.7（番号は付けない）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 微分・積分の考え: 面積 S(t) と定積分、区分求積法の考え"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "fundamental theorem of calculus, part 2": 15,
        "evaluation theorem": 4
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "fundamental-theorem-of-calculus",
    "ja": {
      "term": "微積分学の基本定理",
      "reading": "びせきぶんがくのきほんていり",
      "alt": [
        "微分積分学の基本定理",
        "微分と積分の関係"
      ]
    },
    "en": {
      "term": "fundamental theorem of calculus",
      "alt": [],
      "variants": [
        {
          "term": "fundamental theorem of calculus",
          "register": "both",
          "note": "話し言葉は Khan Academy と MIT OCW が中心、書き言葉は大半が OpenStax Calculus。"
        },
        {
          "term": "FTC",
          "register": "both",
          "note": "略記。書き言葉ではすべて MIT の講義ノートで、OpenStax Calculus は略さない。話し言葉にはほとんど出てこない（MIT OCW だけ）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学II）は「微分と積分の関係」に着目して扱い、定理の名前（微分積分学の基本定理）は出てこない。AP の CED と OpenStax Calculus は定理を名前で呼ぶ。OpenStax Calculus Volume 1 は Part 1（積分を微分すると元に戻る）と Part 2（F(b) − F(a) で計算できる）に分ける。番号の付け方は教科書による。AP の CED は番号を付けず、topic 6.4（accumulation functions）と 6.7（definite integrals）で扱う。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "微分と積分が互いに逆の操作であることを述べる定理。d/dx ∫ₐˣ f(t)dt = f(x) と ∫ₐᵇ f(x)dx = F(b) − F(a) の 2 つからなる。",
    "definition_en": "The theorem that differentiation and integration undo each other: d/dx ∫ₐˣ f(t) dt = f(x), and ∫ₐᵇ f(x) dx = F(b) − F(a).",
    "latex": "\\frac{d}{dx}\\int_a^x f(t)\\,dt = f(x)",
    "spoken_en": "the derivative with respect to x of the integral from a to x of f of t dt equals f of x",
    "examples": [
      {
        "en": "By the fundamental theorem, you just plug in the top and the bottom and subtract.",
        "ja": "基本定理を使えば、上端と下端を代入して引くだけ。",
        "register": "spoken"
      },
      {
        "en": "By the Fundamental Theorem of Calculus, ∫₁³ 2x dx = 3² − 1² = 8.",
        "ja": "微積分学の基本定理より ∫₁³ 2x dx = 3² − 1² = 8。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "by the fundamental theorem of calculus",
        "ja": "微積分学の基本定理より"
      }
    ],
    "pitfalls": [
      "講義ノートでは FTC と略して書くことがある。",
      "答案で使うときは by the Fundamental Theorem of Calculus と名前を出して書く。"
    ],
    "related": [
      "definite-integral",
      "antiderivative",
      "accumulation-function",
      "reverse-of-differentiation",
      "fundamental-theorem-of-calculus-part-1",
      "fundamental-theorem-of-calculus-part-2"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "微分積分学の基本定理",
        "en": "Fundamental theorem of calculus"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.4、6.7（番号 Part 1 ／ Part 2 は使わない）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 微分・積分の考え: 微分と積分の関係"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "fundamental theorem of calculus": 118,
        "FTC": 3
      },
      "written": {
        "fundamental theorem of calculus": 89,
        "FTC": 23
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "gaussian-elimination",
    "ja": {
      "term": "ガウスの消去法",
      "reading": "がうすのしょうきょほう",
      "alt": [
        "掃き出し法"
      ]
    },
    "en": {
      "term": "Gaussian elimination",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "行基本変形で拡大係数行列を行階段形にし、後ろから順に代入して連立 1 次方程式を解く方法。",
    "definition_en": "Solving a system of linear equations by row-reducing its augmented matrix to row-echelon form and then back-substituting.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Gaussian elimination is the same as elimination by hand, just with the numbers in a matrix.",
        "ja": "ガウスの消去法は手で消去するのと同じで、数を行列に並べただけ。",
        "register": "spoken"
      },
      {
        "en": "Solve the system by Gaussian elimination: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.",
        "ja": "ガウスの消去法で連立方程式 x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "簡約階段形まで進めるものは Gauss-Jordan elimination（掃き出し法）と呼び分けることがある。",
      "行階段形で止めて後ろから代入するか、既約行階段形（reduced row-echelon form）まで進めるかは教科書による。Nicholson は Gauss-Jordan の名前を使わず（0 件）、1.2 Gaussian Elimination の節で行階段形と既約行階段形の両方を扱う。"
    ],
    "related": [
      "augmented-matrix",
      "row-operations",
      "row-echelon-form",
      "system-of-three-equations",
      "reduced-row-echelon-form",
      "pivot",
      "system-of-linear-equations"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ガウスの消去法",
        "en": "Gaussian elimination"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "1.2 Gaussian Elimination"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "Gaussian elimination": 6
      },
      "written": {
        "Gaussian elimination": 27
      },
      "sources": [
        "mit-18.06",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "general-addition-rule",
    "ja": {
      "term": "確率の加法定理",
      "reading": "かくりつのかほうていり",
      "alt": [
        "和事象の確率"
      ]
    },
    "en": {
      "term": "addition rule",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「加法定理」は、確率の加法定理（和の法則）を、事象が互いに排反なら確率の和に分けられるという式とし、重なりがあるときは包除原理によると書く（「数学 (教科)」も数学A の項目に「排反と加法定理」を挙げる）。学習指導要領解説（数学A）も、互いに排反なときの P(A ∪ B) = P(A) + P(B) を確率の性質や法則に挙げる。OpenStax Introductory Statistics（3.3）の addition rule は、重なりを引く一般の式 P(A OR B) = P(A) + P(B) − P(A AND B) を指す（IM Geometry（8.6 The Addition Rule）は同じ式を P(A or B) = P(A) + P(B) − P(A and B) と書く）。AP Statistics の CED（topic 2.7）は同じ式に名前を付けない。",
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "AP Statistics",
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2",
        "Intro Statistics",
        "Geometry"
      ]
    },
    "definition_ja": "和事象の確率を求める式 P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。A, B が互いに排反なら P(A ∪ B) = P(A) + P(B)。",
    "definition_en": "The rule for the probability of a union, P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events, P(A or B) = P(A) + P(B).",
    "latex": "P(A \\cup B) = P(A) + P(B) - P(A \\cap B)",
    "spoken_en": "the probability of A or B equals the probability of A plus the probability of B minus the probability of A and B",
    "examples": [
      {
        "en": "Don't forget to subtract the overlap, or you'll count it twice.",
        "ja": "重なりの分を引くのを忘れると、2 回数えてしまう。",
        "register": "spoken"
      },
      {
        "en": "Use the addition rule to find P(A or B) if P(A) = 0.4, P(B) = 0.5, and P(A and B) = 0.2.",
        "ja": "P(A) = 0.4, P(B) = 0.5, P(A かつ B) = 0.2 のとき、重なりを引く和事象の確率の式で P(A または B) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉の addition rule はほとんどが Professor Leonard。",
      "場合の数の「和の法則」（addition principle）とは別。"
    ],
    "related": [
      "general-multiplication-rule",
      "basic-properties-of-probability",
      "mutually-exclusive-events",
      "addition-principle",
      "union-of-events"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.3 Two Basic Rules of Probability（addition rule）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 8.6 The Addition Rule"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.7（P(A ∪ B) = P(A) + P(B) − P(A ∩ B)。addition rule の名前は無い）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A (2) 場合の数と確率: 事象A,Bが互いに排反であるときP(A∪B)=P(A)+P(B)"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「加法定理」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8A%A0%E6%B3%95%E5%AE%9A%E7%90%86",
        "note": "数え上げおよび確率の加法定理（和の法則）: 排反なら和を分配、交わりがあれば包除原理"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学A: 排反と加法定理"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成29年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20180601/https://www.dnc.ac.jp/albums/abm.php?f=abm00009412.pdf&n=%E6%95%B0%E5%AD%A6%E2%85%A0%E3%83%BB%E6%95%B0%E5%AD%A6%EF%BC%A1.pdf",
        "note": "和事象の確率（ja.alt）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "addition rule": 20
      },
      "written": {
        "addition rule": 9
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "general-angle",
    "ja": {
      "term": "一般角",
      "reading": "いっぱんかく",
      "alt": []
    },
    "en": {
      "term": "coterminal angles",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「一般角」に当たる決まった名前は、用例コーパスと参照にほとんど出てこない（OpenStax Calculus Volume 1 の 1.3 Trigonometric Functions に for a general angle θ が 1 回）。回転の量として 360° を超える角や負の角を考えることは英語でも同じで、OpenStax Algebra and Trigonometry（7.1 Angles）は positive angle ／ negative angle と呼び、IM Algebra 2（6.10・6.11）は angle of rotation と呼ぶ。θ + 360°n の形の角は coterminal angles（動径が同じ角）と呼ぶ（OpenStax Algebra and Trigonometry 7.1）。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "動径の回転の向きと量で表した角。360° を超える角や負の角も含み、動径が同じ角は θ + 360°n（n は整数）と表せる。",
    "definition_en": "An angle measured by the rotation of a ray, including angles over 360° and negative angles; angles with the same terminal side are θ + 360°n (n an integer), called coterminal angles in English.",
    "latex": "\\theta + 360^\\circ n\\quad(n\\ \\text{is an integer})",
    "spoken_en": null,
    "examples": [
      {
        "en": "400 degrees and 40 degrees are coterminal — they end up in the same place.",
        "ja": "400° と 40° は動径が同じ。同じところで止まる。",
        "register": "spoken"
      },
      {
        "en": "Among the coterminal angles of −750°, find the one between 0° and 360°.",
        "ja": "−750° と動径が同じ角のうち、0° 以上 360° 未満のものを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語版 Wikipedia「一般角」は英語を general angle と書き、OpenStax Calculus Volume 1（1.3）も for a general angle θ と使うが、用例コーパスと参照ではほとんど使わない。",
      "動径が同じ角どうしの関係そのもの（30° と 390°）は coterminal angle（エントリ coterminal-angle）。OpenStax Algebra and Trigonometry（7.1）は、360° の整数倍を足し引きしてできる角をまとめて coterminal angles と呼ぶ。"
    ],
    "related": [
      "terminal-side",
      "initial-side",
      "radian-measure",
      "counterclockwise",
      "coterminal-angle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles（coterminal angles、positive ／ negative angle）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.3 Trigonometric Functions（a general angle θ）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 6.10 Beyond 2π・6.11 Extending the Domain of Trigonometric Functions（angle of rotation）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「一般角」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%80%E8%88%AC%E8%A7%92",
        "note": "英: general angle"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "coterminal angles": 9
      },
      "written": {
        "coterminal angles": 31
      },
      "sources": [
        "openstax-algtrig",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "general-form-of-a-circle",
    "ja": {
      "term": "円の一般形",
      "reading": "えんのいっぱんけい",
      "alt": []
    },
    "en": {
      "term": "general form of the equation of a circle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円の方程式を展開した形 x² + y² + lx + my + n = 0。平方完成すると中心と半径がわかる。",
    "definition_en": "The expanded form x² + y² + lx + my + n = 0 of a circle's equation; completing the square gives the center and radius.",
    "latex": "x^2 + y^2 + lx + my + n = 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "This is the general form of the equation of a circle, so complete the square in x and in y to get it back into standard form.",
        "ja": "これは円の方程式の一般形なので、x と y それぞれで平方完成して標準形に戻そう。",
        "register": "spoken"
      },
      {
        "en": "Find the center and radius of the circle x² + y² − 4x + 6y − 3 = 0.",
        "ja": "円 x² + y² − 4x + 6y − 3 = 0 の中心と半径を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Intermediate Algebra（11.1）・Algebra and Trigonometry は円の一般形を x² + y² + ax + by + c = 0 と書く（この a, b は係数で、エントリ equation-of-a-circle の (x − a)² + (y − b)² = r² の中心の座標ではない）。D, E, F は 2 次曲線の一般形 Ax² + Bxy + Cy² + Dx + Ey + F = 0 の係数（Algebra and Trigonometry 12.4）。",
      "x² + y² + lx + my + n = 0 が円を表さない（点になる、図形がない）場合があることにも注意する。"
    ],
    "related": [
      "equation-of-a-circle",
      "completing-the-square"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "12.1・12.4（x² + y² + ax + by + c = 0、Ax² + Bxy + Cy² + Dx + Ey + F = 0）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "11.1 Distance and Midpoint Formulas; Circles（General Form of the Equation of a Circle）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "general form of the equation of a circle": 6
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 general form of the equation of a circle（本文 6 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "general-multiplication-rule",
    "ja": {
      "term": "確率の乗法定理",
      "reading": "かくりつのじょうほうていり",
      "alt": []
    },
    "en": {
      "term": "general multiplication rule",
      "alt": [
        "multiplication rule"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "AP Statistics",
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2",
        "Intro Statistics",
        "Geometry"
      ]
    },
    "definition_ja": "積事象の確率を条件付き確率で表す式 P(A ∩ B) = P(A) P(B | A)。A, B が独立なら P(A ∩ B) = P(A) P(B)。",
    "definition_en": "The rule P(A and B) = P(A) P(B | A) for the probability of an intersection; for independent events, P(A and B) = P(A) P(B).",
    "latex": "P(A \\cap B) = P(A)\\,P(B \\mid A)",
    "spoken_en": "the probability of A and B equals the probability of A times the probability of B given A",
    "examples": [
      {
        "en": "Multiply the probability of the first draw by the probability of the second, given the first.",
        "ja": "1 回目の確率に、1 回目の結果のもとでの 2 回目の確率を掛ける。",
        "register": "spoken"
      },
      {
        "en": "Two cards are drawn without replacement. Use the general multiplication rule to find the probability that both are hearts.",
        "ja": "2 枚のカードを戻さずに引く。乗法定理を使って、2 枚ともハートである確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（数学A）は条件付き確率を P_A(B) と書き、P(A ∩ B) = P(A)P_A(B) を「一般的な確率の乗法定理」、独立な試行の P(C) = P(A)P(B) を「確率の乗法定理」と呼ぶ。AP Statistics の CED（topic 2.6）と OpenStax Introductory Statistics は P(B | A) と書き（OpenStax は P(A | B) と合わせて 44 件）、B given A と読む。",
      "場合の数の「積の法則」（multiplication principle）とは別。"
    ],
    "related": [
      "general-addition-rule",
      "tree-diagram",
      "conditional-probability",
      "independence-of-events",
      "multiplication-principle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.3 Two Basic Rules of Probability（multiplication rule、P(A | B)）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.6（general multiplication rule、P(A ∩ B) = P(A)·P(B|A)）、Unit 2 の概要"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A (2) 場合の数と確率: P(A∩B)=P(A)PA(B)（一般的な確率の乗法定理）、独立な試行の P(C)=P(A)P(B)（確率の乗法定理）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 8.8 Conditional Probability（multiplication rule）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "multiplication rule": 6
      },
      "written": {
        "multiplication rule": 4
      },
      "sources": [
        "khan-middle",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 6 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは AP Statistics の CED の呼び方 general multiplication rule（topic 2.6）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "general-solution",
    "ja": {
      "term": "一般解",
      "reading": "いっぱんかい",
      "alt": []
    },
    "en": {
      "term": "general solution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "方程式の解をすべて、整数 n や任意定数 C を含む式で表したもの。三角方程式では θ = π/6 + 2nπ など、微分方程式では任意定数を含む解。",
    "definition_en": "All solutions of an equation, written with an integer n or arbitrary constants: θ = π/6 + 2nπ for a trigonometric equation, a solution with arbitrary constants for a differential equation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For the general solution, add 2π n, where n is any integer.",
        "ja": "一般解にするには 2πn を足す（n は任意の整数）。",
        "register": "spoken"
      },
      {
        "en": "Find the general solution of the equation 2 sin θ = √3.",
        "ja": "方程式 2 sin θ = √3 の一般解を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの general solution の多くは微分方程式（MIT 18.03・OpenStax Calculus）。三角方程式のすべての解を、OpenStax Algebra and Trigonometry（Solving Trigonometric Equations）は general solution と呼ばず all possible solutions と書き、θ = π/3 ± 2kπ（k is an integer）の形で示す（同書の general solution は連立一次方程式の解）。"
    ],
    "related": [
      "trigonometric-equation",
      "particular-solution",
      "differential-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "4.1 Basics of Differential Equations（general solution、particular solution）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Solving Trigonometric Equations（all possible solutions）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 7.6（general solutions）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "general solution": 129
      },
      "written": {
        "general solution": 213
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "general-term",
    "ja": {
      "term": "一般項",
      "reading": "いっぱんこう",
      "alt": []
    },
    "en": {
      "term": "general term",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "sequences"
    ],
    "level": {
      "jp": [
        "数II",
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "数列や展開式の項を、番号（n や r）を使って 1 つの式で表したもの。二項定理では nCr aⁿ⁻ʳ bʳ。",
    "definition_en": "A formula for a typical term of a sequence or an expansion in terms of its index: aₙ as a formula in n for a sequence, C(n, r) aⁿ⁻ʳ bʳ for a binomial expansion.",
    "latex": "{}_n\\mathrm{C}_r\\, a^{n-r} b^r",
    "spoken_en": null,
    "examples": [
      {
        "en": "Write down the (r + 1)th term first, then choose r so the power of x is 4.",
        "ja": "まず第 r + 1 項（一般項）を書いて、x の次数が 4 になるように r を決める。",
        "register": "spoken"
      },
      {
        "en": "Find the general term of the geometric sequence 5, 20, 80, 320, …",
        "ja": "等比数列 5, 20, 80, 320, … の一般項を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数列の一般項 aₙ は、OpenStax Algebra and Trigonometry（13.1）・Intermediate Algebra（12.1）でも general term (nth term)。n の式そのものは explicit formula（漸化式の recursive formula と対。エントリ find-the-nth-term）とも呼ぶ。",
      "二項定理の展開式の一般項 ₙCᵣ aⁿ⁻ʳ bʳ を、OpenStax Algebra and Trigonometry（Binomial Theorem）は the (r+1)th term of the binomial expansion、OpenStax Intermediate Algebra（Binomial Theorem）は the (r + 1)st term と書き、general term とは呼ばない。",
      "用例コーパスの書き言葉は OpenStax Intermediate Algebra が中心で、次が OpenStax Algebra and Trigonometry（数列と級数）。"
    ],
    "related": [
      "expansion",
      "nth-term",
      "find-the-nth-term",
      "multinomial-theorem",
      "binomial-theorem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Sequences and Their Notations・Binomial Theorem（general term、the (r+1)th term）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "試験の例題 Question 4 と採点基準（the general term of the Maclaurin series、General term）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Sequences・Binomial Theorem（general term、the (r + 1)st term）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences・5.2 Infinite Series（general term）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "general term": 45
      },
      "written": {
        "general term": 182
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "generating-function",
    "ja": {
      "term": "母関数",
      "reading": "ぼかんすう",
      "alt": [
        "生成関数"
      ]
    },
    "en": {
      "term": "generating function",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics",
      "sequences"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "数列 a₀, a₁, a₂, … の各項を係数にもつべき級数 a₀ + a₁x + a₂x² + ⋯ のこと。数列についての問題を、関数の計算に置き換えて解くのに使う。",
    "definition_en": "The power series a₀ + a₁x + a₂x² + ⋯ whose coefficients are the terms of a sequence a₀, a₁, a₂, …. It turns questions about the sequence into algebra with functions.",
    "latex": "A(x) = \\sum_{n=0}^{\\infty} a_n x^n",
    "spoken_en": "A of x equals the sum from n equals zero to infinity of a sub n x to the n",
    "examples": [
      {
        "en": "The generating function for 1, 1, 1, 1, and so on is one over one minus x.",
        "ja": "1, 1, 1, 1, … の母関数は 1/(1 − x) だ。",
        "register": "spoken"
      },
      {
        "en": "Find the generating function for the sequence 1, 2, 4, 8, … and write it in closed form.",
        "ja": "数列 1, 2, 4, 8, … の母関数を求め、分数式の形で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語では「母関数」と「生成関数」の 2 つの名前があるが、英語はどちらも generating function。",
      "母関数では x に値を代入せず、係数を並べる入れ物として扱うことが多い。収束するかどうかは問題にしない（形式的べき級数）。",
      "確率・統計の moment generating function（積率母関数）は別物。用例コーパスでは書き言葉に多く（すべて MIT の講義ノート）、話し言葉にはほとんど出てこない。Levin の本では 6.1 Generating Functions に 95 件。"
    ],
    "related": [
      "recurrence-relation",
      "linear-recurrence-relation",
      "binomial-theorem",
      "power-series",
      "sequence"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "母関数",
        "en": "Generating function"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "generating function": 3
      },
      "written": {
        "generating function": 102
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "geometric-distribution",
    "ja": {
      "term": "幾何分布",
      "reading": "きかぶんぷ",
      "alt": []
    },
    "en": {
      "term": "geometric distribution",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "成功の確率が p の試行を独立に繰り返したとき、初めて成功するまでの回数の分布。P(X = k) = (1 − p)^{k−1} p。",
    "definition_en": "The distribution of the number of independent trials, each with success probability p, needed to get the first success: P(X = k) = (1 − p)^{k−1} p.",
    "latex": "P(X = k) = (1-p)^{k-1}p",
    "spoken_en": null,
    "examples": [
      {
        "en": "How many tries until the first success? That's a geometric distribution.",
        "ja": "最初に成功するまで何回かかるか。それが幾何分布。",
        "register": "spoken"
      },
      {
        "en": "A player makes 70% of free throws. Find the probability that the first miss is on the fourth attempt.",
        "ja": "フリースローの成功率が 70% の選手が、4 回目で初めて失敗する確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "期待値は 1/p。"
    ],
    "related": [
      "binomial-distribution",
      "probability-distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "幾何分布",
        "en": "Geometric distribution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "geometric distribution": 27
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "geometric-mean",
    "ja": {
      "term": "相乗平均",
      "reading": "そうじょうへいきん",
      "alt": [
        "幾何平均"
      ]
    },
    "en": {
      "term": "geometric mean",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "n 個の正の数の積の n 乗根。2 数 a, b では √(ab)。",
    "definition_en": "The nth root of the product of n positive numbers; for two numbers a and b, √(ab).",
    "latex": "\\sqrt{ab}",
    "spoken_en": "the square root of a b",
    "examples": [
      {
        "en": "The geometric mean of 2 and 8 is the square root of 16, which is 4.",
        "ja": "2 と 8 の相乗平均は √16 で 4。",
        "register": "spoken"
      },
      {
        "en": "Show that the geometric mean of two positive numbers never exceeds their arithmetic mean.",
        "ja": "2 つの正の数の相乗平均は相加平均を超えないことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "IM Geometry（3.13 Using the Pythagorean Theorem and Similarity）は、直角三角形の斜辺に下ろした高さが斜辺の 2 つの部分の相乗平均になること（3 と 7 なら高さ √21）を相似で扱う（geometric mean の語は使わない）。CK-12 Geometry（4.6 の練習問題）は、三角形の高さを the geometric mean で求めさせる。",
      "用例コーパスの話し言葉は大半が The Organic Chemistry Tutor。書き言葉にはほとんど出てこない。"
    ],
    "related": [
      "arithmetic-mean",
      "am-gm-inequality",
      "geometric-middle-term",
      "right-triangle-similarity"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "幾何平均",
        "en": "Geometric mean"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 3.13 Using the Pythagorean Theorem and Similarity（斜辺に下ろした高さ √21）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.6 Area and Perimeter of Triangles（the geometric mean）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "geometric mean": 39
      },
      "written": {
        "geometric mean": 1
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "geometric-middle-term",
    "ja": {
      "term": "等比中項",
      "reading": "とうひちゅうこう",
      "alt": []
    },
    "en": {
      "term": "geometric mean",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスでは geometric mean of … and … の形も少なく、書き言葉には出てこない）。3 数 a, b, c が等比数列をなすときの b に名前はつけず、b is the geometric mean of a and c（b² = ac）と相乗平均で言う。相乗平均（geometric mean）は正の数 √(ac) についての言い方なので、b が負のとき（2, −6, 18）や a, c が負のときには当たらない。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2"
      ]
    },
    "definition_ja": "3 つの数 a, b, c がこの順に等比数列をなすときの b。b² = ac を満たす。",
    "definition_en": "The number b when a, b, c form a geometric sequence in that order, so that b² = ac.",
    "latex": "b^2 = ac",
    "spoken_en": null,
    "examples": [
      {
        "en": "For a geometric sequence, the middle term squared equals the product of its neighbors.",
        "ja": "等比数列では、真ん中の項の 2 乗が両隣の積に等しい。",
        "register": "spoken"
      },
      {
        "en": "If 2, x, and 18 form a geometric sequence, find all possible values of x.",
        "ja": "2、x、18 がこの順に等比数列をなすとき、x の値をすべて求めよ。",
        "register": "written"
      },
      {
        "en": "Since 2, 6, 18 is geometric, 6 is the geometric mean of 2 and 18: six squared is 2 times 18.",
        "ja": "2, 6, 18 は等比数列なので、6 は 2 と 18 の等比中項（6² = 2 × 18）。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語版 Wikipedia「Geometric mean theorem」は、直角三角形の斜辺に下ろした高さについての定理をこの名前で呼ぶ。相乗平均（geometric-mean）とも同じ英語。",
      "b² = ac からは b = ±√(ac) の 2 通りが出る。"
    ],
    "related": [
      "geometric-sequence",
      "arithmetic-middle-term",
      "am-gm-inequality",
      "geometric-mean"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Geometric_mean_theorem",
        "note": "記事「Geometric mean theorem」（冒頭で geometric mean theorem を太字）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「等比数列」",
        "url": "https://ja.wikipedia.org/wiki/%E7%AD%89%E6%AF%94%E6%95%B0%E5%88%97",
        "note": "脚注（a, b, c が 0 でないとき b を等比中項と呼ぶ）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 1.3 Different Types of Sequences・1.8（等比数列の間の項を求める）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "geometric mean of … and": 5
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 5 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "geometric-multiplicity",
    "ja": {
      "term": "幾何的重複度",
      "reading": "きかてきちょうふくど",
      "alt": []
    },
    "en": {
      "term": "geometric multiplicity",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "固有値 λ に対応する固有空間の次元。1 次独立な固有ベクトルが何本とれるかを表す。",
    "definition_en": "The dimension of the eigenspace of an eigenvalue λ: how many linearly independent eigenvectors it has.",
    "latex": "\\dim E_\\lambda",
    "spoken_en": null,
    "examples": [
      {
        "en": "The eigenvalue repeats twice but has only one independent eigenvector, so the geometric multiplicity is one.",
        "ja": "固有値は 2 重だが 1 次独立な固有ベクトルは 1 本なので、幾何的重複度は 1。",
        "register": "spoken"
      },
      {
        "en": "Show that the geometric multiplicity of λ = 1 is less than its algebraic multiplicity.",
        "ja": "λ = 1 の幾何的重複度が代数的重複度より小さいことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉にわずかに出てくるだけ。Nicholson はこの名前を使わず、固有空間の次元 dim E_λ と書く。"
    ],
    "related": [
      "algebraic-multiplicity",
      "eigenspace",
      "diagonalizable"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors",
        "note": "記事「Eigenvalues and eigenvectors」の節 Eigenspaces, geometric multiplicities, and eigenbasis for a matrix で geometric multiplicity を太字で定義（数学カテゴリから 3 段）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "evidence": {
      "spoken": {
        "geometric multiplicity": 2
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "geometric-sequence",
    "ja": {
      "term": "等比数列",
      "reading": "とうひすうれつ",
      "alt": [
        "幾何数列"
      ]
    },
    "en": {
      "term": "geometric sequence",
      "alt": [
        "geometric progression"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "隣り合う項の比が一定の数列。その一定の比を公比という。",
    "definition_en": "A sequence in which the ratio of each term to the one before it is always the same number, the common ratio.",
    "latex": "a_n = a_1 r^{\\,n-1}",
    "spoken_en": "a sub n equals a sub one times r to the n minus one",
    "examples": [
      {
        "en": "Each term is half the one before, so it's geometric with ratio one half.",
        "ja": "前の項の半分になっていくから、公比 1/2 の等比数列。",
        "register": "spoken"
      },
      {
        "en": "Find the general term of the geometric sequence 3, 6, 12, 24, ….",
        "ja": "等比数列 3, 6, 12, 24, … の一般項を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "geometric progression は話し言葉・書き言葉とも少ない。"
    ],
    "related": [
      "common-ratio",
      "arithmetic-sequence",
      "sum-of-a-geometric-sequence",
      "infinite-geometric-series",
      "geometric-middle-term",
      "sequence"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "等比数列",
        "en": "Geometric progression"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "geometric sequence": 20,
        "geometric progression": 3
      },
      "written": {
        "geometric sequence": 182,
        "geometric progression": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "glide-reflection",
    "ja": {
      "term": "映進",
      "reading": "えいしん",
      "alt": [
        "グライド反射"
      ]
    },
    "en": {
      "term": "glide reflection",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直線に関する対称移動と、その直線に平行な向きの平行移動を続けて行う移動。2 つの移動はどちらを先に行っても同じ図形に移る。",
    "definition_en": "A transformation made of a reflection across a line and a translation in a direction parallel to that line. Doing the two steps in either order gives the same result.",
    "latex": "(x, y) \\to (x + 4, -y)",
    "spoken_en": "x, y maps to x plus 4, negative y",
    "examples": [
      {
        "en": "Think of footprints in the sand: each step is flipped over and slid forward, and that's a glide reflection.",
        "ja": "砂浜の足跡を思い浮かべよう。1 歩ごとに裏返って前にずれている。これが映進だ。",
        "register": "spoken"
      },
      {
        "en": "Triangle ABC has vertices A(1, 2), B(3, 5), and C(4, 1). Find the vertices of its image under a glide reflection: a translation 5 units right followed by a reflection across the x-axis.",
        "ja": "△ABC の頂点は A(1, 2), B(3, 5), C(4, 1) である。△ABC を x 軸方向に 5 だけ平行移動し、さらに x 軸に関して対称移動した図形の頂点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の中学・高校の教科書では「映進」という名前を使わず、対称移動と平行移動の組み合わせとして扱う。",
      "平行移動の向きが対称の軸と平行でなければ glide reflection とは呼ばない。",
      "用例コーパスには出てこない。CK-12 Geometry では 8.17 Composite Transformations に 8 件、IM では Geometry 6.4 Distances and Circles に 2 件。"
    ],
    "related": [
      "reflection",
      "translation",
      "rigid-motion",
      "composition-of-transformations"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.17 Composite Transformations"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 glide reflection（CK-12 Geometry 8.17 Composite Transformations・計 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "goodness-of-fit-test",
    "ja": {
      "term": "適合度検定",
      "reading": "てきごうどけんてい",
      "alt": []
    },
    "en": {
      "term": "goodness-of-fit test",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "1 つのカテゴリ変数の度数が、仮定した分布（割合）に合っているかどうかを χ² で検定する方法。",
    "definition_en": "A chi-square test of whether the counts of one categorical variable fit a hypothesized distribution.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A goodness-of-fit test asks: does the data match the distribution we claimed?",
        "ja": "適合度検定は「データが主張した分布に合っているか」を問う。",
        "register": "spoken"
      },
      {
        "en": "Conduct a goodness-of-fit test to see whether the colors are equally likely.",
        "ja": "色の出方が同様に確からしいか、適合度検定を行え。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics（節の名前 Goodness-of-Fit Test）。AP Statistics の CED（2026 年版）には出てこない（CED のカイ二乗検定は同質性と独立性だけ）。",
      "chi-square goodness-of-fit test とも言う。見出しを中に含むので候補には置かなかった。"
    ],
    "related": [
      "chi-square-test",
      "expected-count",
      "observed-count"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "goodness-of-fit test": 29
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "gradient",
    "ja": {
      "term": "勾配",
      "reading": "こうばい",
      "alt": [
        "グラディエント"
      ]
    },
    "en": {
      "term": "gradient",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "多変数関数の偏導関数を並べたベクトル ∇f = (f_x, f_y)。関数が最も速く増える向きを指す。",
    "definition_en": "The vector ∇f = (f_x, f_y) of partial derivatives, which points in the direction in which the function increases fastest.",
    "latex": "\\nabla f = \\langle f_x, f_y \\rangle",
    "spoken_en": "del f equals the vector f sub x, f sub y",
    "examples": [
      {
        "en": "The gradient is always perpendicular to the level curves.",
        "ja": "勾配はいつも等高線に垂直。",
        "register": "spoken"
      },
      {
        "en": "Find the gradient of f(x, y, z) = xy + yz² at the point (1, −1, 2).",
        "ja": "f(x, y, z) = xy + yz² の点 (1, −1, 2) における勾配を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "∇f は del f または the gradient of f と読む（symbols の nabla を参照）。",
      "日本語の「勾配」は直線の傾き（slope）の意味でも使うが、英語の gradient は多変数の勾配ベクトルを指す。"
    ],
    "related": [
      "directional-derivative",
      "partial-derivative",
      "level-curve",
      "conservative-vector-field",
      "potential-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "gradient": 246
      },
      "written": {
        "gradient": 147
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "gram-schmidt-process",
    "ja": {
      "term": "グラム・シュミットの直交化法",
      "reading": "ぐらむ・しゅみっとのちょっこうかほう",
      "alt": [
        "グラム・シュミットの直交化"
      ]
    },
    "en": {
      "term": "Gram-Schmidt algorithm",
      "alt": [
        "Gram-Schmidt process",
        "Gram-Schmidt orthogonalization"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Gram-Schmidt algorithm ／ Gram-Schmidt process ／ Gram-Schmidt orthogonalization は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "1 次独立なベクトルの組から、同じ空間を張る直交（正規直交）なベクトルの組を順に作る方法。前のベクトルへの正射影を引いていく。",
    "definition_en": "A procedure that turns a linearly independent set of vectors into an orthogonal (or orthonormal) set spanning the same space, by subtracting projections onto the earlier vectors one at a time.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The Gram-Schmidt algorithm: take the next vector and subtract off its projections onto the ones you already have.",
        "ja": "グラム・シュミット法では、次のベクトルから、もうあるベクトルへの射影を引いていく。",
        "register": "spoken"
      },
      {
        "en": "Apply the Gram-Schmidt process to the vectors to obtain an orthonormal basis.",
        "ja": "ベクトルにグラム・シュミットの直交化法を適用し、正規直交基底を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Nicholson は Gram-Schmidt algorithm（16 件）と Gram-Schmidt process（8 件）の両方を書く。用例コーパスでは Gram-Schmidt は話し言葉にわずかに出てくるだけ。"
    ],
    "related": [
      "orthonormal-basis",
      "orthogonal-projection",
      "qr-factorization"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.1 Orthogonal Complements and Projections"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 Gram-Schmidt algorithm（10.2 Orthogonal Sets of Vectors・8.1 Orthogonal Complements and Projections・8.2 Orthogonal Diagonalization）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "graph-coloring",
    "ja": {
      "term": "グラフ彩色",
      "reading": "ぐらふさいしょく",
      "alt": [
        "彩色"
      ]
    },
    "en": {
      "term": "vertex coloring",
      "alt": [
        "graph coloring"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（vertex coloring ／ graph coloring は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフの各頂点に色を割り当て、辺で結ばれた 2 頂点が必ず異なる色になるようにすること。必要な色の最小の個数を彩色数という。",
    "definition_en": "An assignment of colors to the vertices of a graph so that any two vertices joined by an edge get different colors. The smallest number of colors needed is the chromatic number.",
    "latex": "\\chi(G)",
    "spoken_en": "chi of G",
    "examples": [
      {
        "en": "In a vertex coloring, neighbors can't share a color, so a triangle already needs three colors.",
        "ja": "彩色では隣り合う頂点に同じ色を使えないので、三角形だけでもう 3 色必要になる。",
        "register": "spoken"
      },
      {
        "en": "Find a vertex coloring of the graph below that uses as few colors as possible, and state the chromatic number of the graph.",
        "ja": "下のグラフを、できるだけ少ない色の数で彩色せよ。また、このグラフの彩色数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "見出しの vertex coloring は Levin の本の呼び方（10 件、2.5 Coloring 7・2.1 Problems and Definitions 3）。graph coloring は Levin で 5 件。用例コーパスでは話し言葉（MIT OCW）に出てくるが、書き言葉にはほとんど出てこない。",
      "辺に色を付ける edge coloring（辺彩色）は別の問題。彩色数は chromatic number で、χ(G)（chi of G）と書く。",
      "平面グラフはどれも 4 色以内で彩色できる（四色定理、four color theorem）。"
    ],
    "related": [
      "bipartite-graph",
      "planar-graph",
      "graph-network",
      "vertex-graph"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "グラフ彩色",
        "en": "Graph coloring"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.5 Coloring・2.1 Problems and Definitions"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "graph coloring": 7
      },
      "written": {
        "graph coloring": 1
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは Levin の呼び方 vertex coloring（2.5 Coloring・2.1 Problems and Definitions）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "graph-network",
    "ja": {
      "term": "グラフ（離散数学）",
      "reading": "ぐらふ",
      "alt": [
        "離散グラフ",
        "ネットワーク"
      ]
    },
    "en": {
      "term": "graph",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "いくつかの点（頂点）と、それらを結ぶ線（辺）の組で、つながり方だけを表す図形。",
    "definition_en": "A set of points (vertices) together with lines (edges) joining some pairs of them; only how they are connected matters.",
    "latex": "G = (V, E)",
    "spoken_en": null,
    "examples": [
      {
        "en": "In graph theory, a graph is just dots and the connections between them; the picture's shape doesn't matter.",
        "ja": "グラフ理論のグラフは点とそのつながりだけで、絵の形は関係ない。",
        "register": "spoken"
      },
      {
        "en": "Draw a connected graph with 5 vertices and 4 edges.",
        "ja": "頂点が 5 個、辺が 4 本の連結なグラフをかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "関数のグラフ（graph of a function）と同じ語。",
      "台帳の日本語「離散グラフ」は教科書の語ではない。数学 C は単に「グラフ」と呼ぶ。"
    ],
    "related": [
      "vertex-graph",
      "edge-of-a-graph",
      "path-in-a-graph",
      "tree",
      "adjacency-matrix"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.1 Problems and Definitions"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "simple graph | connected graph | directed graph | complete graph | bipartite graph | planar graph | graph with … vertices | graph on … vertices": 50
      },
      "written": {
        "simple graph | connected graph | directed graph | complete graph | bipartite graph | planar graph | graph with … vertices | graph on … vertices": 196
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "graph",
    "ja": {
      "term": "グラフ",
      "reading": "ぐらふ",
      "alt": []
    },
    "en": {
      "term": "graph",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "x と y の関係を満たす点 (x, y) を、座標平面上に集めてできる図。関数で y が x につれてどう変わるかが、目で見て分かる。",
    "definition_en": "All the points (x, y) of the coordinate plane whose coordinates satisfy an equation or a function, drawn as one picture. It shows how y changes with x.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The graph of y = 2x is a straight line through the origin: every time you go right 1, it goes up 2.",
        "ja": "y = 2x のグラフは原点を通る直線で、右に 1 進むごとに 2 上がる。",
        "register": "spoken"
      },
      {
        "en": "The graph of y = a/x passes through the point (2, 6). Find the value of a.",
        "ja": "y = a/x のグラフが点 (2, 6) を通るとき、a の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「y = 2x のグラフ」は the graph of y = 2x。前置詞は of を使う。",
      "同じ graph が動詞で「グラフをかく」の意味になる（Graph y = 2x − 1.）。エントリ draw-a-graph。",
      "離散数学の graph（頂点と辺でできた図、エントリ graph-network）も同じ語で、別の概念。"
    ],
    "related": [
      "function",
      "coordinate-plane",
      "draw-a-graph",
      "sketch-the-graph",
      "graph-network"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "graph of | the graph !theory": 992
      },
      "written": {
        "graph of | the graph !theory": 3399
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "greater-than-or-equal-to",
    "ja": {
      "term": "〜以上",
      "reading": "いじょう",
      "alt": []
    },
    "en": {
      "term": "greater than or equal to",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある数が別の数より大きいか、または等しいこと。「x は 5 以上」は x ≧ 5 と書き、5 を含む。",
    "definition_en": "Either greater than or equal to another number. The symbol is ≥, so x ≥ 5 includes 5 itself.",
    "latex": "a \\geq b",
    "spoken_en": "a is greater than or equal to b",
    "examples": [
      {
        "en": "See the line under the sign? That means greater than or equal to, so 3 itself is included.",
        "ja": "記号の下に線があるよね。これは「以上」の意味だから、3 そのものも含まれる。",
        "register": "spoken"
      },
      {
        "en": "Find the values of x for which 2x − 1 is greater than or equal to 7.",
        "ja": "2x − 1 が 7 以上となる x の値の範囲を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "記号は日本では ≧（共通テスト・センター試験の問題文、学習指導要領解説）、米国の教科書（OpenStax）では ≥。",
      "文章題では「5 以上」を at least 5 と言う。at least は「少なくとも」の意味の日常語でもある。",
      "「以上」は境の数を含む。含まない「〜より大きい」「〜を超える」は greater than（エントリ greater-than）。"
    ],
    "related": [
      "greater-than",
      "less-than-or-equal-to",
      "linear-inequality",
      "properties-of-inequalities",
      "at-least-one"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "問題文の不等号 ≧"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "不等号 ≦ ≧"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "greater than or equal to": 374
      },
      "written": {
        "greater than or equal to": 114
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "greater-than",
    "ja": {
      "term": "〜より大きい",
      "reading": "よりおおきい",
      "alt": [
        "大きい",
        "〜を超える"
      ]
    },
    "en": {
      "term": "greater than",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある数が別の数よりも大きいこと。a が b より大きいことを a > b と書き、等しい場合は含まない。",
    "definition_en": "Larger in value than another number; on a number line, the greater number is farther to the right. The symbol is >, as in 7 > 4.",
    "latex": "a > b",
    "spoken_en": "a is greater than b",
    "examples": [
      {
        "en": "Negative 2 is greater than negative 5 because it's farther to the right on the number line.",
        "ja": "−2 は −5 より大きい。数直線で右にあるからね。",
        "register": "spoken"
      },
      {
        "en": "List all integers that are greater than −3 and less than 2.",
        "ja": "−3 より大きく 2 より小さい整数を、すべて答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「5 more than x」は x + 5 で、大小の比較ではなく足し算。「x は 5 より大きい」は x is greater than 5（x > 5）。",
      "「〜より大きい」「〜を超える」は境の数を含まない。境を含む「〜以上」は greater than or equal to（エントリ greater-than-or-equal-to）。",
      "不等号は左から読む。3 < x は 3 is less than x と読み、x > 3（x is greater than 3）と同じ意味。"
    ],
    "related": [
      "less-than",
      "greater-than-or-equal-to",
      "number-line",
      "linear-inequality",
      "compare"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "greater than !or": 1019
      },
      "written": {
        "greater than !or": 353
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "greatest-common-divisor",
    "ja": {
      "term": "最大公約数",
      "reading": "さいだいこうやくすう",
      "alt": []
    },
    "en": {
      "term": "greatest common divisor",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "GCF",
          "register": "both",
          "note": "greatest common factor の略。話し言葉では大半が The Organic Chemistry Tutor。書き言葉のほうが多く、すべて OpenStax（Elementary Algebra・Algebra and Trigonometry・Prealgebra ほか）。"
        },
        {
          "term": "greatest common factor",
          "register": "both",
          "note": "代数の言い方。話し言葉では Khan Academy の中学の講義が最も多く、The Organic Chemistry Tutor・Professor Leonard・patrickJMT が続く。書き言葉はすべて OpenStax（Elementary Algebra・Prealgebra・Algebra and Trigonometry ほか）。"
        },
        {
          "term": "GCD",
          "register": "both",
          "note": "greatest common divisor の略。話し言葉はほとんどが MIT OCW、書き言葉はすべて MIT の講義ノート。"
        },
        {
          "term": "greatest common divisor",
          "register": "both",
          "note": "数論の言い方。話し言葉は MIT OCW が中心で、Khan Academy の中学の講義にも出てくる。書き言葉はすべて MIT の講義ノート。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語には divisor 系（greatest common divisor ／ GCD）と factor 系（greatest common factor ／ GCF）の 2 系統がある。用例コーパスでは divisor 系は MIT（講義と講義ノート）、factor 系は OpenStax（Elementary Algebra・Prealgebra・Algebra and Trigonometry ほか）と The Organic Chemistry Tutor などに出る。Levin の 6.2 Introduction to Number Theory は GCD と greatest common divisor が中心で、greatest common factor も使う。factor 系は単項式・多項式の共通因数にも使い、factor out the GCF は「共通因数でくくり出す」に当たる。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "小学校",
        "数A"
      ],
      "us": [
        "Discrete Math",
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "2 つ以上の整数に共通する約数のうち、最も大きいもの。",
    "definition_en": "The largest integer that divides each of two or more given integers.",
    "latex": "\\gcd(a, b)",
    "spoken_en": "the GCD of a and b",
    "examples": [
      {
        "en": "The greatest common divisor of 12 and 18 is 6, since 6 is the biggest number that goes into both.",
        "ja": "12 と 18 の最大公約数は 6。両方を割り切る最大の数が 6 だからね。",
        "register": "spoken"
      },
      {
        "en": "Before you do anything else, pull out the GCF, which here is 4x.",
        "ja": "まずは共通因数をくくり出そう。ここでは 4x だ。",
        "register": "spoken"
      },
      {
        "en": "Find gcd(84, 120) using prime factorization.",
        "ja": "素因数分解を用いて 84 と 120 の最大公約数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "GCD（greatest common divisor）は大学の数論（Levin 6.2）、GCF（greatest common factor）は学校の数学（IM Grade 6 は整数の最大公約数、OpenStax は多項式の因数分解の共通因数）で使う。",
      "共通テスト・センター試験の問題文は最大公約数を言葉で書き、記号 gcd は出てこない（日本語版 Wikipedia「最大公約数」は gcd(a, b) と書く）。英語版 Wikipedia「Greatest common divisor」は gcd(a, b) と書き、(a, b) を使う著者もあると注記する。"
    ],
    "related": [
      "least-common-multiple",
      "relatively-prime",
      "euclidean-algorithm",
      "common-factor",
      "prime-factorization",
      "divisor"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "最大公約数",
        "en": "Greatest common divisor"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2 Introduction to Number Theory（GCD、greatest common divisor）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「最大公約数」",
        "url": "https://ja.wikipedia.org/wiki/%E6%9C%80%E5%A4%A7%E5%85%AC%E7%B4%84%E6%95%B0",
        "note": "gcd(a, b) の記号"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Greatest_common_divisor",
        "note": "記事「Greatest common divisor」の節 Definition（gcd(a, b)、(a, b) と書く著者もあるという注）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A 数と式（素因数分解）: 小学校算数科で最大公約数・最小公倍数を学習"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 7.16 Common Factors（greatest common factor）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成31年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20200601/https://www.dnc.ac.jp/albums/abm.php?f=abm00036453.pdf&n=k+19+suugaku1A.pdf",
        "note": "a と a+1 の最大公約数は 1（最大公約数を言葉で書く）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Greatest Common Factor and Factor by Grouping（GCF）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "2.5 Prime Factorization and the Least Common Multiple・Introduction to Factoring Polynomials"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "greatest common divisor": 94,
        "GCD": 30,
        "greatest common factor": 86,
        "GCF": 39
      },
      "written": {
        "GCF": 217,
        "greatest common factor": 127,
        "greatest common divisor": 22,
        "GCD": 80
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "greens-theorem",
    "ja": {
      "term": "グリーンの定理",
      "reading": "ぐりーんのていり",
      "alt": []
    },
    "en": {
      "term": "Green's theorem",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "平面の閉曲線 C に沿った線積分を、C で囲まれた領域 D 上の二重積分で表す定理。",
    "definition_en": "The theorem that turns a line integral around a closed curve C in the plane into a double integral over the region D it encloses.",
    "latex": "\\oint_C P\\,dx + Q\\,dy = \\iint_D \\left( \\frac{\\partial Q}{\\partial x} - \\frac{\\partial P}{\\partial y} \\right) dA",
    "spoken_en": null,
    "examples": [
      {
        "en": "Green's theorem lets you trade a hard line integral for an easier double integral.",
        "ja": "グリーンの定理を使えば、難しい線積分を易しい二重積分に取りかえられる。",
        "register": "spoken"
      },
      {
        "en": "Use Green's theorem to evaluate ∮_C (x² − y) dx + (x + y²) dy, where C is the circle x² + y² = 4 oriented counterclockwise.",
        "ja": "グリーンの定理を用いて、C が反時計回りの円 x² + y² = 4 のときの ∮_C (x² − y) dx + (x + y²) dy を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "曲線の向きは counterclockwise（正の向き）を仮定する。"
    ],
    "related": [
      "line-integral",
      "double-integral",
      "stokes-theorem",
      "curl-vector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "グリーンの定理",
        "en": "Green's theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "Green's theorem": 55
      },
      "written": {
        "Green's theorem": 166
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "grouped-sequence",
    "ja": {
      "term": "群数列",
      "reading": "ぐんすうれつ",
      "alt": []
    },
    "en": {
      "term": "grouped sequence",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。日本語版 Wikipedia「数学 (教科)」は数学B の数列の項目に群数列を挙げる（学習指導要領解説に群数列の語は出てこない。項をいくつかごとに群に区切る考えは、数学III の課題学習〈積分法〉の例に、調和級数の発散を示す方法として出てくる）。英語で説明するなら group the terms like this: (1), (2, 3), (4, 5, 6), … と式で見せる。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": []
    },
    "definition_ja": "数列の項を、ある規則で区切った組（群）に分けて考えるもの。第何群の何番目かで項を特定する。",
    "definition_en": "A sequence whose terms are split into groups by a rule, so that a term is located by its group and its place in the group.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Split the terms into groups: one term, then two, then three, and so on.",
        "ja": "項を 1 個、2 個、3 個、… と群に分けよう。",
        "register": "spoken"
      },
      {
        "en": "In the grouped sequence 1 | 2, 3 | 4, 5, 6 | …, find the first term of the 10th group.",
        "ja": "群数列 1 | 2, 3 | 4, 5, 6 | … で、第 10 群の最初の項を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "sequence",
      "summation-formulas"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学B の数列の項目（群数列）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B (1) 数列: 群数列は出てこない／数学III 課題学習〈積分法〉: 調和級数の項を群に区切り、群ごとに和を取る"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "grow-exponentially",
    "ja": {
      "term": "指数関数的に増加する",
      "reading": "しすうかんすうてきにぞうかする",
      "alt": []
    },
    "en": {
      "term": "grow exponentially",
      "alt": [],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1",
        "Integrated Math 3",
        "Algebra 1",
        "Calculus I"
      ]
    },
    "definition_ja": "量が一定の時間ごとに一定の倍率で増える。",
    "definition_en": "To increase by the same factor over each equal time interval.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The bacteria grow exponentially, so the count doubles every hour.",
        "ja": "細菌は指数関数的に増えるので、1 時間ごとに 2 倍になる。",
        "register": "spoken"
      },
      {
        "en": "A population grows exponentially from 500 to 2000 in 6 years. Find the population after 9 years.",
        "ja": "ある個体数が 6 年で 500 から 2000 に指数関数的に増加した。9 年後の個体数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "減るほうは decay exponentially（指数関数的に減少する）。"
    ],
    "related": [
      "exponential-growth",
      "exponential-decay",
      "exponential-model"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "6.8 Exponential Growth and Decay（grows exponentially）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 4.6 Writing Equations for Exponential Functions・Algebra 2 4.13 Exponential Functions with Base $e$・Algebra 1 5.13 Reasoning about Exponential Graphs (Part 2)"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "grow exponentially": 7
      },
      "written": {
        "grow exponentially": 8
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは IM の呼び方 grow exponentially（Algebra 2 4.6 Writing Equations for Exponential Functions・Algebra 2 4.13 Exponential Functions with Base $e$・Algebra 1 5.13 Reasoning about Exponential Graphs (Part 2)・計 15 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "growth-rate",
    "ja": {
      "term": "成長率",
      "reading": "せいちょうりつ",
      "alt": [
        "増加率"
      ]
    },
    "en": {
      "term": "growth rate",
      "alt": [],
      "variants": [
        {
          "term": "rate of growth",
          "register": "both",
          "note": "話し言葉ではこちらが首位（3Blue1Brown・MIT OCW・Khan Academy）。書き言葉でも growth rate に次いで使う。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "量が一定の時間ごとに、もとの量の何倍分ずつ増えるかを表す割合。指数関数的な増加 y = a(1 + r)ᵗ の r のことで、年 3 % の増加なら r = 0.03。",
    "definition_en": "The percent by which a quantity increases in each time period, written as a decimal. In y = a(1 + r)^t, the growth rate is r.",
    "latex": "y = a(1 + r)^{t}",
    "spoken_en": "y equals a times the quantity one plus r, to the t",
    "examples": [
      {
        "en": "The population goes up 4% a year, so the growth rate is 0.04, and the growth factor is 1.04.",
        "ja": "人口は年に 4 % ずつ増えるから、成長率は 0.04 で、増加の倍率は 1.04 だ。",
        "register": "spoken"
      },
      {
        "en": "A town has 12,000 residents, and its population increases by 3% each year. Write a function for the population after t years and state the growth rate.",
        "ja": "ある町の人口は 12,000 人で、毎年 3 % ずつ増える。t 年後の人口を表す関数を書き、成長率を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "growth rate は r、growth factor は 1 + r（1 期間ごとに何倍になるか）で、別の量。年 3 % なら growth rate は 0.03、growth factor は 1.03。",
      "式に入れるときは百分率を小数に直す。3 % の増加で r = 3 とはしない。",
      "微積分やロジスティック成長の P′ = rP の r も growth rate と呼ぶ（エントリ exponential-growth、logistic-growth）。"
    ],
    "related": [
      "exponential-growth",
      "exponential-decay",
      "exponential-function",
      "compound-interest",
      "percent-change"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "growth rate": 7,
        "rate of growth": 17
      },
      "written": {
        "growth rate": 72,
        "rate of growth": 32
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "half-angle-formulas",
    "ja": {
      "term": "半角の公式",
      "reading": "はんかくのこうしき",
      "alt": []
    },
    "en": {
      "term": "half-angle formulas",
      "alt": [
        "half-angle identities"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "sin(α/2)、cos(α/2)、tan(α/2) を cos α で表す公式（2 乗の形 sin²(α/2) = (1 − cos α)/2 でも使う）。2 倍角の公式から得られる。",
    "definition_en": "The formulas expressing sin(α/2), cos(α/2), and tan(α/2) in terms of cos α, such as sin(α/2) = ±√((1 − cos α)/2) or, squared, sin²(α/2) = (1 − cos α)/2; they come from the double-angle formulas.",
    "latex": "\\sin^2\\frac{\\alpha}{2} = \\frac{1 - \\cos\\alpha}{2}",
    "spoken_en": "sine squared of alpha over two equals one minus cosine alpha over two",
    "examples": [
      {
        "en": "To get sin 22.5 degrees, use the half-angle formula with 45 degrees.",
        "ja": "sin 22.5° は、45° で半角の公式を使えば出る。",
        "register": "spoken"
      },
      {
        "en": "Find the exact value of cos(π/8) with a half-angle formula.",
        "ja": "半角の公式を用いて cos(π/8) の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は half-angle formulas を sin(α/2) = ±√((1 − cos α)/2) と平方根の形で書き（± は α/2 の象限で決める）、2 乗の形 sin²θ = (1 − cos 2θ)/2 は reduction formulas（power-reducing formulas）として別に挙げる。日本語版 Wikipedia「三角関数の公式の一覧」も半角の公式を平方根の形で載せる。講義では、積分で次数を下げる 2 乗の形も half angle formula と呼ぶ。"
    ],
    "related": [
      "double-angle-formulas",
      "angle-addition-formulas"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Double-Angle, Half-Angle, and Reduction Formulas"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角関数の公式の一覧」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0%E3%81%AE%E5%85%AC%E5%BC%8F%E3%81%AE%E4%B8%80%E8%A6%A7",
        "note": "節「倍角・三倍角・半角の公式」（平方根の形）、節「べき乗」"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "half-angle formulas": 28,
        "half-angle identities": 6
      },
      "written": {
        "half-angle formulas": 25,
        "half-angle identities": 4
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "half-life",
    "ja": {
      "term": "半減期",
      "reading": "はんげんき",
      "alt": []
    },
    "en": {
      "term": "half-life",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Calculus I"
      ]
    },
    "definition_ja": "ある量が、もとの量の半分に減るまでにかかる時間。最初の量 A₀、半減期 h のとき、t 時間後の量は A₀(1/2)^{t/h}。",
    "definition_en": "The time it takes for an amount to decrease to half of what it was. If the starting amount is A₀ and the half-life is h, the amount left after time t is A₀(1/2)^(t/h).",
    "latex": "A = A_0\\left(\\frac{1}{2}\\right)^{t/h}",
    "spoken_en": "A equals A naught times one half, to the t over h",
    "examples": [
      {
        "en": "The half-life is 5 hours, so after 10 hours you've got a quarter of it left.",
        "ja": "半減期が 5 時間だから、10 時間たつと 4 分の 1 が残る。",
        "register": "spoken"
      },
      {
        "en": "A 200-milligram sample of a substance has a half-life of 8 days. How much of the sample remains after 24 days?",
        "ja": "半減期が 8 日の物質の試料が 200 mg ある。24 日後に残っている量を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語はハイフンを入れて half-life と書く。複数形は half-lives。",
      "半減期の 2 倍の時間がたつと、0 ではなく 4 分の 1 が残る。減り方は一定の量ずつではなく、一定の割合ずつ（指数関数的減少、エントリ exponential-decay）。",
      "half-life は時間の長さで、残る量ではない。"
    ],
    "related": [
      "exponential-decay",
      "exponential-function",
      "exponential-growth",
      "growth-rate",
      "logarithm"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学特論 (4) 数学と生活や社会との関わり: 放射性物質の半減期に関する数学的モデル"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "half-life": 66
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "hamiltonian-path",
    "ja": {
      "term": "ハミルトン路",
      "reading": "はみるとんろ",
      "alt": [
        "ハミルトン経路"
      ]
    },
    "en": {
      "term": "Hamiltonian path",
      "alt": [
        "Hamilton path"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフのすべての頂点をちょうど 1 回ずつ通る経路のこと。始点に戻るものはハミルトン閉路という。",
    "definition_en": "A path in a graph that visits every vertex exactly once. If it also returns to its starting vertex, it is called a Hamiltonian cycle.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "An Euler trail uses every edge once, but a Hamiltonian path visits every vertex once, and that's a much harder thing to check.",
        "ja": "一筆書きはすべての辺を 1 回ずつ通るが、ハミルトン路はすべての頂点を 1 回ずつ通る。こちらのほうがずっと判定が難しい。",
        "register": "spoken"
      },
      {
        "en": "Does the graph below have a Hamiltonian path? If so, list the vertices in order. If not, explain why.",
        "ja": "下のグラフにハミルトン路はあるか。あれば通る頂点を順に書き、なければその理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "一筆書き（Euler trail）は辺を、ハミルトン路は頂点を 1 回ずつ通る。取り違えやすい。",
      "一筆書きには頂点の次数で判定できる簡単な条件があるが、ハミルトン路があるかどうかを判定する簡単な条件は知られていない。",
      "人名なので大文字の H で書く。用例コーパスでは Hamiltonian path が話し言葉（MIT OCW）・書き言葉（MIT の講義ノート）とも使われる。Levin の本は Hamilton path と書く（18 件。Hamiltonian path は 1 件、2.4 Euler Trails and Circuits）。"
    ],
    "related": [
      "eulerian-path",
      "path-in-a-graph",
      "vertex-graph",
      "graph-network"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ハミルトン路",
        "en": "Hamiltonian path"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "Hamiltonian path": 16
      },
      "written": {
        "Hamiltonian path": 12
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "handshaking-lemma",
    "ja": {
      "term": "握手補題",
      "reading": "あくしゅほだい",
      "alt": []
    },
    "en": {
      "term": "handshaking lemma",
      "alt": [
        "handshake lemma"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（handshaking lemma ／ handshake lemma は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフのすべての頂点の次数の和は、辺の本数の 2 倍に等しいという定理。とくに、次数が奇数の頂点は偶数個ある。",
    "definition_en": "The theorem that the degrees of all the vertices of a graph add up to twice the number of edges. In particular, every graph has an even number of vertices of odd degree.",
    "latex": "\\sum_{v \\in V} \\deg(v) = 2|E|",
    "spoken_en": "the sum over all vertices v of the degree of v equals two times the number of edges",
    "examples": [
      {
        "en": "By the handshaking lemma, the degrees have to add up to an even number, so there's no graph with degrees 3, 3, and 3.",
        "ja": "握手補題より次数の和は偶数になるはずだから、次数が 3, 3, 3 のグラフは存在しない。",
        "register": "spoken"
      },
      {
        "en": "Use the handshaking lemma to show that every graph has an even number of vertices of odd degree.",
        "ja": "握手補題を用いて、どのグラフでも次数が奇数の頂点は偶数個であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "名前はパーティーの握手のたとえから来ている。全員の握手の回数を足すと、1 回の握手を 2 人が 1 回ずつ数えるので、合計は必ず偶数になる。",
      "1 本の辺は両端の 2 頂点の次数に 1 ずつ数えられる、というのが証明の中心。日本語では定理の名前を出さずに「次数の和は辺の数の 2 倍」と書くこともある。",
      "用例コーパスでは書き言葉（すべて MIT の講義ノート）にわずかに出てくるだけで、話し言葉には出てこない。Levin の本では handshaking lemma ／ handshake lemma を合わせて 6 件（2.1 Problems and Definitions 5・1.5 Proofs about Discrete Structures 1）。"
    ],
    "related": [
      "degree-vertex",
      "vertex-graph",
      "graph-network",
      "edge-of-a-graph"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.1 Problems and Definitions・1.5 Proofs about Discrete Structures"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "handshaking lemma": 5
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは Levin の呼び方 handshaking lemma（2.1 Problems and Definitions・1.5 Proofs about Discrete Structures）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "harmonic-series",
    "ja": {
      "term": "調和級数",
      "reading": "ちょうわきゅうすう",
      "alt": []
    },
    "en": {
      "term": "harmonic series",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "級数 1 + 1/2 + 1/3 + ⋯ = Σ1/n。項は 0 に近づくが、和は発散する。",
    "definition_en": "The series 1 + 1/2 + 1/3 + ⋯ = Σ1/n, which diverges even though its terms approach 0.",
    "latex": "\\sum_{n=1}^{\\infty} \\frac{1}{n}",
    "spoken_en": "the sum from n equals one to infinity of one over n",
    "examples": [
      {
        "en": "The harmonic series is the classic example: terms go to zero, but it still diverges.",
        "ja": "調和級数が典型例。項は 0 に近づくのに発散する。",
        "register": "spoken"
      },
      {
        "en": "Show that the harmonic series diverges by grouping its terms.",
        "ja": "項をまとめることで、調和級数が発散することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "符号を交互にした alternating harmonic series（交代調和級数）は収束する（条件収束）。"
    ],
    "related": [
      "p-series",
      "nth-term-test",
      "alternating-series",
      "conditional-convergence"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "調和級数",
        "en": "Harmonic series (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.5（harmonic series）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "harmonic series": 35
      },
      "written": {
        "harmonic series": 69
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "has-a-local-extremum",
    "ja": {
      "term": "極値をもつ",
      "reading": "きょくちをもつ",
      "alt": []
    },
    "en": {
      "term": "has a local extremum",
      "alt": [],
      "variants": [
        {
          "term": "has a relative extremum",
          "register": "spoken",
          "note": "Khan Academy の言い方で、件数は少ない。話し言葉だけ（すべて Khan Academy）で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数がどこかで極大または極小になる。",
    "definition_en": "Of a function: to have a local maximum or a local minimum somewhere.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If f prime never changes sign, f has no local extremum.",
        "ja": "f′ の符号が一度も変わらなければ、f は極値をもたない。",
        "register": "spoken"
      },
      {
        "en": "Find the values of a for which f(x) = x³ + ax has a local extremum.",
        "ja": "f(x) = x³ + ax が極値をもつような a の値の範囲を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「極値をもたない」は has no local extremum（has no local extrema）。"
    ],
    "related": [
      "local-extremum",
      "has-a-local-maximum-at",
      "critical-point"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "has a relative extremum": 3
      },
      "written": {
        "has a local extremum": 16
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "has-a-local-maximum-at",
    "ja": {
      "term": "極大値をとる",
      "reading": "きょくだいちをとる",
      "alt": []
    },
    "en": {
      "term": "has a local maximum at",
      "alt": [],
      "variants": [
        {
          "term": "has a relative maximum at",
          "register": "spoken",
          "note": "Khan Academy の言い方で、件数は少ない。話し言葉だけ（すべて Khan Academy）で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数が x = a で極大になる。",
    "definition_en": "Of a function: to reach a local maximum at x = a.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "So f has a relative maximum at x = −1.",
        "ja": "だから f は x = −1 で極大値をとる。",
        "register": "spoken"
      },
      {
        "en": "f has a local maximum at x = 2, and the local maximum value is 5.",
        "ja": "f は x = 2 で極大となり、極大値は 5 である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「x = 2 で極大値 5 をとる」は f has a local maximum value of 5 at x = 2 とも書く。"
    ],
    "related": [
      "local-maximum",
      "local-extremum"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "has a relative maximum at": 3
      },
      "written": {
        "has a local maximum at": 23
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "height",
    "ja": {
      "term": "高さ",
      "reading": "たかさ",
      "alt": []
    },
    "en": {
      "term": "height",
      "alt": [
        "altitude"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "図形の底辺や底面から、それに垂直な向きに測った長さ。三角形では頂点から底辺（またはその延長）に下ろした垂線の長さ、角柱・円柱では 2 つの底面の間の距離をいう。",
    "definition_en": "The distance from the base measured at right angles to it. For a triangle it is how far a vertex is from the opposite side (or its extension); for a prism or cylinder it is the distance between the two bases.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The height has to be perpendicular to the base, so it's not the slanted side.",
        "ja": "高さは底辺に垂直じゃないといけないから、斜めの辺の長さじゃないよ。",
        "register": "spoken"
      },
      {
        "en": "The area of a triangle is 24 cm² and its base is 8 cm. Find the height of the triangle.",
        "ja": "面積が 24 cm²、底辺が 8 cm の三角形の高さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "三角形の頂点から対辺に下ろした垂線（またはその長さ）は altitude とも言う。",
      "高さは底辺・底面に垂直に測る。斜めの辺の長さや、円錐の母線の長さ（slant height、エントリ slant-height）とは違う。",
      "鈍角三角形で鈍角をはさむ辺を底辺にすると、高さは三角形の外に出て、底辺の延長に垂線を下ろすことになる。"
    ],
    "related": [
      "perpendicular",
      "slant-height",
      "area-of-a-triangle",
      "area-of-the-base",
      "volume"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "height": 1540,
        "altitude": 29
      },
      "written": {
        "height": 1245,
        "altitude": 43
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "herons-formula",
    "ja": {
      "term": "ヘロンの公式",
      "reading": "へろんのこうしき",
      "alt": []
    },
    "en": {
      "term": "Heron's formula",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "三角形の 3 辺の長さ a, b, c だけから面積を求める公式。s = (a + b + c)/2 として S = √(s(s − a)(s − b)(s − c))。",
    "definition_en": "A formula for the area of a triangle from its three side lengths a, b, c alone: with s = (a + b + c)/2, the area is √(s(s − a)(s − b)(s − c)).",
    "latex": "S = \\sqrt{s(s-a)(s-b)(s-c)}, \\quad s = \\frac{a+b+c}{2}",
    "spoken_en": "the square root of s times s minus a times s minus b times s minus c, where s is half the perimeter",
    "examples": [
      {
        "en": "If all you know are the three sides, Heron's formula gets you the area without finding any angles.",
        "ja": "3 辺しかわかっていないなら、ヘロンの公式で角を求めずに面積が出せる。",
        "register": "spoken"
      },
      {
        "en": "Find the area of the triangle with sides 5, 6, and 7 by Heron's formula.",
        "ja": "3 辺の長さが 5, 6, 7 の三角形の面積を、ヘロンの公式を用いて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は s を周の長さの半分（one half of the perimeter）と説明し、sometimes called the semi-perimeter と書く（2 件。semiperimeter と続けて綴るのは英語版 Wikipedia）。"
    ],
    "related": [
      "area-of-a-triangle",
      "area-formula-with-sine",
      "law-of-cosines",
      "inscribed-circle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ヘロンの公式",
        "en": "Heron's formula"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6%20%28%E6%95%99%E7%A7%91%29",
        "note": "数学I 図形と計量: 三角形の面積・ヘロンの公式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Heron's formula": 1
      },
      "written": {
        "Heron's formula": 16
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "hexadecimal",
    "ja": {
      "term": "16 進法",
      "reading": "じゅうろくしんほう",
      "alt": [
        "十六進法",
        "16 進数"
      ]
    },
    "en": {
      "term": "hexadecimal",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（hexadecimal は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "16 を基数とする位取り記数法。0〜9 と A〜F（10〜15 を表す）の 16 個の数字を使う。",
    "definition_en": "The base-16 place-value system, which uses the sixteen digits 0–9 and A–F, where A through F stand for 10 through 15.",
    "latex": "\\mathrm{2F}_{(16)} = 2 \\cdot 16 + 15 = 47",
    "spoken_en": "two F in base sixteen is two times sixteen plus fifteen, which is forty-seven",
    "examples": [
      {
        "en": "In hexadecimal, once you run out of digits at 9, you keep going with A, B, C, D, E, F.",
        "ja": "16 進法では、9 の次は A, B, C, D, E, F と続ける。",
        "register": "spoken"
      },
      {
        "en": "Convert the decimal number 200 to hexadecimal.",
        "ja": "10 進法で表された数 200 を 16 進法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "センター試験（平成 28 年度 本試験 数学I・数学A 第 4 問）は 11011₍₂₎ のように括弧つきの添字で基数を書く。英語版 Wikipedia「Hexadecimal」は、数学では基数を括弧なしの添字で示し（2F₁₆ の形）、C などのプログラミングでは接頭辞 0x を付ける（0x2F）と書く。",
      "hexadecimal は名詞にも形容詞にも使う（in hexadecimal ／ a hexadecimal number）。「16 進数」も同じ語で言える。"
    ],
    "related": [
      "base-n",
      "binary",
      "numeral-system",
      "digit",
      "decimal-system"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.2 Combining Outcomes"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成28年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "note": "第4問: 11011(2) の括弧つきの添字"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Hexadecimal",
        "note": "記事「Hexadecimal」の冒頭と節 Identification（添字、接頭辞 0x。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「十六進法」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8D%81%E5%85%AD%E9%80%B2%E6%B3%95",
        "note": "十六進法（16 進数）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "hexadecimal": 26
      },
      "sources": [
        "khan-middle"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "higher-order-derivative",
    "ja": {
      "term": "高次導関数",
      "reading": "こうじどうかんすう",
      "alt": [
        "高階導関数"
      ]
    },
    "en": {
      "term": "higher-order derivative",
      "alt": [],
      "variants": [
        {
          "term": "higher derivative",
          "register": "spoken",
          "note": "大学（MIT OCW）の言い方で、件数は少ない。話し言葉はすべて MIT OCW、書き言葉はすべて MIT の講義ノート。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "導関数をさらに微分してできる第 2 次・第 3 次…の導関数をまとめた呼び方。",
    "definition_en": "A derivative of a derivative: the second, third, and later derivatives.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For higher derivatives, just keep differentiating.",
        "ja": "高次導関数は、微分を繰り返すだけ。",
        "register": "spoken"
      },
      {
        "en": "Find all higher-order derivatives of f(x) = x⁴.",
        "ja": "f(x) = x⁴ の高次導関数をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "second-derivative",
      "nth-derivative"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 3.6（higher-order derivatives）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "higher derivative": 6,
        "higher-order derivative": 1
      },
      "written": {
        "higher-order derivative": 19,
        "higher derivative": 5
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "histogram",
    "ja": {
      "term": "ヒストグラム",
      "reading": "ひすとぐらむ",
      "alt": [
        "柱状グラフ"
      ]
    },
    "en": {
      "term": "histogram",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "度数分布表を、階級の幅を横、度数を高さとする隣り合った長方形（柱）で表したグラフ。",
    "definition_en": "A graph of a frequency table made of adjacent bars whose widths are the class intervals and whose heights are the frequencies.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Notice the bars of a histogram touch, unlike a bar chart.",
        "ja": "棒グラフと違って、ヒストグラムの柱はくっついていることに注意。",
        "register": "spoken"
      },
      {
        "en": "Draw a histogram of the data and describe the shape of the distribution.",
        "ja": "データのヒストグラムをかき、分布の形を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "棒グラフはカテゴリ変数の度数を離れた棒で表し、ヒストグラムは量的変数の階級を隣り合った柱で表す。OpenStax Introductory Statistics は棒グラフを bar graph と書く（24 件）。"
    ],
    "related": [
      "frequency-table",
      "frequency-polygon",
      "class",
      "box-plot"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ヒストグラム",
        "en": "Histogram"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "histogram": 57
      },
      "written": {
        "histogram": 144
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "hl-congruence",
    "ja": {
      "term": "斜辺と他の 1 辺がそれぞれ等しい",
      "reading": "しゃへんとたのいっぺんがそれぞれひとしい",
      "alt": [
        "HL"
      ]
    },
    "en": {
      "term": "HL",
      "alt": [
        "hypotenuse-leg"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの直角三角形で、斜辺と他の 1 辺がそれぞれ等しければ合同であるという、直角三角形の合同条件。",
    "definition_en": "A congruence criterion for right triangles: if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, the triangles are congruent.",
    "latex": "\\angle C = \\angle F = 90^\\circ,\\ AB = DE,\\ AC = DF \\implies \\triangle ABC \\cong \\triangle DEF",
    "spoken_en": "if angle C and angle F are both ninety degrees, AB equals DE, and AC equals DF, then triangle ABC is congruent to triangle DEF",
    "examples": [
      {
        "en": "Both of these are right triangles, the hypotenuses match, and one pair of legs matches, so we can use HL.",
        "ja": "どちらも直角三角形で、斜辺が等しく、他の 1 辺も等しいから、斜辺と他の 1 辺がそれぞれ等しいという合同条件が使える。",
        "register": "spoken"
      },
      {
        "en": "In right triangles ABC and DEF, ∠C = ∠F = 90°, AB = DE, and AC = DF. Prove that △ABC ≅ △DEF.",
        "ja": "∠C = ∠F = 90° の直角三角形 ABC と DEF で、AB = DE, AC = DF のとき、△ABC ≡ △DEF であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "by HL",
        "ja": "斜辺と他の 1 辺がそれぞれ等しいから"
      }
    ],
    "pitfalls": [
      "日本の直角三角形の合同条件は 2 つある。斜辺と 1 つの鋭角がそれぞれ等しい場合は、英語では hypotenuse-angle（HA）と呼ぶ教科書もあり、AAS として示すこともできる。もう 1 つがこの HL。",
      "HL は 2 辺とその間にない角（SSA）の形だが、その角が直角のときに限って合同条件になる。答案では、2 つの三角形が直角三角形であることを先に書く。",
      "英語の leg は直角をはさむ辺（斜辺でない辺）で、日本語の「他の 1 辺」に当たる。用例コーパスにも IM にも出てこない。CK-12 Geometry は HL と呼ぶ（4.16 HL ほか。by HL ／ HL congruence などの形で 8 件、hypotenuse-leg は 4 件）。"
    ],
    "related": [
      "sss-congruence",
      "sas-congruence",
      "asa-congruence",
      "aas-congruence",
      "congruent"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.16 HL・4.12 Congruent Triangles"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 by HL | HL congruence | (HL) congruence | HL theorem | HL postulate | HL criterion（CK-12 Geometry 4.16 HL・CK-12 Geometry 4.12 Congruent Triangles・計 8 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "hold",
    "ja": {
      "term": "成り立つ",
      "reading": "なりたつ",
      "alt": []
    },
    "en": {
      "term": "hold",
      "alt": [
        "hold true"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "「成り立つ」に当たる動詞として hold（hold true）を使う（The equation holds when x = 3.）。同じことを is true（エントリ true）や works とも言う。「等号が成り立つ」は equality holds（エントリ equality-holds）。",
    "domains": [
      "algebra",
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "等式・不等式・性質などが、ある条件のもとで正しいこと。例えば、等式 2x + 1 = 7 は x = 3 のとき成り立つ。",
    "definition_en": "For an equation, inequality, or rule to be true under the given conditions. For example, the equation 2x + 1 = 7 holds when x = 3.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Does the equation hold when x = 3? Plug it in and see.",
        "ja": "x = 3 のとき、この等式は成り立つかな。代入して確かめてみよう。",
        "register": "spoken"
      },
      {
        "en": "Show that the equation 2(x + 3) = 2x + 6 holds for every value of x.",
        "ja": "等式 2(x + 3) = 2x + 6 が、x がどんな値でも成り立つことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "hold のふつうの意味は「持つ・つかむ」。主語が等式・不等式・性質・定理のときに「成り立つ」の意味になる（The inequality holds.）。",
      "「〜から成り立つ」（〜で構成される）は hold ではなく consist of ／ be made up of。",
      "「値が方程式を満たす」（satisfy、エントリ satisfy）と「方程式がその値で成り立つ」（hold）は主語が逆になる。x = 3 satisfies the equation. ／ The equation holds when x = 3."
    ],
    "related": [
      "true",
      "equality-holds",
      "satisfy",
      "identity",
      "equation"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "holds for | hold for | holds true | hold true | still holds | also holds | equation holds | inequality holds | equality holds": 54
      },
      "written": {
        "holds for | hold for | holds true | hold true | still holds | also holds | equation holds | inequality holds | equality holds": 93
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "homogeneous-system",
    "ja": {
      "term": "同次方程式系",
      "reading": "どうじほうていしきけい",
      "alt": [
        "斉次連立一次方程式",
        "同次連立一次方程式"
      ]
    },
    "en": {
      "term": "homogeneous system",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "定数項がすべて 0 の連立一次方程式 Ax = 0。つねに x = 0（自明な解）を解にもつ。",
    "definition_en": "A linear system whose constant terms are all zero, Ax = 0; it always has the trivial solution x = 0.",
    "latex": "A\\vec{x} = \\vec{0}",
    "spoken_en": null,
    "examples": [
      {
        "en": "A homogeneous system always has at least one solution: all zeros.",
        "ja": "同次方程式系には少なくとも 1 つ解がある。全部 0 の解だ。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the homogeneous system has a nontrivial solution.",
        "ja": "同次方程式系が自明でない解をもつか判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "homogeneous equation は微分方程式の同次方程式（MIT 18.03 の話し言葉）がほとんどで、この語の候補にしなかった。",
      "用例コーパスの話し言葉はすべて MIT 18.02 と 18.03 で、18.02 のほうが多い。Nicholson は homogeneous system（36 件）と書く。"
    ],
    "related": [
      "system-of-linear-equations",
      "null-space"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "1.3 Homogeneous Equations"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "homogeneous system": 21
      },
      "written": {
        "homogeneous system": 1
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "horizontal-asymptote",
    "ja": {
      "term": "水平漸近線",
      "reading": "すいへいぜんきんせん",
      "alt": []
    },
    "en": {
      "term": "horizontal asymptote",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "x → ∞ または x → −∞ のとき、グラフが限りなく近づく水平な直線 y = b。分数関数では、分子と分母の次数を比べて求める。",
    "definition_en": "A horizontal line y = b that a graph approaches as x → ∞ or x → −∞. For a rational function, you find it by comparing the degrees of the numerator and the denominator.",
    "latex": "\\lim_{x \\to \\infty} f(x) = b",
    "spoken_en": "the limit as x approaches infinity of f of x equals b",
    "examples": [
      {
        "en": "Same degree on top and bottom, so the horizontal asymptote is the ratio of the leading coefficients: y = 2/3.",
        "ja": "分子と分母の次数が同じだから、水平漸近線は最高次の係数の比で y = 2/3。",
        "register": "spoken"
      },
      {
        "en": "Find the horizontal asymptote of f(x) = (4x² − 1)/(2x² + 3x).",
        "ja": "f(x) = (4x² − 1)/(2x² + 3x) の水平漸近線を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "グラフは水平漸近線と交わることもある。交わらないのは垂直漸近線（エントリ vertical-asymptote）。",
      "分数関数では、分母の次数のほうが高ければ y = 0、等しければ最高次の係数の比。分子の次数が分母よりちょうど 1 高いときは斜め漸近線（エントリ slant-asymptote）になる。",
      "分数関数だけでなく指数関数にもある。y = 2ˣ の水平漸近線は y = 0。"
    ],
    "related": [
      "asymptote",
      "vertical-asymptote",
      "slant-asymptote",
      "rational-function",
      "limit-at-infinity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "horizontal asymptote": 143
      },
      "written": {
        "horizontal asymptote": 134
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "horizontal-line-test",
    "ja": {
      "term": "水平線テスト",
      "reading": "すいへいせんてすと",
      "alt": []
    },
    "en": {
      "term": "horizontal line test",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「水平線テスト」は学習指導要領解説に無い、本プロジェクトの訳語。日本の数III では「単調増加（減少）なら逆関数がある」のように説明し、判定法に名前を付けない。英語では one-to-one（エントリ one-to-one）かどうかをグラフで確かめる方法として horizontal line test の名前で呼ぶ。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "関数のグラフに水平な直線を引き、どの水平線ともグラフが 2 点以上で交わらなければ、その関数は 1 対 1 で逆関数をもつと判定する方法。",
    "definition_en": "A way to tell from a graph whether a function is one-to-one: if no horizontal line crosses the graph more than once, the function passes and has an inverse function.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "y = x² fails the horizontal line test, because the line y = 4 hits it at x = 2 and at x = −2.",
        "ja": "y = x² は水平線テストに通らない。直線 y = 4 と x = 2 と x = −2 の 2 点で交わるからね。",
        "register": "spoken"
      },
      {
        "en": "Use the horizontal line test to determine whether f(x) = x³ + 1 has an inverse function.",
        "ja": "水平な直線とグラフの交わり方を調べて、f(x) = x³ + 1 が逆関数をもつかどうかを判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「通る／通らない」は pass ／ fail the horizontal line test と言う。",
      "OpenStax Algebra and Trigonometry は、三角関数のグラフは horizontal line test に通らない（fail）と書く。逆関数を考えるには定義域を制限する（エントリ restricted-domain）。",
      "関数かどうかを判定する vertical line test（エントリ vertical-line-test）とは別の判定法。日本語の「水平線」は海の水平線（horizon）の意味にもなるが、ここでは水平な直線のこと。"
    ],
    "related": [
      "one-to-one",
      "inverse-function",
      "vertical-line-test",
      "restricted-domain",
      "find-the-inverse"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "horizontal line test": 9
      },
      "written": {
        "horizontal line test": 35
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "hydrostatic-force",
    "ja": {
      "term": "静水圧による力",
      "reading": "せいすいあつによるちから",
      "alt": []
    },
    "en": {
      "term": "hydrostatic force",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "日本語の見出し「静水圧による力」は学習指導要領解説に無い、本プロジェクトの訳語（静水圧は hydrostatic pressure で、板全体が受ける力とは別の量）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "静止した液体の圧力によって、水中の板などが受ける力の大きさ。深さによって圧力が変わるので、積分で求める。",
    "definition_en": "The total force that a fluid at rest exerts on a submerged surface; since pressure grows with depth, it is found by integrating.",
    "latex": "F = \\int_a^b \\rho g\\,h(y)\\,w(y)\\,dy",
    "spoken_en": "F equals the integral from a to b of rho g times h of y times w of y, dy",
    "examples": [
      {
        "en": "Pressure depends on depth, so chop the plate into horizontal strips and add up the hydrostatic force on each.",
        "ja": "圧力は深さで変わるから、板を横に細長く切って、それぞれにかかる力を足し合わせる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "pressure（圧力。単位面積あたり）と force（力）を区別する。「静水圧」だけだと hydrostatic pressure。",
      "高等学校学習指導要領解説（数学編 理数編）には出てこない。OpenStax Calculus Volume 1 は積分の応用の節（Physical Applications）で扱う。",
      "OpenStax Calculus Volume 1 の Physical Applications の節は、ポンド・フィートの例（水の重さは 1 立方フィートあたり約 62.4 ポンド）と、メートル・ニュートンの例の両方を載せる。"
    ],
    "related": [
      "center-of-mass",
      "definite-integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「静水圧による力」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "hydrostatic force": 10
      },
      "written": {
        "hydrostatic force": 8
      },
      "sources": [
        "openstax-calculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "hyperbola",
    "ja": {
      "term": "双曲線",
      "reading": "そうきょくせん",
      "alt": []
    },
    "en": {
      "term": "hyperbola",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "数C"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Precalculus"
      ]
    },
    "definition_ja": "反比例 y = a/x のグラフのように、2 つに分かれた曲線が組になった形の曲線。二次曲線としては、2 つの定点（焦点）からの距離の差が一定である点の集まり（日本語版 Wikipedia「双曲線」。学習指導要領解説（数学C）は双曲線を二次式で表される曲線として扱う）。",
    "definition_en": "A curve with two separate branches, like the graph of y = a/x. In Precalculus it is defined by two fixed points (the foci): a point is on the hyperbola when the difference of its distances to the foci is constant.",
    "latex": "\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1",
    "spoken_en": "x squared over a squared minus y squared over b squared equals one",
    "examples": [
      {
        "en": "The graph of y = 6/x is a hyperbola, so it comes in two separate pieces: one in Quadrant I and one in Quadrant III.",
        "ja": "y = 6/x のグラフは双曲線だから、第 1 象限と第 3 象限の 2 つの部分に分かれている。",
        "register": "spoken"
      },
      {
        "en": "Find the vertices and foci of the hyperbola x²/9 − y²/16 = 1.",
        "ja": "双曲線 x²/9 − y²/16 = 1 の頂点と焦点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "中 1 の反比例 y = a/x のグラフも、数 C の二次曲線 x²/a² − y²/b² = 1 も、同じ hyperbola。複数形は hyperbolas。",
      "形容詞 hyperbolic は「双曲線の」の意味。hyperbolic functions（双曲線関数、エントリ hyperbolic-functions）は hyperbola とは別の語。",
      "読み方は hy-PER-bo-la（第 2 音節を強く読む。Merriam-Webster: hī-ˈpər-bə-lə）。綴りの似た hyperbole（誇張法。hī-ˈpər-bə-lē）は別の語。"
    ],
    "related": [
      "inverse-proportion",
      "conic-section",
      "asymptote",
      "focus",
      "hyperbolic-functions"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "双曲線",
        "en": "Hyperbola"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「双曲線」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%B7%9A",
        "note": "2 焦点からの距離の差 |PF − PF′| は 2a"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学C 平面上の曲線と複素数平面: 二次曲線（双曲線）"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「hyperbola」",
        "url": "https://www.merriam-webster.com/dictionary/hyperbola",
        "note": "発音 hī-ˈpər-bə-lə。hyperbole は hī-ˈpər-bə-(ˌ)lē（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "hyperbola": 68
      },
      "written": {
        "hyperbola": 390
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "hyperbolic-functions",
    "ja": {
      "term": "双曲線関数",
      "reading": "そうきょくせんかんすう",
      "alt": []
    },
    "en": {
      "term": "hyperbolic functions",
      "alt": [],
      "variants": [
        {
          "term": "hyperbolic trig functions",
          "register": "spoken",
          "note": "大学の講義（MIT OCW）の言い方。話し言葉は大半が MIT OCW で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "sinh x = (eˣ − e⁻ˣ)/2、cosh x = (eˣ + e⁻ˣ)/2 などで定まる関数。",
    "definition_en": "Functions such as sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2.",
    "latex": "\\sinh x = \\frac{e^x - e^{-x}}{2}, \\quad \\cosh x = \\frac{e^x + e^{-x}}{2}",
    "spoken_en": "hyperbolic sine of x equals e to the x minus e to the negative x, over two",
    "examples": [
      {
        "en": "The hyperbolic functions are built from e to the x and e to the negative x.",
        "ja": "双曲線関数は eˣ と e⁻ˣ からつくられる。",
        "register": "spoken"
      },
      {
        "en": "Show that cosh²x − sinh²x = 1.",
        "ja": "cosh²x − sinh²x = 1 を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校では扱わない。sinh は hyperbolic sine と読む。"
    ],
    "related": [
      "inverse-hyperbolic-functions",
      "base-of-the-natural-logarithm"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "双曲線関数",
        "en": "Hyperbolic functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "hyperbolic trig functions": 13,
        "hyperbolic functions": 2
      },
      "written": {
        "hyperbolic functions": 52
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "hypergeometric-distribution",
    "ja": {
      "term": "超幾何分布",
      "reading": "ちょうきかぶんぷ",
      "alt": []
    },
    "en": {
      "term": "hypergeometric distribution",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "2 種類のものが入った有限の集まりから、戻さずに n 個取り出したとき、一方の種類の個数が従う確率分布。",
    "definition_en": "The distribution of the number of items of one kind when n items are drawn without replacement from a finite group containing two kinds.",
    "latex": "P(X = k) = \\frac{\\binom{K}{k}\\binom{N-K}{n-k}}{\\binom{N}{n}}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Without replacement, the draws aren't independent, so it's hypergeometric, not binomial.",
        "ja": "戻さないので試行は独立でない。二項分布ではなく超幾何分布だ。",
        "register": "spoken"
      },
      {
        "en": "A committee of 5 is chosen from 8 women and 6 men. Find the probability that exactly 3 women are chosen.",
        "ja": "女性 8 人・男性 6 人から 5 人の委員を選ぶ。女性がちょうど 3 人選ばれる確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics（節の名前 Hypergeometric Distribution）。"
    ],
    "related": [
      "binomial-distribution",
      "probability-distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "超幾何分布",
        "en": "Hypergeometric distribution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "hypergeometric distribution": 5
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 hypergeometric distribution（Hypergeometric Distribution）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "hypotenuse",
    "ja": {
      "term": "斜辺",
      "reading": "しゃへん",
      "alt": []
    },
    "en": {
      "term": "hypotenuse",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直角三角形で、直角の向かいにある辺。3 辺のうちでいちばん長い。",
    "definition_en": "In a right triangle, the side across from the right angle. It is always the longest of the three sides.",
    "latex": "a^2 + b^2 = c^2",
    "spoken_en": "a squared plus b squared equals c squared",
    "examples": [
      {
        "en": "Find the little square that marks the right angle; the side straight across from it is the hypotenuse.",
        "ja": "直角を表す小さな四角の印を探して。そのまっすぐ向かいにある辺が斜辺だよ。",
        "register": "spoken"
      },
      {
        "en": "The legs of a right triangle are 6 cm and 8 cm long. Find the length of the hypotenuse.",
        "ja": "直角三角形の直角をはさむ 2 辺の長さが 6 cm と 8 cm である。斜辺の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "発音は「ハイポテニュース」に近く、pot の音節を強く読む（Merriam-Webster: hī-ˈpä-tə-ˌnüs）。",
      "直角をはさむ 2 辺は legs（1 本なら a leg）と呼ぶ。三平方の定理 a² + b² = c² では a と b が legs、c が hypotenuse。",
      "hypotenuse は直角三角形の辺の名前。直角三角形でない三角形のいちばん長い辺は hypotenuse とは呼ばない。"
    ],
    "related": [
      "right-triangle",
      "congruence-criteria-for-right-triangles",
      "hl-congruence",
      "sohcahtoa",
      "adjacent-side"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "斜辺",
        "en": "Hypotenuse"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「hypotenuse」",
        "url": "https://www.merriam-webster.com/dictionary/hypotenuse",
        "note": "発音 hī-ˈpä-tə-ˌnüs（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "hypotenuse": 417
      },
      "written": {
        "hypotenuse": 147
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "hypothesis-testing",
    "ja": {
      "term": "仮説検定",
      "reading": "かせつけんてい",
      "alt": []
    },
    "en": {
      "term": "hypothesis test",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "significance test",
          "register": "spoken",
          "note": "話し言葉はすべて Khan Academy で、用例コーパスの書き言葉（OpenStax Introductory Statistics）には出てこない。AP Statistics の CED は significance test とも書く（6 件）。"
        },
        {
          "term": "hypothesis test",
          "register": "written",
          "note": "hypothesis testing を含めて、話し言葉では少なく、書き言葉に多い（すべて OpenStax Introductory Statistics）。OpenStax と AP Statistics の CED は数えられる名詞 hypothesis test（a hypothesis test）を主に使う。"
        }
      ],
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I",
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "母集団についての仮説を立て、標本のデータがその仮説のもとでどれだけ起こりにくいかから、仮説を棄却するかどうかを判断する方法。",
    "definition_en": "A method of deciding whether to reject a hypothesis about a population, based on how unlikely the sample data would be if the hypothesis were true.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is a significance test, so we start by writing down the null and alternative hypotheses.",
        "ja": "これは仮説検定なので、帰無仮説と対立仮説を書き出すところから始める。",
        "register": "spoken"
      },
      {
        "en": "Using a 5% significance level, carry out a hypothesis test of the claim.",
        "ja": "有意水準 5% で、その主張の仮説検定を行え。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉の significance test はすべて Khan Academy（AP Statistics）。書き言葉の hypothesis test(ing) はすべて OpenStax。",
      "結論は「帰無仮説を棄却する」（reject the null hypothesis）か「棄却しない」（fail to reject）で述べる。用例コーパスでは fail to reject が多く、accept the null（帰無仮説を採択する）は少ない。"
    ],
    "related": [
      "null-hypothesis",
      "alternative-hypothesis",
      "significance-level",
      "reject",
      "fail-to-reject"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "仮説検定",
        "en": "Statistical hypothesis test"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Unit 3（hypothesis test。significance test は 6 件）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B 統計的な推測: 仮説検定の方法"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "significance test": 36,
        "hypothesis test": 7
      },
      "written": {
        "hypothesis test": 245
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "hypothesis",
    "ja": {
      "term": "仮定",
      "reading": "かてい",
      "alt": [
        "与えられた条件"
      ]
    },
    "en": {
      "term": "hypothesis",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "証明の「仮定」は、論理・定理の文脈では hypothesis（p ならば q の p）、米国の Geometry の答案では Given（2 列証明の最初の行の見出し）と言い、1 語に決まらない。CK-12 Geometry は hypothesis と呼ぶ（2.11 If Then Statements・2.12 Converse Inverse and Contrapositive Statements、hypothesis and conclusion などの形で 8 件）。",
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "命題「p ならば q」の p の部分。証明では、正しいものとして出発点に使ってよい条件を指す。",
    "definition_en": "The \"if\" part p of a statement \"if p, then q.\" In a proof, it is the condition you are allowed to assume from the start.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "You're allowed to use the hypothesis, so start by writing down what's given.",
        "ja": "仮定は使っていいんだから、まず与えられていることを書き出そう。",
        "register": "spoken"
      },
      {
        "en": "In the statement \"If a triangle is equilateral, then it is isosceles,\" identify the hypothesis and the conclusion.",
        "ja": "命題「三角形が正三角形ならば、二等辺三角形である」の仮定と結論を答えよ。",
        "register": "written"
      },
      {
        "en": "Given: AB ≅ CD and BC ≅ DA. Prove: △ABC ≅ △CDA.",
        "ja": "仮定：AB = CD, BC = DA　結論：△ABC ≡ △CDA",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "統計の「仮説」（数I データの分析の仮説検定）も英語では hypothesis で、別の語。用例コーパスの hypothesis 単独はほとんどがこの統計の意味。",
      "米国の Geometry の 2 列証明では、仮定を Given:、示すことを Prove: という見出しで書き、hypothesis とは書かない。",
      "証明の仮定の意味の hypothesis は、用例コーパスでは hypothesis and conclusion などの形で話し言葉にわずかに出てくるだけで、書き言葉には出てこない。",
      "統計の「仮説」（null hypothesis などの hypothesis）は別の見出し statistical-hypothesis。英語はどちらも hypothesis なので、証明の話か統計の検定の話かで見分ける。"
    ],
    "related": [
      "conditional-statement",
      "proof",
      "prove",
      "two-column-proof",
      "proposition",
      "statistical-hypothesis"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.11 If Then Statements・2.12 Converse Inverse and Contrapositive Statements"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "hypothesis and conclusion | hypothesis and the conclusion | hypotheses of the theorem | hypothesis of the theorem | hypothesis of the conditional | hypothesis of the implication | hypothesis of an implication": 2
      },
      "sources": [
        "mit-18.01"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 hypothesis and conclusion | hypothesis and the conclusion | hypotheses of the theorem | hypothesis of the theorem | hypothesis of the conditional | hypothesis of the implication | hypothesis of an implication（CK-12 Geometry 2.11 If Then Statements・CK-12 Geometry 2.12 Converse Inverse and Contrapositive Statements・計 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "identity-matrix",
    "ja": {
      "term": "単位行列",
      "reading": "たんいぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "identity matrix",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "対角成分がすべて 1 で、ほかの成分がすべて 0 の正方行列。E（米国の教科書では I）と書く。",
    "definition_en": "The square matrix with 1s on the main diagonal and 0s elsewhere, written E in Japanese textbooks and I in US ones.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Multiplying by the identity matrix doesn't change anything, just like multiplying by one.",
        "ja": "単位行列を掛けても何も変わらない。1 を掛けるのと同じ。",
        "register": "spoken"
      },
      {
        "en": "Show that AI = IA = A for the 2 × 2 identity matrix I.",
        "ja": "2 × 2 の単位行列 I について AI = IA = A を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の教科書は単位行列を E で表すが、英語の教科書は I で表す（OpenStax Algebra and Trigonometry は次数を添えて Iₙ）。"
    ],
    "related": [
      "matrix",
      "inverse-matrix",
      "zero-matrix"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "単位行列",
        "en": "Identity matrix"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "identity matrix": 75
      },
      "written": {
        "identity matrix": 24
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "identity",
    "ja": {
      "term": "恒等式",
      "reading": "こうとうしき",
      "alt": []
    },
    "en": {
      "term": "identity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus"
      ]
    },
    "definition_ja": "文字にどんな値を代入しても成り立つ等式。",
    "definition_en": "An equation that holds for every value of the variables.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This isn't an equation to solve — it's an identity, so it's true for every x.",
        "ja": "これは解く方程式じゃなくて恒等式。どんな x でも成り立つ。",
        "register": "spoken"
      },
      {
        "en": "Determine the constants a and b so that (x + 1)(x + a) = x² + 3x + b is an identity.",
        "ja": "(x + 1)(x + a) = x² + 3x + b が x についての恒等式となるように、定数 a, b の値を定めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "is an identity",
        "ja": "恒等式である"
      }
    ],
    "pitfalls": [
      "identity は identity matrix（単位行列）・identity function（恒等関数）にも使う。OpenStax Prealgebra（Properties of Identity, Inverses, and Zero）の Identity Property of Addition（a + 0 = a）の identity も別の意味。",
      "「x についての恒等式」は is an identity、または true for all values of x。IM Algebra 2（2.23 Polynomial Identities (Part 1)）と OpenStax Algebra and Trigonometry（Linear Equations in One Variable）は、恒等式を変数のどんな値でも成り立つ等式（true for all values of the variable(s)）と定義する。"
    ],
    "related": [
      "compare-coefficients",
      "substituting-values",
      "proving-an-identity",
      "trigonometric-identities",
      "equation",
      "hold"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "恒等式",
        "en": "Identity (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Linear Equations in One Variable（identity equation）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 2.23・2.24 Polynomial Identities（glossary: identity）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Properties of Identity, Inverses, and Zero（Identity Property of Addition）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "an identity": 36
      },
      "written": {
        "an identity": 60
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "image",
    "ja": {
      "term": "像",
      "reading": "ぞう",
      "alt": [
        "移した図形"
      ]
    },
    "en": {
      "term": "image",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の中学では「移した図形」「移動後の図形」と言い、像という言葉は写像（大学）で使うことが多い。英語の Geometry では変換で移った点や図形を image と呼び、点 A の像を A′（A prime）と書く。IM の Grade 8 と Geometry の glossary の見出し。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "移動や拡大などの変換によって、点や図形が移った先の点や図形。点 A の像は A′ と書く。",
    "definition_en": "The point or figure that results when a transformation is applied to a point or figure. The image of point A is usually labeled A′.",
    "latex": "\\triangle ABC \\to \\triangle A'B'C'",
    "spoken_en": "triangle A B C maps to triangle A prime B prime C prime",
    "examples": [
      {
        "en": "After we rotate the triangle, the image is labeled with primes, so B prime is the image of B.",
        "ja": "三角形を回転させたら、移った図形にはダッシュを付けて表す。だから B′ は B が移った点だ。",
        "register": "spoken"
      },
      {
        "en": "Translate triangle ABC 4 units left and 2 units up. Give the coordinates of the vertices of its image.",
        "ja": "△ABC を x 軸方向に −4、y 軸方向に 2 だけ平行移動する。移した図形の頂点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語では A′ を「A ダッシュ」と読むが、英語では A prime と読む。",
      "日常語の mirror image（鏡に映った姿）は変換の image とは別の言い方。",
      "大学では関数 f による集合 X の像 f(X) の意味でも使う（Levin では 5.2 Functions に 29 件、Nicholson では 7.2 Kernel and Image of a Linear Transformation に 11 件）。",
      "線形代数の「像」（行列 A の像 = 列空間、Im A）は別の見出し column-space。こちらは図形の変換で移った点・図形。"
    ],
    "related": [
      "preimage",
      "rigid-motion",
      "translation",
      "reflection",
      "rotation",
      "column-space"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "像 (数学)",
        "en": "Image (mathematics)"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "!mirror image of | image point | image points | its image": 27
      },
      "written": {
        "!mirror image of | image point | image points | its image": 35
      },
      "sources": [
        "khan-middle",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "imaginary-axis",
    "ja": {
      "term": "虚軸",
      "reading": "きょじく",
      "alt": []
    },
    "en": {
      "term": "imaginary axis",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "複素数平面の縦軸。純虚数を表す点（と原点）が並ぶ。",
    "definition_en": "The vertical axis of the complex plane, on which the pure imaginary numbers (and 0) lie.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Anything purely imaginary sits right on the imaginary axis.",
        "ja": "純虚数はどれも虚軸の上にある。",
        "register": "spoken"
      },
      {
        "en": "Find the value of a for which the point representing (a + i)(1 + 2i) lies on the imaginary axis.",
        "ja": "(a + i)(1 + 2i) を表す点が虚軸上にあるような a の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "real-axis",
      "complex-plane",
      "pure-imaginary-number"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "imaginary axis": 25
      },
      "written": {
        "imaginary axis": 13
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "imaginary-number",
    "ja": {
      "term": "虚数",
      "reading": "きょすう",
      "alt": []
    },
    "en": {
      "term": "imaginary number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「虚数」は、実数でない複素数 a + bi（b ≠ 0）を虚数とする（センター試験 平成 27 年度 数学II も 1 + 2i を虚数と呼ぶ）。IM Algebra 2（3.10 の用語集）と OpenStax Precalculus（3.1 Complex Numbers）は imaginary number を bi の形の数とし（英語版 Wikipedia「Imaginary number」も実数と i の積と定義する）、OpenStax Algebra and Trigonometry（2.4）は bi を pure imaginary number と呼ぶ。OpenStax Intermediate Algebra（8.8）は b ≠ 0 の a + bi を imaginary number と呼び、日本語と同じ範囲になる。実数でない a + bi を、IM Algebra 2（3.17・3.19）は non-real、Nicholson は nonreal complex number と書く。",
    "domains": [
      "algebra",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "実数でない複素数。a + bi で b ≠ 0 のもの。",
    "definition_en": "A complex number that is not real, a + bi with b ≠ 0 (the Japanese 虚数). IM Algebra 2 and OpenStax Precalculus use imaginary number only for numbers of the form bi, while OpenStax Intermediate Algebra uses it for every a + bi with b ≠ 0.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The square root of negative 9 isn't a real number — it's the imaginary number 3i.",
        "ja": "√−9 は実数ではなく、虚数 3i。",
        "register": "spoken"
      },
      {
        "en": "Find all imaginary numbers z such that z² = −16.",
        "ja": "z² = −16 を満たす虚数 z をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「虚数解」（実数でない解）は、OpenStax Algebra and Trigonometry・Intermediate Algebra は complex solution（エントリ imaginary-solution）、IM Algebra 2（3.17・3.19）は non-real solution と書く。−1 ± 2i のような解は、IM Algebra 2・OpenStax Precalculus の意味の imaginary number（bi）ではない。",
      "実数でない複素数（日本語の虚数）をはっきり言うときは、Nicholson は nonreal complex number、IM Algebra 2（3.17・3.19）は non-real と書く。"
    ],
    "related": [
      "complex-number",
      "pure-imaginary-number",
      "imaginary-unit",
      "imaginary-solution",
      "square-root-of-a-negative-number",
      "real-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "虚数",
        "en": "Imaginary number",
        "note": "英語版の記事は bi（実数と i の積）の意味。日本語版は b ≠ 0 の a + bi"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "2.4 Complex Numbers（pure imaginary number）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「虚数」",
        "url": "https://ja.wikipedia.org/wiki/%E8%99%9A%E6%95%B0",
        "note": "実数でない複素数（b ≠ 0）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成27年度 大学入試センター試験 本試験 数学Ⅱ（問題）",
        "url": "https://web.archive.org/web/20160601/https://www.dnc.ac.jp/albums/abm.php?f=abm00004812.pdf&n=2701-0501+suugaku2.pdf",
        "note": "1 + 2i を虚数と呼ぶ"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 3.10（glossary: imaginary number）・3.17・3.19（non-real）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "Appendix A Complex Numbers（nonreal complex number）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "3.1 Complex Numbers（imaginary number: bi）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "8.8 Use the Complex Number System（imaginary number: a + bi, b ≠ 0）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "imaginary number": 43
      },
      "written": {
        "imaginary number": 36
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "imaginary-part",
    "ja": {
      "term": "虚部",
      "reading": "きょぶ",
      "alt": []
    },
    "en": {
      "term": "imaginary part",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "複素数 a + bi の b。bi ではなく実数 b を指す。",
    "definition_en": "The real number b in the complex number a + bi. Some US textbooks, such as IM Algebra 2 and OpenStax Precalculus, call bi the imaginary part instead.",
    "latex": "\\operatorname{Im}(a + bi) = b",
    "spoken_en": null,
    "examples": [
      {
        "en": "Careful — some books call bi the imaginary part, but Im(z) is the real number b.",
        "ja": "注意。bi を虚部と呼ぶ教科書もあるが、Im(z) は実数 b。",
        "register": "spoken"
      },
      {
        "en": "Find all real x for which the imaginary part of (x + i)² is 4.",
        "ja": "(x + i)² の虚部が 4 となる実数 x をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "虚部は実数 b（日本語版 Wikipedia「複素数」、英語版 Wikipedia「Complex number」、OpenStax Algebra and Trigonometry）。IM Algebra 2（3.12 Arithmetic with Complex Numbers）と OpenStax Precalculus は bi を imaginary part と呼ぶので、授業の教科書の定義を確かめる（Im(z) と書けば b）。"
    ],
    "related": [
      "real-part",
      "complex-number",
      "pure-imaginary-number",
      "equality-of-complex-numbers"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Complex Numbers（the imaginary part is 3）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 3.12 Arithmetic with Complex Numbers（bi is the imaginary part）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "Complex Numbers（the imaginary part is 3i）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Complex_number",
        "note": "記事「Complex number」（the real number b, not the complex number bi）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「複素数」",
        "url": "https://ja.wikipedia.org/wiki/%E8%A4%87%E7%B4%A0%E6%95%B0",
        "note": "虚部は実数 b"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "imaginary part": 137
      },
      "written": {
        "imaginary part": 37
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "imaginary-solution",
    "ja": {
      "term": "虚数解",
      "reading": "きょすうかい",
      "alt": []
    },
    "en": {
      "term": "complex solution",
      "alt": [],
      "variants": [
        {
          "term": "complex solution",
          "register": "both",
          "note": "話し言葉はほとんどが MIT OCW。書き言葉はすべて OpenStax（Intermediate Algebra・Algebra and Trigonometry）。書き言葉ではこれが最も多い。"
        },
        {
          "term": "complex root",
          "register": "spoken",
          "note": "多項式や特性方程式の根として言うとき。話し言葉はほとんどが MIT OCW。書き言葉では少ない（MIT の講義ノート・OpenStax Algebra and Trigonometry）。"
        },
        {
          "term": "imaginary solution",
          "register": "spoken",
          "note": "日本語に近い言い方。話し言葉は大半が The Organic Chemistry Tutor で、書き言葉には出てこない。"
        },
        {
          "term": "non-real solution",
          "register": "written",
          "note": "IM Algebra 2（3.17・3.19 Real and Non-Real Solutions）の言い方。用例コーパスには出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "OpenStax Algebra and Trigonometry・Precalculus（Zeros of Polynomial Functions）は find all complex solutions (real and non-real) のように、complex solution に実数解も含める。「虚数解」（実数でない解）をはっきり言うときは、IM Algebra 2（3.17・3.19 Real and Non-Real Solutions）は non-real solution と書く。OpenStax Algebra and Trigonometry と Intermediate Algebra は complex solutions と書く。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "方程式の解のうち、虚数（実数でない複素数）であるもの。",
    "definition_en": "A solution of an equation that is not a real number.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The discriminant is negative, so we get two complex solutions, and they're conjugates.",
        "ja": "判別式が負なので、虚数解が 2 つ出て、互いに共役。",
        "register": "spoken"
      },
      {
        "en": "Find the values of k for which x² + kx + 4 = 0 has two non-real solutions.",
        "ja": "x² + kx + 4 = 0 が異なる 2 つの虚数解をもつような k の値の範囲を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "two complex solutions",
        "ja": "2 つの虚数解"
      }
    ],
    "pitfalls": [
      "微分方程式の講義（MIT 18.03）で complex solution と言うのは、多くが微分方程式の複素数値の解（実部と虚部をとる関数の解）で、方程式の実数でない解のことではない。書き言葉では complex root より complex solution のほうが多い。",
      "「実数解と虚数解」は real and non-real solutions（IM Algebra 2 3.19。OpenStax Precalculus は real and nonreal solutions）。"
    ],
    "related": [
      "discriminant",
      "complex-conjugate",
      "quadratic-formula",
      "solution",
      "real-solution",
      "fundamental-theorem-of-algebra",
      "conjugate-roots"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 3.17・3.19 Real and Non-Real Solutions（non-real solution）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "Zeros of Polynomial Functions（complex solutions (real and non-real)）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-28",
      "human": null
    },
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "complex root": 22,
        "complex solution": 31,
        "imaginary solution": 17
      },
      "written": {
        "complex solution": 17,
        "complex root": 6
      },
      "sources": [
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "imaginary-unit",
    "ja": {
      "term": "虚数単位",
      "reading": "きょすうたんい",
      "alt": [
        "i"
      ]
    },
    "en": {
      "term": "imaginary unit",
      "alt": [],
      "variants": [
        {
          "term": "square root of negative one",
          "register": "spoken",
          "note": "i を説明するときの言い方。話し言葉では 3Blue1Brown が半分以上を占め、書き言葉には出てこない。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 乗すると −1 になる数 i。",
    "definition_en": "The number i whose square is −1.",
    "latex": "i^2 = -1",
    "spoken_en": "i squared equals negative one",
    "examples": [
      {
        "en": "i is the square root of negative one, so i squared is negative one.",
        "ja": "i は √−1 なので、i² は −1。",
        "register": "spoken"
      },
      {
        "en": "Let i be the imaginary unit. Compute i^{2027}.",
        "ja": "i を虚数単位とする。i²⁰²⁷ を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では、i を名前（imaginary unit）で呼ぶより、the square root of negative one と説明する言い方のほうが多い。",
      "電気工学では i の代わりに j を使う。"
    ],
    "related": [
      "complex-number",
      "imaginary-number",
      "square-root-of-a-negative-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "虚数単位",
        "en": "Imaginary unit"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "square root of negative one": 34,
        "imaginary unit": 6
      },
      "written": {
        "imaginary unit": 4
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "implicit-differentiation",
    "ja": {
      "term": "陰関数の微分",
      "reading": "いんかんすうのびぶん",
      "alt": [
        "陰関数微分"
      ]
    },
    "en": {
      "term": "implicit differentiation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "ja.alt の「陰関数微分」は学習指導要領解説に無い（日本語版 Wikipedia「対数微分法」にはある）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x と y の関係式の両辺を x で微分し、y を x の関数とみて dy/dx を求める方法。",
    "definition_en": "Finding dy/dx by differentiating both sides of an equation in x and y with respect to x, treating y as a function of x.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "With implicit differentiation, every time you differentiate a y, tack on a dy/dx.",
        "ja": "陰関数の微分では、y を微分するたびに dy/dx を掛ける。",
        "register": "spoken"
      },
      {
        "en": "Use implicit differentiation to find dy/dx for x² + y² = 25.",
        "ja": "陰関数の微分を用いて、x² + y² = 25 の dy/dx を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "implicit-function",
      "chain-rule",
      "logarithmic-differentiation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 3.2（implicit differentiation）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「対数微分法」",
        "url": "https://ja.wikipedia.org/wiki/%E5%AF%BE%E6%95%B0%E5%BE%AE%E5%88%86%E6%B3%95",
        "note": "「陰関数微分をすると」"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「陰関数微分」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "implicit differentiation": 111
      },
      "written": {
        "implicit differentiation": 49
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "implicit-function",
    "ja": {
      "term": "陰関数",
      "reading": "いんかんすう",
      "alt": []
    },
    "en": {
      "term": "implicitly defined function",
      "alt": [
        "implicit function"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "F(x, y) = 0 の形の式によって、y が x の関数として間接的に定まるもの。",
    "definition_en": "A function y of x given indirectly by an equation F(x, y) = 0 rather than by a formula y = f(x).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "x squared plus y squared equals 25 isn't a function, but near most points it defines y implicitly.",
        "ja": "x² + y² = 25 は関数ではないが、ほとんどの点の近くでは y を陰関数として定める。",
        "register": "spoken"
      },
      {
        "en": "The equation x² + xy + y² = 7 defines y as an implicitly defined function of x.",
        "ja": "方程式 x² + xy + y² = 7 は、y を x の陰関数として定める。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は topic 3.2・5.12 で implicitly defined function と書く（Unit 3 の概要では implicit functions とも）。"
    ],
    "related": [
      "implicit-differentiation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "陰関数",
        "en": "Implicit function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 3.2・5.12（implicitly defined function）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「陰関数」は出てこない（数学III 微分法は合成関数の導関数まで。level.jp 数III はその応用として。DECISIONS）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "implicitly defined function": 2,
        "implicit function": 3
      },
      "written": {
        "implicitly defined function": 2,
        "implicit function": 3
      },
      "sources": [
        "mit-18.01",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-10-09"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは CED の呼び方 implicitly defined function（topic 3.2・topic 5.12）。register は主張しない。",
        "raised": "2026-10-09"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "impossible",
    "ja": {
      "term": "決して起こらない",
      "reading": "けっしておこらない",
      "alt": []
    },
    "en": {
      "term": "impossible",
      "alt": [],
      "uk": null
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "あることがらが、どんなときにも決して起こらないこと。決して起こらないことがらの確率は 0 である。",
    "definition_en": "Never able to happen. An event that is impossible has a probability of 0.",
    "latex": "P(A) = 0",
    "spoken_en": "P of A equals zero",
    "examples": [
      {
        "en": "You can't roll an 8 on a regular die, so that's impossible, and the probability is 0.",
        "ja": "ふつうのさいころで 8 の目は出ないから、それは決して起こらないことで、確率は 0。",
        "register": "spoken"
      },
      {
        "en": "Two dice are rolled. Explain why getting a sum of 13 is impossible, and find its probability.",
        "ja": "2 個のさいころを投げる。出た目の和が 13 になることが決して起こらない理由を説明し、その確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "どんな確率 p も 0 ≦ p ≦ 1。計算した確率が負になったり 1 を超えたりしたら、どこかが間違っている。",
      "確率が小さくて起こりにくいこと（unlikely）と、決して起こらないこと（impossible、確率 0）を区別する。",
      "日本語の「決して起こらない」は動詞の形だが、英語の impossible は形容詞で、It is impossible to roll an 8. ／ The event is impossible. のように使う。"
    ],
    "related": [
      "certain",
      "likelihood",
      "probability",
      "event",
      "die"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "impossible event | event is impossible | impossible to happen | impossible to occur": 1
      },
      "written": {
        "impossible event | event is impossible | impossible to happen | impossible to occur": 3
      },
      "sources": [
        "openstax-algtrig",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは OpenStax の呼び方 impossible event | event is impossible | impossible to happen | impossible to occur（本文 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "improper-integral",
    "ja": {
      "term": "広義積分",
      "reading": "こうぎせきぶん",
      "alt": []
    },
    "en": {
      "term": "improper integral",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "積分区間が無限に広がるか、区間内で被積分関数が有界でない積分。極限として定義する。",
    "definition_en": "An integral over an infinite interval, or of a function that blows up on the interval, defined as a limit.",
    "latex": "\\int_1^{\\infty} \\frac{dx}{x^2} = \\lim_{b \\to \\infty} \\int_1^{b} \\frac{dx}{x^2} = 1",
    "spoken_en": "the integral from one to infinity of one over x squared dx equals the limit as b goes to infinity of the integral from one to b of one over x squared dx, which is one",
    "examples": [
      {
        "en": "Since the limit is finite, this improper integral converges.",
        "ja": "極限が有限だから、この広義積分は収束する。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the improper integral ∫₁^∞ (1/x) dx converges or diverges.",
        "ja": "広義積分 ∫₁^∞ (1/x) dx が収束するか発散するかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "収束する converges、発散する diverges。",
      "学習指導要領解説に「広義積分」の語は出てこない。数学III の課題学習〈積分法〉の例が、調和級数の発散を調べるのに、部分和と定積分 ∫₁ⁿ (1/x)dx を比べ、n を大きくしたときの定積分の値を観察することを挙げるだけ（∫₁^∞ の形では書かない）。"
    ],
    "related": [
      "definite-integral",
      "convergence-of-improper-integrals",
      "comparison-test-for-integrals",
      "integral-test"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "広義積分",
        "en": "Improper integral"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 積分法: 広義積分の語は無い（課題学習〈積分法〉の例に、調和級数の部分和と ∫₁ⁿ (1/x)dx を比べ、n を大きくしたときの値を観察すること）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "improper integral": 60
      },
      "written": {
        "improper integral": 52
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "incenter",
    "ja": {
      "term": "内心",
      "reading": "ないしん",
      "alt": []
    },
    "en": {
      "term": "incenter",
      "alt": [],
      "uk": "incentre"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "三角形の 3 つの内角の二等分線が交わる点。3 辺から等しい距離にあり、内接円の中心になる。",
    "definition_en": "The point where the bisectors of the three interior angles of a triangle meet. It is the same distance from all three sides, so it is the center of the inscribed circle.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The incenter is where the three angle bisectors meet, and it's the center of the inscribed circle.",
        "ja": "内心は 3 つの角の二等分線の交点で、内接円の中心になる。",
        "register": "spoken"
      },
      {
        "en": "Let I be the incenter of triangle ABC. If ∠A = 70°, find ∠BIC.",
        "ja": "△ABC の内心を I とする。∠A = 70° のとき、∠BIC を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "内心は 3 辺から等距離（頂点から等距離なのは外心）。外心と取り違えない。",
      "英語版 Wikipedia「Incircle and excircles」は、内接円の半径を inradius と書く（節 Radius）。内心は必ず三角形の内部にある。",
      "英国式のつづりは incentre。"
    ],
    "related": [
      "inscribed-circle",
      "circumcenter",
      "excenter",
      "the-five-centers-of-a-triangle",
      "angle-bisector"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary・Geometry 7.7 Circles in Triangles・Geometry 7.6 A Special Point"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences#re_er",
        "note": "記事「American and British English spelling differences」の節 -re, -er（英国の綴りは centre、米国は center）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Incircle_and_excircles",
        "note": "記事「Incircle and excircles」の冒頭（incircle・incenter）と節 Radius（inradius）。数学カテゴリから 5 段"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "incenter": 1
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 incenter（Geometry glossary・Geometry 7.7 Circles in Triangles・Geometry 7.6 A Special Point・計 27 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "inclusion-exclusion-principle",
    "ja": {
      "term": "包除原理",
      "reading": "ほうじょげんり",
      "alt": [
        "包含排除原理",
        "包含と排除の原理"
      ]
    },
    "en": {
      "term": "principle of inclusion-exclusion",
      "alt": [
        "inclusion-exclusion principle"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（principle of inclusion-exclusion ／ inclusion-exclusion principle は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "combinatorics",
      "probability"
    ],
    "level": {
      "jp": [
        "数A",
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "和集合の要素の個数（和事象の確率）を、各集合の個数の和から共通部分の個数を引き、さらに足し戻して求める原理。",
    "definition_en": "Counting a union (or finding the probability of a union of events) by adding the sizes of the sets, subtracting their pairwise intersections, adding back the triple intersections, and so on.",
    "latex": "|A \\cup B| = |A| + |B| - |A \\cap B|",
    "spoken_en": null,
    "examples": [
      {
        "en": "You counted the overlap twice, so subtract it once: that's inclusion-exclusion.",
        "ja": "重なりを 2 回数えたので 1 回引く。それが包除原理だ。",
        "register": "spoken"
      },
      {
        "en": "Use the principle of inclusion-exclusion to count the integers from 1 to 100 divisible by 2, 3, or 5.",
        "ja": "包除原理を用いて、1 から 100 までの整数で 2, 3, 5 のいずれかで割り切れるものの個数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数 A の n(A ∪ B) = n(A) + n(B) − n(A ∩ B) は 2 つの集合の場合。日本の高校の教科書はこの原理に名前を付けない。",
      "Levin は Principle of Inclusion/Exclusion と書き、PIE と略す（3.8 の節の名前）。"
    ],
    "related": [
      "general-addition-rule"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.3 Non-Disjoint Outcomes・3.8 Advanced Counting Using PIE"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "inclusion-exclusion principle": 7
      },
      "sources": [
        "mit-6.042"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Levin の呼び方 principle of inclusion-exclusion（3.8 Advanced Counting Using PIE・3.3 Non-Disjoint Outcomes）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "increase",
    "ja": {
      "term": "増加する",
      "reading": "ぞうかする",
      "alt": []
    },
    "en": {
      "term": "increase",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数量が大きくなること。関数で、x の値が増えるにつれて y の値も大きくなるとき、「y は増加する」という。",
    "definition_en": "To become larger in value. In a function, y increases when its values get larger as x gets larger.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "As x increases, y increases too, so this line goes up from left to right.",
        "ja": "x が増加すると y も増加するから、この直線は右上がりだね。",
        "register": "spoken"
      },
      {
        "en": "For y = 3x, how much does y increase when x increases by 1?",
        "ja": "y = 3x で、x が 1 増加するとき、y はいくつ増加するか。",
        "register": "written"
      },
      {
        "en": "The price of a ticket increased from $8 to $10. By what percent did the price increase?",
        "ja": "チケットの値段が 8 ドルから 10 ドルに上がった。値段は何 % 増加したか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "increase by 3 は「3 増える」（増えた分）、increase to 3 は「3 になる」（増えた後の値）。by と to で答えが変わる。",
      "「関数がある範囲で増加する」（グラフが右上がり）は is increasing の形で言う（エントリ is-monotonically-increasing）。"
    ],
    "related": [
      "decrease",
      "is-monotonically-increasing",
      "increasing-and-decreasing",
      "rate-of-change",
      "change-together"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "increase": 1438
      },
      "written": {
        "increase": 1180
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "increasing-and-decreasing",
    "ja": {
      "term": "関数の増減",
      "reading": "かんすうのぞうげん",
      "alt": []
    },
    "en": {
      "term": "increasing or decreasing",
      "alt": [
        "increasing and decreasing"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「増減」は名詞だが、AP の CED（topic 5.3）と OpenStax Calculus は where f is increasing or decreasing のように形容詞で言い、1 語の名詞は用例コーパスと参照にほとんど出てこない。問題文は Determine the intervals on which f is increasing or decreasing. の形になる。",
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Algebra 2",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x が増えるにつれて関数の値が増えるか減るか、またその区間。f′(x) の符号で調べる。",
    "definition_en": "Whether a function's values go up or down as x increases, and on which intervals; found from the sign of f′(x).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Where f prime is positive, f is increasing, and where it's negative, f is decreasing.",
        "ja": "f′ が正のところで f は増加し、負のところで減少する。",
        "register": "spoken"
      },
      {
        "en": "Determine the intervals on which f(x) = x³ − 3x is increasing or decreasing.",
        "ja": "f(x) = x³ − 3x が増加する区間と減少する区間を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "微分を使って不等式 f(x) > g(x) を証明するとき（h(x) = f(x) − g(x) の増減を調べる）は、英語の決まった書き方は用例コーパスと参照には出てこない。書くなら Let h(x) = f(x) − g(x). と置いて、h′ の符号から h の増減を示し、最小値が正であることを言う（本プロジェクトの書き方の例）。"
    ],
    "related": [
      "determine-where-the-function-is-increasing-and-decreasing",
      "first-derivative-test",
      "is-monotonically-increasing",
      "sign-chart",
      "make-a-sign-chart"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.3（increasing or decreasing）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "increasing and decreasing": 7,
        "increasing or decreasing": 23
      },
      "written": {
        "increasing or decreasing": 60,
        "increasing and decreasing": 13
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "indefinite-integral",
    "ja": {
      "term": "不定積分",
      "reading": "ふていせきぶん",
      "alt": []
    },
    "en": {
      "term": "indefinite integral",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学II・III）には「不定積分」だけが出てきて、「原始関数」は出てこない。OpenStax Calculus Volume 1 は、微分すると f になる関数 1 つを antiderivative、+C を付けた全体（∫f(x)dx が表すもの。the most general antiderivative とも）を indefinite integral と呼び分ける。「不定積分を求めよ」は Find the indefinite integral ／ Evaluate ∫ f(x) dx と書かれ、OpenStax Calculus と講義（Khan Academy ほか）では find the antiderivative とも言う。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "微分すると f(x) になる関数全体を、積分定数 C を付けて F(x) + C の形で表したもの。記号 ∫f(x)dx で書く。",
    "definition_en": "The whole family of antiderivatives of f, written ∫ f(x) dx = F(x) + C.",
    "latex": "\\int f(x)\\,dx = F(x) + C",
    "spoken_en": "the integral of f of x dx equals F of x plus C",
    "examples": [
      {
        "en": "An indefinite integral gives you a whole family of functions, so don't drop the plus C.",
        "ja": "不定積分は関数の族全体を表すから、+C を落とさないように。",
        "register": "spoken"
      },
      {
        "en": "The indefinite integral of 3x² is x³ + C.",
        "ja": "3x² の不定積分は x³ + C である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "find the indefinite integral",
        "ja": "不定積分を求める"
      }
    ],
    "pitfalls": [
      "「ただし C は積分定数」に当たる一文は、英語で書くなら where C is a constant。",
      "∫(1/x)dx は OpenStax Calculus では ln|x| + C と書く。OpenStax Algebra and Trigonometry は底を省いた log x を常用対数（底 10）と定める（学習指導要領解説には ln が出てこない）ので、英語の答案では ln を使う。"
    ],
    "related": [
      "antiderivative",
      "constant-of-integration",
      "definite-integral",
      "integrate",
      "integration-formulas",
      "evaluate-the-integral"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "不定積分",
        "en": "Indefinite integral"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.8（antiderivatives と indefinite integrals）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 微分・積分の考え・数学III 積分法: 不定積分（原始関数は出てこない）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "底を省いた log x は常用対数（底 10）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "indefinite integral": 114
      },
      "written": {
        "indefinite integral": 64
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "independence-of-events",
    "ja": {
      "term": "事象の独立",
      "reading": "じしょうのどくりつ",
      "alt": [
        "独立"
      ]
    },
    "en": {
      "term": "independent events",
      "alt": [
        "independence of events"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "数A",
        "理数"
      ],
      "us": [
        "Algebra 2",
        "Intro Statistics",
        "Discrete Math",
        "Integrated Math 2",
        "AP Statistics",
        "Geometry"
      ]
    },
    "definition_ja": "一方の事象が起こるかどうかが、もう一方の事象の起こる確率に影響しないこと。P(A ∩ B) = P(A)P(B) が成り立つ。",
    "definition_en": "Two events are independent when whether one happens does not change the probability of the other; then P(A and B) = P(A)P(B).",
    "latex": "P(A \\cap B) = P(A)\\,P(B)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Getting heads on the first flip and getting heads on the second flip are independent events: the first flip doesn't change the second.",
        "ja": "1 回目に表が出ることと 2 回目に表が出ることは独立な事象だ。1 回目は 2 回目に影響しない。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the events A and B are independent.",
        "ja": "事象 A と B が独立かどうか判定せよ。",
        "register": "written"
      },
      {
        "en": "A coin toss and a die roll are independent trials, so P(heads and a 6) = (1/2)(1/6) = 1/12.",
        "ja": "硬貨を投げることとさいころを振ることは独立な試行なので、P(表かつ 6) = (1/2)(1/6) = 1/12。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "排反（mutually exclusive）と取り違えない。確率が 0 でない 2 つの事象が排反なら、独立ではない。",
      "学習指導要領解説（数学A）は、独立な試行（2 つの試行の結果が互いに影響を及ぼさない試行）の確率を P(C) = P(A)P(B) で扱い、事象の独立・従属は理数数学I の発展・拡充の内容に挙げる。確率変数の独立は independent-random-variables。",
      "独立でない事象は dependent events（従属な事象）と言う。"
    ],
    "related": [
      "general-multiplication-rule",
      "independent-random-variables",
      "test-for-independence",
      "repeated-trials",
      "mutually-exclusive-events",
      "conditional-probability"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.7"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A (2): 独立な試行の確率 P(C)=P(A)P(B)。事象の独立や従属は理数数学I の発展・拡充"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 8.7 Related Events〜8.10 Using Probability to Determine Whether Events Are Independent"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "independent events | events are independent | independence of events | event … independent of": 22,
        "independence of events": 1
      },
      "written": {
        "independent events | events are independent | independence of events | event … independent of": 61
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "independent-random-variables",
    "ja": {
      "term": "独立な確率変数",
      "reading": "どくりつなかくりつへんすう",
      "alt": []
    },
    "en": {
      "term": "independent random variables",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "一方の確率変数の値が、他方の確率変数の値のとり方に影響しないような確率変数の組。",
    "definition_en": "Random variables such that the value of one does not affect the probabilities of the values of the other.",
    "latex": "P(X = a,\\ Y = b) = P(X = a)\\,P(Y = b)",
    "spoken_en": null,
    "examples": [
      {
        "en": "You can only add the variances like this because the two random variables are independent.",
        "ja": "2 つの確率変数が独立だから、こうして分散を足せる。",
        "register": "spoken"
      },
      {
        "en": "Show that X and Y are independent random variables.",
        "ja": "X と Y が独立な確率変数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はほとんどが MIT の講義ノート（離散数学）。"
    ],
    "related": [
      "sum-of-random-variables",
      "random-variable",
      "independence-of-events"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "5.2 Double Integrals over General Regions（independent random variables）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Independence_(probability_theory)",
        "note": "節 Two random variables（independent random variables の定義）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学II (8) 統計的な推測（発展・拡充）: 二つの確率変数の独立、確率変数の和、互いに独立であるときの分散の性質"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "independent random variables": 11
      },
      "written": {
        "independent random variables": 15
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "independent-system",
    "ja": {
      "term": "独立な連立方程式",
      "reading": "どくりつなれんりつほうていしき",
      "alt": []
    },
    "en": {
      "term": "independent system",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「独立な連立方程式」は学習指導要領解説に無い、本プロジェクトの訳語。日本では「解がただ 1 組」「解が無数にある（不定）」「解がない（不能）」と言い、連立方程式の種類に名前を付けない。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "連立一次方程式のうち、解がちょうど 1 組あるもの。2 つの式のグラフの直線は 1 点で交わる。",
    "definition_en": "A system of linear equations that has exactly one solution. The graphs of the equations are lines that intersect at one point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These two lines have different slopes, so they cross exactly once, which makes this an independent system with one solution.",
        "ja": "この 2 本の直線は傾きが違うから、ちょうど 1 回交わる。だから解がただ 1 組の連立方程式だね。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the system 2x + y = 5, x − y = 1 is an independent system, a dependent system, or an inconsistent system.",
        "ja": "連立方程式 2x + y = 5、x − y = 1 について、解がただ 1 組か、無数にあるか（不定）、解がないか（不能）を判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は、解が無数にあるものを dependent system、解がないものを inconsistent system と呼び、解があるもの（independent と dependent）を consistent と呼ぶ。",
      "OpenStax Elementary Algebra は independent を「2 つの式のグラフが別の直線」の意味で使い、平行な 2 直線を inconsistent and independent と分類する。教科書によって分類の仕方が違う。",
      "線形代数の linearly independent（エントリ linearly-independent）とは別の用語。"
    ],
    "related": [
      "system-of-linear-equations",
      "solving-by-graphing",
      "infinitely-many-solutions",
      "no-solution",
      "linearly-independent"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "independent system": 2
      },
      "written": {
        "independent system": 8
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは OpenStax の呼び方 independent system（本文 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "indeterminate-form",
    "ja": {
      "term": "不定形",
      "reading": "ふていけい",
      "alt": []
    },
    "en": {
      "term": "indeterminate form",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "極限を求めるとき、0/0・∞/∞・∞ − ∞ などの形になって、そのままでは値が決まらないもの。",
    "definition_en": "A form such as 0/0, ∞/∞, or ∞ − ∞ that a limit can take, which on its own does not determine the value.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Plugging in gives zero over zero, an indeterminate form, so factor and cancel.",
        "ja": "代入すると 0/0 の不定形になるので、因数分解して約分する。",
        "register": "spoken"
      },
      {
        "en": "The limit lim_{x→∞} (x² + 1)/(2x²) is of the indeterminate form ∞/∞.",
        "ja": "極限 lim_{x→∞} (x² + 1)/(2x²) は ∞/∞ の不定形である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "0/0 は zero over zero、∞/∞ は infinity over infinity、∞ − ∞ は infinity minus infinity と読む。",
      "不定形になっても極限がないとは限らない。約分・有理化、または L'Hôpital's rule で求める。"
    ],
    "related": [
      "limit",
      "find-the-limit"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "不定形",
        "en": "Indeterminate form"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 4.7（indeterminate form）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "indeterminate form": 71
      },
      "written": {
        "indeterminate form": 41
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "indirect-measurement",
    "ja": {
      "term": "測量",
      "reading": "そくりょう",
      "alt": [
        "間接測定"
      ]
    },
    "en": {
      "term": "indirect measurement",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "高等学校学習指導要領解説は、数学A「数学と人間の活動」で三角比を用いた測量の方法を題材にすると書き、数学I「図形と計量」では直接測定できない 2 地点間の距離などを三角比で求める活動として書く（「測量」の語は数学A）。中学校学習指導要領解説（第 3 学年）は相似・三平方の定理で高さや距離を求める学習を挙げる。英語の授業の単元では indirect measurement（CK-12 Geometry 7.5 Indirect Measurement・7.6 Indirect Measurement Applications ほか、8 件。IM は 3 件）や angle of elevation の問題として出てくる。用例コーパスには出てこない。",
    "domains": [
      "geometry",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "中3",
        "数I",
        "数A"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直接測れない高さや距離を、測れる長さや角から相似や三角比を使って計算で求めること。",
    "definition_en": "Finding a height or distance that cannot be measured directly by calculating it from lengths and angles that can be measured, using similar triangles or trigonometric ratios.",
    "latex": "h = d \\tan \\theta",
    "spoken_en": "h equals d times tangent theta",
    "examples": [
      {
        "en": "We can't climb the flagpole, so we'll use indirect measurement: measure the two shadows and set up similar triangles.",
        "ja": "旗ざおには登れないので、測量で高さを求めよう。2 つの影の長さを測って相似な三角形をつくる。",
        "register": "spoken"
      },
      {
        "en": "A 1.5 m post casts a 2 m shadow at the same time that a building casts a 24 m shadow. Use indirect measurement to find the height of the building.",
        "ja": "長さ 1.5 m の棒の影が 2 m のとき、同じ時刻に建物の影は 24 m であった。建物の高さを求めよ。",
        "register": "written"
      },
      {
        "en": "From a point 40 m from the base of a tower, the angle of elevation to the top is 35°. Find the height of the tower to the nearest meter.",
        "ja": "塔の根もとから 40 m 離れた地点で塔の頂上を見上げた仰角は 35° であった。塔の高さを、小数第 1 位を四捨五入して求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry・Precalculus の問題文は「測量」という語を使わず、angle of elevation（各 44 件）／ angle of depression と長さを与えて高さや距離を求めさせる形になる。"
    ],
    "related": [
      "angle-of-elevation",
      "angle-of-depression",
      "similar-triangles",
      "trigonometric-ratio"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "7.5 Indirect Measurement・7.6 Indirect Measurement Applications・4.28 Basics of Pythagorean Theorem"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 数学と人間の活動: 図形の性質や三角比を用いた測量の方法"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Geometry 4.1 Angles and Steepness ほか: indirect measurement（3 件）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形: 相似・三平方の定理で高さや距離を求める"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 indirect measurement（CK-12 Geometry 7.5 Indirect Measurement・CK-12 Geometry 7.6 Indirect Measurement Applications・CK-12 Geometry 4.28 Basics of Pythagorean Theorem・計 8 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "inductive-hypothesis",
    "ja": {
      "term": "帰納法の仮定",
      "reading": "きのうほうのかてい",
      "alt": []
    },
    "en": {
      "term": "induction hypothesis",
      "alt": [],
      "variants": [
        {
          "term": "induction hypothesis",
          "register": "both",
          "note": "話し言葉・書き言葉とも最も多いが、すべて MIT 6.042（講義とそのテキスト）。"
        },
        {
          "term": "inductive hypothesis",
          "register": "spoken",
          "note": "同じく MIT 6.042 の講義で使う。話し言葉（MIT OCW）・書き言葉（MIT の講義ノート）とも induction hypothesis より少ない。Levin（4.5・4.6）は inductive hypothesis と書く。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "数学的帰納法で「n = k のとき成り立つ」とおく仮定。",
    "definition_en": "In a proof by induction, the assumption that the statement holds for n = k.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Now use the induction hypothesis to replace the first k terms.",
        "ja": "ここで帰納法の仮定を使って、最初の k 項を置き換える。",
        "register": "spoken"
      },
      {
        "en": "By the induction hypothesis, 1 + 3 + ⋯ + (2k − 1) = k².",
        "ja": "帰納法の仮定より、1 + 3 + ⋯ + (2k − 1) = k²。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「n = k のとき成り立つと仮定する」は Assume that P(k) holds for an arbitrary k ≥ 1. のように書く（Levin 4.5 は k を置くときに arbitrary を付ける）。"
    ],
    "related": [
      "proof-by-induction",
      "base-case",
      "prove-by-induction",
      "inductive-step",
      "strong-induction"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Mathematical_induction",
        "note": "記事「Mathematical induction」の節 Description で induction hypothesis と inductive hypothesis を太字で定義（数学カテゴリから 3 段）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.5 Proof by Induction・4.6 Strong Induction（inductive hypothesis、Assume P(k) is true）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "induction hypothesis": 53,
        "inductive hypothesis": 20
      },
      "written": {
        "induction hypothesis": 30,
        "inductive hypothesis": 6
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inductive-reasoning",
    "ja": {
      "term": "帰納的推論",
      "reading": "きのうてきすいろん",
      "alt": []
    },
    "en": {
      "term": "inductive reasoning",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof",
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "いくつかの具体例に共通する規則性から、一般的な結論を推測する考え方。推測した結論は、証明するまで正しいとは限らない。",
    "definition_en": "Reasoning that looks at specific examples, notices a pattern, and makes a general conjecture. The conclusion is only a guess until it is proved.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Every example so far works, but that's just inductive reasoning. We still need a proof.",
        "ja": "ここまでの例は全部成り立つけど、それは帰納的推論にすぎない。まだ証明が要る。",
        "register": "spoken"
      },
      {
        "en": "Use inductive reasoning to predict the next number in the pattern 3, 7, 11, 15, ….",
        "ja": "帰納的推論を用いて、3, 7, 11, 15, … の次の数を予想せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数学的帰納法（proof by induction）は名前に「帰納」が付くが演繹的な証明であり、帰納的推論ではない。",
      "帰納的推論で得た予想は、反例（counterexample）が 1 つあれば否定される。",
      "用例コーパスには inductive reasoning は出てこない。CK-12 Geometry は 21 件（2.4 Reasoning Types・2.5 Inductive Reasoning from Patterns ほか）、Levin の本では 4.5 Proof by Induction に 1 件。"
    ],
    "related": [
      "deductive-reasoning",
      "proof-by-induction",
      "counterexample"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.5 Proof by Induction"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.5 Inductive Reasoning from Patterns・2.4 Reasoning Types・2.6 Conjectures and Counterexamples"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 inductive reasoning（CK-12 Geometry 2.5 Inductive Reasoning from Patterns・CK-12 Geometry 2.4 Reasoning Types・CK-12 Geometry 2.6 Conjectures and Counterexamples・計 21 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "inductive-step",
    "ja": {
      "term": "帰納段階",
      "reading": "きのうだんかい",
      "alt": [
        "n = k + 1 のとき"
      ]
    },
    "en": {
      "term": "inductive step",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「帰納段階」は学習指導要領解説に無い（日本語版 Wikipedia「再帰」は数学的帰納法の証明の段を「基底段階」「帰納段階」と呼ぶ）。日本の答案では「n = k のとき成り立つと仮定すると、n = k + 1 のとき …」と書き、段階に名前をつけない。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Discrete Math"
      ]
    },
    "definition_ja": "数学的帰納法で、n = k のとき成り立つと仮定して、n = k + 1 のときも成り立つことを示す段。",
    "definition_en": "The part of a proof by induction that assumes the statement for n = k and proves it for n = k + 1.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For the inductive step, assume it's true for k and show it's true for k plus 1.",
        "ja": "帰納段階では、k で成り立つと仮定して、k + 1 でも成り立つことを示す。",
        "register": "spoken"
      },
      {
        "en": "Complete the inductive step: assuming 1 + 2 + … + k = k(k + 1)/2, prove the formula for n = k + 1.",
        "ja": "1 + 2 + … + k = k(k + 1)/2 を仮定して、n = k + 1 のときも公式が成り立つことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは MIT 6.042（講義と講義ノート）だけ。OpenStax 6 冊にも台帳の Wikipedia にもない。"
    ],
    "related": [
      "base-case",
      "inductive-hypothesis",
      "proof-by-induction"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Mathematical_induction",
        "note": "記事「Mathematical induction」の節 Description で inductive step（と induction step）を太字で定義（数学カテゴリから 3 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「再帰」",
        "url": "https://ja.wikipedia.org/wiki/%E5%86%8D%E5%B8%B0",
        "note": "数学的帰納法による証明の「基底段階」「帰納段階」"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "flags": [],
    "evidence": {
      "spoken": {
        "inductive step": 34
      },
      "written": {
        "inductive step": 40
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inequality-sign",
    "ja": {
      "term": "不等号",
      "reading": "ふとうごう",
      "alt": []
    },
    "en": {
      "term": "inequality sign",
      "alt": [],
      "variants": [
        {
          "term": "inequality sign",
          "register": "written",
          "note": "書き言葉では inequality symbol と同じくらい使う（OpenStax Elementary Algebra に多い）。話し言葉ではどちらも少ない。"
        },
        {
          "term": "inequality symbol",
          "register": "written",
          "note": "教科書（OpenStax Prealgebra・Elementary Algebra）の書き方。AP Statistics の CED もこの言い方。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数や式の大小関係を表す記号 <、>、≦、≧ のこと。",
    "definition_en": "Any of the symbols <, >, ≤, or ≥, which compare two quantities by showing which one is greater.",
    "latex": "<,\\ >,\\ \\le,\\ \\ge",
    "spoken_en": "less than, greater than, less than or equal to, greater than or equal to",
    "examples": [
      {
        "en": "You divided both sides by negative 2, so don't forget to flip the inequality sign.",
        "ja": "両辺を −2 で割ったんだから、不等号の向きを変えるのを忘れないで。",
        "register": "spoken"
      },
      {
        "en": "Write the correct inequality sign in the blank: −3 ___ −5.",
        "ja": "□ に当てはまる不等号を書け。−3 □ −5",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の ≦ ≧（共通テスト・センター試験の問題文、学習指導要領解説）は、米国の教科書（OpenStax）では ≤ ≥ と書く。",
      "両辺に負の数を掛けたり、両辺を負の数で割ったりしたときは、不等号の向きを変える。英語では flip the inequality sign ／ reverse the inequality sign と言う（エントリ properties-of-inequalities）。",
      "a < b は a is less than b、a > b は a is greater than b と読む（エントリ less-than、greater-than）。"
    ],
    "related": [
      "inequality",
      "properties-of-inequalities",
      "less-than",
      "greater-than",
      "less-than-or-equal-to"
    ],
    "sources": [
      {
        "type": "wikidata",
        "qid": "Q53045737"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "問題文の不等号 ≦ ≧"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "不等号 ≦ ≧"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.5（inequality symbol）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "inequality sign": 7,
        "inequality symbol": 3
      },
      "written": {
        "inequality sign": 13,
        "inequality symbol": 11
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inequality",
    "ja": {
      "term": "不等式",
      "reading": "ふとうしき",
      "alt": []
    },
    "en": {
      "term": "inequality",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "不等号（<、>、≦、≧）を使って、2 つの数や式の大小関係を表した式。文字をふくむ不等式の解は、x > 3 のように範囲で表す。",
    "definition_en": "A mathematical statement that compares two expressions using <, >, ≤, or ≥. The solution of an inequality with a variable is usually a range of values, such as x > 3.",
    "latex": "2x - 1 > 5",
    "spoken_en": "two x minus one is greater than five",
    "examples": [
      {
        "en": "When you solve an inequality, you usually get a whole range of answers, like x is greater than 3.",
        "ja": "不等式を解くと、ふつうは x > 3 みたいに、答えが範囲になるよ。",
        "register": "spoken"
      },
      {
        "en": "Solve the inequality 2x − 1 > 5 and graph the solution on a number line.",
        "ja": "不等式 2x − 1 > 5 を解き、解を数直線上に表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の ≦ ≧（共通テスト・センター試験の問題文、学習指導要領解説）は、米国の教科書（OpenStax）では ≤ ≥ と書く。読み方は less than or equal to ／ greater than or equal to（エントリ less-than-or-equal-to、greater-than-or-equal-to）。",
      "三角不等式（triangle inequality）の inequality も同じ語で、不等式で表される性質の名前になっている（エントリ triangle-inequality）。"
    ],
    "related": [
      "equation",
      "inequality-sign",
      "linear-inequality",
      "properties-of-inequalities",
      "triangle-inequality"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "不等式",
        "en": "Inequality (mathematics)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "問題文の不等号 ≦ ≧"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "不等号 ≦ ≧"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "inequality": 445
      },
      "written": {
        "inequality": 878
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inferential-statistics",
    "ja": {
      "term": "推測統計",
      "reading": "すいそくとうけい",
      "alt": [
        "推測統計学",
        "統計的推測"
      ]
    },
    "en": {
      "term": "statistical inference",
      "alt": [
        "inferential statistics"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "標本のデータから、確率を使って母集団について推定や検定を行う統計の方法。",
    "definition_en": "Using sample data and probability to draw conclusions about a population, by estimation and hypothesis testing.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Descriptive statistics describe the sample; inference goes from the sample to the population.",
        "ja": "記述統計は標本を記述し、推測は標本から母集団へ進む。",
        "register": "spoken"
      },
      {
        "en": "Explain why a confidence interval is an example of statistical inference.",
        "ja": "信頼区間が統計的推測の例である理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数学 B の単元名は「統計的な推測」。",
      "用例コーパスでは inferential statistics も statistical inference も少ない。AP Statistics の CED（2026 年版）は statistical inference と書く（単元 3 の概要ほか）。"
    ],
    "related": [
      "statistics",
      "descriptive-statistics",
      "confidence-interval",
      "hypothesis-testing"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "単元 3 の概要・Topic 3.2〜3.4"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "statistical inference": 1,
        "inferential statistics": 3
      },
      "written": {
        "statistical inference": 4,
        "inferential statistics": 4
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは AP Statistics の CED の呼び方 statistical inference（topic 3.2・topic 3.3・topic 3.4・topic 3.5・topic 3.6・topic 3.7・topic 3.8・topic 3.9・topic 3.10・topic 3.11・topic 3.12・topic 3.13・topic 3.14・topic 4.1・topic 4.2・topic 4.3・topic 4.4・topic 4.5・topic 4.6・topic 4.7・topic 4.8・topic 4.9）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "infinite-discontinuity",
    "ja": {
      "term": "無限不連続",
      "reading": "むげんふれんぞく",
      "alt": []
    },
    "en": {
      "term": "infinite discontinuity",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「無限不連続」は学習指導要領解説に無い（日本語版 Wikipedia「不連続性の分類」には「無限不連続点」がある）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "片側極限の少なくとも一方が ±∞ になる不連続点。そこに垂直な漸近線がある。",
    "definition_en": "A discontinuity where at least one one-sided limit is ∞ or −∞; there is a vertical asymptote there.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "At x = 0, one over x has an infinite discontinuity.",
        "ja": "1/x は x = 0 で無限不連続になる。",
        "register": "spoken"
      },
      {
        "en": "The function f(x) = 1/(x − 3) has an infinite discontinuity at x = 3.",
        "ja": "関数 f(x) = 1/(x − 3) は x = 3 で無限不連続となる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "discontinuous",
      "infinite-limit",
      "removable-discontinuity",
      "jump-discontinuity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「不連続性の分類」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%8D%E9%80%A3%E7%B6%9A%E6%80%A7%E3%81%AE%E5%88%86%E9%A1%9E",
        "note": "無限不連続点（infinite discontinuity）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「無限不連続」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "infinite discontinuity": 5
      },
      "written": {
        "infinite discontinuity": 16
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "infinite-geometric-sequence",
    "ja": {
      "term": "無限等比数列",
      "reading": "むげんとうひすうれつ",
      "alt": []
    },
    "en": {
      "term": "infinite geometric sequence",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "項が限りなく続く等比数列 a, ar, ar², …。",
    "definition_en": "A geometric sequence a, ar, ar², … that goes on forever.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "An infinite geometric sequence with ratio one half goes to zero.",
        "ja": "公比 1/2 の無限等比数列は 0 に収束する。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the infinite geometric sequence 3, 3/2, 3/4, … converges.",
        "ja": "無限等比数列 3, 3/2, 3/4, … が収束するかどうか調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数列 {rⁿ} が収束するのは −1 < r ≦ 1 のとき。無限等比級数（infinite geometric series）の収束条件 |r| < 1 と混同しない。英語では sequence（数列）と series（級数）を区別する。"
    ],
    "related": [
      "infinite-series",
      "convergence"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "infinite geometric sequence": 1
      },
      "written": {
        "infinite geometric sequence": 3
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは OpenStax の呼び方 infinite geometric sequence（本文 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "infinite-geometric-series",
    "ja": {
      "term": "無限等比級数",
      "reading": "むげんとうひきゅうすう",
      "alt": [
        "無限幾何級数"
      ]
    },
    "en": {
      "term": "infinite geometric series",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "等比数列の項を限りなく足した級数。|r| < 1 のとき収束して、和は a/(1 − r)。",
    "definition_en": "The series formed by adding all the terms of a geometric sequence; it converges to a/(1 − r) when |r| < 1.",
    "latex": "\\sum_{n=1}^{\\infty} ar^{n-1} = \\frac{a}{1-r} \\quad (|r| < 1)",
    "spoken_en": "the sum from n equals one to infinity of a r to the n minus one equals a over one minus r, when the absolute value of r is less than one",
    "examples": [
      {
        "en": "As long as the ratio is between −1 and 1, the infinite geometric series converges.",
        "ja": "公比が −1 と 1 の間にあれば、無限等比級数は収束する。",
        "register": "spoken"
      },
      {
        "en": "Find the sum of the infinite geometric series 8 − 4 + 2 − 1 + ⋯.",
        "ja": "無限等比級数 8 − 4 + 2 − 1 + ⋯ の和を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED（topic 10.2）と OpenStax Calculus Volume 2（5.2）は単に geometric series と呼ぶ。geometric series は有限の和にも使う。",
      "初項が 0 でなく、公比の絶対値が 1 以上（|r| ≥ 1）なら発散する。"
    ],
    "related": [
      "geometric-sequence",
      "common-ratio",
      "infinite-series",
      "sum-of-the-series",
      "sum-of-a-geometric-sequence",
      "repeating-decimal"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.4 Series and Their Notations（infinite geometric series）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.2（geometric series）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.2 Infinite Series（geometric series）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "infinite geometric series": 28
      },
      "written": {
        "infinite geometric series": 54
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-notes",
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "infinite-limit",
    "ja": {
      "term": "無限大の極限",
      "reading": "むげんだいのきょくげん",
      "alt": []
    },
    "en": {
      "term": "infinite limit",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「無限大の極限」は学習指導要領解説に無い（日本語版 Wikipedia「数列の極限」には節「無限大の極限」がある）。日本語では同じことを「正の無限大（負の無限大）に発散する」と言う（日本語版 Wikipedia「極限」）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "変数を近づけたとき関数の値が限りなく大きく（または小さく）なる極限。lim f(x) = ∞ と書く。",
    "definition_en": "A limit in which f(x) grows without bound (toward ∞ or −∞), written lim f(x) = ∞.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "That's an infinite limit, so there's a vertical asymptote at x = 2.",
        "ja": "無限大の極限なので、x = 2 に垂直な漸近線がある。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the infinite limit lim_{x→0} 1/x².",
        "ja": "極限 lim_{x→0} 1/x² を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "diverge-to-positive-infinity",
      "limit-at-infinity",
      "infinite-discontinuity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.14（infinite limits）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「極限」",
        "url": "https://ja.wikipedia.org/wiki/%E6%A5%B5%E9%99%90",
        "note": "正の無限大に発散する"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数列の極限」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%88%97%E3%81%AE%E6%A5%B5%E9%99%90",
        "note": "節「無限大の極限」"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「無限大の極限」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "infinite limit": 1
      },
      "written": {
        "infinite limit": 36
      },
      "sources": [
        "mit-18.01",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "infinite-series",
    "ja": {
      "term": "無限級数",
      "reading": "むげんきゅうすう",
      "alt": [
        "級数"
      ]
    },
    "en": {
      "term": "infinite series",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "sequences"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "数列の項を限りなく加えた式 a₁ + a₂ + a₃ + ⋯。",
    "definition_en": "An endless sum a₁ + a₂ + a₃ + ⋯ of the terms of a sequence.",
    "latex": "\\sum_{n=1}^{\\infty} a_n",
    "spoken_en": "the sum from n equals one to infinity of a sub n",
    "examples": [
      {
        "en": "An infinite series is really the limit of its partial sums.",
        "ja": "無限級数とは、実は部分和の極限のこと。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the infinite series ∑ 1/2ⁿ converges.",
        "ja": "無限級数 ∑ 1/2ⁿ が収束するかどうか調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "series は単複同形（a series ／ two series）。sequence（数列）と取り違えない。",
      "日本語版 Wikipedia「級数」は級数を無限項の和として定義し（有限個の和は残りの項を 0 とみる場合）、学習指導要領解説は「無限級数」「無限等比級数」の形で使う。英語の series は有限個の項の和にも使い、OpenStax Algebra and Trigonometry は初項から第 n 項までの和を arithmetic series と呼ぶ。"
    ],
    "related": [
      "sum-of-the-series",
      "convergence",
      "infinite-geometric-sequence",
      "partial-sum",
      "infinite-geometric-series",
      "converge",
      "diverge",
      "nth-term-test",
      "integral-test",
      "comparison-test",
      "ratio-test",
      "alternating-series",
      "power-series"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "級数",
        "en": "Series (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.1（infinite series）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「級数」",
        "url": "https://ja.wikipedia.org/wiki/%E7%B4%9A%E6%95%B0",
        "note": "級数を無限項の和として定義（有限和は特別な場合）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 極限: 無限級数・無限等比級数"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "arithmetic series（初項から第 n 項までの和）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "infinite series": 101
      },
      "written": {
        "infinite series": 71
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "infinitely-many-solutions",
    "ja": {
      "term": "解が無数にある",
      "reading": "かいがむすうにある",
      "alt": [
        "従属な連立方程式"
      ]
    },
    "en": {
      "term": "infinitely many solutions",
      "alt": [
        "dependent system"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "連立方程式の 2 つの式が同じ直線を表し、それを満たす x, y の値の組が限りなくあること。解いていくと 0 = 0 のような、いつでも成り立つ式になる。",
    "definition_en": "A system has infinitely many solutions when both equations describe the same line, so every point on that line is a solution. Solving it ends in a statement that holds for every x and y, like 0 = 0.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Everything cancels and you get 0 = 0, which is always true, so there are infinitely many solutions.",
        "ja": "全部消えて 0 = 0 になる。これはいつでも成り立つから、解が無数にある。",
        "register": "spoken"
      },
      {
        "en": "The second equation is just the first one times 3, so they're the same line. It's a dependent system.",
        "ja": "2 つ目の式は 1 つ目の式を 3 倍しただけだから、同じ直線だ。この連立方程式は解が無数にある。",
        "register": "spoken"
      },
      {
        "en": "Show that the system x + 2y = 4, 3x + 6y = 12 has infinitely many solutions.",
        "ja": "連立方程式 x + 2y = 4、3x + 6y = 12 の解が無数にあることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry（dependent system 18 件）は、解が無数にある連立方程式そのものを dependent system と呼ぶ。「解が無数にある」という答えは infinitely many solutions と書く。",
      "解なし（エントリ no-solution）と区別する。式を整理して x も y も消えたとき、残る式が 0 = 0 なら解が無数にあり、0 = 5 なら解なし。",
      "解が無数にあっても、どんな x, y でもよいわけではない。2 つの式が表す同じ直線の上の点の座標だけが解になる。"
    ],
    "related": [
      "no-solution",
      "system-of-linear-equations",
      "coincide",
      "linear-equation-in-two-variables",
      "solution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "dependent system（18 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "infinitely many solutions": 23,
        "dependent system": 6
      },
      "written": {
        "infinitely many solutions": 63,
        "dependent system": 19
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "infinitude-of-primes",
    "ja": {
      "term": "素数の無限性",
      "reading": "そすうのむげんせい",
      "alt": [
        "素数が無限に存在すること"
      ]
    },
    "en": {
      "term": "Euclid's theorem",
      "alt": [
        "infinitude of primes",
        "infinitely many primes"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Euclid's theorem ／ infinitude of primes ／ infinitely many primes は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory",
      "proof"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "definition_ja": "素数は無限に多く存在するという定理。素数が有限個しかないと仮定し、それらすべての積に 1 を足した数を考えて矛盾を導く背理法の証明がよく知られている。",
    "definition_en": "The theorem that there are infinitely many prime numbers. The classic proof assumes there are only finitely many, multiplies them all together, adds 1, and reaches a contradiction.",
    "latex": "N = p_1 p_2 \\cdots p_n + 1",
    "spoken_en": "N equals p sub 1 times p sub 2 times dot dot dot times p sub n, plus 1",
    "examples": [
      {
        "en": "Euclid's theorem says the primes never run out, and its proof is a classic example of proof by contradiction.",
        "ja": "ユークリッドの定理は素数が尽きないことを言っていて、その証明は背理法の代表例だ。",
        "register": "spoken"
      },
      {
        "en": "Prove Euclid's theorem: there are infinitely many primes.",
        "ja": "素数は無限に存在することを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "証明で作る p₁p₂⋯pₙ + 1 そのものが素数とは限らない。2 × 3 × 5 × 7 × 11 × 13 + 1 = 30031 = 59 × 509。",
      "「素数は無限にある」を infinite primes とすると「無限大の素数」と読めるので、there are infinitely many primes と言う。"
    ],
    "related": [
      "prime-factorization",
      "proof-by-contradiction"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Euclid%27s_theorem",
        "note": "記事名 Euclid's theorem（en.term のリダイレクト先。見出しを決めた参照）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "infinitely many primes": 2
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは Wikipedia の記事名 Euclid's theorem（英語版「Euclid's theorem」、en.term のリダイレクト先）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "infinity",
    "ja": {
      "term": "無限大",
      "reading": "むげんだい",
      "alt": []
    },
    "en": {
      "term": "infinity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "どんな実数よりも大きいことを表す記号 ∞ と、その考え。実数ではない。",
    "definition_en": "The idea, written ∞, of being larger than every real number; it is not a real number.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Infinity isn't a number, so you can't just plug it in.",
        "ja": "無限大は数ではないので、そのまま代入することはできない。",
        "register": "spoken"
      },
      {
        "en": "As x → ∞, 1/x → 0.",
        "ja": "x → ∞ のとき 1/x → 0 である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "−∞ は negative infinity（口頭では minus infinity とも）。"
    ],
    "related": [
      "limit",
      "as-x-approaches-infinity",
      "diverge-to-positive-infinity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "infinity": 2308
      },
      "written": {
        "infinity": 219
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inflection-point",
    "ja": {
      "term": "変曲点",
      "reading": "へんきょくてん",
      "alt": []
    },
    "en": {
      "term": "inflection point",
      "alt": [],
      "variants": [
        {
          "term": "point of inflection",
          "register": "both",
          "note": "少数の言い方だが、AP の CED は topic 5.6 でこちらを使う。話し言葉は大半が Khan Academy、書き言葉は大半が OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線の凹凸が入れ替わる点。",
    "definition_en": "A point where a graph changes concavity.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The inflection point is where it switches from concave down to concave up.",
        "ja": "変曲点は、上に凸から下に凸へ切り替わるところ。",
        "register": "spoken"
      },
      {
        "en": "Find the inflection points of f(x) = x⁴ − 6x².",
        "ja": "f(x) = x⁴ − 6x² の変曲点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "f″(c) = 0 でも凹凸が入れ替わらなければ変曲点ではない（x⁴ の x = 0）。"
    ],
    "related": [
      "concavity",
      "second-derivative",
      "changes-sign"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "変曲点",
        "en": "Inflection point"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.6（point of inflection）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "inflection point": 153,
        "point of inflection": 25
      },
      "written": {
        "inflection point": 41,
        "point of inflection": 7
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "influential-point",
    "ja": {
      "term": "影響点",
      "reading": "えいきょうてん",
      "alt": []
    },
    "en": {
      "term": "influential point",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「影響点」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "取り除くと回帰直線の傾きや切片が大きく変わる点。x の値が大きく外れた点が多い。",
    "definition_en": "A point whose removal would change the slope or intercept of the regression line a lot; often a point far out in the x-direction.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "That point is way off to the right, so it's probably influential.",
        "ja": "あの点は右にずっと離れているので、たぶん影響点。",
        "register": "spoken"
      },
      {
        "en": "Identify any influential points and describe how removing them would change the regression line.",
        "ja": "影響点があれば挙げ、それを取り除くと回帰直線がどう変わるか述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "外れ値（outlier）がいつも影響点になるわけではない。"
    ],
    "related": [
      "outlier",
      "regression-line",
      "residual-plot"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "influential point": 13
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "initial-condition",
    "ja": {
      "term": "初期条件",
      "reading": "しょきじょうけん",
      "alt": []
    },
    "en": {
      "term": "initial condition",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II"
      ]
    },
    "definition_ja": "微分方程式の解を 1 つに決めるために与える、ある点での関数の値などの条件。",
    "definition_en": "A condition such as y(0) = 2, giving the function's value at a single point, that picks out one particular solution of a differential equation.",
    "latex": "y(0) = 2",
    "spoken_en": "y of zero equals two",
    "examples": [
      {
        "en": "The initial condition tells you which curve in the family you're on.",
        "ja": "初期条件で、どの曲線の上にいるかが決まる。",
        "register": "spoken"
      },
      {
        "en": "Solve dy/dt = −0.2y subject to the initial condition y(0) = 50.",
        "ja": "初期条件 y(0) = 50 のもとで dy/dt = −0.2y を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "初期条件を付けた微分方程式の問題は initial value problem（初期値問題、IVP）。"
    ],
    "related": [
      "particular-solution",
      "differential-equation",
      "slope-field"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 7.7–7.9（initial condition）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "initial condition": 347
      },
      "written": {
        "initial condition": 120
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "initial-point",
    "ja": {
      "term": "始点",
      "reading": "してん",
      "alt": []
    },
    "en": {
      "term": "initial point",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "書き言葉は initial point（すべて OpenStax）。話し言葉では矢印の尾の意味で the tail（of the vector）とも言うが、用例コーパスの tail は分布の裾や硬貨の裏（tails）がほとんどなので数えていない。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "有向線分（ベクトル）の出発点。",
    "definition_en": "The starting point of a directed line segment (vector).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Put the tail of the second vector at the tip of the first.",
        "ja": "2 つ目のベクトルの始点を 1 つ目の終点に置く。",
        "register": "spoken"
      },
      {
        "en": "The vector v has initial point (1, 0) and terminal point (4, 3). Write v in component form.",
        "ja": "ベクトル v の始点は (1, 0)、終点は (4, 3) である。v を成分で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "曲線の媒介変数表示でも、始まりの点を initial point と言う。"
    ],
    "related": [
      "terminal-point",
      "directed-segment",
      "vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "initial point": 1
      },
      "written": {
        "initial point": 138
      },
      "sources": [
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "initial-side",
    "ja": {
      "term": "始線",
      "reading": "しせん",
      "alt": []
    },
    "en": {
      "term": "initial side",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "角をはかる基準の半直線。座標平面では x 軸の正の部分にとる。",
    "definition_en": "The ray from which an angle is measured; for an angle in standard position, the positive x-axis.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In standard position, the initial side is always the positive x-axis.",
        "ja": "標準の位置では、始線はいつも x 軸の正の部分。",
        "register": "spoken"
      },
      {
        "en": "Draw the angle −120° with its initial side on the positive x-axis.",
        "ja": "始線を x 軸の正の部分にとって、−120° の角を図示せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry（7.1）は、頂点が原点にあり始線が x 軸の正の部分にある角を an angle in standard position（標準の位置の角）と呼ぶ。",
      "極座標（学習指導要領解説 数学C）の始線は、英語では polar axis（エントリ polar-coordinates）。"
    ],
    "related": [
      "terminal-side",
      "general-angle",
      "coterminal-angle",
      "polar-coordinates"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles（initial side、standard position）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学C 平面上の曲線: 半直線 OX を始線、角 θ を偏角"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "initial side": 11
      },
      "written": {
        "initial side": 17
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inner-function",
    "ja": {
      "term": "内側の関数",
      "reading": "うちがわのかんすう",
      "alt": []
    },
    "en": {
      "term": "inner function",
      "alt": [],
      "variants": [
        {
          "term": "inside function",
          "register": "spoken",
          "note": "話し言葉で最も多い言い方。話し言葉では Professor Leonard が最も多く、Khan Academy・NancyPi・MIT OCW が続く。書き言葉では少ない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "合成関数 f(g(x)) で、先に施す内側の関数 g。",
    "definition_en": "In a composite function f(g(x)), the function g that is applied first.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Don't forget to multiply by the derivative of the inside function.",
        "ja": "内側の関数の導関数を掛けるのを忘れないこと。",
        "register": "spoken"
      },
      {
        "en": "In sin(x²), the inner function is x² and the outer function is sin u.",
        "ja": "sin(x²) では、内側の関数は x²、外側の関数は sin u である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "outer-function",
      "chain-rule",
      "composite-function"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "inside function": 99,
        "inner function": 20
      },
      "written": {
        "inside function": 7,
        "inner function": 25
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inscribed-angle-theorem",
    "ja": {
      "term": "円周角の定理",
      "reading": "えんしゅうかくのていり",
      "alt": []
    },
    "en": {
      "term": "inscribed angle theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "1 つの弧に対する円周角は、同じ弧に対する中心角の半分の大きさになるという定理。このことから、同じ弧に対する円周角はすべて等しい。",
    "definition_en": "The theorem that an inscribed angle has half the measure of the central angle intercepting the same arc. So every inscribed angle on the same arc has the same measure.",
    "latex": "\\angle APB = \\frac{1}{2}\\angle AOB",
    "spoken_en": "angle APB equals one half of angle AOB",
    "examples": [
      {
        "en": "By the inscribed angle theorem, this angle is half of 100, so it's 50 degrees.",
        "ja": "円周角の定理より、この角は 100° の半分だから 50° だ。",
        "register": "spoken"
      },
      {
        "en": "Points A, B, and C are on circle O, and C is on major arc AB. If ∠AOB = 136°, use the inscribed angle theorem to find ∠ACB.",
        "ja": "円 O の周上に 3 点 A、B、C があり、C は長い方の弧 AB 上にある。∠AOB = 136° のとき、円周角の定理を用いて ∠ACB の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "円周角は inscribed angle（エントリ inscribed-angle）、中心角は central angle（エントリ central-angle）。定理の名前は円周角の側から inscribed angle theorem と呼ぶ。",
      "「円周角は中心角の半分」は、同じ弧に対する角どうしの関係。円周上の点が短い方の弧の上にあるときは、長い方の弧に対する中心角（360° から中心角 ∠AOB を引いた角）の半分になる。",
      "4 点が同じ円周上にあることを示す逆の定理は、エントリ converse-of-the-inscribed-angle-theorem。"
    ],
    "related": [
      "inscribed-angle",
      "central-angle",
      "converse-of-the-inscribed-angle-theorem",
      "angle-inscribed-in-a-semicircle",
      "intercepted-arc"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円周角",
        "en": "Inscribed angle"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.14 Inscribed Angles in Circles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "inscribed angle theorem": 8
      },
      "sources": [
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 8 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 inscribed angle theorem（CK-12 Geometry 6.14 Inscribed Angles in Circles・CK-12 Geometry 6.13 Segments from Chords・CK-12 Geometry 6.19 Intersecting Secants Theorem・計 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "inscribed-angle",
    "ja": {
      "term": "円周角",
      "reading": "えんしゅうかく",
      "alt": [
        "内接角"
      ]
    },
    "en": {
      "term": "inscribed angle",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円周上の 1 点を頂点とし、その点を通る 2 本の弦がつくる角。同じ弧に対する円周角はすべて等しく、その弧に対する中心角の半分になる。",
    "definition_en": "An angle whose vertex is on a circle and whose sides are chords of the circle. It equals half the central angle on the same arc, so all inscribed angles on one arc are equal.",
    "latex": "\\angle APB = \\frac{1}{2}\\angle AOB",
    "spoken_en": "angle APB equals one half of angle AOB",
    "examples": [
      {
        "en": "Every inscribed angle that cuts off the same arc has the same measure, so these two angles are equal.",
        "ja": "同じ弧に対する円周角は等しいので、この 2 つの角は等しい。",
        "register": "spoken"
      },
      {
        "en": "If the chord is a diameter, the inscribed angle is always 90 degrees.",
        "ja": "弦が直径なら、円周角はいつも 90°。",
        "register": "spoken"
      },
      {
        "en": "Points A, B, and P lie on a circle with center O, and P is on the major arc AB. If ∠AOB = 110°, find the inscribed angle ∠APB.",
        "ja": "円 O の周上に点 A, B, P があり、P は長い方の弧 AB 上にある。∠AOB = 110° のとき、円周角 ∠APB の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "measure of an inscribed angle",
        "ja": "円周角の大きさ"
      }
    ],
    "pitfalls": [
      "中心角は central angle。円周角は同じ弧に対する中心角の半分。"
    ],
    "related": [
      "angle",
      "chord",
      "cyclic-quadrilateral",
      "arc-length"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円周角",
        "en": "Inscribed angle"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "inscribed angle": 11
      },
      "sources": [
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inscribed-circle",
    "ja": {
      "term": "内接円",
      "reading": "ないせつえん",
      "alt": []
    },
    "en": {
      "term": "inscribed circle",
      "alt": [
        "incircle"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "三角形（多角形）のすべての辺に接する円。三角形の内接円の中心（内心）は 3 つの内角の二等分線の交点で、半径 r と面積 S の間に S = (1/2)r(a + b + c) が成り立つ。",
    "definition_en": "The circle inside a triangle or polygon that is tangent to every side. For a triangle, its center is where the angle bisectors meet, and its radius r satisfies Area = (1/2)r(a + b + c).",
    "latex": "S = \\frac{1}{2}r(a+b+c)",
    "spoken_en": "the area equals one half r times the perimeter",
    "examples": [
      {
        "en": "Connect the center of the inscribed circle to the three vertices, and the triangle splits into three smaller triangles, each with height r.",
        "ja": "内接円の中心と 3 つの頂点を結ぶと、三角形は高さ r の小さな三角形 3 つに分かれる。",
        "register": "spoken"
      },
      {
        "en": "Find the radius r of the inscribed circle of the triangle with sides 5, 6, and 7.",
        "ja": "3 辺の長さが 5, 6, 7 の三角形の内接円の半径 r を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "inscribed angle は円周角で、内接円とは別の概念。",
      "英語版 Wikipedia「Incircle and excircles」（langlink の記事の転送先）は内接円の半径を inradius、中心（内心）を incenter と呼ぶ（IM Geometry・CK-12 Geometry は incenter だけ）。面積 S = r × (半周の長さ s) で、s はヘロンの公式と同じ（OpenStax は semi-perimeter と綴る）。",
      "OpenStax Calculus Volume 3（Curvature）の inscribed circle は曲率円の意味（3 件）で、三角形の内接円ではない。三角形の内接円は IM Geometry（7.7 Circles in Triangles ほか、14 件）と CK-12 Geometry（4.22 Concurrence and Constructions、11 件）に出る。"
    ],
    "related": [
      "circumscribed-circle",
      "herons-formula",
      "area-of-a-triangle",
      "inscribed-angle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "内接円",
        "en": "Inscribed circle"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.7 Circles in Triangles・Geometry 8.2 Playing with Probability・Geometry 7.6 A Special Point"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.22 Concurrence and Constructions: inscribed circle（7 件）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 図形の性質: 外心・内心・重心の性質"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "平成 29・31 年度・令和 3 年度 数学I・A: 3 辺から内接円の半径"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Incircle_and_excircles",
        "note": "langlink の Inscribed circle の転送先: inradius・incenter"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "semi-perimeter の綴り（Heron's formula）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Curvature: inscribed circle は曲率円の意味（3 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 inscribed circle of the triangle | inscribed circle of a triangle | circle inscribed in a triangle | circle inscribed in the triangle | triangle's inscribed circle | its inscribed circle | inscribed circle for the triangle | inscribed circle is tangent | inscribed circle's radius | an inscribed circle（Geometry 7.7 Circles in Triangles・Geometry 8.2 Playing with Probability・Geometry 7.6 A Special Point・計 12 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "inscribed-polygon",
    "ja": {
      "term": "内接多角形",
      "reading": "ないせつたかくけい",
      "alt": []
    },
    "en": {
      "term": "inscribed polygon",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "すべての頂点が 1 つの円の周上にある多角形。このとき、その円は多角形の外接円である。",
    "definition_en": "A polygon whose vertices all lie on a single circle. That circle is the circumscribed circle of the polygon.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Since the square is an inscribed polygon, its diagonal goes through the center, so the diagonal is a diameter of the circle.",
        "ja": "正方形は円に内接しているから、対角線は中心を通る。つまり対角線は円の直径だ。",
        "register": "spoken"
      },
      {
        "en": "A regular hexagon is an inscribed polygon in a circle with radius 5 cm. Find the perimeter of the hexagon.",
        "ja": "半径 5 cm の円に正六角形が内接している。正六角形の周の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "内接円（inscribed circle）とは向きが逆。inscribed polygon は多角形が内側で、頂点が円周上にある。inscribed circle は円が内側で、多角形の辺に接する。",
      "日本の数A では「円に内接する四角形」の性質（向かい合う角の和が 180°）として習う。円に内接する四角形は cyclic quadrilateral とも言う。",
      "CK-12 Geometry で inscribed polygon は 3 件（6.15 Inscribed Quadrilaterals in Circles）。用例コーパスでは話・書ともほとんど出てこない。"
    ],
    "related": [
      "circumscribed-polygon",
      "circumscribed-circle",
      "cyclic-quadrilateral",
      "be-inscribed-in"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.15 Inscribed Quadrilaterals in Circles"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 inscribed polygon（CK-12 Geometry 6.15 Inscribed Quadrilaterals in Circles・計 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "instantaneous-rate-of-change",
    "ja": {
      "term": "瞬間変化率",
      "reading": "しゅんかんへんかりつ",
      "alt": []
    },
    "en": {
      "term": "instantaneous rate of change",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「瞬間変化率」は学習指導要領解説に無いが、日本語版 Wikipedia「微分」は微分係数を x = a における瞬間変化率と説明する。学習指導要領解説（数学II）は同じ量を微分係数と呼び、瞬間の速さや接線の傾きと関連付ける。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "ある瞬間での変化の割合。区間の幅を 0 に近づけたときの平均変化率の極限で、微分係数に等しい。",
    "definition_en": "The rate of change at a single instant: the limit of the average rate of change as the interval shrinks to zero, equal to the derivative.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The instantaneous rate of change at x = 1 is the slope of the tangent line there.",
        "ja": "x = 1 での瞬間変化率は、そこでの接線の傾き。",
        "register": "spoken"
      },
      {
        "en": "Find the instantaneous rate of change of f(x) = x² at x = 3.",
        "ja": "x = 3 における f(x) = x² の瞬間変化率（微分係数）を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "average-rate-of-change",
      "rate-of-change",
      "derivative-at-a-point",
      "instantaneous-velocity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 2.1（instantaneous rate of change）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「微分」",
        "url": "https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86",
        "note": "微分係数は瞬間変化率"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 微分・積分の考え: 微分係数、瞬間の速さ・接線の傾き"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "instantaneous rate of change": 32
      },
      "written": {
        "instantaneous rate of change": 43
      },
      "sources": [
        "khan-ap-calc",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "instantaneous-velocity",
    "ja": {
      "term": "瞬間の速さ",
      "reading": "しゅんかんのはやさ",
      "alt": []
    },
    "en": {
      "term": "instantaneous velocity",
      "alt": [],
      "variants": [
        {
          "term": "instantaneous speed",
          "register": "spoken",
          "note": "少数の言い方。話し言葉では MIT OCW と Khan Academy に少しあり、書き言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "数II の「瞬間の速さ」は位置の変化率で負にもなるので、英語では velocity（向きを含む）に当たる。英語の speed は velocity の絶対値（数III の「速さ」）を指す。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "動く点の位置を時刻 t の関数で表したときの、ある瞬間での位置の変化率。",
    "definition_en": "How fast, and in which direction, an object's position is changing at one instant; in symbols, v(t) = s′(t).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The instantaneous velocity at t = 2 is s prime of 2.",
        "ja": "t = 2 での瞬間の速さは s′(2)。",
        "register": "spoken"
      },
      {
        "en": "Find the instantaneous velocity of the ball at t = 3 seconds.",
        "ja": "t = 3 秒におけるボールの瞬間の速さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "rate-of-change",
      "average-rate-of-change",
      "derivative"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "instantaneous speed": 5,
        "instantaneous velocity": 18
      },
      "written": {
        "instantaneous velocity": 77,
        "instantaneous speed": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integer-part",
    "ja": {
      "term": "整数部分",
      "reading": "せいすうぶぶん",
      "alt": []
    },
    "en": {
      "term": "integer part",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスには integer part は出てこない。説明的な greatest integer less than or equal to が書き言葉（OpenStax Calculus）にわずかに出てくるだけ）。正の数の整数部分は床関数の値 ⌊x⌋（the floor of x）と同じで、OpenStax Calculus Volume 1 は ⌊x⌋ を the floor function と呼び、Volume 2 は [x] を the greatest integer less than or equal to x と説明する。英語版 Wikipedia「Floor and ceiling functions」は integer part のほか integral part・greatest integer とも呼ぶ。英語版 Wikipedia「Floor and ceiling functions」（en.term のリダイレクト先）は冒頭で integer part を太字で定義する。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Calculus I",
        "Calculus II"
      ]
    },
    "definition_ja": "実数 x を超えない最大の整数。正の数では、小数点より前の部分に当たる。",
    "definition_en": "The greatest integer that does not exceed a real number x; for a positive number it is the part before the decimal point.",
    "latex": "\\lfloor x \\rfloor",
    "spoken_en": "the floor of x",
    "examples": [
      {
        "en": "Since 2 < √7 < 3, the integer part of √7 is 2.",
        "ja": "2 < √7 < 3 だから、√7 の整数部分は 2。",
        "register": "spoken"
      },
      {
        "en": "Let a be the integer part of √11. Find a.",
        "ja": "√11 の整数部分を a とする。a を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "負の数では意味が分かれる。⌊x⌋ で考えると −2.5 の整数部分は −3 になるが、英語版 Wikipedia「Floor and ceiling functions」が書くように、integer part を 0 に向かう切り捨て（truncation）の意味に使って −2 とする流儀もある。負の数のときはどちらの意味かを書く。"
    ],
    "related": [
      "floor-function",
      "fractional-part",
      "real-number",
      "nested-radical"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Floor_and_ceiling_functions",
        "note": "記事名 Floor and ceiling functions（en.term integer part のリダイレクト先。冒頭で integer part を太字で定義。数学カテゴリから 2 段）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Mean Value Theorem: the floor function ⌊x⌋"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "級数の練習問題: [x] は the greatest integer less than or equal to x"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "integer-solution",
    "ja": {
      "term": "整数解",
      "reading": "せいすうかい",
      "alt": []
    },
    "en": {
      "term": "integer solution",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math",
        "Pre-Algebra"
      ]
    },
    "definition_ja": "方程式の解のうち、値が整数であるもの。未知数が 2 つ以上なら、どれも整数である組をいう。",
    "definition_en": "A solution of an equation in which the unknowns all take integer values.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We only want integer solutions here, so x = 1.5 doesn't count.",
        "ja": "ここで欲しいのは整数解だけだから、x = 1.5 は数えない。",
        "register": "spoken"
      },
      {
        "en": "Find all integer solutions of x² − y² = 7.",
        "ja": "x² − y² = 7 の整数解をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「自然数解」は positive integer solution。0 を含めるかどうかは英語でも問題文で決める（nonnegative integer solution なら 0 を含む）。Levin は自然数に 0 を含めるので、Levin の natural number solutions は 0 を含む。",
      "未知数が 2 つの整数解は、(x, y) の組（pairs of integers）で答える（エントリ diophantine-equation の例文）。"
    ],
    "related": [
      "diophantine-equation",
      "linear-diophantine-equation",
      "real-solution",
      "solution"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2 Introduction to Number Theory"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Solve Equations Using Integers（integer solutions）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "integer solution": 2
      },
      "written": {
        "integer solution": 2
      },
      "sources": [
        "mit-6.042",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは OpenStax の呼び方 integer solution（本文 2 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "integer",
    "ja": {
      "term": "整数",
      "reading": "せいすう",
      "alt": []
    },
    "en": {
      "term": "integer",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "0 と、0 に 1 ずつ足したり 1 ずつ引いたりしてできる数。…, −2, −1, 0, 1, 2, … のことで、小数や分数の部分をもたない。",
    "definition_en": "One of the numbers …, −3, −2, −1, 0, 1, 2, 3, …, that is, the whole numbers and their opposites. Integers have no fraction or decimal part.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Is 3.5 an integer? No, it's got a decimal part, so it's not.",
        "ja": "3.5 は整数？ 違うね、小数の部分があるから。",
        "register": "spoken"
      },
      {
        "en": "List all integers x that satisfy −3 < x ≤ 2.",
        "ja": "−3 < x ≦ 2 を満たす整数 x をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "whole number は 0, 1, 2, …（負の数を含まない）で、integer とは別の語。「整数」を whole number と訳すと負の整数が抜ける。",
      "integer の g は「ジ」の音で、「インティジャー」に近い（「インテガー」ではない）。"
    ],
    "related": [
      "natural-number",
      "positive-and-negative-numbers",
      "opposite",
      "real-number",
      "number-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "整数",
        "en": "Integer"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "integer": 425
      },
      "written": {
        "integer": 1282
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integral-involving-absolute-value",
    "ja": {
      "term": "絶対値を含む積分",
      "reading": "ぜったいちをふくむせきぶん",
      "alt": []
    },
    "en": {
      "term": "integral of the absolute value",
      "alt": [
        "integral involving an absolute value"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "被積分関数に絶対値を含む定積分。絶対値の中の符号が変わる点で区間を分けて計算する。",
    "definition_en": "A definite integral whose integrand contains an absolute value; you split the interval where the expression inside changes sign.",
    "latex": "\\int_0^2 |x-1|\\,dx = 1",
    "spoken_en": "the integral from zero to two of the absolute value of x minus one, dx, equals one",
    "examples": [
      {
        "en": "With an absolute value inside, figure out where x minus one is negative and split the integral there.",
        "ja": "絶対値があるときは、x − 1 が負になるところを調べて、そこで区間を分ける。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "|x − 1| は the absolute value of x minus one と読む。"
    ],
    "related": [],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.6（a definite integral of the absolute value of the difference of two functions）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "integral of the absolute value": 7
      },
      "written": {
        "integral of the absolute value": 1
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CED の呼び方 integral of the absolute value（topic 8.6）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "integral-test",
    "ja": {
      "term": "積分判定法",
      "reading": "せきぶんはんていほう",
      "alt": []
    },
    "en": {
      "term": "integral test",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "正で減少する関数 f について、級数 Σf(n) と広義積分 ∫₁^∞ f(x) dx の収束・発散が一致するという判定法。",
    "definition_en": "For a positive decreasing f, the series Σf(n) and the improper integral ∫₁^∞ f(x) dx either both converge or both diverge.",
    "latex": "\\sum_{n=1}^{\\infty} f(n) \\text{ and } \\int_1^{\\infty} f(x)\\,dx",
    "spoken_en": null,
    "examples": [
      {
        "en": "The integral test only works if the function is positive and decreasing.",
        "ja": "積分判定法が使えるのは、関数が正で減少するときだけ。",
        "register": "spoken"
      },
      {
        "en": "Use the integral test to determine whether Σ 1/(n ln n) converges.",
        "ja": "積分判定法を用いて Σ 1/(n ln n) が収束するかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "収束しても、級数の和と積分の値は一致しない。"
    ],
    "related": [
      "p-series",
      "improper-integral",
      "comparison-test",
      "nth-term-test"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "積分判定法",
        "en": "Integral test for convergence"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.4（integral test）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "integral test": 49
      },
      "written": {
        "integral test": 23
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integral",
    "ja": {
      "term": "インテグラル",
      "reading": "いんてぐらる",
      "alt": []
    },
    "en": {
      "term": "integral",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「積分記号」は、記号 ∫ を「積分記号」または単に「インテグラル」と呼び、口頭では「インテグラル」と読み、「積分する」と説明することもあると書く。英語の integral は積分の式やその値そのものを指す名詞（the integral of x² from 0 to 1）で、記号 ∫ だけを指すときは integral sign と言う。",
    "domains": [
      "calculus",
      "notation"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "積分記号 ∫ の読み方。英語の integral は、積分の式やその値そのものを指す。",
    "definition_en": "An expression such as ∫ f(x) dx, or its value; the symbol ∫ by itself is the integral sign.",
    "latex": "\\int_0^1 x^2\\,dx",
    "spoken_en": "the integral from zero to one of x squared dx",
    "respelling": "IN-tuh-grul",
    "examples": [
      {
        "en": "This integral looks nasty, but a substitution cleans it up.",
        "ja": "この積分は面倒そうに見えるけど、置換すればすっきりする。",
        "register": "spoken"
      },
      {
        "en": "The integral ∫₀¹ x² dx equals 1/3.",
        "ja": "∫₀¹ x² dx の値は 1/3 である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the integral sign",
        "ja": "積分記号 ∫"
      }
    ],
    "pitfalls": [
      "形容詞の integral には「整数の」の意味もある（Nicholson の an integral multiple of 2π ＝ 2π の整数倍）。",
      "発音は第 1 音節に強勢を置く読み（Merriam-Webster: ˈin-ti-grəl。数学ではふつうこちら）のほかに、第 2 音節に置く読み（in-ˈte-grəl）もある。"
    ],
    "related": [
      "integration",
      "integrate",
      "definite-integral",
      "indefinite-integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「積分記号」",
        "url": "https://ja.wikipedia.org/wiki/%E7%A9%8D%E5%88%86%E8%A8%98%E5%8F%B7",
        "note": "呼び方: 積分記号、インテグラル"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "an integral multiple of 2π（形容詞 integral ＝ 整数の）"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「integral」",
        "url": "https://www.merriam-webster.com/dictionary/integral",
        "note": "発音 ˈin-ti-grəl（usually so in mathematics）、in-ˈte-grəl（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "integral": 3922
      },
      "written": {
        "integral": 2366
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integrals-giving-inverse-trig-functions",
    "ja": {
      "term": "逆三角関数の積分",
      "reading": "ぎゃくさんかくかんすうのせきぶん",
      "alt": []
    },
    "en": {
      "term": "integrals resulting in inverse trigonometric functions",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。見出しの integrals resulting in inverse trigonometric functions は OpenStax Calculus Volume 1 の節の名前。学習指導要領解説には逆三角関数が出てこず、数学III の置換積分法で主に扱う置き換えは ax + b = t と x = a sin θ（〔内容の取扱い〕）。逆三角関数を使わずに求めるなら、∫₀¹ dx/(1 + x²) のような定積分は x = tan θ と置換して値を出す。OpenStax Calculus Volume 1（Calc I）は arcsin・arctan を扱い、∫dx/(1 + x²) = arctan x + C、∫dx/√(1 − x²) = arcsin x + C を公式として使う。",
    "domains": [
      "calculus",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "積分した結果が逆三角関数になる積分。1/(1 + x²) や 1/√(1 − x²) の積分がその代表。",
    "definition_en": "Integrals whose antiderivatives are inverse trigonometric functions, such as ∫ dx/(1 + x²) = arctan x + C.",
    "latex": "\\int \\frac{dx}{1+x^2} = \\arctan x + C",
    "spoken_en": "the integral of one over one plus x squared, dx, equals inverse tangent of x plus C",
    "examples": [
      {
        "en": "One over one plus x squared? That's the derivative of inverse tangent, so the integral is arctan x plus C.",
        "ja": "1/(1 + x²)？ それは arctan の導関数だから、積分は arctan x + C。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "sin⁻¹x は逆関数の arcsin x のことで、1/sin x ではない（1/sin x は csc x）。答案では誤解のない arcsin x と書くのが安全。",
      "arctan x は inverse tangent of x、arctangent of x、arc tan x と読む。"
    ],
    "related": [
      "trigonometric-integrals",
      "integration-by-substitution",
      "partial-fraction-decomposition"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "逆三角関数が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "integrals resulting in inverse trigonometric functions": 1
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 1 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "integrals-of-even-and-odd-functions",
    "ja": {
      "term": "偶関数・奇関数の積分",
      "reading": "ぐうかんすう・きかんすうのせきぶん",
      "alt": [
        "偶関数の定積分",
        "奇関数の定積分"
      ]
    },
    "en": {
      "term": "integrals of even and odd functions",
      "alt": [
        "integrating even and odd functions",
        "integral of an even function",
        "integral of an odd function"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "原点に関して対称な区間 −a ≦ x ≦ a での偶関数・奇関数の定積分の性質。偶関数なら 0 から a までの積分の 2 倍、奇関数なら 0 になる。",
    "definition_en": "On an interval from −a to a, an even function integrates to twice its integral from 0 to a, and an odd function integrates to 0.",
    "latex": "\\int_{-a}^{a} f(x)\\,dx = \\begin{cases} 2\\int_0^a f(x)\\,dx & (f \\text{ が偶関数}) \\\\ 0 & (f \\text{ が奇関数}) \\end{cases}",
    "spoken_en": "the integral from negative a to a of f of x dx equals two times the integral from zero to a if f is even, and zero if f is odd",
    "examples": [
      {
        "en": "Before you grind through this one, check whether the integrand is odd. If it is, the answer is zero.",
        "ja": "計算に入る前に、被積分関数が奇関数か確かめて。奇関数なら答えは 0。",
        "register": "spoken"
      },
      {
        "en": "x squared is even, so the integral from negative two to two is just twice the integral from zero to two.",
        "ja": "x² は偶関数だから、−2 から 2 までの積分は 0 から 2 までの積分の 2 倍。",
        "register": "spoken"
      },
      {
        "en": "Since sin x is an odd function, ∫₋π^π sin x dx = 0.",
        "ja": "sin x は奇関数なので ∫₋π^π sin x dx = 0 である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "条件の「−a から a まで」は、英語で from negative a to a、または an interval symmetric about 0 と言う。",
      "偶関数は even function（グラフが y 軸に関して対称、symmetric about the y-axis）、奇関数は odd function（原点に関して対称、symmetric about the origin）。",
      "積分が 0 でも、面積が 0 とは限らない。"
    ],
    "related": [
      "properties-of-integrals",
      "integrand"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "integrals of even and odd functions": 1,
        "integrating even and odd functions": 1
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは OpenStax の呼び方 integrals of even and odd functions（本文 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "integrand",
    "ja": {
      "term": "被積分関数",
      "reading": "ひせきぶんかんすう",
      "alt": []
    },
    "en": {
      "term": "integrand",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "積分記号 ∫ と dx のあいだにある、積分される関数。",
    "definition_en": "The function being integrated: what sits between the integral sign and dx.",
    "latex": null,
    "spoken_en": null,
    "respelling": "IN-tuh-grand",
    "examples": [
      {
        "en": "The integrand has a square root in it, so try a substitution.",
        "ja": "被積分関数にルートが入っているから、置換してみよう。",
        "register": "spoken"
      },
      {
        "en": "Since the integrand is odd and the interval is symmetric about 0, the integral equals 0.",
        "ja": "被積分関数が奇関数で区間が 0 に関して対称だから、積分の値は 0 である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the integrand is odd",
        "ja": "被積分関数が奇関数である"
      }
    ],
    "pitfalls": [
      "integral（積分）と綴りも音も近いので聞き分ける。integrand は語尾が -and。"
    ],
    "related": [
      "definite-integral",
      "indefinite-integral",
      "integrals-of-even-and-odd-functions",
      "variable-of-integration"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "integrand": 75
      },
      "written": {
        "integrand": 86
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integrate-by-parts",
    "ja": {
      "term": "部分積分する",
      "reading": "ぶぶんせきぶんする",
      "alt": []
    },
    "en": {
      "term": "use integration by parts",
      "alt": [
        "integrate by parts"
      ],
      "variants": [
        {
          "term": "use integration by parts",
          "register": "both",
          "note": "話し言葉でも書き言葉でも最も多い。話し言葉では Khan Academy と MIT OCW が中心で、書き言葉はほとんどが OpenStax Calculus。"
        },
        {
          "term": "do integration by parts",
          "register": "spoken",
          "note": "YouTube の解説（NancyPi など）に多い。話し言葉では NancyPi が半分以上で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "部分積分法を使って積分すること。",
    "definition_en": "To evaluate an integral with the integration by parts formula.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We can't substitute our way out of x times cosine x, so let's do integration by parts.",
        "ja": "x cos x は置換ではどうにもならないから、部分積分しよう。",
        "register": "spoken"
      },
      {
        "en": "Use integration by parts with u = ln x and dv = x dx.",
        "ja": "u = log x、dv = x dx として部分積分せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞の integrate by parts もあるが、用例コーパスでは use ／ do integration by parts のほうが多い。"
    ],
    "related": [
      "integration-by-parts",
      "which-one-to-differentiate"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "use integration by parts": 27,
        "do integration by parts": 11,
        "integrate by parts": 6
      },
      "written": {
        "use integration by parts": 20,
        "integrate by parts": 2
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integrate",
    "ja": {
      "term": "積分する",
      "reading": "せきぶんする",
      "alt": []
    },
    "en": {
      "term": "integrate",
      "alt": [],
      "variants": [
        {
          "term": "take the integral",
          "register": "spoken",
          "note": "口頭の言い方。話し言葉では Khan Academy が半分を占め、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数の原始関数や定積分を求めること。",
    "definition_en": "To find an antiderivative or a definite integral of a function.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Go ahead and integrate term by term; each piece is just the power rule.",
        "ja": "項ごとに積分していいよ。どの項もべき乗の公式でできる。",
        "register": "spoken"
      },
      {
        "en": "Integrating both sides with respect to x gives y = x³ + C.",
        "ja": "両辺を x で積分すると y = x³ + C を得る。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "integrate both sides",
        "ja": "両辺を積分する"
      },
      {
        "en": "integrate with respect to x",
        "ja": "x で積分する"
      }
    ],
    "pitfalls": [
      "「x で積分する」は integrate with respect to x。by x や about x とは言わない。",
      "目的語は直接とる: integrate x squared（x² を積分する）。名詞で言うなら take the integral of x squared。"
    ],
    "related": [
      "integration",
      "antiderivative",
      "indefinite-integral",
      "definite-integral",
      "integral",
      "evaluate-the-integral"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "integrate": 1223,
        "take the integral": 75
      },
      "written": {
        "integrate": 409
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integrating-factor",
    "ja": {
      "term": "積分因子",
      "reading": "せきぶんいんし",
      "alt": []
    },
    "en": {
      "term": "integrating factor",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "1 階線形微分方程式 y′ + P(x)y = Q(x) の両辺に掛けて、左辺を (μy)′ の形にする関数 μ(x) = e^{∫P(x)dx}。",
    "definition_en": "The function μ(x) = e^{∫P(x)dx} that turns the left side of y′ + P(x)y = Q(x) into (μy)′ when both sides are multiplied by it.",
    "latex": "\\mu(x) = e^{\\int P(x)\\,dx}",
    "spoken_en": "mu of x equals e to the integral of P of x dx",
    "examples": [
      {
        "en": "Multiply both sides by the integrating factor, and the left side becomes a product rule in reverse.",
        "ja": "両辺に積分因子を掛けると、左辺は積の微分を逆にした形になる。",
        "register": "spoken"
      },
      {
        "en": "Find an integrating factor for y′ − (1/x)y = x² and solve the equation.",
        "ja": "y′ − (1/x)y = x² の積分因子を求め、方程式を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉はほとんどが MIT OCW（18.03）。"
    ],
    "related": [
      "first-order-linear-differential-equation",
      "differential-equation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "積分因子",
        "en": "Integrating factor"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "integrating factor": 63
      },
      "written": {
        "integrating factor": 35
      },
      "sources": [
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integration-by-completing-the-square",
    "ja": {
      "term": "平方完成による積分",
      "reading": "へいほうかんせいによるせきぶん",
      "alt": []
    },
    "en": {
      "term": "integration by completing the square",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。AP の CED は topic 6.10 で long division と並べて completing the square を使う積分を扱うが、手順を説明する句で名前ではない。平方完成してから積分する手順は日米共通だが、結果の扱いが違う。OpenStax Calculus と AP の CED の範囲では ∫dx/(x² + 2x + 5) を (x + 1)² + 4 と平方完成して arctan で答える。学習指導要領解説には逆三角関数が出てこない。逆三角関数を使わずに求めるなら、定積分は x + 1 = 2 tan θ と置換して値を出す。見出しの「平方完成による積分」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "被積分関数の分母などを平方完成し、置換や既知の公式が使える形に直してから積分する方法。",
    "definition_en": "Integrating by first completing the square, so that a substitution or a known formula such as the arctangent integral applies.",
    "latex": "\\int \\frac{dx}{x^2+2x+5} = \\frac{1}{2}\\arctan\\frac{x+1}{2} + C",
    "spoken_en": "the integral of one over x squared plus two x plus five, dx, equals one half arctan of the quantity x plus one over two, plus C",
    "examples": [
      {
        "en": "Complete the square in the denominator and it turns into an arctan integral.",
        "ja": "分母を平方完成すると、arctan の形の積分になる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "completing-the-square",
      "integrals-giving-inverse-trig-functions",
      "integration-by-substitution"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.10（integrating functions using long division and completing the square）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "逆三角関数が出てこない・「平方完成による積分」は出てこない"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "3.4 Partial Fractions ほか（complete the square）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "integration-by-long-division",
    "ja": {
      "term": "長除法による積分",
      "reading": "ちょうじょほうによるせきぶん",
      "alt": []
    },
    "en": {
      "term": "integration by long division",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。「長除法による積分」も integration by long division も確立した名前ではない。分子の次数を分母より下げてから積分する手順を、AP の CED は topic 6.10 で long division を使う積分として扱う。多項式の割り算そのものは、IM と OpenStax では polynomial long division と呼ぶ。見出しの「長除法による積分」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "分子の次数が分母以上の分数関数を、多項式の割り算で商と余りに分けてから積分する方法。",
    "definition_en": "Integrating a rational function whose numerator has degree at least that of the denominator by first dividing the polynomials.",
    "latex": "\\int \\frac{x^2+1}{x+1}\\,dx = \\int \\left(x - 1 + \\frac{2}{x+1}\\right)dx",
    "spoken_en": "the integral of x squared plus one over x plus one, dx, equals the integral of x minus one plus two over x plus one, dx",
    "examples": [
      {
        "en": "The top's degree is higher than the bottom's, so divide first, then integrate each piece.",
        "ja": "分子の次数のほうが高いから、先に割り算して、それから各項を積分する。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「整式の割り算」は、OpenStax では long division と言い、筆算の形で書く。"
    ],
    "related": [
      "partial-fraction-decomposition"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.10（Integrating Functions Using Long Division and Completing the Square）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「長除法による積分」は出てこない"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 2)",
        "url": "https://im.kendallhunt.com/HS/students/3/index.html",
        "note": "Algebra 2 2.13 Polynomial Division (Part 2)（polynomial long division）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Dividing Polynomials（long division）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "integration-by-parts",
    "ja": {
      "term": "部分積分法",
      "reading": "ぶぶんせきぶんほう",
      "alt": [
        "部分積分"
      ]
    },
    "en": {
      "term": "integration by parts",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "積の微分法を逆に使う積分の方法。∫f(x)g′(x)dx = f(x)g(x) − ∫f′(x)g(x)dx。",
    "definition_en": "A method of integration that runs the product rule backwards: ∫ u dv = uv − ∫ v du.",
    "latex": "\\int u\\,dv = uv - \\int v\\,du",
    "spoken_en": "the integral of u dv equals u v minus the integral of v du",
    "examples": [
      {
        "en": "Integration by parts works well when one factor gets simpler after you differentiate it, like x times e to the x.",
        "ja": "部分積分は、x eˣ のように微分すると簡単になる因数があるときに効く。",
        "register": "spoken"
      },
      {
        "en": "Using integration by parts with u = x and dv = eˣ dx, we get ∫ x eˣ dx = x eˣ − eˣ + C.",
        "ja": "u = x、dv = eˣ dx として部分積分すると ∫ x eˣ dx = x eˣ − eˣ + C。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "use integration by parts",
        "ja": "部分積分を使う"
      }
    ],
    "pitfalls": [
      "日本語版 Wikipedia「部分積分」の ∫u v′ dx = uv − ∫u′ v dx（本エントリの定義の f g′ と同じ形）に対して、OpenStax Calculus Volume 2 は u と dv を決めて ∫u dv = uv − ∫v du と書く。u には微分すると簡単になるほうを選ぶ（OpenStax Calculus Volume 2 は覚え方として LIATE を挙げる）。",
      "動詞で言うときは integrate by parts（部分積分する）。"
    ],
    "related": [
      "integration-by-substitution",
      "integration",
      "integrate-by-parts",
      "which-one-to-differentiate",
      "liate"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "部分積分",
        "en": "Integration by parts"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「部分積分」",
        "url": "https://ja.wikipedia.org/wiki/%E9%83%A8%E5%88%86%E7%A9%8D%E5%88%86",
        "note": "∫u(x)v′(x) dx = [u(x)v(x)] − ∫u′(x)v(x) dx"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "integration by parts": 161
      },
      "written": {
        "integration by parts": 60
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integration-by-substitution",
    "ja": {
      "term": "置換積分法",
      "reading": "ちかんせきぶんほう",
      "alt": [
        "置換積分"
      ]
    },
    "en": {
      "term": "u-substitution",
      "alt": [
        "u-sub",
        "substitution rule"
      ],
      "variants": [
        {
          "term": "integration by substitution",
          "register": "written",
          "note": "正式な名前。コーパスでは少ない。話し言葉は MIT OCW だけ、書き言葉はほとんどが OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の置換積分法は、g(x) = t とおく形と x = g(t) とおく形の両方を含む（学習指導要領解説の数学III は ax + b = t と x = a sin θ の置き換えを挙げる）。OpenStax Calculus の u-substitution は u = g(x) とおく形で、x = a sin θ のように x を置き換える形は trigonometric substitution として別の節（Volume 2）で扱う。OpenStax Calculus Volume 1（5.5 Substitution）は節の名前を Substitution とし、本文で the substitution rule for indefinite integrals とも書く。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "式の一部を新しい変数でおき換えて積分する方法。u = g(x) とおくと ∫f(g(x))g′(x)dx = ∫f(u)du となる。",
    "definition_en": "A method of integration that replaces part of the integrand with a new variable: with u = g(x), ∫ f(g(x)) g′(x) dx = ∫ f(u) du.",
    "latex": "\\int f(g(x))\\,g'(x)\\,dx = \\int f(u)\\,du",
    "spoken_en": "the integral of f of g of x times g prime of x dx equals the integral of f of u du",
    "examples": [
      {
        "en": "This is a classic u-substitution: let u be x squared plus one, and then du is two x dx.",
        "ja": "典型的な置換積分。u = x² + 1 とおけば du = 2x dx。",
        "register": "spoken"
      },
      {
        "en": "With the substitution u = x² + 1, du = 2x dx, the integral becomes ∫ u⁵ du.",
        "ja": "u = x² + 1 と置換すると du = 2x dx で、積分は ∫ u⁵ du になる。",
        "register": "written"
      },
      {
        "en": "The substitution rule is really just the chain rule run backwards.",
        "ja": "置換法則は、要するに連鎖律を逆に使っているだけ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus と AP の CED は新しい変数に u を使う（名前も u-substitution）。",
      "口頭では u-sub と略すこともある。",
      "dx を du で書き直すのを忘れない（du = g′(x) dx）。"
    ],
    "related": [
      "substitution-in-a-definite-integral",
      "integration-by-parts",
      "antiderivative",
      "trigonometric-substitution",
      "let-u-equal",
      "write-dx-in-terms-of-du",
      "change-the-limits-of-integration",
      "change-of-variables"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "置換積分",
        "en": "Integration by substitution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.9（substitution による積分）。Unit 6 の概要では u-substitution"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 積分法: 置換積分法は ax + b = t、x = a sin θ と置き換えるもの"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "integration by substitution": 2,
        "u-substitution": 122,
        "substitution rule": 6,
        "u-sub": 2
      },
      "written": {
        "u-substitution": 25,
        "substitution rule": 2,
        "integration by substitution": 6
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integration-formulas",
    "ja": {
      "term": "積分公式",
      "reading": "せきぶんこうしき",
      "alt": [
        "不定積分の公式",
        "積分表"
      ]
    },
    "en": {
      "term": "integration formulas",
      "alt": [],
      "variants": [
        {
          "term": "integration formulas",
          "register": "written",
          "note": "書き言葉はすべて OpenStax Calculus。話し言葉にはほとんど出てこない。"
        },
        {
          "term": "table of integrals",
          "register": "both",
          "note": "公式を一覧にした表の名前。話し言葉はほとんどが NancyPi。話し言葉・書き言葉とも少なく、書き言葉はすべて OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III",
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I",
        "Calculus II"
      ]
    },
    "definition_ja": "基本的な関数（べき関数・指数関数・三角関数など）の不定積分をまとめた公式。∫xⁿ dx = xⁿ⁺¹/(n + 1) + C、∫eˣ dx = eˣ + C など。一覧にした表を積分表と呼ぶ。",
    "definition_en": "The basic antiderivative rules for power, exponential and trigonometric functions, such as ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C (n ≠ −1) and ∫ eˣ dx = eˣ + C; a list of them is a table of integrals.",
    "latex": "\\int x^n\\,dx = \\frac{x^{n+1}}{n+1} + C \\quad (n \\neq -1)",
    "spoken_en": "the integral of x to the n dx equals x to the n plus one over n plus one, plus C, for n not equal to negative one",
    "examples": [
      {
        "en": "You'll get a sheet of basic integration formulas, but know the power rule by heart.",
        "ja": "基本的な積分公式の一覧は配るけど、べき乗の公式は覚えておいて。",
        "register": "spoken"
      },
      {
        "en": "e to the x is the easy one. Its integral is itself, plus C.",
        "ja": "eˣ は簡単。積分しても eˣ のままで、+C を付けるだけ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "∫xⁿ dx の公式は、OpenStax Calculus では power rule for integrals と呼ぶ。",
      "n = −1 のときは ln|x| + C（学習指導要領解説の書き方では log|x| + C。解説は自然対数を底を書かずに log と書く）。",
      "∫aˣ dx = aˣ/log a + C の log a（自然対数）は、OpenStax Calculus では ln a と書く。OpenStax Algebra and Trigonometry は底を省いた log を常用対数（底 10）とする（学習指導要領解説には ln が出てこない）ので、英語の答案では ln を使う。",
      "OpenStax Calculus Volume 2 は、積分表や計算ソフト（computer algebra system, CAS）を使う方法も扱う。"
    ],
    "related": [
      "indefinite-integral",
      "antiderivative",
      "trigonometric-integrals",
      "numerical-integration",
      "partial-fraction-decomposition"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "ln が出てこない"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "底を省いた log x は常用対数（底 10）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "integration formulas": 4,
        "table of integrals": 8
      },
      "written": {
        "integration formulas": 37,
        "table of integrals": 7
      },
      "sources": [
        "mit-18.01",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "integration",
    "ja": {
      "term": "積分",
      "reading": "せきぶん",
      "alt": []
    },
    "en": {
      "term": "integration",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「積分法」は「積分」という語が原始関数（不定積分）を指すこともあると書き、「積分記号」は ∫ で書いた式を「f(x) の積分」と呼ぶ。英語では、OpenStax Calculus は求める操作を integration（Volume 1 5.2 の the process of integration）、∫ で書いた式やその値を integral（this integral、the integral of …。エントリ integral・definite-integral・indefinite-integral）と呼び分ける。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数から原始関数や定積分を求める操作。微分の逆の操作にあたる。",
    "definition_en": "The process of finding antiderivatives or definite integrals; it undoes differentiation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Differentiation is mechanical, but integration takes some guesswork.",
        "ja": "微分は機械的にできるけど、積分はある程度見当をつけないといけない。",
        "register": "spoken"
      },
      {
        "en": "This problem requires integration by parts twice.",
        "ja": "この問題は部分積分を 2 回行う必要がある。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "do the integration",
        "ja": "積分を実行する"
      },
      {
        "en": "techniques of integration",
        "ja": "積分の手法"
      }
    ],
    "pitfalls": [
      "英語では、操作は integration、∫ で書かれた式そのものは an integral と言い分ける（take the integral ／ this integral diverges）。"
    ],
    "related": [
      "integrate",
      "indefinite-integral",
      "definite-integral",
      "reverse-of-differentiation",
      "integral"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "積分法",
        "en": "Integral calculus"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "5.2 The Definite Integral（the process of integration）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「積分法」",
        "url": "https://ja.wikipedia.org/wiki/%E7%A9%8D%E5%88%86%E6%B3%95",
        "note": "冒頭: 「積分」（integral）という語は原始関数（不定積分）を指すこともある（mapping_note）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「積分記号」",
        "url": "https://ja.wikipedia.org/wiki/%E7%A9%8D%E5%88%86%E8%A8%98%E5%8F%B7",
        "note": "使用例: ∫ で書いた式を「集合 D 上での f(x) の積分」と呼ぶ（mapping_note）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 6（integration）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "integration": 587
      },
      "written": {
        "integration": 429
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "intercepted-arc",
    "ja": {
      "term": "切片の弧",
      "reading": "せっぺんのこ",
      "alt": [
        "角に対する弧"
      ]
    },
    "en": {
      "term": "intercepted arc",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「切片の弧」は学習指導要領解説に無い、本プロジェクトの訳語。日本の数A は「弧 AB に対する円周角」と言い、角の側から弧を「切り取られた弧」と呼ばない。CK-12 Geometry は intercepted arc と呼ぶ（6.14 Inscribed Angles in Circles・6.16 Angles On and Inside a Circle ほか、20 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円周角や中心角の 2 辺が円から切り取る弧のうち、角の内側にある部分。",
    "definition_en": "The arc of a circle that lies inside an angle, cut off by the two sides of the angle, such as the arc between the sides of an inscribed angle or a central angle.",
    "latex": "m\\angle ABC = \\tfrac{1}{2}\\, m\\overset{\\frown}{AC}",
    "spoken_en": "the measure of angle ABC equals one half the measure of arc AC",
    "examples": [
      {
        "en": "An inscribed angle is half its intercepted arc, so if the arc is 80 degrees, the angle is 40.",
        "ja": "円周角は、それに対する弧の中心角の半分だから、弧が 80° なら角は 40° だ。",
        "register": "spoken"
      },
      {
        "en": "Inscribed angle ABC intercepts arc AC, and the measure of arc AC is 110°. Find m∠ABC.",
        "ja": "弧 AC に対する中心角が 110° のとき、弧 AC に対する円周角 ∠ABC を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では「円周角は intercepted arc の半分」のように弧の度数で言える（arc measure）。日本では同じことを「同じ弧に対する中心角の半分」と中心角で言う。",
      "動詞は intercept。The angle intercepts arc AC.（その角は弧 AC を切り取る＝弧 AC に対する角）。",
      "用例コーパスでは話し言葉（The Organic Chemistry Tutor）・書き言葉（OpenStax Algebra and Trigonometry）とも少ない。IM は 0 件。"
    ],
    "related": [
      "inscribed-angle",
      "arc-measure",
      "chord",
      "arc-length"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.14 Inscribed Angles in Circles・6.16 Angles On and Inside a Circle・6.17 Angles Outside a Circle"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "intercepted arc": 2
      },
      "written": {
        "intercepted arc": 4
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは CK-12 の呼び方 intercepted arc（CK-12 Geometry 6.14 Inscribed Angles in Circles・CK-12 Geometry 6.16 Angles On and Inside a Circle・CK-12 Geometry 6.17 Angles Outside a Circle・計 20 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "interior-angle",
    "ja": {
      "term": "内角",
      "reading": "ないかく",
      "alt": []
    },
    "en": {
      "term": "interior angle",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "多角形のとなり合う 2 辺がつくる角のうち、多角形の内側にある角。n 角形には内角が n 個ある。",
    "definition_en": "An angle inside a polygon, formed where two sides meet at a vertex. A polygon with n sides has n interior angles.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a regular hexagon, every interior angle is 120 degrees.",
        "ja": "正六角形では、内角はどれも 120°。",
        "register": "spoken"
      },
      {
        "en": "Find the measure of each interior angle of a regular octagon.",
        "ja": "正八角形の 1 つの内角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "内角をすべて足したものは sum of the interior angles（エントリ sum-of-the-interior-angles）。「1 つの内角」は each interior angle と言う。",
      "1 つの頂点で、内角と外角（exterior angle、エントリ exterior-angle）の和は 180°。",
      "錯角（alternate interior angles）や同側内角（same-side interior angles）の interior は「平行な 2 直線の間」のことで、多角形の interior angle の「図形の内側」とは指す場所が違う。"
    ],
    "related": [
      "polygon",
      "sum-of-the-interior-angles",
      "exterior-angle",
      "regular-polygon",
      "triangle"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.27 Interior Angles in Convex Polygons"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "interior angle": 88
      },
      "written": {
        "interior angle": 3
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "intermediate-value-theorem",
    "ja": {
      "term": "中間値の定理",
      "reading": "ちゅうかんちのていり",
      "alt": []
    },
    "en": {
      "term": "intermediate value theorem",
      "alt": [],
      "variants": [
        {
          "term": "IVT",
          "register": "written",
          "note": "略称。話し言葉にはほとんど出てこず、書き言葉のほうが多い。AP の CED も topic 1.16 で IVT と略す。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "f が [a, b] で連続で、k が f(a) と f(b) の間の値なら、f(c) = k となる c が a と b の間に少なくとも 1 つあるという定理。",
    "definition_en": "If f is continuous on [a, b] and k is between f(a) and f(b), then f(c) = k for at least one c between a and b.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "By the intermediate value theorem, there has to be a root between 1 and 2.",
        "ja": "中間値の定理より、1 と 2 の間に解が必ずある。",
        "register": "spoken"
      },
      {
        "en": "Use the intermediate value theorem to show that x³ + x − 1 = 0 has a solution in (0, 1).",
        "ja": "中間値の定理を用いて、x³ + x − 1 = 0 が (0, 1) に解をもつことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は topic 1.16 で扱う。CED は定理を f が閉区間 [a, b] で連続という条件から述べる。答案でもその条件（f is continuous on [a, b]）を先に書く（本プロジェクトの書き方の例）。"
    ],
    "related": [
      "continuous-function",
      "extreme-value-theorem",
      "mean-value-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "中間値の定理",
        "en": "Intermediate value theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.16（intermediate value theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "intermediate value theorem": 53,
        "IVT": 2
      },
      "written": {
        "intermediate value theorem": 34,
        "IVT": 7
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "interpolation",
    "ja": {
      "term": "補間",
      "reading": "ほかん",
      "alt": [
        "内挿"
      ]
    },
    "en": {
      "term": "interpolation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "データの範囲の内側にある、観測していない値を、モデルなどを使って推定すること。",
    "definition_en": "Estimating an unobserved value inside the range of the data, for example from a model.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Predicting inside the range of the data is interpolation, and that's usually pretty safe.",
        "ja": "データの範囲の内側を予測するのが補間で、ふつうはわりと安全。",
        "register": "spoken"
      },
      {
        "en": "Use the model to estimate the value in 2015. Is this interpolation or extrapolation?",
        "ja": "モデルを使って 2015 年の値を推定せよ。これは補間か外挿か。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "interpolation は、線形代数や数値計算では、与えられた点をすべて通る関数（多項式）を作ることも指す（Nicholson 3.2 の Polynomial Interpolation、6.5 の Lagrange interpolation expansion）。日本語版 Wikipedia「内挿」も、データ列の区間の内側を埋める値や関数を求めることとし、最小二乗近似とは別のものとする。"
    ],
    "related": [
      "extrapolation",
      "regression-line",
      "predict"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "内挿",
        "en": "Interpolation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 5.3（interpolation）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "12.5 Prediction（interpolation）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "2.4 Fitting Linear Models to Data（Recognizing Interpolation or Extrapolation）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.2（Polynomial Interpolation）・6.5（Lagrange interpolation expansion）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "interpolation": 21
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "interquartile-range",
    "ja": {
      "term": "四分位範囲",
      "reading": "しぶんいはんい",
      "alt": []
    },
    "en": {
      "term": "IQR",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "interquartile range",
          "register": "spoken",
          "note": "話し言葉では大半が Khan Academy。書き言葉はすべて OpenStax Introductory Statistics で、IQR より少ない。"
        },
        {
          "term": "IQR",
          "register": "both",
          "note": "話し言葉では interquartile range と並び、書き言葉ではこちらが多い（すべて OpenStax Introductory Statistics）。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "第 3 四分位数から第 1 四分位数を引いた値（Q3 − Q1）。データの中央付近の半分が散らばる幅を表す。",
    "definition_en": "The third quartile minus the first quartile, Q3 − Q1: the width of the middle half of the data.",
    "latex": "\\mathrm{IQR} = Q_3 - Q_1",
    "spoken_en": "the IQR is Q three minus Q one",
    "examples": [
      {
        "en": "The interquartile range tells you how spread out the middle half of the data is.",
        "ja": "四分位範囲は、データの真ん中の半分がどれだけ散らばっているかを表す。",
        "register": "spoken"
      },
      {
        "en": "Find the IQR and use it to determine whether 95 is an outlier.",
        "ja": "IQR を求め、95 が外れ値かどうかを判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語は略称の IQR（アイキューアール）を書き言葉でも見出し並みに使う。用例コーパスの書き言葉は IQR が interquartile range よりずっと多く、話し言葉は interquartile range と IQR が並ぶ。",
      "外れ値の目安は Q1 − 1.5 × IQR より小さい値と Q3 + 1.5 × IQR より大きい値（1.5 × IQR rule）。"
    ],
    "related": [
      "quartile",
      "box-plot",
      "outlier",
      "quartile-deviation",
      "spread"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "四分位範囲",
        "en": "Interquartile range"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "interquartile range": 38,
        "IQR": 24
      },
      "written": {
        "IQR": 74,
        "interquartile range": 18
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "intersect",
    "ja": {
      "term": "交わる",
      "reading": "まじわる",
      "alt": [
        "共有点をもつ"
      ]
    },
    "en": {
      "term": "intersect",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "英語の intersect は共有点をもつことで、接する場合も含む: IM Geometry と CK-12 Geometry は接線を円と exactly one point で intersect する直線と定義し、OpenStax Algebra and Trigonometry も接線が放物線と intersects … at exactly one point と書く。共通テストの問題文は「2 点で交わる」「接する」「共有点をもたない」を分けて言う（令和 3 年度 第 1 日程 数学II）。日本語の「交わる」だけを言うなら intersect at two points、「接する」は is tangent to。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの図形（直線・曲線）が共有点をもつ。英語では、接する場合（共有点が 1 つ）も含めて言う。",
    "definition_en": "Of two lines or curves: to have at least one point in common.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These two lines intersect at (1, 3), so that's our solution.",
        "ja": "この 2 直線は (1, 3) で交わるので、それが答え。",
        "register": "spoken"
      },
      {
        "en": "Find the values of k for which the line y = x + k intersects the parabola y = x² at two points.",
        "ja": "直線 y = x + k が放物線 y = x² と 2 点で交わるような k の値の範囲を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "intersect at",
        "ja": "〜で交わる"
      }
    ],
    "pitfalls": [
      "「共有点をもつ」も intersect で言える（接する場合も intersect に含まれる。mapping_note）。「接する」とはっきり言うなら is tangent to。",
      "cross（the line crosses the x-axis）や meet も話し言葉・書き言葉とも使う。"
    ],
    "related": [
      "tangent-line",
      "parallel-lines",
      "perpendicular-lines"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.4 Quadrilaterals in Circles: 接線は円と exactly one point で intersect する直線"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.2 Identify Circle Components: tangent は円と exactly one point で intersect する直線"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和3年度 大学入学共通テスト 第1日程 数学Ⅱ（問題）",
        "url": "https://web.archive.org/web/20220601/https://www.dnc.ac.jp/albums/abm.php?d=88&f=abm00001261.pdf&n=R3_%E6%95%B0%E5%AD%A6%E2%85%A1%EF%BC%88%EF%BC%91%E6%9C%8816%E6%97%A5%E3%83%BB17%E6%97%A5%EF%BC%89%E3%81%AE%E5%95%8F%E9%A1%8C.pdf",
        "note": "第 3 問: 2 点で交わる・接する・共有点をもたない を分ける解答群"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "intersect": 375
      },
      "written": {
        "intersect": 314
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "intersecting-chords-theorem",
    "ja": {
      "term": "交わる弦の定理",
      "reading": "まじわるげんのていり",
      "alt": []
    },
    "en": {
      "term": "intersecting chords theorem",
      "alt": [
        "chord-chord product theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「交わる弦の定理」は学習指導要領解説に無い（日本語版 Wikipedia「方べきの定理」の図の説明にはある）。日本では方べきの定理の 1 つの場合（2 本の弦 AB, CD が円の内部の点 P で交わるとき PA·PB = PC·PD）として扱い、この場合だけの名前は付けない。CK-12 Geometry は intersecting chords theorem と呼ぶ（6.13 Segments from Chords、12 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円の 2 本の弦 AB, CD が円の内部の点 P で交わるとき、PA·PB = PC·PD が成り立つという定理。方べきの定理のうち、点 P が円の内部にある場合にあたる。",
    "definition_en": "If two chords AB and CD of a circle intersect at a point P inside the circle, then PA · PB = PC · PD. In other words, the two pieces of one chord have the same product as the two pieces of the other.",
    "latex": "PA \\cdot PB = PC \\cdot PD",
    "spoken_en": "P A times P B equals P C times P D",
    "examples": [
      {
        "en": "By the intersecting chords theorem, the two pieces of this chord multiply to the same thing as the two pieces of that one, so 4 times 6 has to equal 3 times x.",
        "ja": "交わる弦の定理から、こっちの弦の 2 つの部分の積は、あっちの弦の 2 つの部分の積と等しい。だから 4 × 6 = 3 × x になるはず。",
        "register": "spoken"
      },
      {
        "en": "Chords AB and CD of a circle intersect at point P, with PA = 4, PB = 9, and PC = 6. Use the intersecting chords theorem to find PD.",
        "ja": "円の弦 AB と弦 CD が点 P で交わり、PA = 4, PB = 9, PC = 6 である。方べきの定理を用いて、PD の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本では方べきの定理の 1 つの場合として習い、この場合だけの名前は付けない。CK-12 Geometry は点 P が円の内部にある場合（6.13 Segments from Chords）と、外の点から接線と割線を引く場合（6.20 Tangent Secant Theorem）を別の節で扱う。",
      "式の PA·PB は、1 本の弦が点 P で分けられた 2 つの部分の積。円の外の点から割線を引く場合は「外側の部分 × 割線全体」になるので、内部と外部で式の組み立てを取り違えない。"
    ],
    "related": [
      "power-of-a-point",
      "chord",
      "secant-tangent-theorem"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.13 Segments from Chords"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「方べきの定理」",
        "url": "https://ja.wikipedia.org/wiki/%E6%96%B9%E3%81%B9%E3%81%8D%E3%81%AE%E5%AE%9A%E7%90%86",
        "note": "図 1 の説明「交わる弦の定理」（英語版 Intersecting chords theorem へのリンク）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 intersecting chords theorem（CK-12 Geometry 6.13 Segments from Chords・計 12 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "intersection-of-events",
    "ja": {
      "term": "積事象",
      "reading": "せきじしょう",
      "alt": []
    },
    "en": {
      "term": "intersection of events",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "intersection of events という名前は、米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（4 つとも 0 件。Levin 3.7 に 1 件）。概念は、AP Statistics の CED（topic 2.5）が P(A ∩ B) を joint probability と呼んで intersection of A and B の確率と説明し、OpenStax Introductory Statistics（3.1 Terminology）が A AND B（A Intersection B とも）と書いて扱う。",
    "domains": [
      "combinatorics",
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "事象 A と事象 B がともに起こるという事象。A ∩ B と書く。",
    "definition_en": "The event that both of the events A and B happen, written A ∩ B.",
    "latex": "A \\cap B",
    "spoken_en": "A intersect B; in probability, often read as A and B",
    "examples": [
      {
        "en": "The intersection of two events is where both happen, so think A and B.",
        "ja": "積事象は両方が起こるところだから、A かつ B と考えよう。",
        "register": "spoken"
      },
      {
        "en": "A die is rolled. Let event A be rolling an even number and event B be rolling a number greater than 3. Find the intersection of events A and B.",
        "ja": "さいころを 1 個投げる。偶数の目が出る事象を A、3 より大きい目が出る事象を B とする。積事象 A ∩ B を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP Statistics の CED（topic 2.5）は P(A ∩ B) を P(A intersect B) とも書く。「かつ」は and、「または」は or。"
    ],
    "related": [
      "union-of-events",
      "general-multiplication-rule",
      "independence-of-events",
      "conditional-probability",
      "mutually-exclusive-events",
      "intersection",
      "complementary-event"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.7 Applications to Probability"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.5（the intersection of A and B、joint probability、P(A intersect B)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（A AND B、A Intersection B）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "intersection of two events | intersection of events": 3
      },
      "sources": [
        "mit-6.042"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Levin の呼び方 intersection of two events | intersection of events（3.7 Applications to Probability）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "intersection",
    "ja": {
      "term": "共通部分",
      "reading": "きょうつうぶぶん",
      "alt": []
    },
    "en": {
      "term": "intersection",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "集合 A と集合 B の両方に属する要素全体の集合のこと。A ∩ B と書く。",
    "definition_en": "The set whose elements are exactly the ones that lie in A and also in B, written A ∩ B.",
    "latex": "A \\cap B",
    "spoken_en": "A intersect B",
    "examples": [
      {
        "en": "The intersection is just what the two sets have in common, the part where the circles overlap in the Venn diagram.",
        "ja": "共通部分は 2 つの集合に共通するもの、ベン図で円が重なっている部分だ。",
        "register": "spoken"
      },
      {
        "en": "Let A = {1, 2, 3, 4} and B = {3, 4, 5}. Find the intersection of A and B.",
        "ja": "A = {1, 2, 3, 4}、B = {3, 4, 5} のとき、A と B の共通部分を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "A ∩ B は A intersect B と読む。確率では A and B とも読む（積事象）。「かつ」の ∩（intersection）と「または」の ∪（union）を取り違えない。",
      "共通部分が空集合のとき、A と B は disjoint（互いに素）という。",
      "交点（point of intersection）や 2 直線の交わりの intersection と取り違えない。"
    ],
    "related": [
      "union",
      "set",
      "complement",
      "intersection-of-events",
      "point-of-intersection",
      "venn-diagram",
      "empty-set"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "共通部分 (数学)",
        "en": "Intersection (set theory)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Linear Inequalities: the intersection of two sets"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "intersection of a and b | intersection of sets | intersection of two sets | intersection of the sets | a intersect b | a intersection b": 12
      },
      "written": {
        "intersection of a and b | intersection of sets | intersection of two sets | intersection of the sets | a intersect b | a intersection b": 2
      },
      "sources": [
        "mit-6.042",
        "openstax-algtrig",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "interval-estimation",
    "ja": {
      "term": "区間推定",
      "reading": "くかんすいてい",
      "alt": []
    },
    "en": {
      "term": "interval estimate",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「区間推定」は推定の方法の名前で、英語の interval estimate は推定した区間を指す（AP Statistics の CED topic 3.3 は信頼区間を interval estimate の 1 つとして定め、OpenStax Introductory Statistics 8.1 は confidence interval estimate と書く）。interval estimation は用例コーパスにも参照にも出てこない。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "母数がある確からしさで含まれる区間を求めて推定すること。また、その区間。",
    "definition_en": "Estimating a population parameter by an interval that contains it with a stated level of confidence; also the interval itself.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Instead of one number, an interval estimate gives a range of plausible values.",
        "ja": "区間推定は 1 つの数でなく、ありそうな値の範囲を与える。",
        "register": "spoken"
      },
      {
        "en": "Construct an interval estimate for the population mean at the 90% confidence level.",
        "ja": "信頼度 90% で母平均の区間推定をせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "confidence-interval",
      "point-estimate",
      "estimation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "8.1 A Single Population Mean using the Normal Distribution（confidence interval estimate）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.3（interval estimate）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "interval estimate": 12
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "interval-notation",
    "ja": {
      "term": "区間記法",
      "reading": "くかんきほう",
      "alt": []
    },
    "en": {
      "term": "interval notation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1"
      ]
    },
    "definition_ja": "数の範囲を、両端の数とかっこで [a, b]、(a, b)、[a, ∞) のように表す書き方。端を含むときは [ ]、含まないときは ( ) を使う。",
    "definition_en": "A way to write a set of numbers using its endpoints, with a bracket when an endpoint is included and a parenthesis when it is not.",
    "latex": "(2, 5]",
    "spoken_en": "two to five, not including two but including five",
    "examples": [
      {
        "en": "In interval notation, infinity always gets a parenthesis, because you never actually reach it.",
        "ja": "区間記法では、∞ の側はいつも丸かっこ。実際に ∞ に届くことはないからね。",
        "register": "spoken"
      },
      {
        "en": "Solve 3x − 4 ≥ 5 and write the solution in interval notation.",
        "ja": "不等式 3x − 4 ≧ 5 を解き、解を区間記法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校では 2 < x ≦ 5 のように不等式で書き、この書き方は大学で使う（閉区間 closed-interval、開区間 open-interval）。米国の教科書は Algebra 1（OpenStax Elementary Algebra）から、不等式の解をこの形でも書かせる。",
      "端を含むのは角かっこ [ ]（bracket）、含まないのは丸かっこ ( )（parenthesis）。∞ の側はいつも丸かっこで、[3, ∞] とは書かない。",
      "(2, 5) は点の座標と同じ形なので、区間か点かは文脈で見分ける。"
    ],
    "related": [
      "closed-interval",
      "open-interval",
      "interval",
      "inequality",
      "domain"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "interval notation": 58
      },
      "written": {
        "interval notation": 228
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "interval-of-convergence",
    "ja": {
      "term": "収束区間",
      "reading": "しゅうそくくかん",
      "alt": []
    },
    "en": {
      "term": "interval of convergence",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "べき級数が収束する x の値の範囲。収束半径から出して、両端は別に調べる。",
    "definition_en": "The set of x for which a power series converges; it comes from the radius of convergence, with the endpoints checked separately.",
    "latex": "(a - R,\\, a + R]",
    "spoken_en": null,
    "examples": [
      {
        "en": "Don't forget to test the endpoints — that's where the interval of convergence gets tricky.",
        "ja": "端点を調べるのを忘れずに。収束区間はそこが難しい。",
        "register": "spoken"
      },
      {
        "en": "Find the interval of convergence of Σ (x − 3)ⁿ/n.",
        "ja": "Σ (x − 3)ⁿ/n の収束区間を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉はすべて Khan Academy。"
    ],
    "related": [
      "radius-of-convergence",
      "power-series",
      "ratio-test"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.13・10.15（interval of convergence）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "interval of convergence": 37
      },
      "written": {
        "interval of convergence": 60
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "interval-of-integration",
    "ja": {
      "term": "積分区間",
      "reading": "せきぶんくかん",
      "alt": []
    },
    "en": {
      "term": "limits of integration",
      "alt": [
        "interval of integration"
      ],
      "variants": [
        {
          "term": "bounds of integration",
          "register": "spoken",
          "note": "話し言葉で最も多く、Khan Academy と MIT OCW が使う。書き言葉にはほぼ出ない。話し言葉の半分以上は Khan Academy で、Professor Leonard も使う。"
        },
        {
          "term": "limits of integration",
          "register": "both",
          "note": "教科書の文章ではこれ（書き言葉はほぼ OpenStax Calculus）。話し言葉では patrickJMT が多い。話し言葉全体では bounds of integration と同じくらいで、書き言葉ではこちらがずっと多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の「積分区間」は区間 a ≦ x ≦ b そのものを指すが、英語の授業と教科書（OpenStax Calculus）では、区間より両端の値を limits of integration（話し言葉では bounds of integration も）と呼ぶことが多い。interval of integration は用例コーパスでは話し言葉・書き言葉とも少ない。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "定積分で積分する x の範囲 a ≦ x ≦ b。両端の a を下端、b を上端という。",
    "definition_en": "The interval from a to b over which a definite integral is taken; its endpoints are the limits of integration.",
    "latex": "\\int_0^2 x^2\\,dx",
    "spoken_en": "the integral from zero to two of x squared dx",
    "examples": [
      {
        "en": "What are the bounds of integration here? Zero to two.",
        "ja": "ここの積分区間は？ 0 から 2 だね。",
        "register": "spoken"
      },
      {
        "en": "The limits of integration are x = 0 and x = 2.",
        "ja": "積分区間は 0 ≦ x ≦ 2 である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "change the limits of integration",
        "ja": "積分区間を変える"
      }
    ],
    "pitfalls": [
      "bounds は「上界・下界」の意味でも使う語。積分の話では上端・下端のことを言っている。"
    ],
    "related": [
      "upper-limit",
      "lower-limit",
      "definite-integral",
      "substitution-in-a-definite-integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "limits of integration": 31,
        "bounds of integration": 42,
        "interval of integration": 4
      },
      "written": {
        "limits of integration": 60,
        "interval of integration": 1,
        "bounds of integration": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "interval",
    "ja": {
      "term": "区間",
      "reading": "くかん",
      "alt": []
    },
    "en": {
      "term": "interval",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "definition_ja": "数直線上で、2 つの数の間にある実数全体。端を含むかどうかで閉区間・開区間などに分かれ、片側が限りなく続くものもある。",
    "definition_en": "The set of all real numbers between two given numbers, with or without the numbers themselves; one side may be unbounded.",
    "latex": "[a, b] = \\{ x \\mid a \\leq x \\leq b \\}",
    "spoken_en": "the closed interval from a to b is the set of x such that a is less than or equal to x, which is less than or equal to b",
    "examples": [
      {
        "en": "On the interval from 0 to 2, the function is increasing.",
        "ja": "0 から 2 までの区間で、この関数は増加している。",
        "register": "spoken"
      },
      {
        "en": "Find the maximum value of f(x) = −x² + 2x on the interval [0, 3].",
        "ja": "0 ≦ x ≦ 3 における f(x) = −x² + 2x の最大値を求めよ。",
        "register": "written"
      },
      {
        "en": "Write the solution set of x ≥ 2 in interval notation: [2, ∞).",
        "ja": "x ≧ 2 を区間の記号で表すと [2, ∞) である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高等学校学習指導要領解説（数学II 積分）と共通テストは区間を a ≦ x ≦ b と不等式で書き、閉区間 [a, b] の記号が解説に出るのは数学III の積分だけ。OpenStax（Elementary Algebra から Calculus まで）は区間記法 [a, b]、(a, b)、[a, ∞) でも書く（interval notation。Intermediate Algebra 223 件ほか）。(a, b) は座標の点と同じ書き方なので文脈で区別する。",
      "統計でも interval を使う: 信頼区間は confidence interval。"
    ],
    "related": [
      "closed-interval",
      "open-interval",
      "endpoint",
      "confidence-interval",
      "quadratic-inequality",
      "interval-notation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "区間 (数学)",
        "en": "Interval (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 積分: 区間 a ≦ x ≦ b で f(x) ≧ 0。数学III 積分: 閉区間 [a, b]（1 件）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "平成 30 年度 数学I: ア ≦ x ≦ イ における関数の最小値（区間の記法は無い）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "interval notation（223 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "interval notation（72 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "interval": 1711
      },
      "written": {
        "interval": 2028
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inverse-function",
    "ja": {
      "term": "逆関数",
      "reading": "ぎゃくかんすう",
      "alt": []
    },
    "en": {
      "term": "inverse function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "y = f(x) を x について解いて x と y を入れかえた関数 f⁻¹(x)。f が 1 対 1 のときに存在し、グラフは y = f(x) と直線 y = x に関して対称。",
    "definition_en": "The function f⁻¹(x) obtained by solving y = f(x) for x and swapping x and y; it exists when f is one-to-one, and its graph is the reflection of y = f(x) across y = x.",
    "latex": "f^{-1}(f(x)) = x",
    "spoken_en": "f inverse of f of x equals x",
    "examples": [
      {
        "en": "The inverse function undoes whatever f does.",
        "ja": "逆関数は、f がしたことを元に戻す。",
        "register": "spoken"
      },
      {
        "en": "Find the inverse function of f(x) = (2x + 1)/(x − 3) and state its domain.",
        "ja": "f(x) = (2x + 1)/(x − 3) の逆関数を求め、その定義域を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "f⁻¹(x) は 1/f(x) ではない。f inverse of x と読む。"
    ],
    "related": [
      "find-the-inverse",
      "one-to-one",
      "logarithmic-function",
      "restricted-domain",
      "derivative-of-an-inverse-function",
      "horizontal-line-test",
      "bijection",
      "function-notation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "逆写像",
        "en": "Inverse function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III (1) 極限: 合成関数や逆関数（数学II の対数関数は逆関数を数学III の内容とする）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "inverse function": 115
      },
      "written": {
        "inverse function": 196
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inverse-hyperbolic-functions",
    "ja": {
      "term": "逆双曲線関数",
      "reading": "ぎゃくそうきょくせんかんすう",
      "alt": []
    },
    "en": {
      "term": "inverse hyperbolic functions",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "双曲線関数の逆関数 sinh⁻¹ x、cosh⁻¹ x など。",
    "definition_en": "The inverses of the hyperbolic functions, such as sinh⁻¹ x and cosh⁻¹ x.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The inverse hyperbolic functions can all be written with natural logs.",
        "ja": "逆双曲線関数はどれも自然対数で書ける。",
        "register": "spoken"
      },
      {
        "en": "Show that sinh⁻¹ x = ln(x + √(x² + 1)).",
        "ja": "sinh⁻¹ x = log(x + √(x² + 1)) を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "hyperbolic-functions",
      "derivatives-of-inverse-trig-functions"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "逆双曲線関数",
        "en": "Inverse hyperbolic functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "inverse hyperbolic functions": 1
      },
      "written": {
        "inverse hyperbolic functions": 9
      },
      "sources": [
        "openstax-calculus",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 inverse hyperbolic functions（本文 9 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "inverse-matrix",
    "ja": {
      "term": "逆行列",
      "reading": "ぎゃくぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "inverse matrix",
      "alt": [
        "matrix inverse"
      ],
      "uk": null,
      "variants": [
        {
          "term": "inverse of a matrix",
          "register": "written",
          "note": "話し言葉では少なく、書き言葉では最も多い（ほとんどが OpenStax Algebra and Trigonometry）。"
        }
      ],
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列 A に対して AX = XA = E となる行列 X。A⁻¹ と書く。",
    "definition_en": "For a square matrix A, the matrix X with AX = XA = E (I), written A⁻¹.",
    "latex": "A^{-1}",
    "spoken_en": "A inverse",
    "examples": [
      {
        "en": "If the determinant is zero, the inverse matrix doesn't exist.",
        "ja": "行列式が 0 なら、逆行列は存在しない。",
        "register": "spoken"
      },
      {
        "en": "Find the inverse of the matrix A = [[2, 1], [5, 3]].",
        "ja": "行列 A = [[2, 1], [5, 3]] の逆行列を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "A⁻¹ は A inverse と読む（用例コーパスの話し言葉に多い）。"
    ],
    "related": [
      "identity-matrix",
      "determinant"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正則行列",
        "en": "Invertible matrix"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学特論 (2) 行列とその応用: 行列の積と逆行列、行列を用いた連立一次方程式の解法"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "inverse matrix": 31,
        "inverse of a matrix": 4,
        "matrix inverse": 1
      },
      "written": {
        "matrix inverse": 7,
        "inverse of a matrix": 20,
        "inverse matrix": 8
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inverse-proportion",
    "ja": {
      "term": "反比例",
      "reading": "はんぴれい",
      "alt": [
        "〜に反比例する",
        "逆変化"
      ]
    },
    "en": {
      "term": "inverse variation",
      "alt": [
        "inverse proportion"
      ],
      "variants": [
        {
          "term": "inversely proportional",
          "register": "both",
          "note": "話し言葉で最も多い（Khan Academy と Professor Leonard）。y is inversely proportional to x の形。"
        },
        {
          "term": "varies inversely",
          "register": "both",
          "note": "書き言葉で最も多い（OpenStax Elementary Algebra・Algebra and Trigonometry）。y varies inversely with x の形。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説（第 1 学年）の「反比例」は、a を比例定数として y = a/x または xy = a で表される関係。英語では inverse variation と呼び、「y は x に反比例する」は y varies inversely with x または y is inversely proportional to x と言う。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "2 つの量 x、y の関係が、0 でない定数 a を使って y = a/x の式で表されること。x が 2 倍、3 倍になると y は 1/2 倍、1/3 倍になり、積 xy はいつも a になる。",
    "definition_en": "A relationship between two quantities that can be written as y = a/x, where a is a nonzero constant, so the product xy is always a. When x doubles, y is cut in half.",
    "latex": "y = \\frac{a}{x}",
    "spoken_en": "y equals a over x",
    "examples": [
      {
        "en": "When x doubles, y gets cut in half, so this table shows an inverse variation.",
        "ja": "x が 2 倍になると y は半分になるから、この表は反比例を表しているね。",
        "register": "spoken"
      },
      {
        "en": "y varies inversely with x, and y = 6 when x = 2. Write an equation for this inverse variation, and find y when x = 4.",
        "ja": "y は x に反比例し、x = 2 のとき y = 6 である。この反比例の式を書き、x = 4 のときの y の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "inverse だけでは「反比例」の意味にならない。inverse function（逆関数、エントリ inverse-function）のように「逆」の意味でも使うので、inverse variation ／ varies inversely の形で言う。",
      "反比例のグラフは双曲線で、英語でも hyperbola（エントリ hyperbola）。x = 0 のときの y の値はなく、y が 0 になることもないので、グラフは x 軸とも y 軸とも交わらない。",
      "y = a/x の a も「比例定数」と呼ぶ（エントリ constant-of-proportionality）。"
    ],
    "related": [
      "direct-proportion",
      "constant-of-proportionality",
      "hyperbola",
      "reciprocal",
      "inverse-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "反比例",
        "en": "Proportionality (mathematics)#Inverse proportionality"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 C 関数: 反比例は a を比例定数として y＝a/x または xy＝a で表される関係"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "inverse variation": 14,
        "varies inversely": 19,
        "inversely proportional": 27
      },
      "written": {
        "inversely proportional": 12,
        "inverse variation": 34,
        "varies inversely": 68
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inverse-trigonometric-function",
    "ja": {
      "term": "逆三角関数",
      "reading": "ぎゃくさんかくかんすう",
      "alt": []
    },
    "en": {
      "term": "inverse trigonometric function",
      "alt": [
        "inverse trig function"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "定義域を制限した三角関数の逆関数。arcsin x（sin⁻¹ x）、arccos x、arctan x など。",
    "definition_en": "The inverse of a trigonometric function on a restricted domain: arcsin x (sin⁻¹ x), arccos x, arctan x, and so on.",
    "latex": "y = \\sin^{-1}x \\iff \\sin y = x,\\ -\\frac{\\pi}{2} \\le y \\le \\frac{\\pi}{2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Inverse sine of one half is π/6 — it only gives you the angle between −π/2 and π/2.",
        "ja": "arcsin(1/2) は π/6。−π/2 から π/2 の間の角しか返さない。",
        "register": "spoken"
      },
      {
        "en": "Evaluate sin(cos⁻¹(3/5)) without a calculator.",
        "ja": "電卓を使わずに sin(cos⁻¹(3/5)) の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説には出てこないが、OpenStax Algebra and Trigonometry（Precalculus）で扱い、AP Calculus でも微分する（CED topic 3.4）。",
      "sin⁻¹ x は 1/sin x ではない。inverse sine of x と読み（用例コーパスの話し言葉にも出てくる）、arcsine とも言う。"
    ],
    "related": [
      "derivatives-of-inverse-trig-functions",
      "integrals-giving-inverse-trig-functions",
      "trigonometric-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "逆三角関数",
        "en": "Inverse trigonometric functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 3.4（derivatives of inverse trigonometric functions）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「逆三角関数」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "inverse trig function": 8,
        "inverse trigonometric function": 5
      },
      "written": {
        "inverse trigonometric function": 72
      },
      "sources": [
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "inverse",
    "ja": {
      "term": "裏",
      "reading": "うら",
      "alt": []
    },
    "en": {
      "term": "inverse",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「裏」は学習指導要領解説に無い（日本語版 Wikipedia「裏 (論理学)」にはある。解説（数学I）が挙げるのは対偶・必要条件・十分条件）。",
    "domains": [
      "logic",
      "proof"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "命題「p ならば q」に対して、仮定と結論をともに否定した命題「p でないならば q でない」。",
    "definition_en": "For a conditional statement \"if p, then q,\" the statement \"if not p, then not q,\" formed by negating both the hypothesis and the conclusion.",
    "latex": "\\neg p \\Rightarrow \\neg q",
    "spoken_en": "if not p, then not q",
    "examples": [
      {
        "en": "The inverse just puts a \"not\" on both parts: if it's not raining, then the ground isn't wet.",
        "ja": "裏は両方に「ない」を付けるだけ。雨が降っていないならば、地面は濡れていない。",
        "register": "spoken"
      },
      {
        "en": "Write the converse, the inverse, and the contrapositive of the statement \"If x = 2, then x² = 4.\" Which of them are true?",
        "ja": "命題「x = 2 ならば x² = 4」の逆・裏・対偶を書け。そのうち真であるものはどれか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "逆は converse、裏は inverse、対偶は contrapositive。元の命題が真でも裏は真とは限らない（上の例の裏「x ≠ 2 ならば x² ≠ 4」は x = −2 が反例）。",
      "inverse 単独は逆関数・逆行列の意味が圧倒的に多いので、裏は the inverse of the statement のように statement を添えて言う（CK-12 Geometry 2.12・Levin 1.2）。",
      "裏と逆は互いに対偶の関係なので、真偽が一致する。"
    ],
    "related": [
      "contrapositive",
      "negation",
      "proposition",
      "condition",
      "counterexample",
      "converse",
      "conditional-statement"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "裏 (論理学)",
        "en": "Inverse (logic)"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.2 Implications（Definition 1.2.6）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.12 Converse Inverse and Contrapositive Statements"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 集合と命題: 「裏」は 0 件（対偶・必要条件・十分条件はある）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 the inverse is the statement | contrapositive, and inverse | contrapositive and inverse | converse, inverse | the inverse of this statement | inverse of this statement（CK-12 Geometry 2.12 Converse Inverse and Contrapositive Statements・計 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "invertible-matrix",
    "ja": {
      "term": "正則行列",
      "reading": "せいそくぎょうれつ",
      "alt": [
        "可逆行列"
      ]
    },
    "en": {
      "term": "invertible matrix",
      "alt": [
        "nonsingular matrix"
      ],
      "uk": null,
      "variants": [
        {
          "term": "matrix is invertible",
          "register": "spoken",
          "note": "述語の形（the matrix is invertible）。話し言葉はすべて MIT OCW で、書き言葉にはほとんど出てこない。"
        }
      ],
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "逆行列をもつ正方行列。行列式が 0 でないことと同じ。",
    "definition_en": "A square matrix that has an inverse; equivalently, its determinant is not zero.",
    "latex": "\\det A \\neq 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "If the matrix is invertible, the system has exactly one solution.",
        "ja": "行列が正則なら、連立方程式の解はちょうど 1 つだ。",
        "register": "spoken"
      },
      {
        "en": "Show that A is an invertible matrix by computing its determinant.",
        "ja": "行列式を計算して、A が正則行列であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「正則」は英語で invertible（または nonsingular）。regular とは訳さない。"
    ],
    "related": [
      "inverse-matrix",
      "singular-matrix",
      "determinant",
      "elementary-matrix"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正則行列",
        "en": "Invertible matrix"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "2.4 Matrix Inverses"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "matrix is invertible": 14,
        "invertible matrix": 20
      },
      "written": {
        "invertible matrix": 4,
        "matrix is invertible": 2
      },
      "sources": [
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "irrational-number",
    "ja": {
      "term": "無理数",
      "reading": "むりすう",
      "alt": []
    },
    "en": {
      "term": "irrational number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "分数で表すことのできない数。√2、√3、π などで、小数で表すと循環しない無限小数になる。",
    "definition_en": "A real number that is not the quotient of any two integers; √2, √3, and π are examples. Its decimal expansion neither ends nor falls into a repeating block.",
    "latex": "\\sqrt{2} = 1.41421\\ldots",
    "spoken_en": "the square root of two equals one point four one four two one, and so on",
    "examples": [
      {
        "en": "Pi is an irrational number, so 3.14 is only an estimate; the decimal never ends and never repeats.",
        "ja": "π は無理数だから、3.14 は近似値にすぎない。小数はどこまでも続いて、繰り返しもしない。",
        "register": "spoken"
      },
      {
        "en": "Which of the following is an irrational number? √25, 0.6, √7, −2/3",
        "ja": "次のうち、無理数であるものを選べ。√25、0.6、√7、−2/3",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "√ が付いていれば無理数、とは限らない。√25 = 5、√(4/9) = 2/3 は有理数。",
      "3.14 や 22/7 は π の近似値で、π そのものではない。3.14 も 22/7 も有理数。",
      "無限小数（エントリ non-terminating-decimal）のうち、0.333… = 1/3 のように循環するものは有理数で、循環しないものだけが無理数。有理数と無理数を合わせて実数（real number、エントリ real-number）。"
    ],
    "related": [
      "rational-number",
      "real-number",
      "square-root-of-a-number",
      "pi",
      "non-terminating-decimal"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "無理数",
        "en": "Irrational number"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "irrational number": 57
      },
      "written": {
        "irrational number": 73
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "is-biased",
    "ja": {
      "term": "偏りがある",
      "reading": "かたよりがある",
      "alt": []
    },
    "en": {
      "term": "biased",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "near",
    "mapping_note": "英語の biased は、標本の選び方（OpenStax Introductory Statistics の biased sample）・推定量（エントリ unbiased-estimator の反対）・硬貨の公平さ（biased coin）にまたがる。中学校学習指導要領解説の「偏り」は標本の抽出について言い（偏りなく抽出する）、日本語の「偏りがある」はこの 1 つ目に当たる。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "標本の選び方や推定の方法が、系統的に一方の結果に寄っていること。",
    "definition_en": "Systematically favoring certain outcomes, said of a way of choosing a sample or of estimating.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you only ask people at the gym, your sample is biased.",
        "ja": "ジムにいる人にだけ聞けば、標本に偏りがある。",
        "register": "spoken"
      },
      {
        "en": "Explain why the sampling method is biased.",
        "ja": "この標本抽出の方法に偏りがある理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "biased coin（表と裏の出方が同じでない硬貨）のように「不公平な」の意味でも使う。"
    ],
    "related": [
      "unbiased",
      "sampling",
      "random-sampling"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 標本調査: 偏りなく抽出するための代表的な方法として無作為抽出"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "biased": 65
      },
      "written": {
        "biased": 36
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "is-monotonically-increasing",
    "ja": {
      "term": "単調に増加する",
      "reading": "たんちょうにぞうかする",
      "alt": [
        "単調増加",
        "右上がり"
      ]
    },
    "en": {
      "term": "is increasing",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「単調に増加する」は、英語では monotonically increasing と言うより、単に is increasing と言う（用例コーパスには monotonically increasing は出てこず、is increasing は話し言葉・書き言葉とも多い）。OpenStax Calculus Volume 1（1.1）は増加を f(x₁) ≤ f(x₂) で定めて等号なしを strictly increasing と呼び、等号を含めるかどうかは定義によるので、区別するときは strictly increasing（狭義）と nondecreasing（広義）を使う。",
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "中2",
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I",
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "x が増えると、関数の値も必ず増える。",
    "definition_en": "Of a function: to rise as x increases, over an interval.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Since f prime is positive everywhere, f is increasing on the whole real line.",
        "ja": "f′ がどこでも正なので、f は実数全体で単調に増加する。",
        "register": "spoken"
      },
      {
        "en": "Show that f(x) = x³ + x is increasing on (−∞, ∞).",
        "ja": "f(x) = x³ + x が (−∞, ∞) で単調に増加することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「右上がり」（1 次関数のグラフ）は、英語ではグラフの形ではなく is increasing か the slope is positive と言う。"
    ],
    "related": [
      "increasing-and-decreasing",
      "decreasing"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.3（is increasing）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「一次関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%80%E6%AC%A1%E9%96%A2%E6%95%B0",
        "note": "傾きが正ならグラフは右上がり"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "is increasing": 200
      },
      "written": {
        "is increasing": 190
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "isomorphism",
    "ja": {
      "term": "同型",
      "reading": "どうけい",
      "alt": [
        "同型写像"
      ]
    },
    "en": {
      "term": "isomorphism",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "isomorphic",
          "register": "written",
          "note": "形容詞（be isomorphic to）。話し言葉には出てこず、書き言葉はすべて MIT の講義ノート。"
        }
      ],
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "2 つのベクトル空間のあいだの、全単射な線形写像。そうした写像があるとき 2 つの空間は同型であるという。",
    "definition_en": "A linear map between two vector spaces that is one-to-one and onto; when one exists, the spaces are isomorphic.",
    "latex": "V \\cong W",
    "spoken_en": null,
    "examples": [
      {
        "en": "Any two vector spaces of the same finite dimension are isomorphic.",
        "ja": "有限次元で次元が同じベクトル空間どうしは同型だ。",
        "register": "spoken"
      },
      {
        "en": "Show that the map T is an isomorphism.",
        "ja": "写像 T が同型写像であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉は MIT の講義ノートのグラフの同型（isomorphic graphs）だけ。Levin もグラフの同型（2.1）で使う。"
    ],
    "related": [
      "onto",
      "one-to-one",
      "linear-transformation",
      "dimension"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "同型写像",
        "en": "Isomorphism"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "7.3 Isomorphisms and Composition"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "isomorphism": 1
      },
      "written": {
        "isomorphism": 10,
        "isomorphic": 15
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "isosceles-triangle-theorem",
    "ja": {
      "term": "二等辺三角形の定理",
      "reading": "にとうへんさんかくけいのていり",
      "alt": [
        "二等辺三角形の底角は等しい",
        "底角定理"
      ]
    },
    "en": {
      "term": "base angles theorem",
      "alt": [
        "isosceles triangle theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "二等辺三角形の 2 つの底角は等しいという定理。逆も成り立ち、2 つの角が等しい三角形は二等辺三角形である。",
    "definition_en": "The theorem that the two base angles of an isosceles triangle are congruent. Its converse also holds: a triangle with two congruent angles is isosceles.",
    "latex": "AB = AC \\implies \\angle B = \\angle C",
    "spoken_en": "if AB equals AC, then angle B equals angle C",
    "examples": [
      {
        "en": "AB equals AC, so by the base angles theorem the base angles are equal. That leaves 140 degrees to split, so each one is 70.",
        "ja": "AB = AC だから、二等辺三角形の定理より底角は等しい。残りの 140° を半分に分けて、それぞれ 70° だ。",
        "register": "spoken"
      },
      {
        "en": "In △ABC, AB = AC and m∠A = 40°. Find m∠B.",
        "ja": "△ABC で AB = AC, ∠A = 40° のとき、∠B の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "逆（2 つの角が等しければ二等辺三角形）は converse of the base angles theorem（converse of the isosceles triangle theorem）と呼ぶ。",
      "問題文の m∠B は the measure of angle B（∠B の大きさ）と読む。英語の問題文は角 ∠B とその大きさ m∠B を書き分ける。",
      "用例コーパスには出てこない。CK-12 Geometry は base angles theorem が 10 件（4.4 Isosceles Triangles・4.5 Equilateral Triangles）、isosceles triangle theorem が 7 件（4.4・5.16 Kites）。IM Geometry の isosceles triangle theorem は 1 件（2.8）。"
    ],
    "related": [
      "isosceles-triangle",
      "equal-angles",
      "triangle",
      "congruent"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "2.8 The Perpendicular Bisector Theorem（1 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.4 Isosceles Triangles・4.5 Equilateral Triangles"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 base angles theorem（CK-12 Geometry 4.4 Isosceles Triangles・CK-12 Geometry 4.5 Equilateral Triangles・計 10 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "isosceles-triangle",
    "ja": {
      "term": "二等辺三角形",
      "reading": "にとうへんさんかくけい",
      "alt": []
    },
    "en": {
      "term": "isosceles triangle",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 辺の長さが等しい三角形。等しい 2 辺にはさまれた角を頂角、残りの 2 角を底角といい、2 つの底角は等しい。",
    "definition_en": "A triangle in which two sides have the same length; the angles opposite those two sides (the base angles) are also equal.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's an isosceles triangle, so the two base angles are equal.",
        "ja": "二等辺三角形だから、2 つの底角は等しい。",
        "register": "spoken"
      },
      {
        "en": "In isosceles triangle ABC with AB = AC, the vertex angle A measures 40°. Find the measure of angle B.",
        "ja": "AB = AC の二等辺三角形 ABC で、頂角 ∠A = 40° のとき、∠B の大きさを求めよ。",
        "register": "written"
      },
      {
        "en": "Prove that the base angles of an isosceles triangle are equal.",
        "ja": "二等辺三角形の底角は等しいことを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "base angles of an isosceles triangle",
        "ja": "二等辺三角形の底角"
      }
    ],
    "pitfalls": [
      "isosceles の綴りと発音に注意（アイササリーズに近い。Merriam-Webster: ī-ˈsä-sə-ˌlēz）。",
      "頂角は vertex angle、底角は base angles、底辺は base。等しい 2 辺は legs と呼ぶ。"
    ],
    "related": [
      "triangle",
      "congruent",
      "vertex",
      "angle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二等辺三角形",
        "en": "Isosceles triangle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「isosceles」",
        "url": "https://www.merriam-webster.com/dictionary/isosceles",
        "note": "発音 ī-ˈsä-sə-ˌlēz（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "isosceles triangle": 6
      },
      "written": {
        "isosceles triangle": 25
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "iterated-integral",
    "ja": {
      "term": "累次積分",
      "reading": "るいじせきぶん",
      "alt": [
        "逐次積分"
      ]
    },
    "en": {
      "term": "iterated integral",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "1 変数の積分を順に繰り返して計算する積分。二重積分はふつう累次積分に直して計算する。",
    "definition_en": "An integral computed by doing one-variable integrals one after another; a double integral is usually evaluated as one.",
    "latex": "\\int_a^b \\int_c^d f(x, y)\\,dy\\,dx",
    "spoken_en": null,
    "examples": [
      {
        "en": "In an iterated integral, work from the inside out.",
        "ja": "累次積分は内側から計算する。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the iterated integral ∫₀¹ ∫₀ˣ xy dy dx.",
        "ja": "累次積分 ∫₀¹ ∫₀ˣ xy dy dx を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "double-integral",
      "fubinis-theorem",
      "changing-the-order-of-integration"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "逐次積分",
        "en": "Iterated integral"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "iterated integral": 23
      },
      "written": {
        "iterated integral": 72
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "jacobian",
    "ja": {
      "term": "ヤコビアン",
      "reading": "やこびあん",
      "alt": [
        "ヤコビ行列式"
      ]
    },
    "en": {
      "term": "Jacobian",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "変数変換 x = g(u, v)、y = h(u, v) の偏導関数を並べた行列式 ∂(x, y)/∂(u, v)。面積の拡大率を表す。",
    "definition_en": "The determinant ∂(x, y)/∂(u, v) of the partial derivatives of a change of variables, which measures how areas are stretched.",
    "latex": "\\frac{\\partial(x, y)}{\\partial(u, v)} = \\begin{vmatrix} x_u & x_v \\\\ y_u & y_v \\end{vmatrix}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Don't forget to multiply by the absolute value of the Jacobian.",
        "ja": "ヤコビアンの絶対値を掛けるのを忘れずに。",
        "register": "spoken"
      },
      {
        "en": "Compute the Jacobian of the transformation x = 2u + v, y = u − 3v.",
        "ja": "変換 x = 2u + v、y = u − 3v のヤコビアンを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "偏導関数を並べた行列そのもの（Jacobian matrix）を指すこともある。用例コーパスの話し言葉は大半が MIT 18.03（連立微分方程式のヤコビ行列）。"
    ],
    "related": [
      "change-of-variables",
      "double-integral",
      "partial-derivative"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "Jacobian": 39
      },
      "written": {
        "Jacobian": 34
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "joint-variation",
    "ja": {
      "term": "結合変化",
      "reading": "けつごうへんか",
      "alt": []
    },
    "en": {
      "term": "joint variation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「結合変化」は学習指導要領解説に無い、本プロジェクトの訳語。日本では「z は x と y の積に比例する」と言う。OpenStax Algebra and Trigonometry は、direct variation・inverse variation（エントリ direct-proportion、inverse-proportion）と同じ節で扱う。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "ある量が、2 つ以上の量の積に比例する関係。z = kxy（k は 0 でない定数）の形で表す。",
    "definition_en": "A relationship in which one quantity is a constant multiple of the product of two or more other quantities. For example, z = kxy, where k is a nonzero constant.",
    "latex": "z = kxy",
    "spoken_en": "z equals k x y",
    "examples": [
      {
        "en": "This is joint variation: z equals k times x times y, so z depends on both x and y at once.",
        "ja": "これは結合変化。z = k × x × y だから、z は x と y の両方で決まる。",
        "register": "spoken"
      },
      {
        "en": "The quantity z varies jointly with x and y, and z = 24 when x = 2 and y = 3. Write the joint variation equation, and find z when x = 5 and y = 4.",
        "ja": "z は x と y の積に比例し、x = 2、y = 3 のとき z = 24 である。z を x と y の式で表し、x = 5、y = 4 のときの z の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「z は x と y の積に比例する」は英語で z varies jointly with x and y と言い、教科書（OpenStax Algebra and Trigonometry）ではこの動詞句のほうが名詞の joint variation より多い。product（積）という語を使わずに言える。",
      "式の k は constant of variation（エントリ constant-of-proportionality）。求め方は比例と同じで、与えられた 1 組の値を代入して k を出す。",
      "「x に比例し y に反比例する」（z = kx/y）は積に比例する形ではないので、joint variation とは別の関係。"
    ],
    "related": [
      "constant-of-proportionality",
      "direct-proportion",
      "inverse-proportion"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "joint variation": 9
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 joint variation（本文 9 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "jump-discontinuity",
    "ja": {
      "term": "跳躍不連続",
      "reading": "ちょうやくふれんぞく",
      "alt": []
    },
    "en": {
      "term": "jump discontinuity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「跳躍不連続」は学習指導要領解説に無い（日本語版 Wikipedia「不連続性の分類」には「跳躍不連続点」がある）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "左右の片側極限はどちらも存在するが、値が一致しない不連続点。グラフが跳んでいる。",
    "definition_en": "A discontinuity where both one-sided limits exist but are different, so the graph jumps.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The floor function has a jump discontinuity at every integer.",
        "ja": "床関数（ガウス記号の関数）はすべての整数で跳躍不連続になる。",
        "register": "spoken"
      },
      {
        "en": "f has a jump discontinuity at x = 0 because the limit from the left differs from the limit from the right.",
        "ja": "左側極限と右側極限が異なるので、f は x = 0 で跳躍不連続である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "discontinuous",
      "removable-discontinuity",
      "infinite-discontinuity",
      "one-sided-limit"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「不連続性の分類」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%8D%E9%80%A3%E7%B6%9A%E6%80%A7%E3%81%AE%E5%88%86%E9%A1%9E",
        "note": "跳躍不連続点（jump discontinuity）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「跳躍不連続」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "jump discontinuity": 24
      },
      "written": {
        "jump discontinuity": 26
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "lagrange-error-bound",
    "ja": {
      "term": "ラグランジュの誤差限界",
      "reading": "らぐらんじゅのごさげんかい",
      "alt": []
    },
    "en": {
      "term": "Lagrange error bound",
      "alt": [
        "Lagrange remainder"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「ラグランジュの誤差限界」は学習指導要領解説に無い、本プロジェクトの訳語。日本の大学の教科書はテイラーの定理の剰余項（ラグランジュの剰余項）として扱い、その上からの評価に名前をつけない。AP の CED は評価そのものを Lagrange error bound と呼ぶ（topic 10.12）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "n 次テイラー多項式で近似したときの誤差が、M|x − a|ⁿ⁺¹/(n + 1)! 以下になるという評価。M は (n + 1) 階導関数の絶対値の上限。",
    "definition_en": "The bound M|x − a|ⁿ⁺¹/(n + 1)! on the error of the nth Taylor polynomial, where M bounds the absolute value of the (n + 1)st derivative.",
    "latex": "|R_n(x)| \\le \\frac{M}{(n+1)!}|x-a|^{n+1}",
    "spoken_en": "the absolute value of R sub n of x is at most M over n plus one factorial times the absolute value of x minus a to the n plus one",
    "examples": [
      {
        "en": "For the Lagrange error bound, you need a bound on the next derivative.",
        "ja": "ラグランジュの誤差限界には、次の階数の導関数の上限が要る。",
        "register": "spoken"
      },
      {
        "en": "Use the Lagrange error bound to show that P₄(0.1) approximates e^{0.1} with error less than 10⁻⁶.",
        "ja": "ラグランジュの誤差限界を用いて、P₄(0.1) が e^{0.1} を誤差 10⁻⁶ 未満で近似することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "taylor-polynomial",
      "taylors-theorem",
      "alternating-series-error-bound"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.12（Lagrange error bound）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Lagrange error bound": 2
      },
      "sources": [
        "khan-ap-calc"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CED の呼び方 Lagrange error bound（topic 10.12）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "lagrange-multipliers",
    "ja": {
      "term": "ラグランジュの未定乗数法",
      "reading": "らぐらんじゅのみていじょうすうほう",
      "alt": []
    },
    "en": {
      "term": "Lagrange multipliers",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "条件 g(x, y) = c のもとで f(x, y) の極値を求めるために、∇f = λ∇g を解く方法。",
    "definition_en": "A method for finding the extreme values of f(x, y) subject to g(x, y) = c by solving ∇f = λ∇g.",
    "latex": "\\nabla f = \\lambda \\nabla g, \\quad g(x, y) = c",
    "spoken_en": "del f equals lambda del g",
    "examples": [
      {
        "en": "With Lagrange multipliers, the gradients of f and g have to line up.",
        "ja": "ラグランジュの未定乗数法では、f と g の勾配が同じ向きにそろう。",
        "register": "spoken"
      },
      {
        "en": "Use Lagrange multipliers to find the maximum of f(x, y) = xy subject to x + y = 10.",
        "ja": "ラグランジュの未定乗数法を用いて、x + y = 10 のもとで f(x, y) = xy の最大値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "method of Lagrange multipliers とも言う（書き言葉でよく使う）。λ は Lagrange multiplier。"
    ],
    "related": [
      "gradient",
      "saddle-point",
      "local-extremum",
      "optimization-problem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ラグランジュの未定乗数法",
        "en": "Lagrange multiplier"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "Lagrange multipliers": 10
      },
      "written": {
        "Lagrange multipliers": 43
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "last-term",
    "ja": {
      "term": "末項",
      "reading": "まっこう",
      "alt": []
    },
    "en": {
      "term": "last term",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "有限個の項からなる数列の最後の項。",
    "definition_en": "The final term of a sequence with finitely many terms.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For an arithmetic series, add the first and last terms and multiply by half the number of terms.",
        "ja": "等差数列の和は、初項と末項を足して項数の半分を掛ける。",
        "register": "spoken"
      },
      {
        "en": "An arithmetic sequence has first term 4, last term 58, and 19 terms. Find the common difference.",
        "ja": "初項 4、末項 58、項数 19 の等差数列の公差を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "last term は式の最後の項（(x + 3)² の 3、FOIL の Last）にも使う語。"
    ],
    "related": [
      "first-term",
      "sum-of-an-arithmetic-sequence"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "last term in … series | last term of … sequence": 6
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 last term in … series | last term of … sequence（本文 6 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "lateral-area",
    "ja": {
      "term": "側面積",
      "reading": "そくめんせき",
      "alt": []
    },
    "en": {
      "term": "lateral area",
      "alt": [],
      "variants": [
        {
          "term": "lateral surface area",
          "register": "written",
          "note": "教科書（OpenStax Calculus）の書き方。話し言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "立体の側面全体の面積。底面の面積は入れない。",
    "definition_en": "The total area of the lateral faces or the curved side of a solid, not counting the base or bases.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Lateral area is just the sides, so don't add in the top and the bottom.",
        "ja": "側面積は側面だけだから、上と下の面は足さないでね。",
        "register": "spoken"
      },
      {
        "en": "Find the lateral area of a cylinder with a radius of 3 cm and a height of 10 cm. Leave your answer in terms of π.",
        "ja": "底面の半径が 3 cm、高さが 10 cm の円柱の側面積を求めよ。ただし、円周率は π とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "表面積（エントリ surface-area）= 側面積 + 底面積。角柱・円柱は底面が 2 つあるので底面積を 2 倍して足す。問題が lateral area（側面積）と surface area（表面積）のどちらを聞いているかを確かめる。",
      "円柱の側面を開くと長方形になり、その横の長さは底面の円周 2πr なので、円柱の側面積は 2πrh。",
      "角柱の側面をつなげて開くと、横の長さが底面の周の長さに等しい長方形になる。だから角柱の側面積は、底面の周の長さ × 高さで求められる。"
    ],
    "related": [
      "lateral-face",
      "surface-area",
      "area-of-the-base",
      "net",
      "cylinder"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "lateral area": 28,
        "lateral surface area": 1
      },
      "written": {
        "lateral surface area": 13
      },
      "sources": [
        "openstax-calculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "lateral-face",
    "ja": {
      "term": "側面",
      "reading": "そくめん",
      "alt": []
    },
    "en": {
      "term": "lateral face",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "角柱や角錐の面のうち、底面以外の面。角柱の側面は長方形（傾いた角柱では平行四辺形）、角錐の側面は三角形になる。",
    "definition_en": "A face of a prism or pyramid that is not a base. The lateral faces of a prism are rectangles (parallelograms if the prism is oblique), and those of a pyramid are triangles.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A triangular prism has three lateral faces, one for each side of the triangle.",
        "ja": "三角柱の側面は 3 つで、底面の三角形の辺 1 本につき 1 つずつある。",
        "register": "spoken"
      },
      {
        "en": "How many lateral faces does a hexagonal prism have, and what shape are they?",
        "ja": "六角柱の側面はいくつあるか。また、側面はどんな形か答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "lateral は「横の」という意味で、強勢は最初の音節（LAT-。Merriam-Webster: ˈla-tə-rəl）。円柱・円錐の側面は平らな面（face）ではなく曲面なので、lateral surface と言う。",
      "底面は形で決まり、図の向きでは決まらない。角柱が横に倒れた図では、下に接している面は側面で、両端の合同な 2 つの面が底面になる。"
    ],
    "related": [
      "prism",
      "pyramid",
      "lateral-area",
      "polyhedron",
      "cylinder"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "9.4 Surface Area（lateral faces 1 件）"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「lateral」",
        "url": "https://www.merriam-webster.com/dictionary/lateral",
        "note": "発音 ˈla-tə-rəl（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "lateral face": 1
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは OpenStax の呼び方 lateral face（本文 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "latus-rectum",
    "ja": {
      "term": "通径",
      "reading": "つうけい",
      "alt": []
    },
    "en": {
      "term": "latus rectum",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "焦点を通り、軸（放物線の軸、楕円・双曲線の主軸）に垂直な弦。放物線 y² = 4px では長さ 4|p|。",
    "definition_en": "The chord through a focus perpendicular to the axis (of a parabola, or the major/transverse axis); for y² = 4px its length is 4|p|.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The latus rectum tells you how wide the parabola is at the focus.",
        "ja": "通径は、焦点のところで放物線がどれだけ開いているかを表す。",
        "register": "spoken"
      },
      {
        "en": "Find the focus, directrix, and endpoints of the latus rectum of the parabola y² = 12x.",
        "ja": "放物線 y² = 12x の焦点・準線・通径の両端を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは書き言葉はすべて OpenStax Algebra and Trigonometry。話し言葉には出てこない。"
    ],
    "related": [
      "focus",
      "directrix",
      "conic-section"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "latus rectum": 32
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "law-of-cosines",
    "ja": {
      "term": "余弦定理",
      "reading": "よげんていり",
      "alt": []
    },
    "en": {
      "term": "law of cosines",
      "alt": [
        "generalized Pythagorean theorem"
      ],
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "三角形の 1 辺の長さの 2 乗を、他の 2 辺の長さとその間の角の余弦で表す定理。三平方の定理を一般の三角形に広げたもの。",
    "definition_en": "The theorem giving the square of one side of a triangle from the other two sides and the angle they enclose: a² = b² + c² − 2bc cos A. It extends the Pythagorean theorem to every triangle.",
    "latex": "a^2 = b^2 + c^2 - 2bc\\cos A",
    "spoken_en": "a squared equals b squared plus c squared minus two b c cosine A",
    "examples": [
      {
        "en": "We have two sides and the angle between them, so the law of cosines gives us the third side right away.",
        "ja": "2 辺とその間の角がわかっているから、余弦定理を使えばすぐに残りの辺が出る。",
        "register": "spoken"
      },
      {
        "en": "In triangle ABC, b = 5, c = 8, and A = 60°. Using the law of cosines, find a.",
        "ja": "△ABC において b = 5, c = 8, A = 60° のとき、余弦定理を使って a を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "角を求めるときは cos A = (b² + c² − a²)/(2bc) の形で使う。3 辺がわかれば角が 1 つに決まる。",
      "英語版 Wikipedia「Law of cosines」は cosine rule ／ cosine formula という別名も挙げる。",
      "A = 90° のとき cos A = 0 となり、三平方の定理 a² = b² + c² に一致する。",
      "OpenStax Algebra and Trigonometry・Precalculus は余弦定理を Generalized Pythagorean Theorem（三平方の定理を直角でない三角形に広げたもの）とも呼ぶ。"
    ],
    "related": [
      "law-of-sines",
      "solving-triangles",
      "cosine",
      "herons-formula"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "余弦定理",
        "en": "Law of cosines"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Law_of_cosines",
        "note": "記事「Law of cosines」の冒頭: cosine formula ／ cosine rule とも呼ぶ（地域は書いていない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "law of cosines": 52,
        "generalized Pythagorean theorem": 3
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "law-of-detachment",
    "ja": {
      "term": "肯定式",
      "reading": "こうていしき",
      "alt": [
        "モーダスポネンス"
      ]
    },
    "en": {
      "term": "law of detachment",
      "alt": [
        "modus ponens"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof",
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "「p ならば q」が真で、p も真であるとき、q が真であると結論する推論の規則。",
    "definition_en": "The rule of inference that if p → q is true and p is true, then q is true.",
    "latex": "(p \\Rightarrow q) \\land p \\implies q",
    "spoken_en": "if p implies q and p is true, then q is true",
    "examples": [
      {
        "en": "Any number that ends in 0 is divisible by 10, and 470 ends in 0. So by modus ponens, 470 is divisible by 10.",
        "ja": "0 で終わる数は 10 で割り切れる。470 は 0 で終わる。だから肯定式により、470 は 10 で割り切れる。",
        "register": "spoken"
      },
      {
        "en": "Use the law of detachment to draw a conclusion: If two angles form a linear pair, then they are supplementary. ∠A and ∠B form a linear pair.",
        "ja": "肯定式を使って結論を書け。2 つの角が一直線をなすならば、その 2 角の和は 180° である。∠A と ∠B は一直線をなす。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "米国の Geometry の教科書では law of detachment、論理学・離散数学では modus ponens と呼ぶ。",
      "p ⇒ q と q から p を結論するのは逆向きの推論（affirming the consequent、後件肯定）で、正しくない。",
      "用例コーパスでは modus ponens が話し言葉（MIT OCW）・書き言葉（MIT の講義ノート）にわずかに出てくるだけで、law of detachment は出てこない。CK-12 Geometry は law of detachment と呼ぶ（2.7 Deductive Reasoning、3 件）。Levin の Discrete Mathematics は modus ponens（1.3、1 件）。"
    ],
    "related": [
      "law-of-syllogism",
      "conditional-statement",
      "deductive-reasoning"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.3 Rules of Logic"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "モーダスポネンス",
        "en": "Modus ponens"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.7 Deductive Reasoning"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "modus ponens": 2
      },
      "written": {
        "modus ponens": 3
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは CK-12 の呼び方 law of detachment（CK-12 Geometry 2.7 Deductive Reasoning・計 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "law-of-large-numbers",
    "ja": {
      "term": "大数の法則",
      "reading": "たいすうのほうそく",
      "alt": []
    },
    "en": {
      "term": "law of large numbers",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "中2",
        "数A",
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "試行の回数や標本の大きさを大きくすると、相対度数や標本平均が、真の確率や母平均に近づくという法則。",
    "definition_en": "The principle that as more trials are run or larger samples are taken, the relative frequency or the sample mean settles close to the true probability or the population mean.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Flip a coin ten times and you might get seven heads, but flip it ten thousand times and you'll be close to half.",
        "ja": "コインを 10 回投げるなら表が 7 回出るかもしれないが、1 万回投げれば半分に近くなる。",
        "register": "spoken"
      },
      {
        "en": "Explain how the law of large numbers applies to the results of the simulation.",
        "ja": "シミュレーションの結果に大数の法則がどう当てはまるか説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「大数」は「たいすう」と読む。"
    ],
    "related": [
      "central-limit-theorem",
      "experimental-probability",
      "sample-mean"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "大数の法則",
        "en": "Law of large numbers"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第2学年 D(2) 確率（大数の法則）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A (2) 場合の数と確率（「大数の法則」を基にして理解）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.3（law of large numbers）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "law of large numbers": 15
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "law-of-sines",
    "ja": {
      "term": "正弦定理",
      "reading": "せいげんていり",
      "alt": []
    },
    "en": {
      "term": "law of sines",
      "alt": [],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "三角形 ABC で、各辺の長さとその対角の正弦の比が一定で、外接円の半径 R の 2 倍に等しいという定理。",
    "definition_en": "In any triangle ABC, the ratio of each side to the sine of the opposite angle is the same, and it equals 2R, where R is the radius of the circumcircle.",
    "latex": "\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C} = 2R",
    "spoken_en": "a over sine A equals b over sine B equals c over sine C equals two R",
    "examples": [
      {
        "en": "We know two angles and one side, so this is a job for the law of sines.",
        "ja": "2 つの角と 1 辺がわかっているから、ここは正弦定理の出番だ。",
        "register": "spoken"
      },
      {
        "en": "In triangle ABC, a = 8, A = 45°, and B = 60°. Apply the law of sines to find b.",
        "ja": "△ABC において a = 8, A = 45°, B = 60° のとき、正弦定理を適用して b を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高等学校学習指導要領解説（数学I）の正弦定理は a/sin A = b/sin B = c/sin C = 2R（R は △ABC の外接円の半径）と、= 2R まで含めて書く。OpenStax Algebra and Trigonometry と Calculus Volume 1 は 2R を書かない。英語版 Wikipedia「Circumcircle」は外接円の半径を circumradius と呼ぶ（IM Geometry・CK-12 Geometry は circumcenter だけ。circumradius は参照と用例コーパスに出てこない）。",
      "英語版 Wikipedia「Law of sines」は sine rule ／ sine formula という別名も挙げる。",
      "2 辺とその一方の対角から角を求めると、解が 2 つになることがある（ambiguous case）。"
    ],
    "related": [
      "law-of-cosines",
      "circumscribed-circle",
      "solving-triangles",
      "ambiguous-case",
      "sine"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正弦定理",
        "en": "Law of sines"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Law_of_sines",
        "note": "記事「Law of sines」の冒頭: sine formula ／ sine rule とも呼ぶ（地域は書いていない）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 正弦定理 a/sinA = b/sinB = c/sinC = 2R（R は外接円の半径）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Circumcircle",
        "note": "半径 circumradius（三角形の外接円の記事。数学カテゴリから 5 段）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "circumcenter（23 件。circumradius は 0 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "circumcenter（13 件。circumradius は 0 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "law of sines": 12
      },
      "written": {
        "law of sines": 34
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "yt:nancypi"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "law-of-syllogism",
    "ja": {
      "term": "三段論法",
      "reading": "さんだんろんぽう",
      "alt": []
    },
    "en": {
      "term": "law of syllogism",
      "alt": [
        "hypothetical syllogism"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof",
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "「p ならば q」と「q ならば r」がともに真のとき、「p ならば r」も真であると結論する推論の規則。",
    "definition_en": "The rule of inference that if p → q and q → r are both true, then p → r is also true.",
    "latex": "(p \\Rightarrow q) \\land (q \\Rightarrow r) \\implies (p \\Rightarrow r)",
    "spoken_en": "if p implies q and q implies r, then p implies r",
    "examples": [
      {
        "en": "If it rains, the game is canceled, and if the game is canceled, we have practice. So by the law of syllogism, if it rains, we have practice.",
        "ja": "雨なら試合は中止、試合が中止なら練習がある。だから三段論法で、雨なら練習がある。",
        "register": "spoken"
      },
      {
        "en": "Use the law of syllogism to write a conclusion from the two statements: If x > 5, then x > 3. If x > 3, then x > 1.",
        "ja": "次の 2 つの命題から、三段論法を使って結論を書け。x > 5 ならば x > 3。x > 3 ならば x > 1。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数学で言う三段論法（p ⇒ q、q ⇒ r なら p ⇒ r）は、英語では law of syllogism（論理学では hypothetical syllogism）。",
      "日本語版 Wikipedia の「三段論法」に対応する英語版の記事 Syllogism は、アリストテレスの定言三段論法（categorical syllogism）の記事で、この規則ではない。",
      "用例コーパス・IM・Levin とも law of syllogism は出てこない。CK-12 Geometry は 2.7 Deductive Reasoning に 2 件。"
    ],
    "related": [
      "law-of-detachment",
      "conditional-statement",
      "deductive-reasoning"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.7 Deductive Reasoning"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 law of syllogism（CK-12 Geometry 2.7 Deductive Reasoning・計 2 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "laws-of-exponents",
    "ja": {
      "term": "指数法則",
      "reading": "しすうほうそく",
      "alt": []
    },
    "en": {
      "term": "exponent rules",
      "alt": [],
      "variants": [
        {
          "term": "exponent rules",
          "register": "both",
          "note": "話し言葉では Khan Academy の中学の講義と Professor Leonard が中心。書き言葉はすべて OpenStax Algebra and Trigonometry。"
        },
        {
          "term": "properties of exponents",
          "register": "written",
          "note": "話し言葉では少なく、書き言葉では最も多い（OpenStax Prealgebra・Algebra and Trigonometry・Intermediate Algebra ほか）。"
        },
        {
          "term": "rules of exponents",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry の言い方。話し言葉では少なく（ほとんどが MIT OCW）、書き言葉はすべて OpenStax Algebra and Trigonometry。"
        },
        {
          "term": "laws of exponents",
          "register": "written",
          "note": "OpenStax Calculus Volume 1（1.5）・Elementary Algebra（9.8）の書き方。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1",
        "Integrated Math 2",
        "Algebra 1"
      ]
    },
    "definition_ja": "aᵐ · aⁿ = aᵐ⁺ⁿ、(aᵐ)ⁿ = aᵐⁿ、(ab)ⁿ = aⁿ bⁿ などの累乗の計算規則。0 や負の整数、有理数の指数にも広げられる。",
    "definition_en": "The rules for powers such as aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, and (ab)ⁿ = aⁿ bⁿ, extended to zero, negative, and rational exponents.",
    "latex": "a^m a^n = a^{m+n},\\quad (a^m)^n = a^{mn},\\quad (ab)^n = a^n b^n",
    "spoken_en": null,
    "examples": [
      {
        "en": "By the exponent rules, x squared times x cubed is x to the fifth — same base, so the exponents add.",
        "ja": "指数法則から x²·x³ = x⁵。底が同じなので指数を足す。",
        "register": "spoken"
      },
      {
        "en": "Simplify (a² b⁻³)⁴ · a⁻⁵ b¹⁰ using the rules of exponents.",
        "ja": "指数法則を用いて (a²b⁻³)⁴ · a⁻⁵b¹⁰ を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "laws of exponents（指数法則の直訳）は、OpenStax Calculus Volume 1（1.5 の Rule: Laws of Exponents）や Elementary Algebra（9.8）の書き方。",
      "指数の拡張（0 乗・負の指数・有理数の指数）は、英語でも zero exponent ／ negative exponents ／ rational exponents と個別に言う。"
    ],
    "related": [
      "properties-of-logarithms",
      "zero-exponent",
      "negative-exponent",
      "rational-exponent",
      "exponent"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.5 Exponential and Logarithmic Functions（Rule: Laws of Exponents）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 7.4〜7.8（exponent rules）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "10.2 Use Multiplication Properties of Exponents・10.4 Divide Monomials"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "9.8 Rational Exponents（laws of exponents）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "rules of exponents": 5,
        "laws of exponents": 6,
        "exponent rules": 40,
        "properties of exponents": 6
      },
      "written": {
        "properties of exponents": 102,
        "laws of exponents": 17,
        "exponent rules": 15,
        "rules of exponents": 46
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "lead-to-a-contradiction",
    "ja": {
      "term": "矛盾する",
      "reading": "むじゅんする",
      "alt": []
    },
    "en": {
      "term": "arrive at a contradiction",
      "alt": [
        "lead to a contradiction",
        "get a contradiction",
        "reach a contradiction"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "仮定から推論を進めた結果、矛盾が生じる。背理法で、仮定が誤りだと結論する直前の段階。",
    "definition_en": "To reach, by valid reasoning from an assumption, a statement that conflicts with the assumption or with a known fact, which shows that the assumption must be false.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Assume there's a largest prime. We're going to arrive at a contradiction pretty quickly.",
        "ja": "最大の素数があると仮定しよう。すぐに矛盾が生じる。",
        "register": "spoken"
      },
      {
        "en": "If we plug that back in, we get a contradiction, so that case can't happen.",
        "ja": "それを代入し直すと矛盾する。だからその場合は起こらない。",
        "register": "spoken"
      },
      {
        "en": "Assuming that √2 is rational, we arrive at a contradiction. Hence √2 is irrational.",
        "ja": "√2 が有理数であると仮定すると矛盾する。よって √2 は無理数である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "主語に注意。arrive at ／ get ／ reach a contradiction の主語は we（証明する人）、lead to a contradiction の主語は仮定（This assumption leads to a contradiction）。",
      "「A は B と矛盾する」は A contradicts B（他動詞で目的語が要る）。This contradicts the fact that n is odd. のように使う。",
      "用例コーパスでは話し言葉が get ／ lead to ／ reach ／ arrive at a contradiction の 4 つの言い方に割れ、どれも少ない。"
    ],
    "related": [
      "contradiction",
      "proof-by-contradiction",
      "assume",
      "negate"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "reach a contradiction": 2,
        "lead to a contradiction": 3,
        "arrive at a contradiction": 2,
        "get a contradiction": 5
      },
      "written": {
        "lead to a contradiction": 2,
        "arrive at a contradiction": 2,
        "reach a contradiction": 1
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満 ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 arrive at a contradiction（本文 2 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "leading-coefficient",
    "ja": {
      "term": "最高次の係数",
      "reading": "さいこうじのけいすう",
      "alt": [
        "主係数",
        "最高次係数"
      ]
    },
    "en": {
      "term": "leading coefficient",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "多項式の最高次の項の係数。3x² − 5x + 1 では 3、二次式 ax² + bx + c では a。",
    "definition_en": "The coefficient of the leading term of a polynomial. In 3x² − 5x + 1 it is 3, and in ax² + bx + c it is a.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When the leading coefficient isn't 1, like in 2x squared plus 7x plus 3, factoring takes a few more steps.",
        "ja": "2x² + 7x + 3 みたいに最高次の係数が 1 じゃないときは、因数分解に少し手間がかかる。",
        "register": "spoken"
      },
      {
        "en": "State the leading coefficient of the polynomial −x³ + 4x² − 2.",
        "ja": "多項式 −x³ + 4x² − 2 の最高次の係数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "先頭の項の係数とは限らない。1 + 2x − 5x² の最高次の係数は、最後の項の −5。",
      "係数の 1 は書かないので見落としやすい。x² − 4x + 3 の最高次の係数は 1、−x³ + 4x² − 2 では −1。",
      "二次関数 y = ax² + bx + c のグラフは、最高次の係数 a が正なら下に凸（opens upward、エントリ opens-upward）、負なら上に凸（opens downward、エントリ opens-downward）。"
    ],
    "related": [
      "leading-term",
      "coefficient",
      "degree",
      "polynomial",
      "quadratic-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "leading coefficient": 47
      },
      "written": {
        "leading coefficient": 124
      },
      "sources": [
        "khan-algebra",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "leading-digit",
    "ja": {
      "term": "最高位の数字",
      "reading": "さいこういのすうじ",
      "alt": []
    },
    "en": {
      "term": "first digit",
      "alt": [
        "leading digit"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "整数を 10 進法で書いたときの一番上の位の数字。常用対数の小数部分から求められる。",
    "definition_en": "The digit in the highest place of an integer written in base 10; it can be found from the decimal part of its common logarithm.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The decimal part is 0.35, and 10 to the 0.35 is about 2.2, so the first digit is 2.",
        "ja": "小数部分は 0.35 で、10^0.35 は約 2.2 なので、最高位の数字は 2。",
        "register": "spoken"
      },
      {
        "en": "Using log₁₀ 2 = 0.3010 and log₁₀ 3 = 0.4771, find the first digit of 3^40.",
        "ja": "log₁₀ 2 = 0.3010, log₁₀ 3 = 0.4771 として、3⁴⁰ の最高位の数字を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは leading digit は少なく、話し言葉（Khan Academy の中学の講義）にしか出てこない。first digit は話し言葉（Khan Academy・MIT OCW）・書き言葉（OpenStax Prealgebra ほか）とも出てくるが、多くは位取りや数の並びの説明で、常用対数の問題ではない。"
    ],
    "related": [
      "number-of-digits",
      "common-logarithm",
      "digit"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「常用対数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%B8%B8%E7%94%A8%E5%AF%BE%E6%95%B0",
        "note": "最高位の数字の求め方"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "first digit": 15,
        "leading digit": 5
      },
      "written": {
        "first digit": 11
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "leading-term",
    "ja": {
      "term": "最高次の項",
      "reading": "さいこうじのこう",
      "alt": []
    },
    "en": {
      "term": "leading term",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "多項式の項のうち、次数がいちばん高い項。3x² − 5x + 1 では 3x²。",
    "definition_en": "The term of a polynomial with the highest degree. In 3x² − 5x + 1, the leading term is 3x².",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The leading term is the one with the highest power, so here it's negative x squared, even though it's at the end.",
        "ja": "最高次の項は次数がいちばん高い項だから、ここでは −x²。最後に書いてあってもね。",
        "register": "spoken"
      },
      {
        "en": "Identify the leading term and the degree of the polynomial 4 − 3x + 7x³.",
        "ja": "多項式 4 − 3x + 7x³ の最高次の項と次数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "leading term は「先頭に書いてある項」ではなく、次数が最も高い項。5 + 2x − x² の最高次の項は、最後に書いてある −x²。降べきの順（エントリ write-in-descending-order）に並べると先頭に来る。",
      "係数の符号も含めて答える。−3x² + 5x − 1 の最高次の項は 3x² ではなく −3x²。",
      "多項式のグラフの端のふるまい（x を大きくしたり小さくしたりしたときにグラフがどちらへ向かうか、end behavior。エントリ end-behavior）は、最高次の項で決まる。最高次の項の係数だけを指すときは leading coefficient（エントリ leading-coefficient）。"
    ],
    "related": [
      "leading-coefficient",
      "degree",
      "write-in-descending-order",
      "end-behavior"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "leading term": 60
      },
      "written": {
        "leading term": 70
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "least-common-denominator",
    "ja": {
      "term": "最小公分母",
      "reading": "さいしょうこうぶんぼ",
      "alt": []
    },
    "en": {
      "term": "LCD",
      "alt": [],
      "variants": [
        {
          "term": "least common denominator",
          "register": "both",
          "note": "略さない形。教科書は最初にこの形で導入し、あとは LCD と書く（OpenStax）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "2 つ以上の分数（分数式）の分母の最小公倍数。通分するときの共通の分母に使う。",
    "definition_en": "The least common multiple of the denominators of two or more fractions or rational expressions. It is the smallest denominator you can use to rewrite them with a common denominator.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The LCD of 1/4 and 1/6 is 12, so rewrite both fractions with 12 on the bottom.",
        "ja": "1/4 と 1/6 の最小公分母は 12 だから、どちらの分数も分母を 12 に直そう。",
        "register": "spoken"
      },
      {
        "en": "Find the LCD of 5/(2x) and 3/x², and then add the two fractions.",
        "ja": "5/(2x) と 3/x² の最小公分母を求め、2 つの分数式の和を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "LCD は 1 語として読まず、L-C-D と 1 文字ずつ読む。",
      "最小公倍数は least common multiple（LCM、エントリ least-common-multiple）。LCD は分母どうしの LCM のことで、分母の話をするときに使う。",
      "分数方程式（エントリ rational-equation）では、両辺に LCD をかけて分母を払う。通分（エントリ find-a-common-denominator）で分子にもかけ忘れないようにする。"
    ],
    "related": [
      "find-a-common-denominator",
      "least-common-multiple",
      "rational-equation",
      "clear-the-denominators",
      "denominator"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "LCD": 99,
        "least common denominator": 7
      },
      "written": {
        "least common denominator": 62,
        "LCD": 338
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "least-common-multiple",
    "ja": {
      "term": "最小公倍数",
      "reading": "さいしょうこうばいすう",
      "alt": []
    },
    "en": {
      "term": "least common multiple",
      "alt": [],
      "uk": null,
      "register": "both",
      "variants": [
        {
          "term": "LCM",
          "register": "written",
          "note": "略語。書き言葉はすべて OpenStax（Prealgebra・Elementary Algebra）。話し言葉では少ない（Khan Academy の中学の講義・NancyPi）。書き言葉では least common multiple と並ぶ（OpenStax Prealgebra に頼る。OpenStax Prealgebra を除くと least common multiple のほうが多い）。"
        },
        {
          "term": "least common multiple",
          "register": "both",
          "note": "話し言葉ではこちらがずっと多く、ほとんどが Khan Academy の中学の講義。書き言葉は OpenStax（Elementary Algebra・Prealgebra）で、LCM と並ぶ。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "小学校",
        "数A"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "2 つ以上の正の整数に共通する倍数のうち、最も小さい正のもの。",
    "definition_en": "The smallest positive integer that is a multiple of each of two or more given positive integers.",
    "latex": "\\operatorname{lcm}(a, b)",
    "spoken_en": "the LCM of a and b",
    "examples": [
      {
        "en": "The least common multiple of 4 and 6 is 12, not 24.",
        "ja": "4 と 6 の最小公倍数は 24 じゃなくて 12 だ。",
        "register": "spoken"
      },
      {
        "en": "To add these fractions, use the LCM of the denominators as the common denominator.",
        "ja": "この分数を足すには、分母の最小公倍数を共通の分母にしよう。",
        "register": "spoken"
      },
      {
        "en": "Find the least common multiple of 18 and 30.",
        "ja": "18 と 30 の最小公倍数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "正の整数 a, b について gcd(a, b) × lcm(a, b) = ab。最大公約数から最小公倍数を出すときに使う。",
      "通分は find a common denominator（find-a-common-denominator）。分母の最小公倍数を使った分母は least common denominator（LCD、エントリ least-common-denominator）と言う。"
    ],
    "related": [
      "greatest-common-divisor",
      "multiple",
      "find-a-common-denominator",
      "prime-factorization",
      "least-common-denominator"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "最小公倍数",
        "en": "Least common multiple"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 7.17・7.18 Common Multiples（least common multiple）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A 数と式（素因数分解）: 小学校算数科で最大公約数・最小公倍数を学習"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "least common multiple": 109,
        "LCM": 7
      },
      "written": {
        "least common multiple": 67,
        "LCM": 70
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "least-squares",
    "ja": {
      "term": "最小二乗法",
      "reading": "さいしょうにじょうほう",
      "alt": []
    },
    "en": {
      "term": "least squares",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Linear Algebra",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "データの点と直線（や曲線）との縦のずれ（残差）の 2 乗の和が最小になるように、直線（や曲線）を決める方法。",
    "definition_en": "The method of choosing a line (or curve) that makes the sum of the squared vertical distances (residuals) from the data points as small as possible.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Least squares picks the line that makes the sum of the squared residuals as small as possible.",
        "ja": "最小二乗法は、残差の 2 乗の和がいちばん小さくなる直線を選ぶ。",
        "register": "spoken"
      },
      {
        "en": "Use the method of least squares to fit a line to the data.",
        "ja": "最小二乗法を用いて、データに直線を当てはめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "線形代数では、解のない連立方程式 Ax = b に最も近い解を求める方法として扱う。Nicholson 5.6（Best Approximation and Least Squares）はこれを best approximation と呼び、正規方程式 AᵀA z = Aᵀ b で求める。"
    ],
    "related": [
      "regression-line",
      "scatter-plot",
      "residual"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "最小二乗法",
        "en": "Least squares"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.6 Best Approximation and Least Squares（best approximation、normal equations）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 5.5 Least-Squares Regression"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "least squares": 51
      },
      "written": {
        "least squares": 43
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "left-hand-limit",
    "ja": {
      "term": "左側極限",
      "reading": "ひだりがわきょくげん",
      "alt": []
    },
    "en": {
      "term": "limit from the left",
      "alt": [],
      "variants": [
        {
          "term": "left-hand limit",
          "register": "written",
          "note": "書き言葉ではこちらが多いが、ほぼ OpenStax Precalculus。話し言葉では少ない（ほとんどが MIT OCW）。"
        },
        {
          "term": "left-sided limit",
          "register": "spoken",
          "note": "少数の言い方。話し言葉はほとんどが NancyPi で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x を a より小さい側から a に近づけたときの極限。",
    "definition_en": "The limit of f(x) as x approaches a from values less than a.",
    "latex": "\\lim_{x \\to a^-} f(x)",
    "spoken_en": "the limit as x approaches a from the left of f of x",
    "examples": [
      {
        "en": "From the left, the graph is heading toward 1, so the limit from the left is 1.",
        "ja": "左から近づくとグラフは 1 に向かうので、左側極限は 1。",
        "register": "spoken"
      },
      {
        "en": "Find lim_{x→0⁻} 1/x.",
        "ja": "lim_{x→−0} 1/x を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語版 Wikipedia「片側極限」は x → a⁻ などと並べて x → a − 0 の書き方を挙げる。OpenStax は x → a⁻ と書き、as x approaches a from the left と言う。"
    ],
    "related": [
      "right-hand-limit",
      "one-sided-limit",
      "limit"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.3（limit from the left）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「片側極限」",
        "url": "https://ja.wikipedia.org/wiki/%E7%89%87%E5%81%B4%E6%A5%B5%E9%99%90",
        "note": "lim_{x→a−0} の記法"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "limit from the left": 21,
        "left-hand limit": 4,
        "left-sided limit": 4
      },
      "written": {
        "limit from the left": 9,
        "left-hand limit": 34
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "left-hand-side",
    "ja": {
      "term": "左辺",
      "reading": "さへん",
      "alt": []
    },
    "en": {
      "term": "left-hand side",
      "alt": [],
      "variants": [
        {
          "term": "left side",
          "register": "written",
          "note": "書き言葉ではこちらが多い（教科書 OpenStax・IM・CK-12 の書き方）。話し言葉では left-hand side のほうがずっと多い。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "等式や不等式で、等号や不等号の左側にある式。",
    "definition_en": "The expression to the left of the equal sign or inequality sign in an equation or inequality.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's simplify the left-hand side first, and then we'll look at the right.",
        "ja": "まず左辺を整理して、それから右辺を見よう。",
        "register": "spoken"
      },
      {
        "en": "Expand the left-hand side of 3(x + 2) = 21, and then solve the equation.",
        "ja": "3(x + 2) = 21 の左辺を展開し、方程式を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "略して LHS と書くことがある（右辺は RHS）。",
      "左辺・右辺の「辺」も、三角形の辺も、英語では同じ side。どちらの意味かは文脈で区別する。"
    ],
    "related": [
      "right-hand-side",
      "both-sides",
      "equation",
      "equal-sign",
      "inequality"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "left-hand side": 910,
        "left side": 271
      },
      "written": {
        "left-hand side": 54,
        "left side": 167
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "left-riemann-sum",
    "ja": {
      "term": "左リーマン和",
      "reading": "ひだりりーまんわ",
      "alt": []
    },
    "en": {
      "term": "left Riemann sum",
      "alt": [],
      "variants": [
        {
          "term": "left Riemann sum",
          "register": "spoken",
          "note": "AP の呼び方（CED topic 6.2）。話し言葉はほぼ Khan Academy。書き言葉（OpenStax Calculus）は left-endpoint approximation が多く、こちらは少ない。"
        },
        {
          "term": "left-endpoint approximation",
          "register": "written",
          "note": "OpenStax Calculus の書き方で、話し言葉には出ない。書き言葉では left Riemann sum より多い。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「左リーマン和」は学習指導要領解説に無い（日本語版 Wikipedia「リーマン和」にはある）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "区間をいくつかの小区間に分け、各小区間の左端での関数の値を高さとする長方形の面積の和。小区間の幅はそろっていなくてもよい。",
    "definition_en": "A Riemann sum whose rectangles take their heights from the left endpoint of each subinterval.",
    "latex": "L_n = \\sum_{i=1}^{n} f(x_{i-1})\\,\\Delta x_i",
    "spoken_en": "L sub n equals the sum from i equals one to n of f of x sub i minus one delta x sub i",
    "examples": [
      {
        "en": "For an increasing function, the left Riemann sum is an underestimate.",
        "ja": "増加関数なら、左リーマン和は過小評価になる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "区分求積法の和 Σ_{k=0}^{n−1}（左端）と Σ_{k=1}^{n}（右端）の名前は学習指導要領解説に出てこないが、AP の CED（topic 6.2）は left ／ right ／ midpoint Riemann sums と trapezoidal sums に名前を付けて比べる（解説で台形公式に触れるのは、理数に関する学科の科目「理数数学II」の発展・拡充の例だけ）。",
      "AP の CED（topic 6.2）は、小区間の幅がそろわない分け方（nonuniform partitions）でもリーマン和で近似する。表で与えられた関数では、表の x の値で区切るので幅がそろわないことがある。"
    ],
    "related": [
      "right-riemann-sum",
      "midpoint-riemann-sum",
      "riemann-sum",
      "overestimate-and-underestimate",
      "trapezoidal-rule"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.2（left ／ right ／ midpoint Riemann sums、uniform or nonuniform partitions）。Topic 6.3（the width of the ith subinterval）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「リーマン和」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E5%92%8C",
        "note": "左リーマン和・右リーマン和"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「左リーマン和」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "left Riemann sum": 16
      },
      "written": {
        "left Riemann sum": 6,
        "left-endpoint approximation": 21
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "legs",
    "ja": {
      "term": "直角をはさむ辺",
      "reading": "ちょっかくをはさむへん",
      "alt": []
    },
    "en": {
      "term": "leg",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「直角三角形」は斜辺以外の 2 辺を「直角をはさむ 2 辺」か「隣辺」と呼ぶ（隣辺は三角比では角のとなりの辺の意味にも使う）。英語では 1 本ずつを leg と呼ぶ（2 本で legs）。CK-12 Geometry 4.4 Isosceles Triangles は二等辺三角形の等しい 2 辺も legs と呼ぶ。台形の平行でない 2 辺を leg と呼ぶのは CK-12 5.26 の問題と IM Grade 6 の図の説明にあり、CK-12 5.13 は non-parallel sides と書く。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直角三角形で、直角をはさむ 2 つの辺。斜辺でない方の 2 辺。",
    "definition_en": "In a right triangle, a leg is one of the two sides that form the right angle. The side across from the right angle is the hypotenuse, not a leg.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If one leg is 3 and the other leg is 4, the hypotenuse has to be 5.",
        "ja": "直角をはさむ辺の一方が 3、もう一方が 4 なら、斜辺は 5 だ。",
        "register": "spoken"
      },
      {
        "en": "Right triangle ABC has its right angle at C, with AC = 8 and AB = 17. How long is leg BC?",
        "ja": "∠C = 90° の直角三角形 ABC で、AC = 8、AB = 17 である。直角をはさむ辺 BC の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（中学校）と日本語版 Wikipedia「直角三角形」は「直角をはさむ 2 辺」と 2 辺まとめて書くので、英語では 1 本なら a leg、2 本なら the legs ／ both legs と単数・複数に気をつける。",
      "斜辺 hypotenuse は leg に入らない（エントリ hypotenuse）。",
      "二等辺三角形の legs は等しい 2 辺で、直角をはさむとは限らない（エントリ isosceles-triangle、trapezoid）。"
    ],
    "related": [
      "hypotenuse",
      "right-triangle",
      "pythagorean-theorem",
      "isosceles-triangle",
      "trapezoid"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.4 Isosceles Triangles: 等しい 2 辺を legs と呼ぶ"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 6)",
        "url": "https://im.kendallhunt.com/MS/students/1/index.html",
        "note": "2.8 How Much for One?: 台形の legs"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「直角三角形」",
        "url": "https://ja.wikipedia.org/wiki/%E7%9B%B4%E8%A7%92%E4%B8%89%E8%A7%92%E5%BD%A2",
        "note": "斜辺以外の 2 辺は「直角をはさむ 2 辺」か「隣辺」"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 三平方の定理: 「直角をはさむ 2 辺」"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "leg of a right triangle | legs of a right triangle | leg of the right triangle | legs of the right triangle | the other leg | legs and the hypotenuse | leg and the hypotenuse | hypotenuse and a leg | hypotenuse and one leg | length of the leg | lengths of the legs": 3
      },
      "written": {
        "leg of a right triangle | legs of a right triangle | leg of the right triangle | legs of the right triangle | the other leg | legs and the hypotenuse | leg and the hypotenuse | hypotenuse and a leg | hypotenuse and one leg | length of the leg | lengths of the legs": 48
      },
      "sources": [
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "leibniz-notation",
    "ja": {
      "term": "ライプニッツの記法",
      "reading": "らいぷにっつのきほう",
      "alt": []
    },
    "en": {
      "term": "Leibniz notation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "notation"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "導関数を dy/dx、d/dx f(x) のように分数の形で書く記法。",
    "definition_en": "Writing derivatives as ratios of differentials, such as dy/dx and d/dx f(x).",
    "latex": "\\frac{dy}{dx}",
    "spoken_en": "d y d x",
    "examples": [
      {
        "en": "In Leibniz notation, the derivative is dy/dx, and you say it 'dee y dee x.'",
        "ja": "ライプニッツの記法では導関数は dy/dx で、d y d x と読む。",
        "register": "spoken"
      },
      {
        "en": "Write the chain rule in Leibniz notation: dy/dx = (dy/du)(du/dx).",
        "ja": "合成関数の微分をライプニッツの記法で書くと dy/dx = (dy/du)(du/dx)。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "dy/dx は over を入れずに d y d x と読むことが多い（用例コーパスの話し言葉でも dy dx のほうが dy over dx よりずっと多い）。読み方は symbols/derivative-leibniz。"
    ],
    "related": [
      "prime-notation",
      "derivative",
      "differential"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ライプニッツの記法",
        "en": "Leibniz's notation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 3（Leibniz notation）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Leibniz notation": 5
      },
      "written": {
        "Leibniz notation": 6
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは CED の呼び方 Leibniz notation（unit3・unit9）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "lemma",
    "ja": {
      "term": "補題",
      "reading": "ほだい",
      "alt": []
    },
    "en": {
      "term": "lemma",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math",
        "Linear Algebra"
      ]
    },
    "definition_ja": "大きな定理を証明するための足がかりとして、先に証明しておく補助的な命題。",
    "definition_en": "A smaller result proved first so that it can be used as a step in the proof of a larger theorem.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's knock out a quick lemma first, and then the theorem falls right out.",
        "ja": "まず簡単な補題を片づけよう。そうすれば定理はすぐに出る。",
        "register": "spoken"
      },
      {
        "en": "By Lemma 2, if n² is even, then n is even.",
        "ja": "補題 2 より、n² が偶数ならば n は偶数である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは lemma は話し言葉がすべて MIT OCW、書き言葉がすべて MIT の講義ノートで、高校向けのソースには出てこない。",
      "複数形は lemmas と lemmata（英語版 Wikipedia「Lemma (mathematics)」が冒頭で並べる）。参照（Nicholson）は lemmas。"
    ],
    "related": [
      "corollary",
      "axiom",
      "proof",
      "theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "補題",
        "en": "Lemma (mathematics)"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "lemma（193 件）、lemmas"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "lemma": 81
      },
      "written": {
        "lemma": 132
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "less-than-or-equal-to",
    "ja": {
      "term": "〜以下",
      "reading": "いか",
      "alt": []
    },
    "en": {
      "term": "less than or equal to",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある数が別の数より小さいか、または等しいこと。「x は 5 以下」は x ≦ 5 と書き、5 を含む。",
    "definition_en": "Either less than or equal to another number. The symbol is ≤, so x ≤ 5 includes 5 itself.",
    "latex": "a \\leq b",
    "spoken_en": "a is less than or equal to b",
    "examples": [
      {
        "en": "If the speed limit is 65, your speed has to be less than or equal to 65.",
        "ja": "制限速度が 65 なら、スピードは 65 以下でないといけない。",
        "register": "spoken"
      },
      {
        "en": "Find all positive integers n such that 3n + 2 is less than or equal to 11.",
        "ja": "3n + 2 が 11 以下となる正の整数 n をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "記号は日本では ≦（共通テスト・センター試験の問題文、学習指導要領解説）、米国の教科書（OpenStax）では ≤。",
      "文章題では「5 以下」を at most 5 や no more than 5 と言う。",
      "「以下」は境の数を含む。less than と訳すと境の数が抜けて「未満」になる（エントリ less-than）。"
    ],
    "related": [
      "less-than",
      "greater-than-or-equal-to",
      "linear-inequality",
      "properties-of-inequalities",
      "number-line"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "問題文の不等号 ≦"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "不等号 ≦ ≧"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "less than or equal to": 469
      },
      "written": {
        "less than or equal to": 71
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "less-than",
    "ja": {
      "term": "〜未満",
      "reading": "みまん",
      "alt": [
        "小さい",
        "〜より小さい"
      ]
    },
    "en": {
      "term": "less than",
      "alt": [
        "smaller than"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある数が別の数よりも小さいこと。「x は 5 未満」は x < 5 と書き、5 は含まない。",
    "definition_en": "Smaller in value than another number; on a number line, the smaller number is farther to the left. The symbol is <, as in 3 < 8.",
    "latex": "a < b",
    "spoken_en": "a is less than b",
    "examples": [
      {
        "en": "x is less than 5, so 5 itself doesn't work.",
        "ja": "x は 5 未満だから、5 そのものは当てはまらない。",
        "register": "spoken"
      },
      {
        "en": "Four less than a number x is less than 10. Write an inequality and solve it for x.",
        "ja": "ある数 x から 4 を引いた数は 10 未満である。不等式をつくり、x の値の範囲を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「5 less than x」は x − 5 で、x から 5 を引く。英語の語順（5 が先）と引き算の順序が逆になる。「x は 5 未満」は x is less than 5（x < 5）。",
      "「〜以下」は less than ではなく less than or equal to（エントリ less-than-or-equal-to）。less than は境の数を含まない「未満」。",
      "「〜より小さい」も less than で、「未満」と同じく境の数を含まない。"
    ],
    "related": [
      "greater-than",
      "less-than-or-equal-to",
      "number-line",
      "linear-inequality",
      "properties-of-inequalities"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "less than !or": 1799,
        "smaller than !or": 179
      },
      "written": {
        "less than !or": 824,
        "smaller than !or": 49
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "let-u-equal",
    "ja": {
      "term": "u とおく",
      "reading": "ゆーとおく",
      "alt": [
        "〜を x とする"
      ]
    },
    "en": {
      "term": "let u equal",
      "alt": [],
      "variants": [
        {
          "term": "let u equal",
          "register": "spoken",
          "note": "話し言葉は大半が blackpenredpen。書き言葉にはほとんど出てこない。"
        },
        {
          "term": "set u equal to",
          "register": "spoken",
          "note": "MIT OCW・Khan Academy などに分かれて使われる。書き言葉には出てこない。"
        },
        {
          "term": "let u =",
          "register": "written",
          "note": "答案・教科書では Let u = x² + 1. のように = で書く。書き言葉に多く（ほとんどが OpenStax Calculus）、話し言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "calculus"
    ],
    "level": {
      "jp": [
        "中1",
        "数III"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "式の一部や求める数を、新しい文字（u や x）で表すと宣言すること。置換積分の最初の一歩のほか、文章題で「求める数を x とする」ときにも使う。",
    "definition_en": "To introduce a new letter, such as u or x, for part of an expression or for an unknown number. It is the first step of a substitution, and of setting up an equation in a word problem.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let u equal x squared plus one. Then du is two x dx.",
        "ja": "u = x² + 1 とおこう。すると du = 2x dx。",
        "register": "spoken"
      },
      {
        "en": "Let u = x² + 1. Then du = 2x dx, and the integral becomes ∫ u³ du.",
        "ja": "u = x² + 1 とおくと du = 2x dx で、積分は ∫ u³ du となる。",
        "register": "written"
      },
      {
        "en": "Let x equal the number of adult tickets. Then the student tickets are 50 minus x.",
        "ja": "大人の券の枚数を x としよう。すると学生の券は 50 − x 枚だ。",
        "register": "spoken"
      },
      {
        "en": "Let x = the smaller number. The larger number is x + 7, and the sum of the two numbers is 31. Find both numbers.",
        "ja": "小さいほうの数を x とする。大きいほうの数は x + 7 で、2 数の和は 31 である。2 つの数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「x² + 1 = t とおく」は式が先だが、英語では Let u = x² + 1. と新しい文字を先に書く。",
      "日本語の文章題の「〜を x とする」も同じ形で、文字を先に書いて Let x = the number of students. とする。教科書（OpenStax Prealgebra・Elementary Algebra）は Let n = the number of … の形で書く。"
    ],
    "related": [
      "integration-by-substitution",
      "write-dx-in-terms-of-du",
      "change-the-limits-of-integration",
      "substitute-new-variable"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "文章題の文字の置き方（Let n = …）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Use a Problem-Solving Strategy（Let n = the number of …）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "set u equal to": 11,
        "let u =": 1,
        "let u equal": 14
      },
      "written": {
        "let u =": 88,
        "let u equal": 2
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "level-curve",
    "ja": {
      "term": "等高線",
      "reading": "とうこうせん",
      "alt": []
    },
    "en": {
      "term": "level curve",
      "alt": [
        "contour line"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "2 変数関数 z = f(x, y) で、f(x, y) = k（定数）となる点が xy 平面上に描く曲線。",
    "definition_en": "For z = f(x, y), the curve in the xy-plane along which f(x, y) equals a constant k.",
    "latex": "f(x, y) = k",
    "spoken_en": null,
    "examples": [
      {
        "en": "Level curves bunched together mean the surface is steep there.",
        "ja": "等高線が詰まっているところは、曲面が急。",
        "register": "spoken"
      },
      {
        "en": "Sketch several level curves of f(x, y) = x² + 4y².",
        "ja": "f(x, y) = x² + 4y² の等高線をいくつかかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "等高線を並べた図は contour map（書き言葉で使う）。contour line は話し言葉・書き言葉とも少ない。",
      "3 変数関数で f(x, y, z) = k となる面は level surface。"
    ],
    "related": [
      "function-of-several-variables",
      "gradient",
      "directional-derivative"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "level curve": 61,
        "contour line": 7
      },
      "written": {
        "level curve": 72,
        "contour line": 7
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "levels-of-measurement",
    "ja": {
      "term": "尺度水準",
      "reading": "しゃくどすいじゅん",
      "alt": [
        "尺度"
      ]
    },
    "en": {
      "term": "levels of measurement",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "データの値がどんな操作に意味をもつかによる分類。名義尺度・順序尺度・間隔尺度・比例尺度の 4 つ。",
    "definition_en": "A classification of data by which operations are meaningful: nominal, ordinal, interval and ratio.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Zip codes are numbers, but their level of measurement is nominal.",
        "ja": "郵便番号は数だが、尺度水準は名義尺度だ。",
        "register": "spoken"
      },
      {
        "en": "Identify the level of measurement of each variable: nominal, ordinal, interval, or ratio.",
        "ja": "各変数の尺度水準（名義・順序・間隔・比例）を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics（節の名前 Frequency, Frequency Tables, and Levels of Measurement）。名義尺度（nominal scale）などは 1 つの水準の名前なので候補にしなかった。"
    ],
    "related": [
      "categorical-variable",
      "quantitative-variable",
      "data"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "levels of measurement": 10
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "lexicographic-order",
    "ja": {
      "term": "辞書式順序",
      "reading": "じしょしきじゅんじょ",
      "alt": []
    },
    "en": {
      "term": "lexicographic order",
      "alt": [
        "alphabetical order",
        "dictionary order"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しは英語版 Wikipedia の記事名 Lexicographic order（「辞書式順序」の langlink 先）。OpenStax Prealgebra は alphabetical order と書く（1 件、語を並べる順序として）。Levin の Discrete Mathematics は、1 語の中の文字の順を in alphabetical order と書き（3.3・3.4・3.5）、名前を並べる順にも使う（3.9）。用例コーパスでは alphabetical order も少なく（話し言葉は MIT OCW・Khan Academy）、lexicographic order と dictionary order は参照（CED・OpenStax・IM・CK-12・Levin）にも用例コーパスにも出てこない。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "文字や数字の列を、1 文字目で比べ、同じなら 2 文字目で比べ…と、辞書の見出しと同じ規則で並べる順序。",
    "definition_en": "The order used in a dictionary: compare two strings by their first symbols, and if those are the same, by their second symbols, and so on.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's write them in lexicographic order, the way a dictionary would: ABC, ACB, BAC, and so on.",
        "ja": "辞書式順序で書き出そう。辞書と同じように ABC, ACB, BAC, … だ。",
        "register": "spoken"
      },
      {
        "en": "All arrangements of the letters a, b, c, d, e are listed in alphabetical order. Which arrangement is 50th?",
        "ja": "a, b, c, d, e の 5 文字を並べたものすべてを辞書式順序に並べる。50 番目は何か。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "桁数が同じ数字の並び（123, 132, 213, …）なら、辞書式順序は小さい順と同じ。in increasing order（小さい順）と言い換えられる。",
      "「〇番目の文字列」を求める問題は、先頭の文字ごとに何通りあるか（4! = 24 など）を数えて絞り込む。"
    ],
    "related": [
      "permutation",
      "enumeration",
      "counting",
      "factorial"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.3・3.4・3.5（1 語の中の文字の in alphabetical order）・3.9"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "辞書式順序",
        "en": "Lexicographic order"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "alphabetical order": 5
      },
      "written": {
        "alphabetical order": 1
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-6.042",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 2）。en.term は lexicographic order（英語版 Wikipedia の記事名）、alphabetical order は en.alt。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "lhopitals-rule",
    "ja": {
      "term": "ロピタルの定理",
      "reading": "ろぴたるのていり",
      "alt": []
    },
    "en": {
      "term": "L'Hôpital's rule",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "0/0 や ∞/∞ の不定形の極限を、分子と分母をそれぞれ微分した商の極限で求める定理。",
    "definition_en": "A rule for limits of the indeterminate forms 0/0 and ∞/∞: under the right conditions, lim f/g = lim f′/g′.",
    "latex": "\\lim_{x \\to a} \\frac{f(x)}{g(x)} = \\lim_{x \\to a} \\frac{f'(x)}{g'(x)}",
    "spoken_en": "the limit as x approaches a of f of x over g of x equals the limit of f prime of x over g prime of x",
    "examples": [
      {
        "en": "It's zero over zero, so use L'Hôpital's rule: differentiate the top and the bottom separately.",
        "ja": "0/0 なのでロピタルの定理を使う。分子と分母を別々に微分する。",
        "register": "spoken"
      },
      {
        "en": "Use L'Hôpital's rule to evaluate lim_{x→0} (eˣ − 1)/x.",
        "ja": "ロピタルの定理を用いて lim_{x→0} (eˣ − 1)/x を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "L'Hôpital の h は発音しない（Merriam-Webster: ˌlō-pē-ˈtälz-）。Merriam-Webster の見出しは L'Hopital's rule で、L'Hospital's rule を異綴りとして挙げる。",
      "商の微分（quotient rule）ではない。分子と分母を別々に微分する。",
      "学習指導要領（平成30年告示）の本文に「ロピタル」は無く、解説は数学III の課題学習の例で発展として触れるだけ。AP の CED は topic 4.7 で扱う。"
    ],
    "related": [
      "indeterminate-form",
      "limit",
      "find-the-limit"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ロピタルの定理",
        "en": "L'Hôpital's rule"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 4.7（L'Hôpital's rule）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 課題学習の例: 発展としてロピタルの定理"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「l'Hôpital's rule」",
        "url": "https://www.merriam-webster.com/dictionary/l%27H%C3%B4pital%27s%20rule",
        "note": "発音 ˌlō-pē-ˈtälz-。見出しは L'Hopital's rule、異綴り L'Hospital's rule（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "L'Hôpital's rule": 125
      },
      "written": {
        "L'Hôpital's rule": 88
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "liate",
    "ja": {
      "term": "LIATE",
      "reading": "えるあいえーてぃーいー",
      "alt": []
    },
    "en": {
      "term": "LIATE",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "OpenStax Calculus Volume 2 が紹介する覚え方で、学習指導要領解説には対応するものが出てこない。部分積分で u（微分する側）に選ぶ順を、Logarithmic（対数）・Inverse trig（逆三角）・Algebraic（多項式など）・Trigonometric（三角）・Exponential（指数）の頭文字で表す。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "部分積分で、どちらの因数を u（微分する側）にするかの目安。対数・逆三角・代数・三角・指数の順に、先にあるほうを u にする。",
    "definition_en": "A mnemonic for integration by parts: choose u in the order Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "By LIATE, the log comes first, so let u be ln x and dv be x dx.",
        "ja": "LIATE だと対数が先だから、u = ln x、dv = x dx にする。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "あくまで目安で、いつも正しいとは限らない。"
    ],
    "related": [
      "which-one-to-differentiate",
      "integration-by-parts",
      "integrate-by-parts"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "LIATE に当たる覚え方は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "LIATE": 2
      },
      "written": {
        "LIATE": 7
      },
      "sources": [
        "openstax-calculus",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 LIATE（本文 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "lie-in",
    "ja": {
      "term": "含まれる",
      "reading": "ふくまれる",
      "alt": []
    },
    "en": {
      "term": "lie in",
      "alt": [],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「含まれる」は、直線が平面に含まれる（図形）ときにも、集合が別の集合に含まれるときにも使う。英語では、図形の場合は line ℓ lies in plane P と lie in を使い、集合の場合は A is contained in B ／ A is a subset of B（エントリ subset）と言い分ける。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "直線や点が、ある平面の上にすっかりのっていること。直線 ℓ のすべての点が平面 P 上にあるとき、ℓ は P に含まれるという。",
    "definition_en": "To be completely on a given plane, said of a line or a point. A line lies in a plane when every point of the line is a point of the plane.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Both of these edges lie in the top face, so they're in the same plane.",
        "ja": "この 2 本の辺はどちらも上の面に含まれているから、同じ平面上にある。",
        "register": "spoken"
      },
      {
        "en": "In the rectangular prism shown, which edges lie in plane ABCD?",
        "ja": "図の直方体で、平面 ABCD に含まれる辺をすべて答えよ。",
        "register": "written"
      },
      {
        "en": "True or false: If lines ℓ and m both lie in plane P, then ℓ and m must intersect.",
        "ja": "正しいか誤りか答えよ。直線 ℓ と m がどちらも平面 P に含まれるならば、ℓ と m は必ず交わる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "集合の「含まれる」（A は B に含まれる）は A is a subset of B ／ A is contained in B（エントリ subset）で、lie in は使わない。",
      "平面を主語にすると plane P contains line ℓ（P は ℓ を含む）。主語と目的語が日本語の「含まれる」と逆になる。",
      "点が直線の上にあるときは lie on を使う（point A lies on line ℓ）。"
    ],
    "related": [
      "plane",
      "line",
      "coplanar",
      "subset",
      "line-of-intersection"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "lies in the plane | lie in the plane | lies in a plane | lie in a plane | contained in the plane": 6
      },
      "written": {
        "lies in the plane | lie in the plane | lies in a plane | lie in a plane | contained in the plane": 8
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 6 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは OpenStax の呼び方 lies in the plane | lie in the plane | lies in a plane | lie in a plane | contained in the plane（本文 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "like-radicals",
    "ja": {
      "term": "同類根号",
      "reading": "どうるいこんごう",
      "alt": []
    },
    "en": {
      "term": "like radicals",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「同類根号」は学習指導要領解説に無い、本プロジェクトの訳語。日本の中 3 では「根号の中が同じ数」の和・差として、名前を付けずに計算する。英語では like terms にならって like radicals と呼び、OpenStax Elementary Algebra が扱う。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3",
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "根号の中の数と根号の指数（2 乗根か 3 乗根かなど）がそれぞれ同じ根号の式。2√3 と 5√3 のように、同類項と同じ要領でまとめられる。",
    "definition_en": "Radicals that have the same index and the same number under the radical sign. Like radicals can be combined the same way as like terms.",
    "latex": "2\\sqrt{3} + 5\\sqrt{3} = 7\\sqrt{3}",
    "spoken_en": "two root three plus five root three equals seven root three",
    "examples": [
      {
        "en": "These are like radicals, so just add the numbers in front: 2 root 3 plus 5 root 3 is 7 root 3.",
        "ja": "根号の中が同じだから、前の数を足すだけ。2√3 + 5√3 = 7√3 だ。",
        "register": "spoken"
      },
      {
        "en": "Simplify √20 + 3√5. First rewrite √20 so that the two terms are like radicals.",
        "ja": "√20 + 3√5 を簡単にせよ。まず √20 を変形して、根号の中を同じ数にしてから計算すること。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "√2 と √3 は like radicals ではないので、√2 + √3 は √5 にならず、これ以上まとめられない。",
      "そのままでは違って見えても、根号を簡単にすると like radicals になることがある（√20 = 2√5 なので √20 + 3√5 = 5√5）。",
      "like は like terms（同類項、エントリ like-terms）と同じ like。根号の指数が違う √3 と ∛3 は like radicals ではない。"
    ],
    "related": [
      "like-terms",
      "radical-sign",
      "simplify-radicals",
      "square-root-of-a-number",
      "combine-like-terms"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "like radicals": 1
      },
      "written": {
        "like radicals": 38
      },
      "sources": [
        "openstax-elemalg",
        "openstax-intalg",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "like-terms",
    "ja": {
      "term": "同類項",
      "reading": "どうるいこう",
      "alt": []
    },
    "en": {
      "term": "like terms",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "文字の部分（文字とその指数）が同じ項。3x と −5x、2x² と x² は同類項だが、x と x² は同類項ではない。",
    "definition_en": "Terms that have exactly the same variable part, with the same exponents, such as 3x and −5x. Only their coefficients can be different.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "x and x squared aren't like terms, so you can't put them together into one term.",
        "ja": "x と x² は同類項じゃないから、1 つの項にはまとめられないよ。",
        "register": "spoken"
      },
      {
        "en": "Simplify 4a + 3b − a + 5b by combining like terms.",
        "ja": "同類項をまとめて、4a + 3b − a + 5b を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "like は「〜が好き」ではなく「同じ種類の」という意味の形容詞。項どうしの関係なので、複数形 like terms で言う（3x and 5x are like terms）。",
      "x と x²、xy と x は文字の部分が違うので同類項ではない。係数が同じでも文字の部分が違えばまとめられない。",
      "同類項をまとめることは combine like terms と言う。"
    ],
    "related": [
      "term",
      "coefficient",
      "algebraic-expression",
      "linear-expression",
      "distributive-property"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）〔用語・記号〕",
        "note": "中2 A 数と式: 同類項"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "like terms": 171
      },
      "written": {
        "like terms": 367
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "likelihood",
    "ja": {
      "term": "起こりやすさ",
      "reading": "おこりやすさ",
      "alt": []
    },
    "en": {
      "term": "likelihood",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "あることがらが起こると期待できる程度。その程度を 0 から 1 までの数で表したものが確率である。",
    "definition_en": "How likely it is that something will happen. Probability measures the likelihood of an event with a number from 0 to 1.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A probability close to 1 means a high likelihood, and a probability close to 0 means it's pretty unlikely.",
        "ja": "確率が 1 に近ければ起こりやすく、0 に近ければまず起こらないということ。",
        "register": "spoken"
      },
      {
        "en": "A bag contains 3 red marbles and 5 blue marbles. Compare the likelihood of drawing a red marble with the likelihood of drawing a blue marble by finding each probability.",
        "ja": "袋の中に赤玉 3 個と青玉 5 個が入っている。それぞれの確率を求めて、赤玉が出る起こりやすさと青玉が出る起こりやすさを比べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "統計の最尤法（maximum likelihood）で使う likelihood（尤度）は別の用語。中学の確率で言う likelihood は「起こりやすさ」の意味。",
      "英語では同じ意味で chance とも言う（the chance of rolling a 6）。",
      "形容詞は likely（起こりやすい）。「同様に確からしい」は equally likely（エントリ equally-likely）。"
    ],
    "related": [
      "probability",
      "equally-likely",
      "certain",
      "impossible",
      "event"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "likelihood !function !ratio": 21
      },
      "written": {
        "likelihood !function !ratio": 25
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "limacon",
    "ja": {
      "term": "リマソン",
      "reading": "りまそん",
      "alt": [
        "蝸牛形",
        "パスカルの蝸牛形"
      ]
    },
    "en": {
      "term": "limaçon",
      "alt": [
        "limacon"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "極方程式 r = a + b cos θ（または sin θ）で表される曲線。a と b の大小で、内側にループができるものやくぼみのあるものになる。a = b のときがカージオイド。",
    "definition_en": "The curve r = a + b cos θ (or sin θ); depending on a and b it has an inner loop or a dimple, and a = b gives a cardioid.",
    "latex": "r = a + b\\cos\\theta",
    "spoken_en": "r equals a plus b cosine theta",
    "examples": [
      {
        "en": "When b is bigger than a, the limaçon has an inner loop.",
        "ja": "b が a より大きいと、リマソンには内側のループができる。",
        "register": "spoken"
      },
      {
        "en": "Sketch the limaçon r = 1 + 2 cos θ.",
        "ja": "リマソン r = 1 + 2 cos θ をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ç（セディーユ）付きで書く（OpenStax Algebra and Trigonometry）。"
    ],
    "related": [
      "cardioid",
      "rose-curve",
      "polar-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "limaçon": 29
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "limit-at-infinity",
    "ja": {
      "term": "無限遠での極限",
      "reading": "むげんえんでのきょくげん",
      "alt": []
    },
    "en": {
      "term": "limit at infinity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「無限遠での極限」は学習指導要領解説に無い（日本語版 Wikipedia「拡大実数」には「無限遠における極限」がある）。学習指導要領解説（数学III）は「x の値を限りなく大きくしたときの f(x) の極限」と言う。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x → ∞（または x → −∞）のときの関数の極限。",
    "definition_en": "The limit of f(x) as x → ∞ or as x → −∞.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The limit at infinity is 2, so y = 2 is a horizontal asymptote.",
        "ja": "x → ∞ の極限が 2 なので、y = 2 が水平な漸近線。",
        "register": "spoken"
      },
      {
        "en": "Find the limit at infinity of f(x) = (3x² − x)/(x² + 5).",
        "ja": "x → ∞ のときの f(x) = (3x² − x)/(x² + 5) の極限を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "x → ±∞ でのグラフのふるまいを、OpenStax（Precalculus・Algebra and Trigonometry・Calculus）と AP の CED（topic 1.15）は end behavior と言う。"
    ],
    "related": [
      "as-x-approaches-infinity",
      "infinite-limit",
      "find-the-asymptotes"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.15（limits at infinity）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 極限: x の値を限りなく大きくしたときの f(x) の極限"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「拡大実数」",
        "url": "https://ja.wikipedia.org/wiki/%E6%8B%A1%E5%A4%A7%E5%AE%9F%E6%95%B0",
        "note": "無限遠における極限"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "end behavior"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "end behavior"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "limit at infinity": 16
      },
      "written": {
        "limit at infinity": 42
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "limit-comparison-test",
    "ja": {
      "term": "極限比較判定法",
      "reading": "きょくげんひかくはんていほう",
      "alt": []
    },
    "en": {
      "term": "limit comparison test",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「極限比較判定法」は学習指導要領解説に無い（日本語版 Wikipedia「積分判定法」の関連項目に「極限比較判定法」がある）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "正の項の級数で lim aₙ/bₙ が正の有限の値なら、Σaₙ と Σbₙ の収束・発散が一致するという判定法。",
    "definition_en": "For series of positive terms, if lim aₙ/bₙ is a positive finite number, then Σaₙ and Σbₙ both converge or both diverge.",
    "latex": "\\lim_{n\\to\\infty} \\frac{a_n}{b_n} = c > 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "For the limit comparison test, divide the two terms and take the limit.",
        "ja": "極限比較判定法では、2 つの項を割って極限をとる。",
        "register": "spoken"
      },
      {
        "en": "Use the limit comparison test with Σ 1/n to determine whether Σ 1/(2n + 3) converges.",
        "ja": "Σ 1/n との極限比較判定法を用いて、Σ 1/(2n + 3) が収束するかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "comparison-test",
      "p-series",
      "harmonic-series"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.6（limit comparison test）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「積分判定法」",
        "url": "https://ja.wikipedia.org/wiki/%E7%A9%8D%E5%88%86%E5%88%A4%E5%AE%9A%E6%B3%95",
        "note": "関連項目「極限比較判定法」（英語版 Limit comparison test へのリンク）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "limit comparison test": 44
      },
      "written": {
        "limit comparison test": 24
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "limit-definition-of-the-derivative",
    "ja": {
      "term": "導関数の定義",
      "reading": "どうかんすうのていぎ",
      "alt": []
    },
    "en": {
      "term": "definition of the derivative",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "導関数を f′(x) = lim_{h→0} (f(x + h) − f(x))/h で定めること、またその式。",
    "definition_en": "Defining the derivative as f′(x) = lim_{h→0} (f(x + h) − f(x))/h, or that formula itself.",
    "latex": "f'(x) = \\lim_{h \\to 0} \\frac{f(x+h) - f(x)}{h}",
    "spoken_en": "f prime of x equals the limit as h approaches zero of f of x plus h minus f of x, over h",
    "examples": [
      {
        "en": "Use the definition of the derivative here, not the power rule.",
        "ja": "ここでは xⁿ の微分の公式ではなく、導関数の定義を使う。",
        "register": "spoken"
      },
      {
        "en": "Use the definition of the derivative to find f′(x) for f(x) = x².",
        "ja": "導関数の定義に従って、f(x) = x² の f′(x) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus は limit definition of the derivative とも書く。(f(x + h) − f(x))/h の部分は difference quotient（差分商）と呼ぶ。"
    ],
    "related": [
      "derivative",
      "derivative-at-a-point",
      "limit",
      "average-rate-of-change"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 2.1（definition of the derivative）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "definition of the derivative": 33
      },
      "written": {
        "definition of the derivative": 29
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "limit-laws",
    "ja": {
      "term": "極限の性質",
      "reading": "きょくげんのせいしつ",
      "alt": []
    },
    "en": {
      "term": "algebraic properties of limits",
      "alt": [
        "properties of limits",
        "algebraic limit laws"
      ],
      "variants": [
        {
          "term": "limit laws",
          "register": "written",
          "note": "OpenStax Calculus の書き方（節の名前 The Limit Laws）で、書き言葉で最も多い。話し言葉には出てこない。"
        },
        {
          "term": "limit properties",
          "register": "spoken",
          "note": "Khan Academy に多い言い方。話し言葉は大半が Khan Academy で、書き言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "和・差・積・商・定数倍の極限が、それぞれの極限の和・差・積・商・定数倍になるという性質。",
    "definition_en": "The rules that the limit of a sum, difference, product, quotient, or constant multiple is the sum, difference, product, quotient, or constant multiple of the limits.",
    "latex": "\\lim_{x \\to a} (f(x) + g(x)) = \\lim_{x \\to a} f(x) + \\lim_{x \\to a} g(x)",
    "spoken_en": "the limit of f plus g is the limit of f plus the limit of g",
    "examples": [
      {
        "en": "By the limit properties, you can split this into two limits.",
        "ja": "極限の性質より、これは 2 つの極限に分けられる。",
        "register": "spoken"
      },
      {
        "en": "Use the limit laws to evaluate lim_{x→2} (3x² − x).",
        "ja": "極限の性質を用いて lim_{x→2} (3x² − x) を求めよ。",
        "register": "written"
      },
      {
        "en": "By the limit laws, when both limits exist, you can split the limit of a sum into a sum of limits.",
        "ja": "極限の性質より、両方の極限が存在すれば、和の極限は極限の和に分けられる。",
        "register": "spoken"
      },
      {
        "en": "Using the algebraic properties of limits, the limit of f(x) + g(x) is the limit of f(x) plus the limit of g(x), as long as both limits exist.",
        "ja": "極限の性質から、f(x) + g(x) の極限は、両方の極限が存在するとき f(x) の極限と g(x) の極限の和になる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "商の性質は、分母の極限が 0 でないときだけ使える。",
      "OpenStax Calculus は極限の性質を limit laws と呼び（Volume 1 の節 The Limit Laws ほか、65 件）、Limits at Infinity and Asymptotes（Volume 1）と Sequences（Volume 2）では algebraic limit laws とも書く（7 件）。AP Calculus の CED は Unit 1 の概要と topic 1.5 の題で algebraic properties of limits と書く（4 件）。"
    ],
    "related": [
      "limit",
      "find-the-limit"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.5（algebraic properties of limits）、Unit 1 の概要"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "limit properties": 17,
        "properties of limits": 10
      },
      "written": {
        "properties of limits": 16,
        "limit laws": 59,
        "algebraic limit laws": 7,
        "limit properties": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "limit-of-a-riemann-sum",
    "ja": {
      "term": "リーマン和の極限",
      "reading": "りーまんわのきょくげん"
    },
    "en": {
      "term": "limit of a Riemann sum",
      "alt": [
        "limit of Riemann sums"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III",
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "リーマン和で、区間の分け方を細かくしていった（n → ∞、小区間の幅 → 0）ときの極限。これを定積分の定義とする。",
    "definition_en": "The value a Riemann sum approaches as the partition is refined (n → ∞, subinterval widths → 0); this limit is taken as the definition of the definite integral.",
    "latex": "\\int_a^b f(x)\\,dx = \\lim_{n \\to \\infty} \\sum_{i=1}^{n} f(x_i)\\,\\Delta x",
    "spoken_en": "the integral from a to b of f of x dx equals the limit as n goes to infinity of the sum from i equals one to n of f of x sub i delta x",
    "examples": [
      {
        "en": "The definite integral is defined as the limit of a Riemann sum as n goes to infinity.",
        "ja": "定積分は、n を無限大にしたときのリーマン和の極限として定義される。",
        "register": "spoken"
      },
      {
        "en": "The definite integral is defined as a limit of Riemann sums, and the sample points can be anywhere in each subinterval.",
        "ja": "定積分はリーマン和の極限として定義され、代表点は各小区間のどこにとってもよい。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Δx は delta x、xᵢ は x sub i、xᵢ* は x sub i star と読む。",
      "区分求積法の lim (1/n)Σ f(k/n) は区間を n 等分して右端をとる形。OpenStax Calculus Volume 1（5.2）の定義も等分（regular partition）のままだが、代表点 xᵢ* は各小区間のどこにとってもよい（sample point）。幅のそろわない分け方を使うときは、いちばん広い小区間の幅を 0 に近づける極限をとると書く。"
    ],
    "related": [
      "riemann-sum",
      "definite-integral",
      "summation-notation",
      "as-n-approaches-infinity"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.3（limit of a Riemann sum と定積分の記法）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "5.1 Approximating Areas（sample point）・5.2 The Definite Integral（regular partition、nonregular partition）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "limit of a Riemann sum": 1
      },
      "written": {
        "limit of Riemann sums": 4,
        "limit of a Riemann sum": 1
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは CED の呼び方 limit of a Riemann sum（topic 6.3）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "limit-of-a-sequence",
    "ja": {
      "term": "数列の極限",
      "reading": "すうれつのきょくげん",
      "alt": []
    },
    "en": {
      "term": "limit of a sequence",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "n を限りなく大きくしたときに、数列の項が近づく値。",
    "definition_en": "The value that the terms of a sequence approach as n grows without bound.",
    "latex": "\\lim_{n \\to \\infty} \\frac{2n+1}{n} = 2",
    "spoken_en": "the limit as n goes to infinity of two n plus one over n equals two",
    "examples": [
      {
        "en": "For the limit of a sequence, only the big values of n matter.",
        "ja": "数列の極限では、n が大きいところだけが問題になる。",
        "register": "spoken"
      },
      {
        "en": "Find the limit of the sequence aₙ = (3n² − 1)/(n² + 2).",
        "ja": "数列 aₙ = (3n² − 1)/(n² + 2) の極限を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは limit of a sequence も limit of the sequence も少なく、話し言葉にはほとんど出てこない。授業では the limit as n goes to infinity ／ approaches infinity と式の読みで言う（話し言葉ではこの言い方のほうが多い）。"
    ],
    "related": [
      "limit",
      "converge",
      "diverge",
      "as-n-approaches-infinity",
      "monotone-convergence-theorem",
      "convergence",
      "sequence"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "数列の極限",
        "en": "Limit of a sequence"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences（limit of a sequence・limit of the sequence）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "limit of a sequence | limit of the sequence | limits of sequences": 4
      },
      "written": {
        "limit of a sequence | limit of the sequence | limits of sequences": 19
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "limit-of-sine-x-over-x",
    "ja": {
      "term": "sin x / x の極限",
      "reading": "えっくすぶんのさいんえっくすのきょくげん",
      "alt": []
    },
    "en": {
      "term": "sin x over x",
      "alt": [
        "sine of x over x"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。学習指導要領解説（数学III）は「三角関数の極限」で lim_{θ→0} sin θ / θ = 1 を取り扱うとし、英語では名前を付けず、式をそのまま the limit as x approaches 0 of sine x over x equals 1 と読む。",
    "domains": [
      "calculus",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x → 0 のとき sin x / x → 1 となる極限。三角関数の導関数を求める土台になる。",
    "definition_en": "The limit of sin x / x as x approaches 0, which equals 1; it underlies the derivatives of the trigonometric functions.",
    "latex": "\\lim_{x \\to 0} \\frac{\\sin x}{x} = 1",
    "spoken_en": "the limit as x approaches zero of sine x over x equals one",
    "examples": [
      {
        "en": "The limit of sine x over x as x goes to zero is one. You'll use that a lot.",
        "ja": "x → 0 のとき sin x / x の極限は 1。これはよく使う。",
        "register": "spoken"
      },
      {
        "en": "Using lim_{x→0} (sin x)/x = 1, find lim_{x→0} (sin 3x)/x.",
        "ja": "lim_{x→0} (sin x)/x = 1 を用いて、lim_{x→0} (sin 3x)/x を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "x はラジアンで測る。度数法では 1 にならない。"
    ],
    "related": [
      "squeeze-theorem",
      "derivatives-of-trigonometric-functions",
      "limit"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 極限: 三角関数の極限 lim_{θ→0} sin θ/θ = 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sine of x over x": 6,
        "sin x over x": 1
      },
      "sources": [
        "khan-ap-calc",
        "yt:nancypi"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 7 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "limit",
    "ja": {
      "term": "極限値",
      "reading": "きょくげんち",
      "alt": [
        "極限",
        "関数の極限"
      ]
    },
    "en": {
      "term": "limit",
      "alt": [],
      "variants": [
        {
          "term": "limiting value",
          "register": "both",
          "note": "少数の言い方。話し言葉では NancyPi・3Blue1Brown・MIT OCW などに分かれ、書き言葉では OpenStax Algebra and Trigonometry が多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x を a に限りなく近づけたときに、f(x) が限りなく近づいていく値。",
    "definition_en": "The value that f(x) gets closer and closer to as x gets closer and closer to a.",
    "latex": "\\lim_{x \\to a} f(x) = L",
    "spoken_en": "the limit as x approaches a of f of x equals L",
    "examples": [
      {
        "en": "The limit as x approaches 2 of x squared is 4.",
        "ja": "x を 2 に近づけたときの x² の極限は 4。",
        "register": "spoken"
      },
      {
        "en": "Find lim_{x→3} (x² − 9)/(x − 3).",
        "ja": "lim_{x→3} (x² − 9)/(x − 3) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は値を「極限値」とも「極限」とも言い（日本語版 Wikipedia「極限」は数列が限りなく近づく値を「数列の極限あるいは極限値」と書き、高等学校学習指導要領解説（数学III）は「数列の極限を求める」と書く）、考え方や操作も「極限」と言う。英語はどれも limit（極限値が 4 は the limit is 4）。",
      "lim_{x→a} f(x) は the limit as x approaches a of f of x と、近づけ方（as x approaches a）を先に読む。"
    ],
    "related": [
      "one-sided-limit",
      "right-hand-limit",
      "left-hand-limit",
      "take-the-limit",
      "as-x-approaches-a",
      "infinity"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "極限",
        "en": "Limit (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.2（limit notation）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「極限」",
        "url": "https://ja.wikipedia.org/wiki/%E6%A5%B5%E9%99%90",
        "note": "冒頭: 数列が限りなく近づく値を数列の極限あるいは極限値という（pitfalls[0]）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III (1) 極限: 簡単な数列の極限を求める（pitfalls[0]）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "limit": 3322,
        "limiting value": 25
      },
      "written": {
        "limit": 1671,
        "limiting value": 22
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "line-graph",
    "ja": {
      "term": "折れ線グラフ",
      "reading": "おれせんぐらふ",
      "alt": []
    },
    "en": {
      "term": "line graph",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Pre-Algebra",
        "Intro Statistics"
      ]
    },
    "definition_ja": "時間などの順に並べたデータの点を、線分で結んだグラフ。変化のようすを見るのに使う。",
    "definition_en": "A graph that joins data points plotted in order, usually of time, with line segments, to show change.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A line graph is good for showing how something changes over time.",
        "ja": "折れ線グラフは、時間とともにどう変わるかを示すのに向いている。",
        "register": "spoken"
      },
      {
        "en": "Make a line graph of the monthly temperatures.",
        "ja": "月ごとの気温の折れ線グラフをかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "グラフ理論の line graph（辺を頂点とするグラフ）は別の概念。"
    ],
    "related": [
      "time-series",
      "statistical-graph",
      "bar-chart"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "折れ線グラフ",
        "en": "Line chart"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "line graph": 5
      },
      "written": {
        "line graph": 31
      },
      "sources": [
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "line-integral",
    "ja": {
      "term": "線積分",
      "reading": "せんせきぶん",
      "alt": []
    },
    "en": {
      "term": "line integral",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "関数やベクトル場を曲線に沿って積分したもの。力の場のする仕事などを表す。",
    "definition_en": "The integral of a function or a vector field along a curve, giving, for example, the work done by a force field.",
    "latex": "\\int_C \\mathbf{F} \\cdot d\\mathbf{r}",
    "spoken_en": "the line integral over C of F dot d r",
    "examples": [
      {
        "en": "For a line integral, parametrize the curve first.",
        "ja": "線積分では、まず曲線をパラメータ表示しよう。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the line integral ∫_C (x + y) ds, where C is the segment from (0, 0) to (1, 2).",
        "ja": "C が (0, 0) から (1, 2) への線分のとき、線積分 ∫_C (x + y) ds を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉はほとんどが MIT OCW（18.02）。"
    ],
    "related": [
      "vector-field",
      "conservative-vector-field",
      "greens-theorem",
      "potential-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "線積分",
        "en": "Line integral"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "line integral": 85
      },
      "written": {
        "line integral": 361
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "line-of-intersection",
    "ja": {
      "term": "交線",
      "reading": "こうせん",
      "alt": []
    },
    "en": {
      "term": "line of intersection",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数C"
      ],
      "us": [
        "Geometry",
        "Calculus III"
      ]
    },
    "definition_ja": "2 つの平面が交わってできる直線。交わる 2 平面の交線は、ちょうど 1 本の直線になる。",
    "definition_en": "The line where two planes meet. Two planes that intersect always meet in exactly one line.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Two planes that cross always meet in a line, and we call it the line of intersection.",
        "ja": "交わる 2 つの平面は、必ず直線で交わる。その直線を交線と言うんだ。",
        "register": "spoken"
      },
      {
        "en": "In the rectangular prism ABCD-EFGH shown, which edge is the line of intersection of the top face ABCD and the side face containing B, C, and G?",
        "ja": "図の直方体 ABCD-EFGH で、上の面 ABCD と、B、C、G を含む側面の交線になっている辺を答えよ。",
        "register": "written"
      },
      {
        "en": "Find parametric equations for the line where the planes 2x − y + z = 4 and x + y − z = 1 meet.",
        "ja": "2 平面 2x − y + z = 4、x + y − z = 1 の交線の媒介変数表示を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 直線や 2 曲線が交わるところは点（point of intersection、エントリ point-of-intersection）、2 平面が交わるところは直線（line of intersection）。",
      "集合の「共通部分」も intersection（エントリ intersection）と言う。line of intersection は図形どうしが交わってできる直線のこと。"
    ],
    "related": [
      "point-of-intersection",
      "intersect",
      "plane",
      "equation-of-a-plane",
      "lie-in"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "line of intersection": 12
      },
      "sources": [
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "line-of-reflection",
    "ja": {
      "term": "対称の軸",
      "reading": "たいしょうのじく",
      "alt": []
    },
    "en": {
      "term": "line of symmetry",
      "alt": [
        "line of reflection"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「対称の軸」は、線対称な図形を折り返す直線にも、図形を対称移動するときの直線にも使う。英語では、線対称な図形の直線を line of symmetry、図形を対称移動するときの直線を line of reflection と言い分ける。放物線の軸は axis of symmetry（エントリ axis-of-symmetry）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "線対称な図形で、その直線で折り返すと両側がぴったり重なる直線。図形を対称移動するときに折り返しの基準にする直線もこう呼ぶ。",
    "definition_en": "A line that splits a figure into two halves that match exactly when you fold along it. The line a figure is flipped over in a reflection is called the line of reflection.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The diagonal of a rectangle isn't a line of symmetry — fold along it and the corners don't match up.",
        "ja": "長方形の対角線は対称の軸じゃないよ。対角線で折っても、角がぴったり重ならないでしょ。",
        "register": "spoken"
      },
      {
        "en": "Triangle A′B′C′ is the image of triangle ABC after a reflection. Draw the line of reflection.",
        "ja": "△A′B′C′ は △ABC を対称移動した図形である。対称の軸をかき入れよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "line of symmetry は 1 つの図形の中にある直線で、図形によって本数が違う（長方形は 2 本、正方形は 4 本、円は無数）。line of reflection は、もとの図形と対称移動した像（image）の 2 つの間にある直線。",
      "「対称の軸は何本あるか」は How many lines of symmetry does it have? 複数形は lines of symmetry で、symmetry は単数のまま。",
      "放物線の軸は axis of symmetry（エントリ axis-of-symmetry）。英語で axis とだけ言うと x 軸・y 軸（x-axis, y-axis）とまぎれる。"
    ],
    "related": [
      "line-symmetry",
      "reflection",
      "reflect",
      "axis-of-symmetry",
      "symmetric"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "line of symmetry": 7
      },
      "written": {
        "line of symmetry": 16
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "line-perpendicular-to-a-plane",
    "ja": {
      "term": "直線と平面の垂直",
      "reading": "ちょくせんとへいめんのすいちょく",
      "alt": []
    },
    "en": {
      "term": "perpendicular to the plane",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数A"
      ],
      "us": [
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "definition_ja": "直線が平面と交わり、その交点を通る平面上のすべての直線と垂直であること。平面上の交わる 2 直線と垂直であれば、それで十分である。",
    "definition_en": "A line meets a plane and is perpendicular to every line in the plane through the point of intersection; being perpendicular to two intersecting lines in the plane is enough.",
    "latex": "\\ell \\perp \\alpha",
    "spoken_en": "line ell is perpendicular to plane alpha",
    "examples": [
      {
        "en": "Once the line is perpendicular to the plane, it's perpendicular to every line in that plane through its foot, not just the two we checked.",
        "ja": "直線が平面に垂直なら、確かめた 2 本だけでなく、交点を通る平面上のすべての直線と垂直になる。",
        "register": "spoken"
      },
      {
        "en": "In tetrahedron ABCD, AB ⊥ BC and AB ⊥ BD. Show that AB is perpendicular to the plane BCD.",
        "ja": "四面体 ABCD において AB ⊥ BC, AB ⊥ BD である。AB が平面 BCD に垂直であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "平面上の 1 本の直線と垂直なだけでは平面に垂直とは言えない。交わる 2 直線と垂直であることを示す。",
      "平面に垂直な直線は normal to the plane とも言う（法線ベクトル normal vector の normal）。",
      "用例コーパスでは話し言葉はすべて MIT OCW（大半は 18.06）、書き言葉は少なく、ほとんどが OpenStax Calculus。"
    ],
    "related": [
      "foot-of-the-perpendicular",
      "normal-vector",
      "three-perpendiculars-theorem",
      "equation-of-a-plane",
      "perpendicular-lines",
      "dihedral-angle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.5 Equations of Lines and Planes in Space（perpendicular to the plane）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "4.2 Projections and Planes・4.3 More on the Cross Product（perpendicular to the plane）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 B 図形（空間図形）: 直線と平面の垂直"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "perpendicular to the plane": 13
      },
      "written": {
        "perpendicular to the plane": 8
      },
      "sources": [
        "mit-18.02",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "line-symmetry",
    "ja": {
      "term": "線対称",
      "reading": "せんたいしょう",
      "alt": []
    },
    "en": {
      "term": "reflection symmetry",
      "alt": [
        "line symmetry",
        "reflectional symmetry"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「線対称」は図形の性質の名前で、「この図形は線対称である」と言う。英語ではこれを has line symmetry ／ is symmetric about a line の形で言う。CK-12 Geometry はこの性質を reflection symmetry と呼ぶ。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "図形をある直線で折り返すと、両側がぴったり重なる性質。その直線を対称の軸という。",
    "definition_en": "The property a figure has when you can fold it along a line so that the two halves match exactly. The fold line is called the line of symmetry.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Does the letter A have line symmetry? Yep — fold it straight down the middle and the two sides match.",
        "ja": "A という文字は線対称かな？ そう、真ん中で縦に折ると両側がぴったり重なるね。",
        "register": "spoken"
      },
      {
        "en": "Which of the following figures have reflection symmetry? For each one that does, draw every line of symmetry.",
        "ja": "次の図形のうち、線対称なものはどれか。線対称な図形には、対称の軸をすべてかき入れよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "symmetry は名詞なので、「この図形は線対称だ」は The figure has line symmetry. または The figure is symmetric about a line.（形容詞は symmetric、エントリ symmetric）。is line symmetry とは言わない。",
      "線対称な図形の対称の軸は line of symmetry（エントリ line-of-reflection）。1 本とは限らず、正方形には 4 本ある。",
      "点対称（エントリ point-symmetry）と区別する。線対称は直線で折り返して重なる性質、点対称は点のまわりに 180° 回転して重なる性質。"
    ],
    "related": [
      "line-of-reflection",
      "point-symmetry",
      "symmetric",
      "reflection",
      "rotational-symmetry"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "線対称",
        "en": "Axial symmetry"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.3 Reflection Symmetry"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 reflection symmetry（CK-12 Geometry 8.3 Reflection Symmetry・計 15 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "line",
    "ja": {
      "term": "直線",
      "reading": "ちょくせん",
      "alt": []
    },
    "en": {
      "term": "line",
      "alt": [
        "straight line"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "まっすぐで、両方向に限りなくのびている線。2 点を通る直線はただ 1 本だけある。",
    "definition_en": "A straight path of points that goes on forever in both directions. Through any two points there is exactly one line.",
    "latex": "\\overleftrightarrow{AB}",
    "spoken_en": "line A B",
    "examples": [
      {
        "en": "A line goes on forever in both directions, but a segment stops at its two endpoints.",
        "ja": "直線は両方向にどこまでものびるけど、線分は 2 つの端点で終わる。",
        "register": "spoken"
      },
      {
        "en": "Graph the line y = 2x + 1.",
        "ja": "直線 y = 2x + 1 をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の line は両方向に限りなくのびる直線だけを指す。線分は line segment（エントリ segment）、半直線は ray（エントリ ray）で、別の語。",
      "英語では直線 AB を、AB の上に両向きの矢印を付けて ↔AB と書き、line AB と読む。共通テストの問題文は記号を使わず「直線 AB」と書く。",
      "日本語の「線」は曲線も含むが、英語の line はまっすぐな線。一次関数のグラフも the line y = 2x + 1 と line で呼ぶ。"
    ],
    "related": [
      "segment",
      "ray",
      "pass-through",
      "equation-of-a-line",
      "parallel-lines"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "直線",
        "en": "Line (geometry)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和5年度 大学入学共通テスト 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20240408/https://www.dnc.ac.jp/albums/abm.php?d=432&f=abm00003627.pdf&n=2023_ot_19_sugaku1A.pdf",
        "note": "直線 AB の書き方"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!number !tangent !secant !regression !normal !real !straight line !segment !segments !integral !integrals !graph !graphs": 3327,
        "straight line": 165
      },
      "written": {
        "!number !tangent !secant !regression !normal !real !straight line !segment !segments !integral !integrals !graph !graphs": 3580,
        "straight line": 81
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-approximation",
    "ja": {
      "term": "1 次の近似式",
      "reading": "いちじのきんじしき",
      "alt": [
        "近似式",
        "一次近似",
        "線形近似",
        "線型近似",
        "接線近似"
      ]
    },
    "en": {
      "term": "linear approximation",
      "alt": [],
      "variants": [
        {
          "term": "linearization",
          "register": "both",
          "note": "1 次式 L(x) そのものを指す言い方で、件数は少ない。AP の CED は topic 4.6 で使う。話し言葉は MIT OCW と Khan Academy、書き言葉はほとんどが MIT の講義ノート。"
        },
        {
          "term": "local linearity",
          "register": "spoken",
          "note": "AP の CED（topic 4.6）と Khan Academy の言い方。話し言葉はすべて Khan Academy で、書き言葉には出てこない。"
        },
        {
          "term": "tangent line approximation",
          "register": "both",
          "note": "少数の言い方。話し言葉は MIT OCW だけで、書き言葉は MIT の講義ノートと OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "ja.alt の「接線近似」は学習指導要領解説に無い（日本語版 Wikipedia「線型近似」は接線近似とも呼ぶと書く）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "接線を使って、x = a の近くで f(x) ≈ f(a) + f′(a)(x − a) と近似すること、またその式。",
    "definition_en": "Approximating f(x) near x = a by its tangent line: f(x) ≈ f(a) + f′(a)(x − a).",
    "latex": "f(x) \\approx f(a) + f'(a)(x - a)",
    "spoken_en": "f of x is approximately f of a plus f prime of a times x minus a",
    "examples": [
      {
        "en": "Near x = a, the tangent line is a good estimate. That's the linear approximation.",
        "ja": "x = a の近くでは接線がよい近似になる。それが 1 次の近似式。",
        "register": "spoken"
      },
      {
        "en": "Use a linear approximation to estimate √4.1.",
        "ja": "1 次の近似式を用いて √4.1 の近似値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 1 と AP の CED（topic 4.6）は、1 次式 L(x) = f(a) + f′(a)(x − a) を linearization とも呼ぶ。"
    ],
    "related": [
      "slope-of-the-tangent-line",
      "small-change",
      "differential",
      "taylor-polynomial",
      "tangent-plane"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 4.6（linearization ／ local linearity）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「線型近似」",
        "url": "https://ja.wikipedia.org/wiki/%E7%B7%9A%E5%9E%8B%E8%BF%91%E4%BC%BC",
        "note": "接線近似とも呼ぶ"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「接線近似」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "linear approximation": 78,
        "tangent line approximation": 4,
        "linearization": 10,
        "local linearity": 8
      },
      "written": {
        "linear approximation": 101,
        "linearization": 15,
        "tangent line approximation": 10
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-combination",
    "ja": {
      "term": "一次結合",
      "reading": "いちじけつごう",
      "alt": [
        "線形結合"
      ]
    },
    "en": {
      "term": "linear combination",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "いくつかのベクトルをそれぞれ実数倍して足した形 sa + tb。",
    "definition_en": "A sum of scalar multiples of vectors, such as sa + tb.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Any vector in the plane is a linear combination of i and j.",
        "ja": "平面のどんなベクトルも i と j の一次結合になる。",
        "register": "spoken"
      },
      {
        "en": "Write c = ⟨7, 4⟩ as a linear combination of a = ⟨1, 2⟩ and b = ⟨3, −1⟩.",
        "ja": "c = ⟨7, 4⟩ を a = ⟨1, 2⟩ と b = ⟨3, −1⟩ の一次結合で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はほとんどが MIT の講義ノート。"
    ],
    "related": [
      "linearly-independent",
      "decomposition-of-a-vector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "線型結合",
        "en": "Linear combination"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "linear combination": 146
      },
      "written": {
        "linear combination": 78
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-diophantine-equation",
    "ja": {
      "term": "一次不定方程式",
      "reading": "いちじふていほうていしき",
      "alt": []
    },
    "en": {
      "term": "linear Diophantine equation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（linear Diophantine equation は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "a, b, c を整数とする ax + by = c の形の方程式で、整数解を求めるもの。整数解をもつのは c が a と b の最大公約数の倍数のときに限る。",
    "definition_en": "An equation ax + by = c whose coefficients are integers and whose solutions must be integers. It has integer solutions exactly when gcd(a, b) divides c.",
    "latex": "ax + by = c",
    "spoken_en": null,
    "examples": [
      {
        "en": "For 7x + 4y = 1, one solution is x = −1, y = 2, and every other solution comes from that one.",
        "ja": "7x + 4y = 1 の解の 1 つは x = −1, y = 2 で、ほかの解はすべてこれから得られる。",
        "register": "spoken"
      },
      {
        "en": "Find all integer solutions of the linear Diophantine equation 7x + 4y = 1.",
        "ja": "一次不定方程式 7x + 4y = 1 の整数解をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「不定方程式」を indefinite equation と直訳しない。英語はディオファントスの名前で Diophantine と言い、D は大文字。",
      "一般解 x = −1 + 4k, y = 2 − 7k には「k は整数」（where k is an integer）を添える。"
    ],
    "related": [
      "diophantine-equation",
      "integer-solution",
      "euclidean-algorithm",
      "greatest-common-divisor",
      "relatively-prime",
      "extended-euclidean-algorithm",
      "bezouts-identity"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2.5 Solving Linear Diophantine Equations"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 数学と人間の活動: 一次不定方程式の整数解"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成27年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20160601/https://www.dnc.ac.jp/albums/abm.php?f=abm00004808.pdf&n=2701-0402+suugaku1A.pdf",
        "note": "第4問: 1次不定方程式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Levin の呼び方 linear Diophantine equation（6.2 Introduction to Number Theory）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-equation-in-two-variables",
    "ja": {
      "term": "二元一次方程式",
      "reading": "にげんいちじほうていしき",
      "alt": []
    },
    "en": {
      "term": "linear equation in two variables",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの文字をふくむ一次方程式で、ax + by = c の形に整理できるもの。解は無数にあり、解を座標とする点を集めるとグラフは直線になる。",
    "definition_en": "An equation with two variables that can be written in the form ax + by = c. It has infinitely many solutions, and they make up a line when graphed.",
    "latex": "ax + by = c",
    "spoken_en": "a x plus b y equals c",
    "examples": [
      {
        "en": "x + y = 10 is a linear equation in two variables, so there isn't just one answer: x = 3, y = 7 works, x = 4, y = 6 works, and so on.",
        "ja": "x + y = 10 は二元一次方程式だから、答えは 1 つじゃない。x = 3, y = 7 でも、x = 4, y = 6 でも成り立つ。",
        "register": "spoken"
      },
      {
        "en": "Adult tickets cost $8 and child tickets cost $5, and the total was $61. Using x for the number of adult tickets and y for the number of child tickets, write a linear equation in two variables.",
        "ja": "大人券は 1 枚 8 ドル、子ども券は 1 枚 5 ドルで、合計は 61 ドルだった。大人券を x 枚、子ども券を y 枚として、二元一次方程式をつくれ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「元」は文字（未知数）の数のこと。英語では「二元」を in two variables と言う。",
      "英語の linear equation（一次方程式、エントリ linear-equation）は、文字の数を区別せずに使う。文字が 2 つであることを言うときに in two variables を付ける。",
      "二元一次方程式 1 つだけでは解は 1 組に決まらない。もう 1 つの式と組にした連立方程式（エントリ system-of-linear-equations）にすると決まることがある。"
    ],
    "related": [
      "linear-equation",
      "solution-pair",
      "system-of-linear-equations",
      "linear-function",
      "variable"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "linear equation in two variables": 4
      },
      "written": {
        "linear equation in two variables": 35
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-equation",
    "ja": {
      "term": "一次方程式",
      "reading": "いちじほうていしき",
      "alt": []
    },
    "en": {
      "term": "linear equation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "未知数について 1 次の方程式。未知数が 1 つなら、移項して整理すると ax + b = 0（a ≠ 0）の形になる。",
    "definition_en": "An equation whose variables appear only to the first power and are not multiplied together, such as 3x − 5 = 7 or 2x + y = 5. In one variable, it can be written as ax + b = 0 with a ≠ 0.",
    "latex": "ax + b = 0 \\quad (a \\neq 0)",
    "spoken_en": "a x plus b equals zero, where a is not zero",
    "examples": [
      {
        "en": "3x plus 5 equals 20 is a linear equation, so subtract 5 from both sides, divide by 3, and you get x equals 5.",
        "ja": "3x + 5 = 20 は一次方程式だから、両辺から 5 を引いて 3 で割れば x = 5 だ。",
        "register": "spoken"
      },
      {
        "en": "Solve the linear equation 4(x − 2) = 2x + 6.",
        "ja": "一次方程式 4(x − 2) = 2x + 6 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「一次の」に当たる英語は linear（1 次式は linear expression、エントリ linear-expression）。linear は line（直線）と同じ語源で、2 つの未知数の一次方程式のグラフは直線になる。",
      "未知数が 1 つなら linear equation in one variable、2 つ（二元一次方程式）なら linear equation in two variables と言い分ける。"
    ],
    "related": [
      "equation",
      "linear-expression",
      "solve",
      "linear-inequality",
      "system-of-linear-equations"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "一次方程式",
        "en": "Linear equation"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "linear equation": 201
      },
      "written": {
        "linear equation": 526
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-expression",
    "ja": {
      "term": "一次式",
      "reading": "いちじしき",
      "alt": []
    },
    "en": {
      "term": "linear expression",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "文字について次数が 1 の式。2x + 3 や −5x のように、x の 1 次の項と定数項（無くてもよい）でできている。",
    "definition_en": "An expression in which each variable is raised only to the first power, such as 2x + 3 or −5x. With one variable x, it has the form ax + b, where a ≠ 0.",
    "latex": "ax + b \\quad (a \\neq 0)",
    "spoken_en": "a x plus b, where a is not equal to zero",
    "examples": [
      {
        "en": "2x + 3 is a linear expression because the highest power of x is 1.",
        "ja": "2x + 3 は、x の次数がいちばん高くても 1 だから一次式だね。",
        "register": "spoken"
      },
      {
        "en": "Which of the following is a linear expression? (A) x² + 1 (B) 4x − 9 (C) 3/x",
        "ja": "次のうち、一次式であるものを選べ。(A) x² + 1 (B) 4x − 9 (C) 3/x",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「一次」は英語では linear。一次方程式は linear equation、一次関数は linear function と、同じ linear を使う。",
      "linear は line（直線）から来た語で、一次関数のグラフが直線になることと結びついている。",
      "x² や xy を含む式は、次数が 2 なので一次式ではない。"
    ],
    "related": [
      "algebraic-expression",
      "like-terms",
      "constant-term",
      "linear-inequality",
      "system-of-linear-equations"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "linear expression": 1
      },
      "written": {
        "linear expression": 5
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 linear expression（本文 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-function",
    "ja": {
      "term": "一次関数",
      "reading": "いちじかんすう",
      "alt": []
    },
    "en": {
      "term": "linear function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "y が x の一次式 y = ax + b（a ≠ 0）で表される関数。グラフは直線になる。",
    "definition_en": "A function whose rule is y = mx + b. Its graph is a straight line; m gives the slope, and b is the value of y when x = 0.",
    "latex": "y = ax + b",
    "spoken_en": "y equals a x plus b",
    "examples": [
      {
        "en": "A linear function has a constant rate of change, so its graph is always a straight line.",
        "ja": "一次関数は変化の割合が一定だから、グラフはいつも直線になる。",
        "register": "spoken"
      },
      {
        "en": "Write an equation for the linear function whose graph has a slope of −3 and a y-intercept of 5.",
        "ja": "グラフの傾きが −3、切片が 5 である一次関数の式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では y = mx + b と書く。m が傾き（slope）、b が y 切片（y-intercept）で、中学校学習指導要領解説の y = ax + b の a が m に当たる。",
      "日本語版 Wikipedia「一次関数」は、実際に一次であることを求めるときは a ≠ 0 とし、a = 0（定数関数、エントリ constant-function）も含めて呼ぶ流儀もあると書く。",
      "「一次」は英語で linear（エントリ linear）。"
    ],
    "related": [
      "slope",
      "y-intercept",
      "rate-of-change",
      "constant-function",
      "direct-proportion"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "一次関数",
        "en": "Linear function (calculus)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 関数: 一次関数 y = ax + b"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「一次関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%80%E6%AC%A1%E9%96%A2%E6%95%B0",
        "note": "a ≠ 0 とする場合と、a = 0 を含める場合"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "linear function": 48
      },
      "written": {
        "linear function": 277
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-inequality",
    "ja": {
      "term": "一次不等式",
      "reading": "いちじふとうしき",
      "alt": []
    },
    "en": {
      "term": "linear inequality",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "整理すると ax + b > 0（a ≠ 0）などの形になる、未知数について一次の不等式。",
    "definition_en": "An inequality in which the variable appears only to the first power, such as 3x − 5 < 7; it can be rearranged into a form like ax + b > 0 with a ≠ 0.",
    "latex": "ax + b > 0 \\quad (a \\neq 0)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Solving a linear inequality is just like solving an equation, except you have to watch out when you divide by a negative.",
        "ja": "一次不等式は方程式と同じように解けるが、負の数で割るときだけは気をつけよう。",
        "register": "spoken"
      },
      {
        "en": "Solve the linear inequality 3x − 5 < 2x + 4.",
        "ja": "一次不等式 3x − 5 < 2x + 4 を解け。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "solve a linear inequality",
        "ja": "一次不等式を解く"
      }
    ],
    "pitfalls": [
      "中学校学習指導要領〔用語・記号〕（中1 数と式）は ≦ ≧、OpenStax・IM は ≤ ≥ と書く（慣習差 inequality-symbols）。読みは less than or equal to ／ greater than or equal to。",
      "英語の linear inequality は 1 変数に限らず、y ≤ 2x + 1 のような 2 変数の一次の不等式も指す。そのグラフ（半平面）は、学習指導要領解説では数学II「図形と方程式」の「不等式の表す領域」で扱う（慣習差 inequality-terms-scope、エントリ system-of-linear-inequalities）。"
    ],
    "related": [
      "properties-of-inequalities",
      "system-of-inequalities",
      "solution-set",
      "number-line",
      "polynomial-inequality"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "不等式",
        "en": "Inequality (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 1)",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 2.21・2.22 Graphing Linear Inequalities in Two Variables (Part 1・2)・2.24 Solutions to Systems of Linear Inequalities in Two Variables"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）〔用語・記号〕",
        "note": "中1 A 数と式: ≦ ≧"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 図形と方程式: 不等式の表す領域（8 件）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Linear_inequality",
        "note": "記事「Linear inequality」（数学カテゴリから 4 段。冒頭で太字で定義）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "linear inequality": 9
      },
      "written": {
        "linear inequality": 130
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-pair",
    "ja": {
      "term": "一直線をなす角",
      "reading": "いっちょくせんをなすかく",
      "alt": []
    },
    "en": {
      "term": "linear pair",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「一直線をなす角」は学習指導要領解説に無い、本プロジェクトの訳語。日本では「隣り合う 2 つの角の和が 180°」を性質として扱い、2 角の組に名前を付けない。CK-12 Geometry は linear pair と呼ぶ（1.15 Supplementary Angles ほか、30 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "1 辺を共有し、残りの 2 辺が一直線になる 2 つの角の組。2 つの角の和は 180° になる。",
    "definition_en": "A pair of adjacent angles whose non-shared sides form a straight line. The two angles always add up to 180°.",
    "latex": "m\\angle 1 + m\\angle 2 = 180^\\circ",
    "spoken_en": "the measure of angle one plus the measure of angle two equals one hundred eighty degrees",
    "examples": [
      {
        "en": "These two angles form a linear pair, so they have to add up to 180.",
        "ja": "この 2 つの角は一直線をなす角だから、和は 180° になるはず。",
        "register": "spoken"
      },
      {
        "en": "∠1 and ∠2 form a linear pair. If m∠1 = (3x + 10)° and m∠2 = (2x + 20)°, find x.",
        "ja": "∠1 と ∠2 は一直線をなす角である。∠1 = (3x + 10)°, ∠2 = (2x + 20)° のとき、x を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "linear pair の 2 角は必ず補角（supplementary angles）どうしだが、補角どうしでも隣り合って一直線をなしていなければ linear pair ではない。",
      "用例コーパスでは linear pair は話し言葉に少し出てくるだけで（すべて The Organic Chemistry Tutor）、書き言葉には出てこない。IM にも出てこない。"
    ],
    "related": [
      "adjacent-angles",
      "supplementary-angle",
      "straight-angle",
      "angle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.15 Supplementary Angles・4.18 Exterior Angles and Theorems・2.7 Deductive Reasoning"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "linear pair": 7
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 linear pair（CK-12 Geometry 1.15 Supplementary Angles・CK-12 Geometry 4.18 Exterior Angles and Theorems・CK-12 Geometry 2.7 Deductive Reasoning・計 30 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-programming",
    "ja": {
      "term": "線形計画法",
      "reading": "せんけいけいかくほう",
      "alt": []
    },
    "en": {
      "term": "linear programming",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "OpenStax Algebra and Trigonometry（11.3）・Precalculus（9.3）は連立不等式の解の領域を feasible region と呼ぶが、1 次式の最大・最小を求める問題は扱わない。線形計画法の名前と例は Nicholson 8.10（1 次の目的関数を 1 次不等式の制約のもとで最大にする例を the general linear programming problem の簡単な場合と呼ぶ）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "1 次不等式で表される条件（制約）のもとで、1 次式（目的関数）の最大値・最小値を求める方法。領域の頂点で最大・最小になる。",
    "definition_en": "Finding the maximum or minimum of a linear expression (the objective function) under linear inequality constraints; the extreme values occur at vertices of the feasible region.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is a linear programming problem, so the maximum has to occur at a corner of the feasible region; just check the corners.",
        "ja": "線形計画法の問題なので、最大値は実行可能領域の頂点のどれかでとる。頂点を調べればいい。",
        "register": "spoken"
      },
      {
        "en": "If x and y satisfy x ≥ 0, y ≥ 0, x + 2y ≤ 8, and 3x + y ≤ 9, find the maximum value of x + y.",
        "ja": "x, y が x ≧ 0, y ≧ 0, x + 2y ≦ 8, 3x + y ≦ 9 を満たすとき、x + y の最大値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパス（OpenStax の教科書を含む）には linear programming は出てこない。OpenStax Algebra and Trigonometry は連立不等式の領域を扱うが、この名前では呼ばない。"
    ],
    "related": [
      "region",
      "constraint",
      "feasible-region",
      "objective-function",
      "optimization-problem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "線形計画法",
        "en": "Linear programming"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.3 Systems of Nonlinear Equations and Inequalities: Two Variables（feasible region）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.10 An Application to Constrained Optimization（Example 8.10.3 の後の the general linear programming problem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-recurrence-relation",
    "ja": {
      "term": "線形漸化式",
      "reading": "せんけいぜんかしき",
      "alt": []
    },
    "en": {
      "term": "linear recurrence",
      "alt": [
        "linear recurrence relation"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "discrete"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "数列の各項を、前のいくつかの項の 1 次式で表す漸化式。aₙ₊₂ = paₙ₊₁ + qaₙ のように係数が定数のものは、特性方程式を使って一般項を求められる。",
    "definition_en": "A recurrence that expresses each term as a linear combination of earlier terms, such as a_(n+2) = p a_(n+1) + q a_n. When the coefficients are constants, it can be solved with the characteristic equation.",
    "latex": "a_{n+2} = p\\,a_{n+1} + q\\,a_n",
    "spoken_en": "a sub n plus 2 equals p times a sub n plus 1 plus q times a sub n",
    "examples": [
      {
        "en": "The Fibonacci numbers satisfy a linear recurrence, so we can get a closed form from the characteristic equation.",
        "ja": "フィボナッチ数列は線形漸化式を満たすから、特性方程式から一般項が求められる。",
        "register": "spoken"
      },
      {
        "en": "Solve the linear recurrence aₙ₊₂ = 5aₙ₊₁ − 6aₙ with a₀ = 1 and a₁ = 4.",
        "ja": "漸化式 aₙ₊₂ = 5aₙ₊₁ − 6aₙ（a₀ = 1, a₁ = 4）を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数B では「線形漸化式」という名前を使わないことが多く、aₙ₊₁ = paₙ + q の形や隣接 3 項間の漸化式として個別に扱う。",
      "aₙ₊₁ = aₙ² のように前の項の 2 乗などを含むものは linear ではない。",
      "用例コーパスでは linear recurrence は話し言葉ではすべて MIT OCW、書き言葉ではすべて MIT の講義ノート。Nicholson は linear recurrence を 18 件、linear recurrence relation を 4 件使う（3.4 An Application to Linear Recurrences ほか）。"
    ],
    "related": [
      "recurrence-relation",
      "characteristic-equation",
      "fibonacci-sequence",
      "solve-the-recurrence",
      "generating-function"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "linear recurrence": 15
      },
      "written": {
        "linear recurrence": 39
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-transformation-of-a-random-variable",
    "ja": {
      "term": "確率変数の変換",
      "reading": "かくりつへんすうのへんかん",
      "alt": []
    },
    "en": {
      "term": "linear transformation of a random variable",
      "alt": [
        "transforming random variables"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "確率変数 X を Y = aX + b の形の確率変数に直すこと。E(Y) = aE(X) + b、V(Y) = a²V(X) となる。",
    "definition_en": "Turning a random variable X into Y = aX + b; then E(Y) = aE(X) + b and Var(Y) = a²Var(X).",
    "latex": "E(aX+b) = aE(X)+b,\\quad V(aX+b) = a^2V(X)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Adding a constant shifts the mean but doesn't change the standard deviation.",
        "ja": "定数を足すと平均はずれるが、標準偏差は変わらない。",
        "register": "spoken"
      },
      {
        "en": "If Y = 3X − 2 and E(X) = 5, Var(X) = 4, find E(Y) and Var(Y).",
        "ja": "Y = 3X − 2 で E(X) = 5, Var(X) = 4 のとき、E(Y) と Var(Y) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "分散の記号は日本の教科書が V(X)。英語の授業では the variance of X と言葉で言うことが多い（用例コーパスの話し言葉では、すべて Khan Academy）。",
      "用例コーパスにはどの言い方も出てこない（Khan Academy に scale a random variable が少しあるだけ）。"
    ],
    "related": [
      "random-variable",
      "expected-value",
      "transformation-of-a-variable"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear-transformation",
    "ja": {
      "term": "一次変換",
      "reading": "いちじへんかん",
      "alt": [
        "線形変換",
        "線形写像",
        "線型写像"
      ]
    },
    "en": {
      "term": "linear transformation",
      "alt": [
        "linear map"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "ベクトル x を行列 A を掛けて Ax に移すような変換。和と実数倍を保つ（T(x + y) = T(x) + T(y), T(kx) = kT(x)）。",
    "definition_en": "A transformation that sends a vector x to Ax for a matrix A; it preserves sums and scalar multiples: T(x + y) = T(x) + T(y) and T(kx) = kT(x).",
    "latex": "T(\\vec{x}) = A\\vec{x}",
    "spoken_en": null,
    "examples": [
      {
        "en": "A linear transformation keeps grid lines parallel and evenly spaced, and keeps the origin fixed.",
        "ja": "一次変換では、格子の線は平行で等間隔のままで、原点は動かない。",
        "register": "spoken"
      },
      {
        "en": "Find the matrix of the linear transformation that rotates the plane by 90° counterclockwise.",
        "ja": "平面を反時計回りに 90° 回転させる一次変換の行列を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校（旧課程）は「一次変換」、大学の線形代数は「線形変換」「線形写像」と呼ぶ。英語はどちらも linear transformation（linear map とも）。",
      "用例コーパスの話し言葉では半分ほどが 3Blue1Brown。OpenStax の 6 冊にはほとんど出てこない（2 件）。"
    ],
    "related": [
      "matrix",
      "matrix-multiplication",
      "composition-of-transformations",
      "determinant"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "線型写像",
        "en": "Linear map"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "linear transformation": 136
      },
      "written": {
        "linear transformation": 3
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linear",
    "ja": {
      "term": "一次の",
      "reading": "いちじの",
      "alt": [
        "線形",
        "線型"
      ]
    },
    "en": {
      "term": "linear",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "式や関数が、文字について 1 次であること。一次関数や一次方程式のグラフは直線になる。",
    "definition_en": "Describes an expression, equation, or function in which the variables appear only to the first power. The graph of a linear equation or function is a straight line.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's linear because x is only to the first power, so the graph is a straight line.",
        "ja": "x が 1 乗しかないから一次だね。だからグラフは直線になる。",
        "register": "spoken"
      },
      {
        "en": "Which of the following functions are linear: y = 4x − 7, y = x² + 1, y = 6/x?",
        "ja": "次の関数のうち、一次関数であるものを選べ。y = 4x − 7、y = x² + 1、y = 6/x",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「一次の」は英語で linear。一次方程式は linear equation、一次関数は linear function、一次式は linear expression（エントリ linear-equation、linear-function、linear-expression）。",
      "大学の「線形代数」も linear algebra で同じ語。ja.alt の「線形」「線型」はこの訳。語源はグラフが直線（line）になること。",
      "同じ並びで、「二次の」は quadratic（エントリ quadratic-expression）、「三次の」は cubic（エントリ cubic-function）。"
    ],
    "related": [
      "linear-expression",
      "linear-equation",
      "linear-function",
      "degree",
      "line"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "linear": 1252
      },
      "written": {
        "linear": 1789
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linearity-of-expectation",
    "ja": {
      "term": "期待値の線形性",
      "reading": "きたいちのせんけいせい",
      "alt": []
    },
    "en": {
      "term": "linearity of expectation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "確率変数の和の期待値は、期待値の和に等しいという性質。E(aX + bY) = aE(X) + bE(Y) は X と Y が独立でなくても成り立つ。",
    "definition_en": "The fact that the expected value of a sum of random variables equals the sum of their expected values. The rule E(aX + bY) = aE(X) + bE(Y) holds even when X and Y are not independent.",
    "latex": "E(aX + bY) = aE(X) + bE(Y)",
    "spoken_en": "E of a X plus b Y equals a times E of X plus b times E of Y",
    "examples": [
      {
        "en": "We don't need independence here. Linearity of expectation works for any random variables.",
        "ja": "ここで独立性はいらない。期待値の線形性はどんな確率変数にも使える。",
        "register": "spoken"
      },
      {
        "en": "n guests check their hats, and the hats are returned at random. Use linearity of expectation to find the expected number of guests who get their own hat back.",
        "ja": "n 人の客が帽子を預け、帽子は無作為に返される。期待値の線形性を用いて、自分の帽子が戻ってくる客の人数の期待値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数B では E(X + Y) = E(X) + E(Y) や E(aX + b) = aE(X) + b を公式として扱う。英語の証明では by linearity of expectation と性質の名前で理由を書く。",
      "和の期待値は独立でなくても分けられるが、積の期待値 E(XY) = E(X)E(Y) は X と Y が独立のときしか成り立たない。",
      "用例コーパスでは話し言葉はすべて MIT OCW、書き言葉はすべて MIT の講義ノート。"
    ],
    "related": [
      "expected-value",
      "sum-of-random-variables",
      "random-variable",
      "independent-random-variables"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "linearity of expectation": 19
      },
      "written": {
        "linearity of expectation": 30
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linearly-dependent",
    "ja": {
      "term": "一次従属",
      "reading": "いちじじゅうぞく",
      "alt": [
        "線形従属"
      ]
    },
    "en": {
      "term": "linearly dependent",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "一次独立でないこと。どれか 1 つのベクトルが、ほかのベクトルの一次結合で表せる。",
    "definition_en": "Not linearly independent: at least one of the vectors can be written as a linear combination of the others.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If one of them is a combination of the others, the set is linearly dependent.",
        "ja": "どれかがほかの組み合わせで書けるなら、その組は一次従属。",
        "register": "spoken"
      },
      {
        "en": "Show that the vectors are linearly dependent by finding a nontrivial linear combination equal to 0.",
        "ja": "0 になる自明でない一次結合を見つけて、ベクトルが一次従属であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "linearly-independent",
      "linear-combination"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "linearly dependent": 7
      },
      "written": {
        "linearly dependent": 13
      },
      "sources": [
        "mit-18.06",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "linearly-independent",
    "ja": {
      "term": "一次独立",
      "reading": "いちじどくりつ",
      "alt": [
        "線形独立",
        "線型独立"
      ]
    },
    "en": {
      "term": "linearly independent",
      "alt": [
        "linear independence"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "definition_ja": "ベクトル a, b が sa + tb = 0 ならば s = t = 0 に限るとき、a, b は一次独立であるという。平面では、0 でなく平行でないこと。",
    "definition_en": "Vectors a and b are linearly independent when sa + tb = 0 only for s = t = 0; in the plane, when they are nonzero and not parallel.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These two vectors aren't multiples of each other, so they're linearly independent.",
        "ja": "この 2 つのベクトルは互いの実数倍ではないので、一次独立。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the vectors ⟨1, 2, 0⟩, ⟨0, 1, 1⟩, and ⟨1, 0, −2⟩ are linearly independent.",
        "ja": "ベクトル ⟨1, 2, 0⟩, ⟨0, 1, 1⟩, ⟨1, 0, −2⟩ が一次独立かどうか判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校は「一次独立」、大学の線形代数は「線形独立」と呼ぶことが多いが、英語はどちらも linearly independent（名詞は linear independence）。"
    ],
    "related": [
      "linearly-dependent",
      "linear-combination",
      "decomposition-of-a-vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "linearly independent": 19,
        "linear independence": 2
      },
      "written": {
        "linearly independent": 31,
        "linear independence": 2
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "literal-equation",
    "ja": {
      "term": "リテラル方程式",
      "reading": "りてらるほうていしき",
      "alt": []
    },
    "en": {
      "term": "literal equation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「リテラル方程式」は学習指導要領解説に無い、本プロジェクトの訳語。日本では中 2 で「等式の変形」（エントリ rearranging-an-equation）として扱い、この種の等式に名前はない。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "文字を 2 つ以上含む等式。公式などを、その中の 1 つの文字について解くときにこう呼ぶ。",
    "definition_en": "An equation with two or more variables, such as a formula. Solving a literal equation means solving for one variable in terms of the others.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is a literal equation, so we won't get a number. We're solving for w in terms of A and l.",
        "ja": "これはリテラル方程式だから、答えは数にならない。A と l を使って w について解くんだ。",
        "register": "spoken"
      },
      {
        "en": "Solve the literal equation P = 2l + 2w for l.",
        "ja": "等式 P = 2l + 2w を l について解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "答えは数ではなく、ほかの文字を使った式になる（A = lw を w について解くと w = A/l）。",
      "OpenStax Prealgebra は、公式を特定の文字について解くことを solving a literal equation とも言うと書く。OpenStax Intermediate Algebra は Solving Literal Equations の節で扱う。",
      "literal はここでは「文字（letter）の」という意味で、「文字どおりの」の意味ではない。"
    ],
    "related": [
      "rearranging-an-equation",
      "solve-for",
      "equation",
      "variable",
      "linear-equation-in-two-variables"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "literal equation": 3
      },
      "sources": [
        "openstax-intalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは OpenStax の呼び方 literal equation（本文 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "local-extremum",
    "ja": {
      "term": "極値",
      "reading": "きょくち",
      "alt": [
        "相対極値"
      ]
    },
    "en": {
      "term": "local extremum",
      "alt": [],
      "variants": [
        {
          "term": "relative extremum",
          "register": "spoken",
          "note": "話し言葉ではこちらが多いが、ほぼ Khan Academy。書き言葉では少ない（すべて OpenStax Calculus）。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "ja.alt の「相対極値」は学習指導要領解説に無い、本プロジェクトの訳語（relative extremum の訳）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Calculus III"
      ]
    },
    "definition_ja": "極大値と極小値をまとめた呼び方。",
    "definition_en": "A local maximum or local minimum value.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A relative extremum can only happen where f prime is zero or undefined.",
        "ja": "極値をとりうるのは、f′ が 0 か存在しない点だけ。",
        "register": "spoken"
      },
      {
        "en": "Find all local extrema of f(x) = x³ − 3x.",
        "ja": "f(x) = x³ − 3x の極値をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形は extrema（local extrema）。maximum ／ minimum の複数形も maxima ／ minima。",
      "AP の CED は topic 5.2 で local extrema、Unit 5 の概要で relative extrema と書き、両方を使う。",
      "2 変数関数の極値も local extremum（local maximum ／ minimum）と言う。極値でない停留点は saddle point。"
    ],
    "related": [
      "local-maximum",
      "local-minimum",
      "critical-point",
      "first-derivative-test",
      "find-the-local-extrema",
      "saddle-point"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "極値",
        "en": "Maximum and minimum"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.2（local extrema）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「相対極値」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "relative extremum": 8,
        "local extremum": 2
      },
      "written": {
        "local extremum": 77,
        "relative extremum": 2
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "local-maximum",
    "ja": {
      "term": "極大値",
      "reading": "きょくだいち",
      "alt": [
        "極大",
        "相対的最大値",
        "局所的最大値"
      ]
    },
    "en": {
      "term": "local maximum",
      "alt": [],
      "variants": [
        {
          "term": "relative maximum",
          "register": "spoken",
          "note": "話し言葉ではこちらが最も多いが、ほぼすべて Khan Academy（AP の CED も Unit 5 の概要で relative maximum と書く）。書き言葉では少ない（ほとんど OpenStax Calculus）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "ja.alt の「相対的最大値」「局所的最大値」は学習指導要領解説に無い（日本語版 Wikipedia「最大と最小」にはある。relative maximum ／ local maximum に当たる）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x = a の近くで f(a) が最も大きいときの f(a) の値。",
    "definition_en": "A value f(a) that is the largest among the values of f near x = a.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "f prime changes from positive to negative at x = 1, so there's a relative maximum there.",
        "ja": "x = 1 で f′ が正から負に変わるので、そこで極大になる。",
        "register": "spoken"
      },
      {
        "en": "f has a local maximum value of 4 at x = −1.",
        "ja": "f は x = −1 で極大値 4 をとる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "口頭では local max ／ relative max と略す。",
      "区間全体での最大値は absolute maximum（global maximum）。極大値と取り違えない。"
    ],
    "related": [
      "local-minimum",
      "local-extremum",
      "has-a-local-maximum-at",
      "first-derivative-test"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "極値",
        "en": "Maximum and minimum"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 5（relative maximum）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「相対的最大値」「局所的最大値」は出てこない"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「最大と最小」",
        "url": "https://ja.wikipedia.org/wiki/%E6%9C%80%E5%A4%A7%E3%81%A8%E6%9C%80%E5%B0%8F",
        "note": "極大値を局所的最大値（local maximum）・相対的最大値（relative maximum）とも呼ぶ（ja.alt）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "local maximum": 23,
        "relative maximum": 56
      },
      "written": {
        "local maximum": 102,
        "relative maximum": 6
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "local-minimum",
    "ja": {
      "term": "極小値",
      "reading": "きょくしょうち",
      "alt": [
        "極小"
      ]
    },
    "en": {
      "term": "local minimum",
      "alt": [],
      "variants": [
        {
          "term": "relative minimum",
          "register": "spoken",
          "note": "話し言葉ではこちらが最も多いが、すべて Khan Academy（AP の CED も Unit 5 の概要で relative minimum と書く）。書き言葉では少ない（すべて OpenStax Calculus）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x = a の近くで f(a) が最も小さいときの f(a) の値。",
    "definition_en": "A value f(a) that is the smallest among the values of f near x = a.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The derivative goes from negative to positive, so that's a relative minimum.",
        "ja": "導関数が負から正に変わるので、そこは極小。",
        "register": "spoken"
      },
      {
        "en": "Show that f(x) = x⁴ − 2x² has a local minimum at x = 1.",
        "ja": "f(x) = x⁴ − 2x² が x = 1 で極小となることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "口頭では local min ／ relative min と略す。区間全体での最小値は absolute minimum（global minimum）。"
    ],
    "related": [
      "local-maximum",
      "local-extremum",
      "first-derivative-test"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "極値",
        "en": "Maximum and minimum"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 5（relative minimum）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "local minimum": 25,
        "relative minimum": 44
      },
      "written": {
        "local minimum": 96,
        "relative minimum": 6
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "locus",
    "ja": {
      "term": "軌跡",
      "reading": "きせき",
      "alt": []
    },
    "en": {
      "term": "locus",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "CED・OpenStax・IM・CK-12 に locus の語は出てこず、条件を満たす点全体は the set of all points … such that …（OpenStax Algebra and Trigonometry・Precalculus）／ the set of all points that …（IM・CK-12 Geometry）と書く（用例コーパスでも locus は話・書とも出てこない）。",
    "domains": [
      "geometry",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II",
        "数C"
      ],
      "us": [
        "Geometry",
        "Precalculus"
      ]
    },
    "definition_ja": "ある条件を満たす点全体がつくる図形。",
    "definition_en": "The figure formed by all the points that satisfy a given condition.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "So the locus of z is a circle centered at 1 with radius 2.",
        "ja": "だから z の軌跡は、1 を中心とする半径 2 の円。",
        "register": "spoken"
      },
      {
        "en": "Find the locus of the point z that satisfies |z − 1| = |z + i|.",
        "ja": "|z − 1| = |z + i| を満たす点 z の軌跡を求めよ。",
        "register": "written"
      },
      {
        "en": "Find the locus of points P that are equidistant from A(1, 0) and B(3, 2).",
        "ja": "2 点 A(1, 0), B(3, 2) から等しい距離にある点 P の軌跡を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the locus of z",
        "ja": "点 z の軌跡"
      }
    ],
    "pitfalls": [
      "「点 z が描く図形」も locus of z で言える。"
    ],
    "related": [
      "conic-section",
      "ellipse",
      "complex-plane",
      "apollonian-circle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "軌跡 (数学)",
        "en": "Locus (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "12 章 Analytic Geometry の円錐曲線の定義（the set of all points … such that）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "10 章 Analytic Geometry の円錐曲線の定義（the set of all points … such that）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "the set of all points that …"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "the set of all points that …"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-24。docs/DECISIONS.md「Phase 2 代数 2・Precalculus の単元の前の修正」の、残った ③ を人間が決めた行）。en.term は locus のまま。register は主張しない。",
        "raised": "2026-09-24"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "logarithm",
    "ja": {
      "term": "対数",
      "reading": "たいすう",
      "alt": [
        "log"
      ]
    },
    "en": {
      "term": "logarithm",
      "alt": [],
      "variants": [
        {
          "term": "log",
          "register": "both",
          "note": "短い言い方（the log of 8 base 2）。the log of の形で数えた。話し言葉では the logarithm of よりやや多く、MIT OCW と 3Blue1Brown が中心。書き言葉では the logarithm of のほうがやや多い。"
        },
        {
          "term": "logarithm",
          "register": "both",
          "note": "the logarithm of の形で数えた。話し言葉では the log of よりやや少なく、MIT OCW が中心。書き言葉ではこちらのほうがやや多く、大半が OpenStax Algebra and Trigonometry。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "a > 0, a ≠ 1、M > 0 のとき、aᵖ = M となる p を log_a M と書き、a を底とする M の対数という。",
    "definition_en": "For a > 0, a ≠ 1 and M > 0, the number p with aᵖ = M, written log_a M: the logarithm of M to the base a.",
    "latex": "a^p = M \\iff p = \\log_a M",
    "spoken_en": "a to the p equals M if and only if p equals log base a of M",
    "examples": [
      {
        "en": "A logarithm is just an exponent: log base 2 of 8 is the exponent you put on 2 to get 8, so it's 3.",
        "ja": "対数は指数のこと。log₂ 8 は、2 を何乗すれば 8 になるかの、その指数。だから 3。",
        "register": "spoken"
      },
      {
        "en": "Find the value of log₃ 81 − log₂ (1/8).",
        "ja": "log₃ 81 − log₂(1/8) の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "log₂ 8 は log base 2 of 8 と読む。",
      "OpenStax Algebra and Trigonometry（6.3）と Intermediate Algebra（10.3）は、底を省いた log x を常用対数（底 10）、ln x を自然対数とする（学習指導要領解説には ln が出てこない）。"
    ],
    "related": [
      "common-logarithm",
      "natural-logarithm",
      "argument",
      "properties-of-logarithms",
      "change-of-base-formula",
      "rewrite-in-exponential-form",
      "rewrite-in-logarithmic-form"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "対数",
        "en": "Logarithm"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.3 Logarithmic Functions（common logarithm log x、natural logarithm ln x）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "10.3 Evaluate and Graph Logarithmic Functions（底の書かれない log は底 10）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III: ln は出てこない（〔用語・記号〕は 自然対数, e）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the log of": 62,
        "the logarithm of": 36
      },
      "written": {
        "the logarithm of": 41,
        "the log of": 26
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "logarithmic-differentiation",
    "ja": {
      "term": "対数微分法",
      "reading": "たいすうびぶんほう",
      "alt": [
        "対数微分"
      ]
    },
    "en": {
      "term": "logarithmic differentiation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "両辺の対数をとってから微分し、導関数を求める方法。積・商・累乗の多い式や xˣ に使う。",
    "definition_en": "Differentiating after first taking the natural log of both sides; useful for products, quotients, powers, and functions like xˣ.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For x to the x, use logarithmic differentiation: take ln of both sides first.",
        "ja": "xˣ には対数微分法を使う。まず両辺の log をとる。",
        "register": "spoken"
      },
      {
        "en": "Use logarithmic differentiation to find dy/dx for y = xˣ.",
        "ja": "対数微分法を用いて、y = xˣ の dy/dx を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "derivative-of-the-logarithm",
      "implicit-differentiation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "対数微分法",
        "en": "Logarithmic differentiation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 微分法: 対数微分法を取り扱うことも考えられる"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "logarithmic differentiation": 18
      },
      "written": {
        "logarithmic differentiation": 16
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "logarithmic-equation",
    "ja": {
      "term": "対数方程式",
      "reading": "たいすうほうていしき",
      "alt": []
    },
    "en": {
      "term": "logarithmic equation",
      "alt": [],
      "variants": [
        {
          "term": "log equation",
          "register": "spoken",
          "note": "話し言葉ではすべて NancyPi で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "未知数を対数の真数（や底）に含む方程式。真数条件（底に未知数があれば底の条件も）を確かめて解く。",
    "definition_en": "An equation with the unknown inside a logarithm (or in its base), solved while checking that each argument is positive (and that the base is positive and not 1 when it contains the unknown).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Combine the logs into one, then rewrite it in exponential form.",
        "ja": "対数を 1 つにまとめてから、指数の形に直す。",
        "register": "spoken"
      },
      {
        "en": "Solve the logarithmic equation log₃(x + 2) + log₃ x = 1.",
        "ja": "方程式 log₃(x + 2) + log₃ x = 1 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉の log equation はすべて NancyPi。"
    ],
    "related": [
      "argument",
      "logarithmic-inequality",
      "exponential-equation",
      "rewrite-in-exponential-form",
      "one-to-one-property",
      "extraneous-solution",
      "undo-the-log"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "logarithmic equation": 7,
        "log equation": 12
      },
      "written": {
        "logarithmic equation": 82
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "logarithmic-function",
    "ja": {
      "term": "対数関数",
      "reading": "たいすうかんすう",
      "alt": []
    },
    "en": {
      "term": "logarithmic function",
      "alt": [],
      "variants": [
        {
          "term": "log function",
          "register": "spoken",
          "note": "話し言葉の短い言い方（natural log function の形でもよく使う）。書き言葉ではほとんど使わない。"
        },
        {
          "term": "logarithmic function",
          "register": "both",
          "note": "書き言葉ではこちらがずっと多い（OpenStax Calculus・OpenStax Algebra and Trigonometry・OpenStax Intermediate Algebra）。話し言葉でも使う。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3",
        "AP Calculus AB",
        "AP Calculus BC",
        "Calculus I"
      ]
    },
    "definition_ja": "y = log_a x（a > 0, a ≠ 1）で表される関数。指数関数 y = aˣ の逆関数で、定義域は x > 0。",
    "definition_en": "The function y = log_a x (a > 0, a ≠ 1), the inverse of y = aˣ, defined for x > 0.",
    "latex": "y = \\log_a x",
    "spoken_en": "y equals log base a of x",
    "examples": [
      {
        "en": "The logarithmic function y = log₂ x is the reflection of the exponential function y = 2ˣ across the line y = x.",
        "ja": "対数関数 y = log₂ x は、指数関数 y = 2ˣ を直線 y = x に関して対称移動したもの。",
        "register": "spoken"
      },
      {
        "en": "Sketch the graph of y = log₂(x + 3) and state its domain and asymptote.",
        "ja": "y = log₂(x + 3) のグラフをかき、定義域と漸近線を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "exponential-function",
      "logarithm",
      "inverse-function",
      "asymptote",
      "logarithmic-scale"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "対数",
        "en": "Logarithm"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.12・2.7（logarithmic functions）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "log function": 34,
        "logarithmic function": 18
      },
      "written": {
        "logarithmic function": 175,
        "log function": 5
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "logarithmic-inequality",
    "ja": {
      "term": "対数不等式",
      "reading": "たいすうふとうしき",
      "alt": []
    },
    "en": {
      "term": "logarithmic inequality",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。参照（CED・OpenStax・IM・CK-12）には、対数を含む不等式を解く問題そのものが出てこない（OpenStax Algebra and Trigonometry（6.4 Graphs of Logarithmic Functions）は、真数が正になる x の範囲を不等式で求めて定義域を出すだけ）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": []
    },
    "definition_ja": "未知数を対数の真数に含む不等式。真数条件と、底が 1 より大きいか小さいかに注意して解く。",
    "definition_en": "An inequality with the unknown inside a logarithm, solved while watching the positivity of the arguments and whether the base is greater or less than 1.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The base is bigger than 1, so the log is increasing and the inequality keeps its direction.",
        "ja": "底が 1 より大きいので対数関数は増加し、不等号の向きはそのまま。",
        "register": "spoken"
      },
      {
        "en": "Solve log_{1/2}(x − 2) > −1.",
        "ja": "不等式 log_{1/2}(x − 2) > −1 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "logarithmic-equation",
      "exponential-inequality",
      "argument"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.4 Graphs of Logarithmic Functions（定義域の不等式だけ）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学II: 対数方程式と対数不等式"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成30年度 大学入試センター試験 本試験 数学Ⅱ・数学Ｂ（問題）",
        "url": "https://web.archive.org/web/20190601/https://www.dnc.ac.jp/albums/abm.php?f=abm00033144.pdf&n=h30+suugaku2+02suugaku2B.pdf",
        "note": "第1問〔2〕: 対数を含む不等式（両辺の対数をとり、真数の条件を考えて）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "logarithmic-scale",
    "ja": {
      "term": "対数目盛",
      "reading": "たいすうめもり",
      "alt": []
    },
    "en": {
      "term": "logarithmic scale",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "目盛りが 1 つ進むごとに、値が一定の倍率（10 倍など）で大きくなる目盛り。とても大きな範囲の量を短い目盛りに収められる。",
    "definition_en": "A scale on which each step up multiplies the value by the same factor, such as 10, instead of adding the same amount. It fits a very wide range of values onto a short scale.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The Richter scale is a logarithmic scale, so going up by 1 means the shaking is 10 times bigger.",
        "ja": "マグニチュードは対数目盛だから、1 上がると揺れは 10 倍になる。",
        "register": "spoken"
      },
      {
        "en": "Earthquake magnitude is measured on a base-10 logarithmic scale. How many times greater is the amplitude of a magnitude 6 earthquake than that of a magnitude 4 earthquake?",
        "ja": "地震のマグニチュードは 10 を底とする対数目盛で表される。マグニチュード 6 の地震の振幅は、マグニチュード 4 の地震の振幅の何倍か求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "マグニチュードが 1 大きいと振幅は 10 倍、2 大きいと 100 倍になる（20 倍ではない）。目盛りの差は倍率の指数になる。",
      "log scale とも言う。pH や音の大きさ（デシベル）も対数目盛の例。",
      "目盛りの数字は等間隔に並ぶが、表す値は等間隔ではない。目盛りの値を引き算しても、もとの量の差にはならない。"
    ],
    "related": [
      "logarithm",
      "common-logarithm",
      "logarithmic-function",
      "exponent"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "logarithmic scale": 16
      },
      "written": {
        "logarithmic scale": 8
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "logical-connective",
    "ja": {
      "term": "論理結合子",
      "reading": "ろんりけつごうし",
      "alt": [
        "論理演算子",
        "かつ",
        "または"
      ]
    },
    "en": {
      "term": "logical connective",
      "alt": [
        "connective"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「論理結合子」は学習指導要領解説に無い（日本語版 Wikipedia「命題」にはある。langlink の「論理演算」は論理演算子と書く）。",
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I",
        "大学"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "「かつ」「または」「でない」「ならば」のように、命題どうしを結びつけたり否定したりして新しい命題を作る語や記号のこと。",
    "definition_en": "A word or symbol, such as \"and,\" \"or,\" \"not,\" or \"if … then,\" that joins or modifies statements to build a new statement.",
    "latex": "\\land,\\ \\lor,\\ \\lnot,\\ \\to,\\ \\leftrightarrow",
    "spoken_en": "and, or, not, implies, if and only if",
    "examples": [
      {
        "en": "The main connective in this statement is the \"or,\" so that's the last column you fill in.",
        "ja": "この命題でいちばん外側の結合子は「または」だから、表のいちばん最後の列はそれになるよ。",
        "register": "spoken"
      },
      {
        "en": "Identify the logical connective in the statement \"x > 2 and x < 5,\" and write the statement in symbols.",
        "ja": "命題「x > 2 かつ x < 5」に使われている論理結合子を答え、その命題を記号で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高等学校学習指導要領解説（数学I 集合と命題）は必要条件・十分条件・対偶や簡単な命題の証明を扱い、集合の記号として a ∈ A、A ∩ B、A ∪ B、A ⊂ B、Ā を挙げるが、「論理結合子」の名前は出てこない。真理値表と「p → q」の否定「p ∧ (¬q)」は、理数に関する学科の科目「理数数学I」の発展・拡充の内容に挙がる（数学I の節には無い）。",
      "数学の or は両方が真の場合も真になる（CK-12 Geometry 2.9 は p ∨ q を、p と q の少なくとも一方が真のとき真とし、Levin は inclusive or と書く）。日本語版 Wikipedia「論理和」も、命題の少なくとも一つが真であれば真とする（非排他的な論理和）。",
      "用例コーパスでは書き言葉（MIT の講義ノート）にわずかにあるだけで、話し言葉には出てこない。CK-12 Geometry（2.9 And and Or Statements・2.10 Negative Statements）は logical connective を 18 件、Levin（1.1 Mathematical Statements）は 19 件使う。"
    ],
    "related": [
      "negation",
      "conditional-statement",
      "truth-table",
      "truth-value",
      "biconditional"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "論理演算",
        "en": "Logical connective"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.9 And and Or Statements・2.10 Negative Statements"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「命題」",
        "url": "https://ja.wikipedia.org/wiki/%E5%91%BD%E9%A1%8C",
        "note": "五つの論理結合子"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 集合と命題: 必要条件・十分条件・対偶、集合の記号 a∈A・A∩B・A∪B・A⊂B・Ā（論理結合子の名前は無い）。理数数学I (1) 数と式（発展・拡充）: 真理値表と p∧（¬q）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1 Mathematical Statements: logical connectives（5 connectives: and, or, not, if…then, if and only if）。inclusive or"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「論理和」",
        "url": "https://ja.wikipedia.org/wiki/%E8%AB%96%E7%90%86%E5%92%8C",
        "note": "複数の命題の少なくとも一つが真であることを示す（非排他的な）論理演算"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "logical connective": 1
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CK-12 の呼び方 logical connective（CK-12 Geometry 2.9 And and Or Statements・CK-12 Geometry 2.10 Negative Statements・計 11 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "logistic-differential-equation",
    "ja": {
      "term": "ロジスティック微分方程式",
      "reading": "ろじすてぃっくびぶんほうていしき",
      "alt": []
    },
    "en": {
      "term": "logistic differential equation",
      "alt": [],
      "variants": [
        {
          "term": "logistic equation",
          "register": "written",
          "note": "書き言葉で最も多い。話し言葉では logistic differential equation より少なく、大半が MIT OCW。"
        },
        {
          "term": "logistic differential equation",
          "register": "both",
          "note": "書き言葉は OpenStax Calculus。話し言葉ではすべて Khan Academy。話し言葉ではこちらのほうが多く、書き言葉では logistic equation のほうが多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "個体数などが上限（環境収容力）に近づくにつれて増え方が遅くなる様子を表す微分方程式 dP/dt = kP(1 − P/M)。",
    "definition_en": "The differential equation dP/dt = kP(1 − P/M), for growth that slows as P approaches the carrying capacity M.",
    "latex": "\\frac{dP}{dt} = kP\\left(1 - \\frac{P}{M}\\right)",
    "spoken_en": "d P d t equals k P times the quantity one minus P over M",
    "examples": [
      {
        "en": "In the logistic model, the population grows fastest when it's at half the carrying capacity.",
        "ja": "ロジスティックモデルでは、個体数が環境収容力の半分のときに最も速く増える。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "上限 M は carrying capacity（環境収容力）と呼ぶ。"
    ],
    "related": [
      "differential-equation",
      "separable-differential-equation",
      "eulers-method",
      "exponential-model"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ロジスティック方程式",
        "en": "Logistic differential equation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "logistic equation": 7,
        "logistic differential equation": 20
      },
      "written": {
        "logistic differential equation": 17,
        "logistic equation": 37
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "logistic-growth",
    "ja": {
      "term": "ロジスティック増加",
      "reading": "ろじすてぃっくぞうか",
      "alt": []
    },
    "en": {
      "term": "logistic growth",
      "alt": [],
      "variants": [
        {
          "term": "logistic growth",
          "register": "written",
          "note": "話し言葉には少なく（すべて MIT OCW）、書き言葉のほうが多い（OpenStax Algebra and Trigonometry・OpenStax Calculus）。"
        },
        {
          "term": "logistic model",
          "register": "written",
          "note": "話し言葉には出てこず、書き言葉だけにある（OpenStax Algebra and Trigonometry・OpenStax Calculus）。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「ロジスティック増加」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "はじめは指数関数的に増え、上限（環境収容力）に近づくにつれて増え方が鈍くなる増加。f(t) = c/(1 + ae^(−bt)) で表す。",
    "definition_en": "Growth that is nearly exponential at first and levels off as it approaches a limiting value (the carrying capacity), modeled by f(t) = c/(1 + ae^(−bt)).",
    "latex": "f(t) = \\frac{c}{1 + a e^{-bt}}",
    "spoken_en": "f of t equals c over one plus a e to the negative b t",
    "examples": [
      {
        "en": "Logistic growth looks exponential at first, then flattens out at the carrying capacity.",
        "ja": "ロジスティック増加は、はじめは指数関数的で、やがて環境収容力で平らになる。",
        "register": "spoken"
      },
      {
        "en": "A rumor spreads according to the logistic model N(t) = 1000/(1 + 99e^(−0.5t)). When will 500 people have heard it?",
        "ja": "うわさがロジスティックモデル N(t) = 1000/(1 + 99e^(−0.5t)) に従って広がる。500 人が聞くのはいつか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "微分方程式としての形は logistic-differential-equation を参照。"
    ],
    "related": [
      "exponential-growth",
      "logistic-differential-equation",
      "exponential-model"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "logistic growth": 5
      },
      "written": {
        "logistic growth": 21,
        "logistic model": 16
      },
      "sources": [
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "lower-limit",
    "ja": {
      "term": "下端",
      "reading": "かたん",
      "alt": []
    },
    "en": {
      "term": "lower limit",
      "alt": [
        "lower limit of integration"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "定積分 ∫ₐᵇ f(x)dx で、積分記号の下に書く a。",
    "definition_en": "The number a written at the bottom of the integral sign in ∫ₐᵇ f(x) dx.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Don't forget to subtract the value at the lower limit.",
        "ja": "下端での値を引くのを忘れないで。",
        "register": "spoken"
      },
      {
        "en": "Swapping the upper and lower limits changes the sign of the integral.",
        "ja": "上端と下端を入れ替えると、定積分の符号が変わる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では lower bound や bottom bound とも言う。",
      "数列の和の Σ の下の数（lower limit of summation）や、統計の信頼区間の下限も lower limit と言う。"
    ],
    "related": [
      "upper-limit",
      "interval-of-integration",
      "definite-integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "lower limit of integration | at the lower limit": 10
      },
      "written": {
        "lower limit of integration | at the lower limit": 5
      },
      "sources": [
        "mit-18.03",
        "openstax-calculus",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "lu-decomposition",
    "ja": {
      "term": "LU 分解",
      "reading": "えるゆーぶんかい",
      "alt": []
    },
    "en": {
      "term": "LU factorization",
      "alt": [
        "LU decomposition"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（LU factorization ／ LU decomposition は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列を、下三角行列 L と上三角行列 U の積 A = LU に分けること。ガウスの消去法の手順を行列で記録したものになる。",
    "definition_en": "Writing a matrix as the product A = LU of a lower triangular matrix L and an upper triangular matrix U; it records the steps of Gaussian elimination.",
    "latex": "A = LU",
    "spoken_en": null,
    "examples": [
      {
        "en": "Once you have the LU factorization, solving Ax = b is just two triangular systems.",
        "ja": "LU 分解があれば、Ax = b は三角行列の連立 2 つを解くだけになる。",
        "register": "spoken"
      },
      {
        "en": "Find an LU factorization of the matrix.",
        "ja": "行列の LU 分解を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Nicholson は LU-factorization（ハイフン付き、26 件）と書く。用例コーパスには LU factorization・LU decomposition とも出てこない。"
    ],
    "related": [
      "gaussian-elimination",
      "elementary-matrix",
      "matrix-multiplication"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "2.7 LU-Factorization"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 LU factorization（2.7 LU-Factorization）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "maclaurin-series",
    "ja": {
      "term": "マクローリン級数",
      "reading": "まくろーりんきゅうすう",
      "alt": [
        "マクローリン展開"
      ]
    },
    "en": {
      "term": "Maclaurin series",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "x = 0 を中心とするテイラー級数。",
    "definition_en": "A Taylor series centered at x = 0.",
    "latex": "\\sin x = x - \\frac{x^3}{3!} + \\frac{x^5}{5!} - \\cdots",
    "spoken_en": "sine x equals x minus x cubed over three factorial plus x to the fifth over five factorial minus dot dot dot",
    "examples": [
      {
        "en": "You should know the Maclaurin series for e to the x, sine, and cosine by heart.",
        "ja": "e^x、sin、cos のマクローリン級数は覚えておこう。",
        "register": "spoken"
      },
      {
        "en": "Use the Maclaurin series for cos x to find the Maclaurin series for x cos(x²).",
        "ja": "cos x のマクローリン級数を用いて、x cos(x²) のマクローリン級数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉ではすべて Khan Academy。MIT OCW の講義には Maclaurin series が出てこない（Taylor series はよく出てくる）。"
    ],
    "related": [
      "taylor-series",
      "taylor-polynomial",
      "power-series"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.14・10.15（Maclaurin series）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "Maclaurin series": 67
      },
      "written": {
        "Maclaurin series": 68
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "magnitude",
    "ja": {
      "term": "ベクトルの大きさ",
      "reading": "べくとるのおおきさ",
      "alt": [
        "ノルム",
        "大きさ"
      ]
    },
    "en": {
      "term": "magnitude",
      "alt": [
        "norm"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「大きさ」は一般語で、ベクトルでは長さを指す。英語はベクトルの大きさに magnitude（線形代数では norm、話し言葉では length とも）を使う。",
    "domains": [
      "vectors",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "definition_ja": "ベクトルを表す有向線分の長さ。|a| と書く。成分では √(a₁² + a₂²)。",
    "definition_en": "The length of the directed line segment representing a vector, written |a|; in components, √(a₁² + a₂²).",
    "latex": "|\\vec{a}| = \\sqrt{a_1^2 + a_2^2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The magnitude is just the length of the arrow, so use the Pythagorean theorem.",
        "ja": "大きさは矢印の長さなので、三平方の定理を使う。",
        "register": "spoken"
      },
      {
        "en": "Find the magnitude of the vector ⟨−3, 4⟩.",
        "ja": "ベクトル ⟨−3, 4⟩ の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "線形代数では ‖v‖ と縦線 2 本で書き、norm とも呼ぶ（用例コーパスでは magnitude がずっと多く、norm はほとんど出てこない）。",
      "magnitude は「地震の規模」「桁（order of magnitude）」の意味でも使う。用例コーパスで確かめた文脈にも、その意味のものが少し混じっていた。"
    ],
    "related": [
      "vector",
      "unit-vector",
      "dot-product"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ノルム",
        "en": "Norm (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "magnitude": 306,
        "norm": 3
      },
      "written": {
        "magnitude": 358,
        "norm": 4
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "major-arc",
    "ja": {
      "term": "優弧",
      "reading": "ゆうこ",
      "alt": []
    },
    "en": {
      "term": "major arc",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円周上の 2 点で分けられる 2 つの弧のうち、大きいほうの弧。その中心角は 180° より大きい。",
    "definition_en": "The larger of the two arcs into which two points divide a circle. Its measure is greater than 180°.",
    "latex": "m\\overset{\\frown}{ACB} = 360^\\circ - m\\overset{\\frown}{AB}",
    "spoken_en": "the measure of arc ACB equals 360 degrees minus the measure of arc AB",
    "examples": [
      {
        "en": "For a major arc, use three letters, like arc ACB, so it doesn't get mixed up with minor arc AB.",
        "ja": "優弧は弧 ACB のように 3 文字で書こう。劣弧 AB とまぎれないようにするためだ。",
        "register": "spoken"
      },
      {
        "en": "Points A, B, and C lie on circle O, and m∠AOB = 100°. Find the measure of major arc ACB.",
        "ja": "点 A, B, C は円 O 上にあり、∠AOB = 100° である。優弧 ACB に対する中心角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では優弧を 3 文字（arc ACB）で表し、2 文字の arc AB は劣弧を指す。",
      "ちょうど 180° の弧は半円（semicircle）で、優弧にも劣弧にも入れない。",
      "用例コーパス・IM には出てこず、CK-12 Geometry では 6.9 Arcs in Circles に出てくる。英語版 Wikipedia にも独立した記事は無く、Curve の記事の一節で扱う。"
    ],
    "related": [
      "minor-arc",
      "arc-measure",
      "reflex-angle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.9 Arcs in Circles"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 major arc（CK-12 Geometry 6.9 Arcs in Circles・計 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "major-axis",
    "ja": {
      "term": "長軸",
      "reading": "ちょうじく",
      "alt": []
    },
    "en": {
      "term": "major axis",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "楕円の 2 つの焦点を通る軸。楕円の最も長い径で、長さは 2a。",
    "definition_en": "The longest chord of an ellipse, through both foci; its length is 2a.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The major axis is horizontal here because 25 is under the x squared.",
        "ja": "x² の下が 25 だから、ここでは長軸が横向き。",
        "register": "spoken"
      },
      {
        "en": "Find the length of the major axis of the ellipse 4x² + y² = 36.",
        "ja": "楕円 4x² + y² = 36 の長軸の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "半分の長さ a は semi-major axis（半長軸）。"
    ],
    "related": [
      "minor-axis",
      "ellipse",
      "focus",
      "transverse-axis"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "major axis": 11
      },
      "written": {
        "major axis": 105
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "make-a-sign-chart",
    "ja": {
      "term": "増減表をかく",
      "reading": "ぞうげんひょうをかく",
      "alt": []
    },
    "en": {
      "term": "make a sign chart",
      "alt": [
        "draw a sign chart"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。増減表（x・f′(x)・f(x) の行を 1 つの表にまとめたもの）に当たる定型の表は、AP の CED にも OpenStax Calculus にも出てこない。近い作業は f′ の符号を区間ごとに書き出す sign chart（AP の CED は Unit 9 の概要で答えを見つける道具として挙げる。エントリ sign-chart）で、OpenStax Calculus は同じ作業を、区間ごとの sign of f′(x) を表（table）にまとめる形で示す。見出しの make a sign chart は、CED の呼び方 sign chart に動詞を付けた本プロジェクトの言い方で、用例コーパスには出てこない。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "f′(x) の符号と f(x) の増減を表にまとめる。",
    "definition_en": "To set out, in a table or on a number line, the sign of f′(x) and whether f is increasing or decreasing.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's make a sign chart for f prime: test one point in each interval.",
        "ja": "f′ の符号の表を作ろう。各区間で 1 点ずつ調べる。",
        "register": "spoken"
      },
      {
        "en": "Make a sign chart for f′(x) = (x − 1)(x + 2) and use it to find where f is increasing.",
        "ja": "f′(x) = (x − 1)(x + 2) の符号の表をかき、f が増加する区間を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "sign-chart",
      "increasing-and-decreasing",
      "first-derivative-test"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 9 の概要（sign chart）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "区間ごとの sign of f′(x) を表にまとめる（本文 31 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "markov-chain",
    "ja": {
      "term": "マルコフ連鎖",
      "reading": "まるこふれんさ",
      "alt": []
    },
    "en": {
      "term": "Markov chain",
      "alt": [
        "Markov process"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Markov chain ／ Markov process は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra",
      "probability"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "次の状態に移る確率が今の状態だけで決まるような、状態の移り変わりの列。遷移行列を掛けて次の確率分布を求める。",
    "definition_en": "A sequence of states in which the probability of the next state depends only on the current one; the next distribution is found by multiplying by a transition matrix.",
    "latex": "\\vec{x}_{k+1} = P\\vec{x}_k",
    "spoken_en": null,
    "examples": [
      {
        "en": "In a Markov chain, the past doesn't matter; only where you are now.",
        "ja": "マルコフ連鎖では過去は関係なく、今どこにいるかだけが効く。",
        "register": "spoken"
      },
      {
        "en": "Find the steady-state vector of the Markov chain with the given transition matrix.",
        "ja": "与えられた遷移行列のマルコフ連鎖の定常ベクトルを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスには Markov process も Markov chain もほとんど出てこない。"
    ],
    "related": [
      "eigenvalue",
      "eigenvector",
      "matrix-multiplication",
      "dynamical-system"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "マルコフ連鎖",
        "en": "Markov chain"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "2.9 An Application to Markov Chains"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "Markov chain": 1,
        "Markov process": 2
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 Markov chain（2.9 An Application to Markov Chains・2.8 An Application to Input-Output Economic Models）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "mathematical-model",
    "ja": {
      "term": "数学的モデル",
      "reading": "すうがくてきもでる",
      "alt": [
        "数理モデル"
      ]
    },
    "en": {
      "term": "mathematical model",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics",
      "functions"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "現実の現象を、式やグラフなどの数学の言葉で表したもの。そう表すことをモデル化という。",
    "definition_en": "A description of a real-world situation in mathematical terms, such as an equation or a graph; making one is called modeling.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The exponential model fits the early data well, but it can't keep going forever.",
        "ja": "指数関数のモデルは初めのデータにはよく合うが、いつまでも続くわけではない。",
        "register": "spoken"
      },
      {
        "en": "Find a mathematical model for the population and use it to predict the population in 2030.",
        "ja": "人口の数学的モデルを求め、2030 年の人口を予測せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "IM Algebra 1（3.4 Linear Models、5.11 Modeling Exponential Behavior）と OpenStax Algebra and Trigonometry（4.3・6.7）は、linear model・exponential model のように種類を付けて model と言い、作ることを modeling（モデル化）と言う。",
      "データに当てはめる（fit the data）モデルは、回帰直線（regression line）や指数関数のモデル（exponential model）など。"
    ],
    "related": [
      "regression-line",
      "exponential-model",
      "simulation",
      "predict"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "数理モデル",
        "en": "Mathematical model"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "4.3 Fitting Linear Models to Data・6.7 Exponential and Logarithmic Models"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 1 3.4 Linear Models・5.11 Modeling Exponential Behavior ほか（mathematical model、modeling）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "mathematical model": 1
      },
      "written": {
        "mathematical model": 19
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "matrix-multiplication",
    "ja": {
      "term": "行列の積",
      "reading": "ぎょうれつのせき",
      "alt": []
    },
    "en": {
      "term": "matrix multiplication",
      "alt": [
        "multiply matrices"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列 A の各行と行列 B の各列の成分を掛けて足し、積 AB の成分を作る演算。A の列の数と B の行の数が等しいときだけ定まる。",
    "definition_en": "The operation that forms each entry of AB from a row of A and a column of B by multiplying and adding; defined only when A has as many columns as B has rows.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Matrix multiplication isn't commutative: AB and BA are usually different.",
        "ja": "行列の積は交換法則が成り立たない。AB と BA はふつう違う。",
        "register": "spoken"
      },
      {
        "en": "Find AB and BA, and verify that AB ≠ BA.",
        "ja": "AB と BA を求め、AB ≠ BA であることを確かめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「行列を掛ける」は multiply the matrices。積 AB は A times B と読む。"
    ],
    "related": [
      "matrix",
      "identity-matrix",
      "inverse-matrix"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "行列の乗法",
        "en": "Matrix multiplication"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "matrix multiplication": 69,
        "multiply matrices": 14
      },
      "written": {
        "matrix multiplication": 33,
        "multiply matrices": 3
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "matrix",
    "ja": {
      "term": "行列",
      "reading": "ぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "matrix",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "数や文字を長方形に並べ、かっこでくくったもの。横の並びを行、縦の並びを列という。",
    "definition_en": "A rectangular array of numbers or symbols in brackets; its horizontal lines are rows and its vertical lines are columns.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A 2 by 3 matrix has two rows and three columns.",
        "ja": "2 × 3 行列は 2 行 3 列。",
        "register": "spoken"
      },
      {
        "en": "Write the coefficient matrix of the system of equations.",
        "ja": "連立方程式の係数行列を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形は matrices（matrix の不規則な複数形）。",
      "m × n 行列は m by n matrix と読む（用例コーパスの話し言葉でも two by two はよく出てくる）。"
    ],
    "related": [
      "row",
      "column",
      "entry",
      "determinant",
      "matrix-multiplication"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "行列",
        "en": "Matrix (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "matrix": 3491
      },
      "written": {
        "matrix": 819
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "maximum",
    "ja": {
      "term": "最大値",
      "reading": "さいだいち",
      "alt": []
    },
    "en": {
      "term": "maximum",
      "alt": [
        "maximum value"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics",
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "データや関数のとる値のうち、もっとも大きい値。",
    "definition_en": "The greatest of the values that the data, or a function on its domain, take.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The maximum is 98 and the minimum is 41, so the range is 57.",
        "ja": "最大値が 98、最小値が 41 なので、範囲は 57。",
        "register": "spoken"
      },
      {
        "en": "Find the maximum value of f(x) = −x² + 4x + 1 on 0 ≤ x ≤ 3.",
        "ja": "0 ≤ x ≤ 3 における f(x) = −x² + 4x + 1 の最大値を求めよ。",
        "register": "written"
      },
      {
        "en": "Find the maximum value of y = x² − 4x + 1 on −1 ≤ x ≤ 3, and the value of x where it occurs.",
        "ja": "−1 ≦ x ≦ 3 における y = x² − 4x + 1 の最大値と、そのときの x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "極大値（local maximum）とは別。区間全体での最大値を関数について言うときは absolute maximum とも言う。"
    ],
    "related": [
      "minimum",
      "absolute-extrema",
      "five-number-summary",
      "local-maximum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the maximum value | absolute maximum": 64
      },
      "written": {
        "the maximum value | absolute maximum": 184
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "mean-absolute-deviation",
    "ja": {
      "term": "平均絶対偏差",
      "reading": "へいきんぜったいへんさ",
      "alt": []
    },
    "en": {
      "term": "mean absolute deviation",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "各データと平均値の差の絶対値を求め、それらの平均をとった値。データが平均値のまわりにどれくらい散らばっているかを表す。",
    "definition_en": "The average distance between each data value and the mean, found by averaging the absolute values of the differences. It measures how spread out the data are.",
    "latex": "\\text{MAD} = \\frac{1}{n}\\sum_{i=1}^{n} |x_i - \\bar{x}|",
    "spoken_en": "MAD equals one over n times the sum from i equals 1 to n of the absolute value of x sub i minus x bar",
    "examples": [
      {
        "en": "The mean absolute deviation is 2, so on average the values are 2 away from the mean.",
        "ja": "平均絶対偏差は 2 だから、データは平均して平均値から 2 離れているということ。",
        "register": "spoken"
      },
      {
        "en": "Find the mean absolute deviation of the data set 2, 4, 6, 8.",
        "ja": "データ 2、4、6、8 の平均絶対偏差を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "IM Grade 6 は mean absolute deviation (MAD) として 6 年生で教える。日本の中学・高校では扱わず、散らばりは範囲・四分位範囲・分散・標準偏差（エントリ standard-deviation）で表す。",
      "略して MAD と書き、エム・エー・ディーと 1 文字ずつ読むか、mad と 1 語で読む。",
      "差を 2 乗せず絶対値をとる点が標準偏差と違う。2、4、6、8 なら平均値 5 との差の絶対値は 3、1、1、3 で、平均絶対偏差は 2。"
    ],
    "related": [
      "standard-deviation",
      "absolute-value",
      "arithmetic-mean",
      "deviation",
      "interquartile-range"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 glossary（mean absolute deviation (MAD)）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "mean absolute deviation": 21
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "mean-value-theorem",
    "ja": {
      "term": "平均値の定理",
      "reading": "へいきんちのていり",
      "alt": []
    },
    "en": {
      "term": "mean value theorem",
      "alt": [],
      "variants": [
        {
          "term": "MVT",
          "register": "written",
          "note": "略称。話し言葉にはほとんど出てこない。書き言葉はすべて MIT の講義ノート。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "f が [a, b] で連続、(a, b) で微分可能なら、f′(c) = (f(b) − f(a))/(b − a) となる c が a と b の間にあるという定理。",
    "definition_en": "If f is continuous on [a, b] and differentiable on (a, b), then f′(c) = (f(b) − f(a))/(b − a) for some c between a and b.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The mean value theorem says that somewhere the tangent slope equals the secant slope.",
        "ja": "平均値の定理は、どこかで接線の傾きが割線の傾きに等しくなると言っている。",
        "register": "spoken"
      },
      {
        "en": "Verify that f(x) = x² satisfies the hypotheses of the mean value theorem on [0, 2], and find c.",
        "ja": "f(x) = x² が [0, 2] で平均値の定理の仮定を満たすことを確かめ、c を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "apply the mean value theorem",
        "ja": "平均値の定理を適用する"
      }
    ],
    "pitfalls": [],
    "related": [
      "rolles-theorem",
      "average-rate-of-change",
      "slope-of-the-tangent-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "平均値の定理",
        "en": "Mean value theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.1（mean value theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "mean value theorem": 130,
        "MVT": 1
      },
      "written": {
        "mean value theorem": 81,
        "MVT": 7
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "measure-of-center",
    "ja": {
      "term": "代表値",
      "reading": "だいひょうち",
      "alt": [
        "データの代表値",
        "中心の測度"
      ]
    },
    "en": {
      "term": "measure of center",
      "alt": [],
      "variants": [
        {
          "term": "measure of central tendency",
          "register": "spoken",
          "note": "話し言葉では measure of center より多いが、ほとんどが Khan Academy の AP Statistics。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「代表値」に 1 語で当たる英語の総称はなく、measure of central tendency ／ measure of center（中心の指標）と言う。用例コーパスでは話し言葉は measure of central tendency のほうが多く（すべて Khan Academy）、書き言葉には measure of center だけが出てくる（すべて OpenStax Introductory Statistics）。見出しは AP Statistics の CED（2026 年版。topic 1.7 ほか）の呼び方にした。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データ全体の特徴を 1 つの数で表す値。平均値・中央値・最頻値など。",
    "definition_en": "A single number that describes a typical or central value of the data, such as the mean, median, or mode.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The mean and the median are both measures of central tendency, but they react differently to outliers.",
        "ja": "平均値も中央値も代表値だが、外れ値への反応が違う。",
        "register": "spoken"
      },
      {
        "en": "Which measure of center best describes the data? Explain.",
        "ja": "このデータをもっともよく表す代表値はどれか。説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "arithmetic-mean",
      "median",
      "mode",
      "spread"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.7（measure of center）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "measure of center": 5,
        "measure of central tendency": 15
      },
      "written": {
        "measure of center": 3
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "measure",
    "ja": {
      "term": "測る",
      "reading": "はかる",
      "alt": [
        "長さを測る",
        "角度を測る"
      ]
    },
    "en": {
      "term": "measure",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "定規や分度器などの道具を使って、長さや角の大きさなどを数で調べる。",
    "definition_en": "To find the size of something, such as a length or an angle, as a number of units, usually with a tool like a ruler or a protractor.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Grab your protractor and measure angle B. You should get about 65 degrees.",
        "ja": "分度器を出して、角 B を測ってみて。だいたい 65° になるはず。",
        "register": "spoken"
      },
      {
        "en": "Use a ruler to measure the length of segment AB to the nearest tenth of a centimeter.",
        "ja": "定規を使って、線分 AB の長さを 0.1 cm 単位で測れ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "名詞の measure は「大きさ」の意味で使う。∠A の大きさは the measure of angle A で、m∠A と書く（m∠A = 50°）。",
      "道具は、長さなら ruler（定規・ものさし）、角なら protractor（分度器）。measure … with a ruler、use a protractor to measure … の形で言う（IM と CK-12 Geometry は分度器を use a protractor to measure と書く）。",
      "統計の measure of central tendency（代表値。エントリ measure-of-center）の measure は「指標」の意味で、「測る」とは別。"
    ],
    "related": [
      "angle",
      "degree-measure",
      "unit",
      "indirect-measurement",
      "measure-of-center"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Grade 7 7.3 Nonadjacent Angles・7.5 Using Equations to Solve for Unknown Angles（use a protractor to measure。pitfalls[1]）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.9 Measuring Angles（use a protractor to measure。pitfalls[1]）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "measure the length | measure the angle | measure the height | measure the side | measure each | measure it | measure with": 44
      },
      "written": {
        "measure the length | measure the angle | measure the height | measure the side | measure each | measure it | measure with": 31
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "measurement",
    "ja": {
      "term": "測定値",
      "reading": "そくていち",
      "alt": []
    },
    "en": {
      "term": "measurement",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "長さ・重さ・時間などを測って得た値。測定値は近似値で、真の値との間に誤差がある。",
    "definition_en": "A value obtained by measuring a quantity such as length, weight, or time. A measurement is an approximate value, so it differs a little from the true value.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A measurement is never exact. If your ruler only has millimeter marks, 12.8 centimeters really means somewhere from 12.75 to 12.85.",
        "ja": "測定値はぴったりの値にはならない。定規に 1 mm の目盛りしかなければ、12.8 cm というのは、本当は 12.75 cm から 12.85 cm の間のどこか、ということ。",
        "register": "spoken"
      },
      {
        "en": "The mass of a bag of rice was measured to the nearest 10 g, and the measurement was 2350 g. If the true mass is a grams, write an inequality for a.",
        "ja": "米袋の重さを、10 g 未満を四捨五入して測ったところ、測定値は 2350 g だった。真の値を a g として、a の範囲を不等号を使って表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の measurement は「測った値」のほかに、「測ること」そのもの（動詞は measure、エントリ measure）や、単位の体系（the metric system of measurement）の意味でも使う。",
      "測定値と真の値との差が誤差（エントリ error）。測定値のどの桁まで信頼できるかは有効数字（エントリ significant-figures）で表す。",
      "12.8 cm と 12.80 cm は、測定値としては精度が違う。12.80 cm は 0.01 cm の位まで測った値。"
    ],
    "related": [
      "measure",
      "error",
      "approximate-value",
      "significant-figures",
      "round"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!of !linear measurement !error !errors !of !unit !units !system !systems": 69
      },
      "written": {
        "!of !linear measurement !error !errors !of !unit !units !system !systems": 89
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "median-of-a-triangle",
    "ja": {
      "term": "中線",
      "reading": "ちゅうせん",
      "alt": []
    },
    "en": {
      "term": "median",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "三角形の 1 つの頂点と、その対辺の中点を結ぶ線分のこと。3 本の中線は 1 点（重心）で交わる。",
    "definition_en": "In a triangle, the segment that connects a vertex with the midpoint of the side opposite that vertex. The three medians meet at one point, the centroid.",
    "latex": "AG : GM = 2 : 1",
    "spoken_en": "A G to G M is two to one",
    "examples": [
      {
        "en": "Draw the median from A. It goes to the midpoint of BC, so BM equals MC.",
        "ja": "A から中線を引こう。BC の中点に向かうので BM = MC になる。",
        "register": "spoken"
      },
      {
        "en": "In △ABC, AM is a median and G is the centroid. If AM = 12, find AG.",
        "ja": "△ABC において、AM は中線、G は重心である。AM = 12 のとき、AG を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の median は統計の「中央値」（median）と同じ語。図形の話なら中線、データの話なら中央値と読み分ける。紛れそうなときは the median of a triangle と言う。",
      "中線は頂点と対辺の中点を結ぶ線分。角の二等分線（angle bisector）、垂直二等分線（perpendicular bisector）、頂点から対辺への垂線（altitude）と取り違えない。",
      "IM Geometry（6.16・6.17）と CK-12 Geometry（4.23 Medians）は、the medians of a triangle が 1 点（重心）で交わる、の形で使う。"
    ],
    "related": [
      "centroid",
      "median",
      "midpoint",
      "triangle",
      "the-five-centers-of-a-triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "中線",
        "en": "Median (geometry)"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 6.16・6.17 Lines in Triangles（the medians of a triangle）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.23 Medians"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "median of a triangle | median of the triangle | medians of a triangle | medians of the triangle": 1
      },
      "sources": [
        "mit-18.02"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 median of a triangle | median of the triangle | medians of a triangle | medians of the triangle（Geometry 5.11 Prisms Practice・Geometry 6.17 Lines in Triangles・Geometry 6.16 Weighted Averages in a Triangle・計 5 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "median",
    "ja": {
      "term": "中央値",
      "reading": "ちゅうおうち",
      "alt": [
        "メジアン"
      ]
    },
    "en": {
      "term": "median",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データを小さい順に並べたとき中央にくる値。データの個数が偶数なら中央の 2 つの値の平均。",
    "definition_en": "The middle value of the data in order; with an even number of values, the mean of the two middle values.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "With an even number of values, the median is the average of the two in the middle.",
        "ja": "値の個数が偶数なら、中央値は真ん中の 2 つの平均。",
        "register": "spoken"
      },
      {
        "en": "Find the median and the interquartile range of the data.",
        "ja": "データの中央値と四分位範囲を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "中央値は第 2 四分位数（Q2）でもある。",
      "図形の「中線」（三角形の頂点と向かい合う辺の中点を結ぶ線分）も英語では median と言う。それは別の見出し median-of-a-triangle。"
    ],
    "related": [
      "quartile",
      "arithmetic-mean",
      "mode",
      "measure-of-center",
      "median-of-a-triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "中央値",
        "en": "Median"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "median": 461
      },
      "written": {
        "median": 270
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.02",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "menelauss-theorem",
    "ja": {
      "term": "メネラウスの定理",
      "reading": "めねらうすのていり",
      "alt": []
    },
    "en": {
      "term": "Menelaus's theorem",
      "alt": [
        "Menelaus' theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Menelaus's theorem ／ Menelaus' theorem は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "definition_ja": "△ABC の頂点を通らない 1 本の直線が、直線 BC, CA, AB と P, Q, R で交わるとき、(BP/PC)(CQ/QA)(AR/RB) = 1 が成り立つという定理。逆（P, Q, R のうち 1 個か 3 個が辺の延長上にあり、積が 1 なら 3 点は一直線上にある）も成り立ち、3 点が一直線上にあることの証明に使う。",
    "definition_en": "If a line that passes through no vertex of triangle ABC meets lines BC, CA, and AB at P, Q, and R, then (BP/PC)(CQ/QA)(AR/RB) = 1. The converse also holds when one or all three of P, Q, and R lie on extensions of the sides, so the theorem is a tool for proving that three points are collinear.",
    "latex": "\\frac{BP}{PC}\\cdot\\frac{CQ}{QA}\\cdot\\frac{AR}{RB} = 1",
    "spoken_en": "BP over PC, times CQ over QA, times AR over RB, equals one",
    "examples": [
      {
        "en": "Menelaus's theorem looks just like Ceva's, but here the three points lie on one line instead of the three lines meeting at one point.",
        "ja": "メネラウスの定理はチェバの定理と同じ形だが、こちらは 3 本の直線が 1 点で交わるのではなく、3 つの点が 1 本の直線上にある。",
        "register": "spoken"
      },
      {
        "en": "A line meets side AB at R, side CA at Q, and the extension of BC at P. If AR : RB = 2 : 1 and CQ : QA = 1 : 3, use Menelaus's theorem to find BP : PC.",
        "ja": "直線が辺 AB と R で、辺 CA と Q で、辺 BC の延長と P で交わる。AR : RB = 2 : 1, CQ : QA = 1 : 3 のとき、メネラウスの定理を用いて BP : PC を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Menelaus は英語で「メネレイアス」に近く読む（第 3 音節 LAY に強勢。Merriam-Webster: ˌme-nə-ˈlā-əs）。所有格のつづりは 2 通りあり、英語版 Wikipedia の記事名は Menelaus's theorem、記事が参考文献に挙げる MathWorld の項目名は Menelaus' Theorem。",
      "直線は 3 辺のうち 2 辺と内分点で、1 辺と延長上（外分点）で交わるか、3 辺とも延長上で交わる。比は三角形の周りを一方向にたどって書く。"
    ],
    "related": [
      "cevas-theorem",
      "collinear",
      "point-of-external-division",
      "triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "メネラウスの定理",
        "en": "Menelaus's theorem"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「Menelaus」",
        "url": "https://www.merriam-webster.com/dictionary/Menelaus",
        "note": "発音 ˌme-nə-ˈlā-əs（2026-09-26 に確かめた）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Menelaus%27s_theorem",
        "note": "記事「Menelaus's theorem」の冒頭（Menelaus's theorem を太字で定義。参考文献に MathWorld の Menelaus' Theorem。数学カテゴリから 3 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 Menelaus's theorem（英語版「Menelaus's theorem」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "midline",
    "ja": {
      "term": "振動の中心線",
      "reading": "しんどうのちゅうしんせん",
      "alt": [
        "中央線"
      ]
    },
    "en": {
      "term": "midline",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「振動の中心線」は学習指導要領解説に無い、本プロジェクトの訳語。日本の数II では y = sin θ + 1 のグラフを「y 軸方向に 1 平行移動したもの」として扱い、この線に名前を付けない。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "正弦曲線の最大値と最小値のちょうど中間を通る、x 軸に平行な直線。y = A sin(Bx − C) + D のグラフでは y = D。",
    "definition_en": "The horizontal line halfway between the maximum and minimum values of a sinusoidal function. For y = A sin(Bx − C) + D, the midline is y = D.",
    "latex": "y = D",
    "spoken_en": "y equals D",
    "examples": [
      {
        "en": "The max is 5 and the min is negative 1, so the midline is right in the middle at y equals 2.",
        "ja": "最大値が 5 で最小値が −1 だから、振動の中心線はちょうど真ん中の y = 2。",
        "register": "spoken"
      },
      {
        "en": "Find the amplitude, period, and midline of the graph of y = 3 sin(2x) + 1.",
        "ja": "y = 3 sin 2x + 1 のグラフの振幅、周期、振動の中心線を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "振幅（amplitude）は midline から最大値までの距離で、x 軸から最大値までの距離ではない。y = 3 sin 2x + 1 の振幅は 3。",
      "幾何で三角形の 2 辺の中点を結ぶ線分（エントリ midsegment-theorem）を midline と呼ぶこともある。",
      "midline は直線なので、答えは y = 2 のように式で書く。"
    ],
    "related": [
      "sinusoid",
      "amplitude",
      "period",
      "phase-shift",
      "midsegment-theorem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "midline": 79
      },
      "sources": [
        "openstax-algtrig",
        "openstax-precalculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "midpoint-formula",
    "ja": {
      "term": "中点公式",
      "reading": "ちゅうてんこうしき",
      "alt": []
    },
    "en": {
      "term": "midpoint formula",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「中点公式」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書は「中点の座標」として式を示す。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "座標平面上の線分の中点の座標を、両端の x 座標どうし・y 座標どうしの平均として求める式。",
    "definition_en": "The formula that gives the midpoint of a segment in the coordinate plane by averaging the x-coordinates and the y-coordinates of its endpoints.",
    "latex": "M = \\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)",
    "spoken_en": "M equals x one plus x two over two, comma, y one plus y two over two",
    "examples": [
      {
        "en": "Just use the midpoint formula: add the x's and divide by two, then do the same with the y's.",
        "ja": "中点の公式を使えばいい。x 座標を足して 2 で割り、y 座標も同じようにする。",
        "register": "spoken"
      },
      {
        "en": "Use the midpoint formula to find the midpoint of the segment with endpoints (−3, 5) and (7, −1).",
        "ja": "2 点 (−3, 5), (7, −1) を両端とする線分の中点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスで midpoint formula は書き言葉に多く（OpenStax Intermediate Algebra・OpenStax Algebra and Trigonometry）、話し言葉にはほとんど出てこない。",
      "線分を m : n に内分する点の座標の式で m = n = 1 とした場合に当たる。"
    ],
    "related": [
      "midpoint",
      "point-of-internal-division",
      "coordinates"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "midpoint formula": 1
      },
      "written": {
        "midpoint formula": 26
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "midpoint-riemann-sum",
    "ja": {
      "term": "中点リーマン和",
      "reading": "ちゅうてんりーまんわ",
      "alt": []
    },
    "en": {
      "term": "midpoint Riemann sum",
      "alt": [
        "midpoint sum"
      ],
      "uk": null,
      "variants": [
        {
          "term": "midpoint rule",
          "register": "written",
          "note": "大学の教科書の書き言葉。OpenStax Calculus Volume 2・3 に出てきて、話し言葉コーパスには出てこない。数値積分の方法の名前として使う。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「中点リーマン和」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II"
      ]
    },
    "definition_ja": "各小区間の中点での関数の値を高さとする長方形の面積の和。定積分の近似に使う。",
    "definition_en": "A Riemann sum that uses the midpoint of each subinterval for the heights; as a numerical method it is called the midpoint rule.",
    "latex": "M_n = \\sum_{i=1}^{n} f\\!\\left(\\frac{x_{i-1}+x_i}{2}\\right)\\Delta x",
    "spoken_en": "M sub n equals the sum from i equals one to n of f of the midpoint of each subinterval, times delta x",
    "examples": [
      {
        "en": "With the same number of rectangles, the midpoint rule usually beats the left and right sums.",
        "ja": "同じ長方形の数なら、中点リーマン和はたいてい左や右の和より精度がいい。",
        "register": "spoken"
      },
      {
        "en": "Approximate ∫₀⁴ x² dx with a midpoint Riemann sum using four subintervals.",
        "ja": "4 つの小区間を使って、中点リーマン和で ∫₀⁴ x² dx を近似せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Riemann sum の一つとして習うときは midpoint Riemann sum（midpoint sum）、数値積分の方法としては midpoint rule と呼ぶ。"
    ],
    "related": [
      "left-riemann-sum",
      "right-riemann-sum",
      "riemann-sum",
      "numerical-integration",
      "trapezoidal-rule",
      "simpsons-rule"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.2（left ／ right ／ midpoint Riemann sums）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「中点リーマン和」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "midpoint rule": 44
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-10-09"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "midpoint",
    "ja": {
      "term": "中点",
      "reading": "ちゅうてん",
      "alt": [
        "中点の座標"
      ]
    },
    "en": {
      "term": "midpoint",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Geometry",
        "Algebra 1",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "線分を 2 等分する点。座標平面では両端の座標の平均が中点の座標になる。",
    "definition_en": "The point that divides a segment into two equal parts; its coordinates are the averages of the endpoints' coordinates.",
    "latex": "\\left(\\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2}\\right)",
    "spoken_en": "x one plus x two over two, comma, y one plus y two over two",
    "examples": [
      {
        "en": "The midpoint is just the average of the x's and the average of the y's.",
        "ja": "中点は x 座標の平均と y 座標の平均をとるだけ。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinates of the midpoint of the segment joining (−2, 3) and (4, 7).",
        "ja": "2 点 (−2, 3), (4, 7) を結ぶ線分の中点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "midpoint formula",
        "ja": "中点の公式"
      }
    ],
    "pitfalls": [
      "中点の座標の式を、OpenStax（Intermediate Algebra・Algebra and Trigonometry）と CK-12 Geometry は midpoint formula と名前で呼ぶ。"
    ],
    "related": [
      "point-of-internal-division",
      "perpendicular-bisector",
      "centroid"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "中点",
        "en": "Midpoint"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.5 Midpoint Formula"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Distance and Midpoint Formulas; Circles（midpoint formula）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "midpoint": 110
      },
      "written": {
        "midpoint": 165
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "midsegment-theorem",
    "ja": {
      "term": "中点連結定理",
      "reading": "ちゅうてんれんけつていり",
      "alt": []
    },
    "en": {
      "term": "midsegment theorem",
      "alt": [
        "midpoint theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語では、三角形の 2 辺の中点を結ぶ線分に midsegment という名前があり、この定理はその線分の性質として midsegment theorem と呼ばれる（CK-12 Geometry 4.19 Midsegment Theorem）。CK-12 Geometry 5.13 Trapezoids は、台形の 2 本の脚（legs）の中点を結ぶ線分も midsegment と呼ぶ。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "三角形の 2 辺の中点を結ぶ線分は、残りの辺に平行で、長さはその辺の半分であるという定理。",
    "definition_en": "The theorem that a segment connecting the midpoints of two sides of a triangle runs parallel to the remaining side and has half its length. Such a segment is called a midsegment of the triangle.",
    "latex": "MN \\parallel BC,\\quad MN = \\frac{1}{2}BC",
    "spoken_en": "MN is parallel to BC, and MN equals one half BC",
    "examples": [
      {
        "en": "M and N are midpoints, so the midsegment theorem tells us MN is parallel to BC and half as long.",
        "ja": "M と N は中点だから、中点連結定理より、MN は BC に平行で、長さは BC の半分。",
        "register": "spoken"
      },
      {
        "en": "In triangle ABC, M and N are the midpoints of AB and AC, and BC = 14 cm. Use the midsegment theorem to find MN.",
        "ja": "△ABC で、点 M、N はそれぞれ辺 AB、AC の中点で、BC = 14 cm である。中点連結定理を用いて、MN の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では、2 辺の中点を結ぶ線分そのものを midsegment と呼ぶ。頂点と向かい合う辺の中点を結ぶ線分は中線（median、エントリ median-of-a-triangle）で、別のもの。",
      "midpoint theorem と言っても、中点の座標を求める公式（midpoint formula、エントリ midpoint-formula）とは別のもの。",
      "台形の midsegment の長さは、平行な 2 辺の長さの和の半分。三角形のように 1 辺の半分ではない。"
    ],
    "related": [
      "midpoint",
      "triangle-proportionality-theorem",
      "parallel-lines",
      "trapezoid",
      "median-of-a-triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "中点連結定理",
        "en": "Midpoint theorem (triangle)"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.19 Midsegment Theorem・5.13 Trapezoids"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 midsegment theorem（CK-12 Geometry 4.19 Midsegment Theorem・CK-12 Geometry 5.13 Trapezoids・計 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "minimum",
    "ja": {
      "term": "最小値",
      "reading": "さいしょうち",
      "alt": []
    },
    "en": {
      "term": "minimum",
      "alt": [
        "minimum value"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics",
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "データや関数のとる値のうち、もっとも小さい値。",
    "definition_en": "The least of the values that the data, or a function on its domain, take.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Start the box plot at the minimum and end it at the maximum.",
        "ja": "箱ひげ図は最小値から最大値までかく。",
        "register": "spoken"
      },
      {
        "en": "Find the minimum value of y = x² − 6x + 10 and the value of x at which it occurs.",
        "ja": "y = x² − 6x + 10 の最小値と、そのときの x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "極小値（local minimum）とは別。区間全体での最小値を関数について言うときは absolute minimum とも言う。"
    ],
    "related": [
      "maximum",
      "absolute-extrema",
      "five-number-summary",
      "local-minimum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "the minimum value | absolute minimum": 50
      },
      "written": {
        "the minimum value | absolute minimum": 121
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "minor-arc",
    "ja": {
      "term": "劣弧",
      "reading": "れっこ",
      "alt": []
    },
    "en": {
      "term": "minor arc",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円周上の 2 点で分けられる 2 つの弧のうち、小さいほうの弧。その中心角は 180° より小さい。",
    "definition_en": "The smaller of the two arcs into which two points divide a circle. Its measure is less than 180°.",
    "latex": "m\\overset{\\frown}{AB} < 180^\\circ",
    "spoken_en": "the measure of arc AB is less than 180 degrees",
    "examples": [
      {
        "en": "Arc AB with just two letters means the minor arc, the short way around the circle.",
        "ja": "2 文字だけの弧 AB は劣弧、つまり円を短いほうに回った弧のことだ。",
        "register": "spoken"
      },
      {
        "en": "In circle O, the measure of major arc ACB is 250°. Find the measure of minor arc AB.",
        "ja": "円 O で優弧 ACB に対する中心角は 250° である。劣弧 AB に対する中心角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では 2 文字の arc AB が劣弧を指し、優弧は 3 文字（arc ACB）で書いて区別する。",
      "ちょうど 180° の弧は半円（semicircle）で、劣弧にも優弧にも入れない。",
      "用例コーパス・IM には出てこず、CK-12 Geometry には出てくる（ほとんど 6.9 Arcs in Circles）。英語版 Wikipedia の Minor arc は曖昧さ回避のページ。"
    ],
    "related": [
      "major-arc",
      "arc-measure",
      "chord"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.9 Arcs in Circles・6.12 Chords and Central Angle Arcs"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 minor arc（CK-12 Geometry 6.9 Arcs in Circles・CK-12 Geometry 6.12 Chords and Central Angle Arcs・計 9 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "minor-axis",
    "ja": {
      "term": "短軸",
      "reading": "たんじく",
      "alt": []
    },
    "en": {
      "term": "minor axis",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "楕円の中心を通り、長軸に垂直な軸。長さは 2b。",
    "definition_en": "The chord of an ellipse through its center perpendicular to the major axis; its length is 2b.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The minor axis is the short way across the ellipse.",
        "ja": "短軸は楕円を短い方向に横切る軸。",
        "register": "spoken"
      },
      {
        "en": "The ellipse has a major axis of length 10 and a minor axis of length 6. Write its equation.",
        "ja": "長軸の長さが 10、短軸の長さが 6 の楕円の方程式を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉にはほとんど出てこず、書き言葉（OpenStax）の語。"
    ],
    "related": [
      "major-axis",
      "ellipse"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "minor axis": 3
      },
      "written": {
        "minor axis": 51
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "minor",
    "ja": {
      "term": "小行列式",
      "reading": "しょうぎょうれつしき",
      "alt": []
    },
    "en": {
      "term": "minor",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正方行列から 1 つの行と 1 つの列を除いてできる行列の行列式。",
    "definition_en": "The determinant of the matrix left after deleting one row and one column of a square matrix.",
    "latex": "M_{ij}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Cover up the first row and first column; what's left gives you the minor.",
        "ja": "第 1 行と第 1 列を隠して、残りから小行列式が出る。",
        "register": "spoken"
      },
      {
        "en": "Expand by minors along the first row to evaluate the determinant.",
        "ja": "第 1 行に沿って小行列式で展開し、行列式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "符号 (−1)^{i+j} を掛けたものは余因子（cofactor）。"
    ],
    "related": [
      "cofactor",
      "cofactor-expansion",
      "determinant"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "小行列式",
        "en": "Minor (linear algebra)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "matrix of minors | expand by minors | minor of the entry | the minor of | determinant of the minor": 7
      },
      "written": {
        "matrix of minors | expand by minors | minor of the entry | the minor of | determinant of the minor": 47
      },
      "sources": [
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "mixture-problem",
    "ja": {
      "term": "濃度の問題",
      "reading": "のうどのもんだい",
      "alt": []
    },
    "en": {
      "term": "mixture problem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「濃度の問題」は、食塩水の濃度など、濃さの違うものを混ぜ合わせる文章題を指す。OpenStax Elementary Algebra はこれを mixture problem と呼び、濃度は concentration と言って percent で表す。食塩水は salt solution。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "濃さの違う食塩水などを混ぜ合わせたときの、濃度や量を求める文章題。混ぜる前と後で食塩の量が変わらないことを使って式を立てる。",
    "definition_en": "A word problem about combining mixtures, such as salt solutions with different concentrations. The amount of salt before mixing equals the amount after, and that fact gives the equation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For mixture problems, I always make a table: amount of solution, percent, and amount of salt.",
        "ja": "濃度の問題では、いつも表をつくるよ。食塩水の量、濃度、食塩の量の 3 列。",
        "register": "spoken"
      },
      {
        "en": "Solve this mixture problem: how many grams of a 10% salt solution must be added to 200 g of a 4% salt solution to make a 6% salt solution?",
        "ja": "次の濃度の問題を解け。4% の食塩水 200 g に 10% の食塩水を何 g 混ぜると、6% の食塩水になるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "solution は「溶液」の意味で、方程式の「解」（エントリ solution）と同じ語。salt solution は食塩水。",
      "濃度は計算では小数に直して使う。5% の食塩水 300 g に含まれる食塩は 300 × 0.05 = 15 g。",
      "「濃度」は concentration。「5% の食塩水」は a 5% salt solution。"
    ],
    "related": [
      "percent",
      "solution",
      "word-problem",
      "write-an-equation",
      "motion-problem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "salt solution・concentration"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "mixture problem": 7
      },
      "written": {
        "mixture problem": 6
      },
      "sources": [
        "openstax-elemalg",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 mixture problem（本文 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "mode",
    "ja": {
      "term": "最頻値",
      "reading": "さいひんち",
      "alt": [
        "モード"
      ]
    },
    "en": {
      "term": "mode",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データの中でもっとも多く現れる値。度数分布表では度数がもっとも大きい階級の階級値。",
    "definition_en": "The value that occurs most often in the data; in a frequency table, the midpoint of the class with the greatest frequency.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The mode is just the value that shows up the most.",
        "ja": "最頻値は、いちばん多く出てくる値のこと。",
        "register": "spoken"
      },
      {
        "en": "Find the mean, median, and mode of the data.",
        "ja": "データの平均値・中央値・最頻値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "mode は電卓のモード（degree mode ／ radian mode）の mode と同じ語。",
      "最頻値が 2 つあるデータを bimodal と言う。"
    ],
    "related": [
      "median",
      "arithmetic-mean",
      "measure-of-center",
      "frequency"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "最頻値",
        "en": "Mode (statistics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the mode !menu": 30
      },
      "written": {
        "the mode !menu": 63
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "modular-inverse",
    "ja": {
      "term": "モジュラ逆数",
      "reading": "もじゅらぎゃくすう",
      "alt": [
        "逆元"
      ]
    },
    "en": {
      "term": "modular multiplicative inverse",
      "alt": [
        "modular inverse",
        "inverse modulo",
        "multiplicative inverse modulo"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（modular multiplicative inverse ／ modular inverse ／ inverse modulo ／ multiplicative inverse modulo は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": []
    },
    "definition_ja": "整数 a と法 n について、ax ≡ 1 (mod n) を満たす整数 x のこと。a と n が互いに素のときに限って存在し、n を法としてただ 1 つに定まる。",
    "definition_en": "An integer x with ax ≡ 1 (mod n), for a given integer a and modulus n. It exists exactly when a and n are relatively prime, and it is unique modulo n.",
    "latex": "a x \\equiv 1 \\pmod{n}",
    "spoken_en": "a x is congruent to 1 mod n",
    "examples": [
      {
        "en": "Three times five is fifteen, and fifteen leaves a remainder of one when you divide by seven, so five is the modular multiplicative inverse of three mod seven.",
        "ja": "3 × 5 = 15 で、15 を 7 で割った余りは 1。だから 7 を法とする 3 の逆元は 5 だ。",
        "register": "spoken"
      },
      {
        "en": "Use the extended Euclidean algorithm to find the modular multiplicative inverse of 17 modulo 43.",
        "ja": "拡張ユークリッドの互除法を用いて、43 を法とする 17 の逆元を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「逆数」とあっても分数 1/a ではなく整数である。英語でも multiplicative inverse だけでは実数の逆数 1/a と区別がつかないので、modulo n を付けて言う。",
      "a と n が互いに素でなければ存在しない。たとえば 4 を法とする 2 の逆元は無い（2x は常に偶数）。"
    ],
    "related": [
      "bezouts-identity",
      "extended-euclidean-algorithm",
      "modulus",
      "relatively-prime",
      "euclidean-algorithm",
      "congruence-modulo-n"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Modular_multiplicative_inverse",
        "note": "記事名 Modular multiplicative inverse（ja 記事モジュラ逆数の langlink 先。見出しを決めた参照）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "inverse modulo": 1,
        "multiplicative inverse modulo": 1
      },
      "written": {
        "inverse modulo": 1,
        "multiplicative inverse modulo": 1
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは Wikipedia の記事名 modular multiplicative inverse（英語版「Modular multiplicative inverse」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "modulus-of-a-complex-number",
    "ja": {
      "term": "複素数の絶対値",
      "reading": "ふくそすうのぜったいち",
      "alt": []
    },
    "en": {
      "term": "absolute value of a complex number",
      "alt": [
        "modulus"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II",
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "複素数 z = a + bi の原点からの距離 |z| = √(a² + b²)。",
    "definition_en": "The distance |z| = √(a² + b²) of z = a + bi from the origin of the complex plane.",
    "latex": "|a + bi| = \\sqrt{a^2 + b^2}",
    "spoken_en": "the absolute value of a plus b i equals the square root of a squared plus b squared",
    "examples": [
      {
        "en": "The modulus is just the length of the arrow from 0 to z.",
        "ja": "絶対値は、0 から z への矢印の長さ。",
        "register": "spoken"
      },
      {
        "en": "Find the absolute value of the complex number 5 − 12i.",
        "ja": "複素数 5 − 12i の絶対値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は absolute value と書き、modulus とも呼ぶと説明する。MIT OCW（18.03）の講義は modulus と言う。modulus は合同式の「法」の意味でも使う。",
      "2 点 z、w の距離は |z − w|。"
    ],
    "related": [
      "argument-of-a-complex-number",
      "polar-form",
      "complex-plane",
      "complex-conjugate"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "複素数の絶対値",
        "en": "Absolute value of a complex number"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "modulus of": 5,
        "absolute value of a complex number": 1
      },
      "written": {
        "absolute value of a complex number": 11
      },
      "sources": [
        "mit-18.03",
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "modulus",
    "ja": {
      "term": "法",
      "reading": "ほう",
      "alt": []
    },
    "en": {
      "term": "modulo",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "mod n",
          "register": "written",
          "note": "合同式 a ≡ b (mod n) の書き方。書き言葉では modulo（すべて MIT の講義ノート）と併記。mod n は話し言葉にはほとんど出てこず、書き言葉ではすべて MIT の講義ノート。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "合同式 a ≡ b (mod n) の n のこと。a − b が n で割り切れるとき、a と b は n を法として合同という。",
    "definition_en": "The positive integer n in a congruence a ≡ b (mod n), read \"a is congruent to b modulo n\"; n itself is called the modulus. Two integers are congruent modulo n when n divides their difference.",
    "latex": "a \\equiv b \\pmod{n}",
    "spoken_en": "a is congruent to b modulo n",
    "examples": [
      {
        "en": "17 is congruent to 2 modulo 5, because 17 minus 2 is divisible by 5.",
        "ja": "17 − 2 は 5 で割り切れるので、17 は 5 を法として 2 と合同（17 ≡ 2 (mod 5)）。",
        "register": "spoken"
      },
      {
        "en": "If a ≡ b (mod n), then a + c ≡ b + c (mod n).",
        "ja": "a ≡ b (mod n) ならば a + c ≡ b + c (mod n)。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "法そのもの（名詞）は modulus と言う（Nicholson（8.8）は n を the modulus と呼び、Levin 6.2 も divide the modulus と書く）。複素数の絶対値（modulus of a complex number）と同じ綴り。",
      "≡ は is congruent to と読む。図形の「合同」と同じ単語 congruent を使う。",
      "日本語の「5 を法として」は modulo 5。Levin（6.2）は 8 ≡ 23 (mod 5) を eight is congruent to 23 modulo five と読み、just mod five とも言うと書く。"
    ],
    "related": [
      "remainder",
      "classification-by-remainder",
      "congruent",
      "modulus-of-a-complex-number",
      "euclidean-algorithm",
      "congruence-modulo-n"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.8 An Application to Linear Codes over Finite Fields（We refer to n as the modulus）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2 Introduction to Number Theory（modulo、the modulus）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「整数の合同」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B4%E6%95%B0%E3%81%AE%E5%90%88%E5%90%8C",
        "note": "n を法として、法 n のもとで、mod n で"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "modulo": 87,
        "mod n": 1
      },
      "written": {
        "modulo": 31,
        "mod n": 48
      },
      "sources": [
        "mit-18.03",
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "monomial",
    "ja": {
      "term": "単項式",
      "reading": "たんこうしき",
      "alt": []
    },
    "en": {
      "term": "monomial",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "数と文字の掛け算だけでできている式。3x²、ab のほか、−5 や x のように 1 つの数や文字だけの式も単項式とする。",
    "definition_en": "An expression that is a number, a variable, or a product of numbers and variables, such as 3x², ab, or −5. A monomial has exactly one term.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "5x squared is a monomial, but once you add another term, like 5x squared plus 1, it's not a monomial anymore.",
        "ja": "5x² は単項式。でも 5x² + 1 みたいに項をもう 1 つ足したら、もう単項式じゃない。",
        "register": "spoken"
      },
      {
        "en": "Which of the following expressions are monomials? 6ab, x² − 1, −9, 5x + 2y",
        "ja": "次の式のうち、単項式であるものをすべて選べ。6ab、x² − 1、−9、5x + 2y",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "中学校学習指導要領解説は「単項式と多項式の意味」と対比で挙げ、日本語版 Wikipedia「多項式」は「多項式」を単項式でない整式の意味で用いる流儀があると書く。英語の polynomial は monomial も含む（エントリ polynomial）。",
      "mono- は「1 つの」の意味。項が 2 つの式は binomial（二項式）、3 つの式は trinomial（三項式）と、項の数で呼び名が変わる。",
      "x/3 は (1/3)x なので単項式だが、3/x のように文字で割る式は単項式ではない。"
    ],
    "related": [
      "polynomial",
      "term",
      "coefficient",
      "degree",
      "like-terms"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "単項式",
        "en": "Monomial"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 A 数と式: 単項式と多項式の意味"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「多項式」",
        "url": "https://ja.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F",
        "note": "注意（多項式と整式）: 「多項式」を単項式でない整式の意味で用いる流儀"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "monomial": 15
      },
      "written": {
        "monomial": 268
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "monotone-convergence-theorem",
    "ja": {
      "term": "単調収束定理",
      "reading": "たんちょうしゅうそくていり",
      "alt": [
        "単調数列定理"
      ]
    },
    "en": {
      "term": "monotone convergence theorem",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "数III",
        "大学"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "有界な単調数列は収束するという定理。",
    "definition_en": "The theorem that every bounded monotone sequence converges.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's increasing and bounded above, so by the monotone convergence theorem it has a limit.",
        "ja": "増加して上に有界だから、単調収束定理より極限をもつ。",
        "register": "spoken"
      },
      {
        "en": "Use the monotone convergence theorem to show that the sequence a₁ = 1, aₙ₊₁ = √(2 + aₙ) converges.",
        "ja": "単調収束定理を用いて、a₁ = 1、aₙ₊₁ = √(2 + aₙ) で定まる数列が収束することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉には出てこず、書き言葉はすべて OpenStax Calculus。",
      "数III の教科書では「有界な単調数列は収束する」を定理の名前なしで扱うことがある。"
    ],
    "related": [
      "monotonic",
      "bounded",
      "limit-of-a-sequence",
      "converge"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "monotone convergence theorem": 17
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "monotonic",
    "ja": {
      "term": "単調",
      "reading": "たんちょう",
      "alt": []
    },
    "en": {
      "term": "monotone",
      "alt": [
        "monotonic"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "analysis",
      "sequences"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "増加し続けるか、減少し続けること。",
    "definition_en": "Always increasing or always decreasing.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If a sequence is monotone and bounded, it has to converge.",
        "ja": "数列が単調で有界なら、必ず収束する。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the sequence aₙ = 1/n is monotone.",
        "ja": "数列 aₙ = 1/n が単調かどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「有界で単調な数列は収束する」は、OpenStax Calculus Volume 2 では Monotone Convergence Theorem と呼ぶ。monotonic は用例コーパスには出てこない。"
    ],
    "related": [
      "bounded",
      "is-monotonically-increasing",
      "convergence",
      "monotone-convergence-theorem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Monotonic_function",
        "note": "記事「Monotonic function」（en.alt の monotonic）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 指数関数・対数関数: 2^x が単調に増加する（level.jp 数II）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "monotone": 34
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "motion-problem",
    "ja": {
      "term": "速さの問題",
      "reading": "はやさのもんだい",
      "alt": []
    },
    "en": {
      "term": "uniform motion problem",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "道のり・速さ・時間の関係（道のり = 速さ × 時間）を使って、分からない量を求める文章題。",
    "definition_en": "A word problem about something moving at a constant speed. It is solved with the relationship distance = rate × time (D = rt).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For uniform motion problems, always start with d = rt: distance equals rate times time.",
        "ja": "速さの問題は、いつも d = rt、つまり「道のり = 速さ × 時間」から始めよう。",
        "register": "spoken"
      },
      {
        "en": "Solve this uniform motion problem: two trains leave the same station at the same time and travel in opposite directions at 50 mph and 70 mph. In how many hours will they be 360 miles apart?",
        "ja": "次の速さの問題を解け。2 本の列車が同じ駅を同時に出発し、時速 50 マイルと時速 70 マイルで反対方向に進む。2 本の列車が 360 マイル離れるのは何時間後か。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Elementary Algebra（3.5 Solve Uniform Motion Applications ほか）は節の名前を uniform motion applications とし、本文では速さが一定の動きの文章題を uniform motion problems と呼んで D = rt の式を立てる（3.5・5.4・8.8。Intermediate Algebra は 2.4 の本文で a uniform motion application、4.2・7.5・9.5 で uniform motion problems と書く）。OpenStax の uniform motion は動きそのもの（a uniform motion situation）の言い方で、問題の型は uniform motion problem と言う。",
      "uniform motion の uniform は「一定の」という意味で、速さが変わらない（等速の）動きを指す。",
      "D = rt の r は rate で、速さのこと（エントリ speed）。OpenStax Prealgebra・Elementary Algebra はこの式を the distance, rate, and time formula と呼ぶ。rate は rate of change（変化の割合）のように「割合・率」の意味でも使うので、文脈で読み分ける。",
      "mph は miles per hour の略（時速 … マイル）。",
      "AP の CED（topic 4.2・9.6）や OpenStax Algebra and Trigonometry の motion problems は、位置・速度・加速度の問題（直線運動）や媒介変数表示・放物運動の問題で（エントリ rectilinear-motion・projectile-motion）、一定の速さで D = rt を使うこのエントリの文章題とは別の型。"
    ],
    "related": [
      "speed",
      "word-problem",
      "write-an-equation",
      "mixture-problem",
      "linear-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "3.5 Solve Uniform Motion Applications・5.4 Solve Applications with Systems of Equations・8.8 Solve Uniform Motion and Work Applications（本文は uniform motion problems）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "2.4 Solve Mixture and Uniform Motion Applications（本文は a uniform motion application）・4.2・7.5・9.5（uniform motion problems）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "9.7 Solve a Formula for a Specific Variable（Use the Distance, Rate, and Time Formula。pitfalls[2]・level.us）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 4.2（rectilinear motion problems）・Topic 9.6（motion problems）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "10.7 Parametric Equations: Graphs（projectile motion problems。pitfalls[4]）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A 数と式: (道のり) = (速さ) × (時間) の言葉の式（level.jp 中1）・第 2 学年 A(2) 連立二元一次方程式を具体的な場面で活用する（速さ・道のりの語は無い。level.jp 中2）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "uniform motion problem": 10
      },
      "sources": [
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-10-09"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "move-term-to-other-side",
    "ja": {
      "term": "移項する",
      "reading": "いこうする",
      "alt": [
        "移項"
      ]
    },
    "en": {
      "term": "move a term to the other side",
      "alt": [],
      "variants": [
        {
          "term": "from both sides",
          "register": "both",
          "note": "subtract 3 from both sides の形で。話し言葉で最も多い。Khan Academy（中学の講義と高校向け）と YouTube の解説が中心で、大学の講義（MIT OCW）には少ない。書き言葉では to both sides と同じくらい。"
        },
        {
          "term": "to both sides",
          "register": "both",
          "note": "add 3 to both sides の形で。話し言葉では from both sides と同じくらい多く、Khan Academy と YouTube の解説が中心で、MIT OCW には少ない。書き言葉でも from both sides と同じくらい。"
        },
        {
          "term": "to the other side",
          "register": "both",
          "note": "move the 3x over to the other side ／ bring it over to the other side の形で。話し言葉では両辺の言い方より少ない。大学の講義に寄る（MIT OCW が最も多く、Khan Academy では少ない）。書き言葉には少ない。"
        },
        {
          "term": "isolate x",
          "register": "written",
          "note": "x だけを片側に残す（isolate the variable）言い方。書き言葉のほうが多く（大半が OpenStax Prealgebra）、話し言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "「移項」に当たる一語の名詞は用例コーパスと参照に出てこない。移項の意味の transposition は用例コーパスにも OpenStax にも出てこない（transposition は MIT OCW 18.01 の講義に別の意味でわずかに出てくるだけ）。同じ操作を「両辺から引く／両辺に足す」（subtract 3 from both sides ／ add 3 to both sides）と言うのが話し言葉でも書き言葉でも最も多く、教室では「反対側へ動かす」（move it over to the other side）も使われる。",
    "domains": [
      "algebra",
      "classroom"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "等式や不等式の項を、符号を変えて反対側へ移すこと。",
    "definition_en": "To move a term to the other side of an equation or inequality, changing its sign.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Move the 3x over to the other side so the x's are together.",
        "ja": "3x を移項して x をまとめて。",
        "register": "spoken"
      },
      {
        "en": "Subtract 5 from both sides: 2x = 8.",
        "ja": "5 を移項して 2x = 8",
        "register": "written"
      },
      {
        "en": "When you move a term to the other side, flip its sign: plus 5 becomes minus 5.",
        "ja": "移項するときは符号を変える。+5 は −5 になる。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "get all the x's on one side",
        "ja": "x を片側に集める"
      }
    ],
    "pitfalls": [
      "中学・高校レベルの説明では「両辺から引く／両辺に足す」（from both sides ／ to both sides）が多い（Khan Academy と YouTube の解説が中心）。大学の講義（MIT OCW）では move … over to the other side のほうが from both sides よりずっと多い。答案に書くなら両辺の言い方。",
      "移項を名詞 transposition で言う例は用例コーパスにない。動詞句で言う。"
    ],
    "related": [
      "substitute"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "from both sides": 392,
        "to the other side": 60,
        "to both sides": 323,
        "isolate x": 3
      },
      "written": {
        "to both sides": 92,
        "from both sides": 92,
        "to the other side": 10,
        "isolate x": 14
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "moving-average",
    "ja": {
      "term": "移動平均",
      "reading": "いどうへいきん",
      "alt": []
    },
    "en": {
      "term": "moving average",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（moving average の言い方は CED・OpenStax・IM・CK-12・Nicholson・Levin で 0 件）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": []
    },
    "definition_ja": "時系列データで、一定の期間の値の平均を、期間を 1 つずつずらしながら求めたもの。短期の変動をならして傾向を見るのに使う。",
    "definition_en": "In a time series, the mean of the values over a fixed period, recomputed as the period slides forward one step at a time; it smooths out short-term changes to show the trend.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A seven-day moving average smooths out the ups and downs from day to day.",
        "ja": "7 日間の移動平均で、日ごとの上下がならされる。",
        "register": "spoken"
      },
      {
        "en": "Compute the 3-month moving averages for the data and describe the trend.",
        "ja": "データの 3 か月移動平均を求め、傾向を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉にわずかに出てくるだけで、OpenStax Introductory Statistics には出てこない。"
    ],
    "related": [
      "time-series",
      "trend",
      "arithmetic-mean"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "移動平均",
        "en": "Moving average"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B 数学と社会生活: 移動平均の定義と平均気温のデータの例"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "moving average": 3
      },
      "sources": [
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元 2 の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "multinomial-theorem",
    "ja": {
      "term": "多項定理",
      "reading": "たこうていり",
      "alt": []
    },
    "en": {
      "term": "multinomial theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "(a + b + c)ⁿ のような多項式の累乗を展開したときの各項の係数を与える定理。",
    "definition_en": "The theorem giving the coefficient of each term when a power of a sum of several terms, such as (a + b + c)ⁿ, is expanded.",
    "latex": "\\frac{n!}{p!\\,q!\\,r!}\\,a^p b^q c^r \\quad (p + q + r = n)",
    "spoken_en": "n factorial over p factorial q factorial r factorial, times a to the p b to the q c to the r",
    "examples": [
      {
        "en": "The multinomial theorem is like the binomial theorem, but now you split n into three parts instead of two.",
        "ja": "多項定理は二項定理と同じ考えで、n を 2 つではなく 3 つに分ける。",
        "register": "spoken"
      },
      {
        "en": "Find the coefficient of a²bc³ in the expansion of (a + b + c)^6.",
        "ja": "(a + b + c)⁶ の展開式における a²bc³ の係数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは書き言葉（MIT の講義ノート）にわずかに出てくるだけ。multinomial は CED・OpenStax・IM・CK-12 に出てこない。a^p b^q c^r の係数 n!/(p! q! r!) は、OpenStax Algebra and Trigonometry（Counting Principles）が同じものを含む順列の数（Permutations of n Non-Distinct Objects）として扱う式と同じ。"
    ],
    "related": [
      "expansion",
      "general-term",
      "binomial-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "多項定理",
        "en": "Multinomial theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Counting Principles（Permutations of n Non-Distinct Objects）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "multinomial theorem": 2
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "multiple",
    "ja": {
      "term": "倍数",
      "reading": "ばいすう",
      "alt": []
    },
    "en": {
      "term": "multiple",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "number-theory"
    ],
    "level": {
      "jp": [
        "中2",
        "数A"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Discrete Math"
      ]
    },
    "definition_ja": "ある整数に整数を掛けてできる数。整数 k を使って b = ak と書けるとき、b は a の倍数である。",
    "definition_en": "A number you get by multiplying a given integer by an integer; b is a multiple of a when b = ak for some integer k.",
    "latex": "b = ak \\quad (k \\in \\mathbb{Z})",
    "spoken_en": "b equals a times k for some integer k",
    "examples": [
      {
        "en": "Twelve is a multiple of four, because four times three is twelve.",
        "ja": "12 は 4 の倍数だ。4 × 3 = 12 だから。",
        "register": "spoken"
      },
      {
        "en": "Prove that n³ − n is a multiple of 6 for every integer n.",
        "ja": "すべての整数 n について、n³ − n は 6 の倍数であることを証明せよ。",
        "register": "written"
      },
      {
        "en": "Any multiple of pi works here, so write x equals pi over 4 plus n pi.",
        "ja": "π の整数倍なら何を足してもいいので、x = π/4 + nπ と書く。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "multiple of 3",
        "ja": "3 の倍数"
      },
      {
        "en": "is a multiple of 6",
        "ja": "6 の倍数である"
      }
    ],
    "pitfalls": [
      "「3 の倍数」は a multiple of 3（冠詞 a を付ける）。「〜の倍数である」は is a multiple of 〜。公倍数は common multiple、最小公倍数は least common multiple (LCM)。"
    ],
    "related": [
      "common-factor",
      "prime-factorization",
      "division-algorithm",
      "remainder"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "倍数",
        "en": "Multiple (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "multiple of": 354
      },
      "written": {
        "multiple of": 273
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "multiplication-principle",
    "ja": {
      "term": "積の法則",
      "reading": "せきのほうそく",
      "alt": []
    },
    "en": {
      "term": "multiplication principle",
      "alt": [
        "fundamental counting principle",
        "rule of product",
        "product principle"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "事柄 A の起こり方が m 通りあり、そのそれぞれに対して事柄 B の起こり方が n 通りずつあるとき、A と B がともに起こる場合の数は mn 通りであるという法則。",
    "definition_en": "If a first choice can be made in m ways and, for each of those, a second choice can be made in n ways, then the two choices together can be made in m · n ways.",
    "latex": "|A \\times B| = |A| \\cdot |B|",
    "spoken_en": "the size of A cross B equals the size of A times the size of B",
    "examples": [
      {
        "en": "Three shirts and four pairs of pants, so by the multiplication principle that's 3 times 4, or 12 outfits.",
        "ja": "シャツが 3 枚、ズボンが 4 本だから、積の法則で 3 × 4 = 12 通りの組み合わせになる。",
        "register": "spoken"
      },
      {
        "en": "A license plate has 2 letters followed by 3 digits. Use the multiplication principle to find the number of possible license plates.",
        "ja": "ナンバープレートは、アルファベット 2 文字の後に数字 3 けたを並べたものである。積の法則を用いて、考えられるナンバープレートの数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は multiplication principle を 31 件、fundamental counting principle を 4 件使う。用例コーパスの話し言葉にはどちらも出てこない。",
      "確率の乗法定理（multiplication rule）とは別の語。積の法則は場合の数を数える法則。",
      "使えるのは、2 つ目の選び方の数が 1 つ目の結果によらず同じ n 通りのとき。選べるものの中身は変わってもよい（順列の数え方）。",
      "用例コーパスの MIT OCW 6.042 は積の法則を product rule と呼ぶ（選び方の数が同じなら中身が変わってよい形は generalized product rule）。微分の product rule（product-rule）と同じ名前なので、文脈で区別する。"
    ],
    "related": [
      "addition-principle",
      "counting",
      "permutation",
      "general-multiplication-rule",
      "tree-diagram"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.2 Combining Outcomes（Product Principle）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Rule_of_product",
        "note": "記事「Rule of product」の冒頭（rule of product・multiplication principle。数学カテゴリから 3 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "multiplication principle": 31,
        "fundamental counting principle": 4
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "multiplication",
    "ja": {
      "term": "乗法",
      "reading": "じょうほう",
      "alt": []
    },
    "en": {
      "term": "multiplication",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "数を掛け合わせる計算（かけ算）。結果を積という。",
    "definition_en": "The operation of multiplying numbers together; the answer is called the product.",
    "latex": "a \\times b = ab",
    "spoken_en": "a times b equals a b",
    "examples": [
      {
        "en": "Do the multiplication first: 2 plus 3 times 4 is 2 plus 12, so it's 14, not 20.",
        "ja": "かけ算を先にやろう。2 + 3 × 4 は 2 + 12 で 14。20 じゃないよ。",
        "register": "spoken"
      },
      {
        "en": "Decide the sign of the product before doing the multiplication, and then find the value of (−4) × (−6) × (−1).",
        "ja": "(−4) × (−6) × (−1) について、乗法を計算する前に積の符号を決め、その値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "× は times と読む（3 times 4）。演算の名前は multiplication、動詞は multiply（エントリ multiply）、答えの「積」は product。",
      "乗法の記号は × のほかに中黒 ·（3 · 4）も使い、文字式では記号を省いて 2x、ab と書く。"
    ],
    "related": [
      "multiply",
      "division",
      "the-four-operations",
      "commutative-property",
      "distributive-property"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "乗法",
        "en": "Multiplication"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "multiplication": 571
      },
      "written": {
        "multiplication": 561
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "multiplicity",
    "ja": {
      "term": "重複度",
      "reading": "ちょうふくど",
      "alt": [
        "根の重複度",
        "多重根"
      ]
    },
    "en": {
      "term": "multiplicity",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "多項式を因数分解したとき、因数 (x − a) が現れる回数。これを零点 x = a の重複度という。",
    "definition_en": "The number of times a factor (x − a) appears when a polynomial is completely factored. That number is the multiplicity of the zero x = a.",
    "latex": "f(x) = (x - 2)^2 (x + 1)",
    "spoken_en": "f of x equals the quantity x minus two squared, times the quantity x plus one",
    "examples": [
      {
        "en": "The factor x minus 2 is squared, so the zero at 2 has multiplicity 2, and the graph just touches the axis and bounces back.",
        "ja": "x − 2 が 2 乗されているから、x = 2 の零点は重複度 2。グラフは x 軸に接して跳ね返るだけだよ。",
        "register": "spoken"
      },
      {
        "en": "Find the zeros of f(x) = x(x + 3)³ and state the multiplicity of each.",
        "ja": "f(x) = x(x + 3)³ の零点を求め、それぞれの重複度を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校では重解（エントリ double-root）までで、重複度という語は大学で使う。重解は重複度 2 の解に当たる。",
      "OpenStax Algebra and Trigonometry は、x 切片でのグラフのようすを零点の multiplicity で読む。奇数なら x 軸を横切り、偶数なら x 軸に接して跳ね返る。",
      "線形代数の固有値の重複度（エントリ algebraic-multiplicity）とは別の場面の語。"
    ],
    "related": [
      "double-root",
      "zeros-of-a-polynomial",
      "x-intercept",
      "polynomial-function",
      "algebraic-multiplicity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "!algebraic !geometric multiplicity": 7
      },
      "written": {
        "!algebraic !geometric multiplicity": 119
      },
      "sources": [
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "multiply",
    "ja": {
      "term": "掛ける",
      "reading": "かける",
      "alt": [
        "かける"
      ]
    },
    "en": {
      "term": "multiply",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic",
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数や式どうしの積を求める計算をする。たとえば 3 に 4 を掛けると 12 になる。",
    "definition_en": "To find the product of two or more numbers or expressions. For example, multiplying 3 by 4 gives 12.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When you multiply two negatives, like negative 3 times negative 4, you get positive 12.",
        "ja": "負の数どうしを掛けると、たとえば (−3) × (−4) は正の 12 になる。",
        "register": "spoken"
      },
      {
        "en": "Multiply both sides of the equation by 6 to clear the fractions.",
        "ja": "方程式の両辺に 6 を掛けて、分母をはらう。",
        "register": "written"
      },
      {
        "en": "Multiply 5 by 4 first, and then add the 2.",
        "ja": "先に 5 に 4 を掛けて、それから 2 を足そう。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "multiply A by B は A × B（A に B を掛ける）。「両辺に 3 を掛ける」は multiply both sides by 3 で、前置詞は by。",
      "times は話し言葉の「かける」。3 times 4 は 3 × 4。",
      "名詞の「掛け算・乗法」は multiplication（エントリ multiplication）。掛けた結果の積は product。"
    ],
    "related": [
      "multiplication",
      "add",
      "subtract",
      "divide",
      "distributive-property"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "multiply": 5554
      },
      "written": {
        "multiply": 2948
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "multivariable-chain-rule",
    "ja": {
      "term": "多変数の連鎖律",
      "reading": "たへんすうのれんさりつ",
      "alt": []
    },
    "en": {
      "term": "generalized chain rule",
      "alt": [
        "multivariable chain rule"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "多変数関数の合成関数の微分法。z = f(x, y)、x = x(t)、y = y(t) なら dz/dt = f_x dx/dt + f_y dy/dt。",
    "definition_en": "The chain rule for composite functions of several variables: if z = f(x, y) with x = x(t) and y = y(t), then dz/dt = f_x dx/dt + f_y dy/dt.",
    "latex": "\\frac{dz}{dt} = \\frac{\\partial z}{\\partial x}\\frac{dx}{dt} + \\frac{\\partial z}{\\partial y}\\frac{dy}{dt}",
    "spoken_en": "d z d t equals partial z partial x times d x d t plus partial z partial y times d y d t",
    "examples": [
      {
        "en": "Draw a tree diagram so you don't miss any path in the chain rule.",
        "ja": "樹形図をかいて、連鎖律の経路を落とさないようにしよう。",
        "register": "spoken"
      },
      {
        "en": "Use the chain rule to find dz/dt for z = x²y, x = cos t, y = sin t.",
        "ja": "連鎖律を用いて、z = x²y、x = cos t、y = sin t のときの dz/dt を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "chain-rule",
      "partial-derivative",
      "total-differential"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "generalized chain rule": 7,
        "multivariable chain rule": 1
      },
      "sources": [
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは OpenStax の呼び方 generalized chain rule（本文 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "mutually-exclusive-events",
    "ja": {
      "term": "互いに排反",
      "reading": "たがいにはいはん",
      "alt": [
        "排反事象",
        "排反"
      ]
    },
    "en": {
      "term": "mutually exclusive",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics",
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2",
        "Intro Statistics",
        "AP Statistics"
      ]
    },
    "definition_ja": "2 つの事象が同時には起こらないこと。A ∩ B = ∅ のとき A と B は互いに排反で、P(A ∪ B) = P(A) + P(B) が成り立つ。",
    "definition_en": "Describes events that cannot happen at the same time, so A ∩ B = ∅; for such events, P(A ∪ B) = P(A) + P(B).",
    "latex": "A \\cap B = \\emptyset \\implies P(A \\cup B) = P(A) + P(B)",
    "spoken_en": "if A intersect B is the empty set, then the probability of A or B is the probability of A plus the probability of B",
    "examples": [
      {
        "en": "Rolling a 2 and rolling a 5 on one die are mutually exclusive, so you can just add the probabilities.",
        "ja": "1 個のさいころで 2 が出ることと 5 が出ることは互いに排反だから、確率をそのまま足せばいい。",
        "register": "spoken"
      },
      {
        "en": "Events A and B are mutually exclusive, with P(A) = 0.3 and P(B) = 0.45. Find P(A ∪ B).",
        "ja": "事象 A, B は互いに排反で、P(A) = 0.3、P(B) = 0.45 である。P(A ∪ B) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "排反と独立（independent）を取り違えない。確率が 0 でない 2 つの事象が排反なら、独立ではない。",
      "AP Statistics の CED（topic 2.5）は mutually exclusive, or disjoint と書き、OpenStax Algebra and Trigonometry（13.5）も disjoint events を mutually exclusive と言い換える。集合の言葉の disjoint（共通部分が空）と同じ意味。",
      "用例コーパスの話し言葉はほとんど Professor Leonard。「排反事象」という名詞は mutually exclusive events と言う。"
    ],
    "related": [
      "independence-of-events",
      "general-addition-rule",
      "union-of-events",
      "intersection-of-events",
      "empty-set",
      "complementary-event",
      "event"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles（disjoint events）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.5（mutually exclusive, or disjoint）"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "排反",
        "en": "Mutual exclusivity"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "mutually exclusive": 22
      },
      "written": {
        "mutually exclusive": 99
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "natural-logarithm",
    "ja": {
      "term": "自然対数",
      "reading": "しぜんたいすう",
      "alt": []
    },
    "en": {
      "term": "natural logarithm",
      "alt": [],
      "variants": [
        {
          "term": "natural log",
          "register": "spoken",
          "note": "話し言葉の短い言い方。話し言葉では natural logarithm よりずっと多く、半分余りが Khan Academy。書き言葉では natural logarithm のほうがずっと多い。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "e を底とする対数 log_e x。OpenStax では ln x と書く。",
    "definition_en": "The logarithm to base e, written ln x.",
    "latex": "\\ln x = \\log_e x",
    "spoken_en": "the natural log of x",
    "examples": [
      {
        "en": "Take the natural log of both sides to bring the exponent down.",
        "ja": "両辺の自然対数をとって、指数を前に下ろす。",
        "register": "spoken"
      },
      {
        "en": "Solve e^(2x) = 5 and write the answer using the natural logarithm.",
        "ja": "e^(2x) = 5 を解き、答えを自然対数で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ln x は「エル エヌ x」、または the natural log of x と読む。",
      "OpenStax Algebra and Trigonometry は底を省いた log x を常用対数（底 10）と定める（学習指導要領解説には ln が出てこない）ので、英語の答案では自然対数に ln を使う。"
    ],
    "related": [
      "logarithm",
      "common-logarithm",
      "base-of-the-natural-logarithm",
      "derivative-of-the-logarithm"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "自然対数",
        "en": "Natural logarithm"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "ln が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "natural log": 966,
        "natural logarithm": 54
      },
      "written": {
        "natural logarithm": 96,
        "natural log": 29
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "natural-number",
    "ja": {
      "term": "自然数",
      "reading": "しぜんすう",
      "alt": []
    },
    "en": {
      "term": "natural number",
      "alt": [],
      "variants": [
        {
          "term": "natural number",
          "register": "both",
          "note": "話し言葉で最も多い（Khan Academy と MIT OCW）。書き言葉では 3 つの中で最も少ない。"
        },
        {
          "term": "positive integer",
          "register": "both",
          "note": "書き言葉で最も多い（大半が OpenStax Calculus・Algebra and Trigonometry）。0 を含まないことがはっきりする言い方。"
        },
        {
          "term": "counting number",
          "register": "both",
          "note": "書き言葉は大半が中学・Algebra の教科書（OpenStax Prealgebra・Elementary Algebra）。話し言葉では少ない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の自然数は 1, 2, 3, … で、0 を含まない（日本語版 Wikipedia「自然数」は、日本では高校の教育課程で 0 を入れないとする）。英語の natural number は 0 を含める流儀もあり（大学の離散数学。Levin は ℕ = {0, 1, 2, …} と書く）、counting numbers は 1, 2, 3, …、whole numbers は 0, 1, 2, … を指す。0 を含まないことをはっきりさせるときは positive integer（正の整数）と言う。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "1, 2, 3, … のように、ものを数えるときに使う数。正の整数と同じで、0 を含まない。",
    "definition_en": "A counting number: 1, 2, 3, and so on. Some books, especially in college math, also count 0 as a natural number.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In this class, zero is not a natural number, so the list starts at 1.",
        "ja": "この授業では 0 は自然数に入れないから、1 から数えるよ。",
        "register": "spoken"
      },
      {
        "en": "Find all natural numbers n such that 3n < 14.",
        "ja": "3n < 14 を満たす自然数 n をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「自然数」を whole numbers と訳さない。whole numbers は 0 を含む（0, 1, 2, …）。",
      "問題文の natural number に 0 が入るかは本や授業による。positive integer と言えば 0 を含まないことがはっきりする。"
    ],
    "related": [
      "integer",
      "positive-number",
      "prime-number",
      "real-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "自然数",
        "en": "Natural number"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「自然数」",
        "url": "https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E6%95%B0",
        "note": "日本では高校教育課程においては 0 を入れない"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction (4th ed.)",
        "url": "https://discrete.openmathbooks.org/dmoi4.pdf",
        "note": "0.2: ℕ = {0, 1, 2, …}"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "positive integer": 50,
        "natural number": 89,
        "counting number": 15
      },
      "written": {
        "positive integer": 155,
        "natural number": 42,
        "counting number": 131
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "necessary-and-sufficient-condition",
    "ja": {
      "term": "必要十分条件",
      "reading": "ひつようじゅうぶんじょうけん",
      "alt": []
    },
    "en": {
      "term": "if and only if",
      "alt": [
        "necessary and sufficient condition"
      ],
      "uk": null,
      "variants": [
        {
          "term": "iff",
          "register": "written",
          "note": "if and only if の略記で、書き言葉（証明・講義ノート）で使う。読むときは if and only if と言う。話し言葉にはほとんど出てこず、書き言葉ではすべて MIT の講義ノート。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語では名詞の「必要十分条件」よりも、接続の言い方 p if and only if q（p ⇔ q）で述べる。「A であるための必要十分条件は B である」は A if and only if B。用例コーパスでは if and only if が話し言葉・書き言葉ともよく出てくるのに対し、名詞の necessary and sufficient condition は書き言葉（MIT の講義ノート）にわずかにあるだけ。",
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "p ⇒ q と q ⇒ p がともに真のとき、q を p であるための必要十分条件という。このとき p と q は同値である。",
    "definition_en": "q is a necessary and sufficient condition for p when p implies q and q implies p; in other words, p holds if and only if q holds.",
    "latex": "p \\iff q",
    "spoken_en": "p if and only if q",
    "examples": [
      {
        "en": "A number is divisible by 3 if and only if the sum of its digits is divisible by 3.",
        "ja": "整数が 3 で割り切れるための必要十分条件は、各位の数の和が 3 で割り切れることである。",
        "register": "written"
      },
      {
        "en": "\"If and only if\" means you have to prove both directions: this one implies that one, and that one implies this one.",
        "ja": "「必要十分」ということは両方向を示さないといけない。こっちからあっち、あっちからこっち。",
        "register": "spoken"
      },
      {
        "en": "Prove that n² is odd iff n is odd.",
        "ja": "n² が奇数であることは、n が奇数であるための必要十分条件であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "必要十分条件を示すには、⇒ と ⇐ の両方向を証明する。英語では「両方向」を both directions と言える。",
      "iff は書くときの略記。口で読むときは if and only if。"
    ],
    "related": [
      "necessary-condition",
      "sufficient-condition",
      "equivalent",
      "condition",
      "proposition"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "if and only if": 76,
        "iff": 2
      },
      "written": {
        "if and only if": 114,
        "iff": 84,
        "necessary and sufficient condition": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "necessary-condition",
    "ja": {
      "term": "必要条件",
      "reading": "ひつようじょうけん",
      "alt": []
    },
    "en": {
      "term": "necessary condition",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II",
        "Discrete Math"
      ]
    },
    "definition_ja": "「p ならば q」が真のとき、q を p であるための必要条件という。q が成り立たなければ p は成り立たない。",
    "definition_en": "If \"if p, then q\" is true, q is a necessary condition for p: p cannot hold unless q holds.",
    "latex": "p \\Rightarrow q",
    "spoken_en": "p implies q, so q is necessary for p",
    "examples": [
      {
        "en": "Terms going to zero is a necessary condition for convergence, but it's not enough. Look at the harmonic series.",
        "ja": "項が 0 に近づくことは収束のための必要条件だけど、それだけでは足りない。調和級数を見てごらん。",
        "register": "spoken"
      },
      {
        "en": "Show that x² = 4 is a necessary condition for x = 2 but not a sufficient condition.",
        "ja": "x² = 4 は x = 2 であるための必要条件であるが、十分条件ではないことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "向きに注意。p ⇒ q が真のとき「q は p であるための必要条件」「p は q であるための十分条件」。英語でも q is a necessary condition for p と for で p を指す。"
    ],
    "related": [
      "sufficient-condition",
      "necessary-and-sufficient-condition",
      "condition",
      "harmonic-series"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.2 Implications: necessary condition（6 件）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 10 の 1 件（all necessary conditions are met）は判定法の前提の意味で、用語の用法ではない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!all necessary condition": 3
      },
      "written": {
        "!all necessary condition": 6
      },
      "sources": [
        "khan-ap-stats",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 !all necessary condition（本文 5 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "negate",
    "ja": {
      "term": "否定をとる",
      "reading": "ひていをとる",
      "alt": [
        "否定をつくる"
      ]
    },
    "en": {
      "term": "negate",
      "alt": [],
      "uk": null
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "命題の否定をつくる。「すべての x について P」の否定は「ある x について P でない」になり、その逆も成り立つ。",
    "definition_en": "To form the negation of a statement; negating \"for all x, P(x)\" gives \"there exists an x such that P(x) is false,\" and vice versa.",
    "latex": "\\neg\\,(\\forall x,\\ P(x)) \\iff \\exists x,\\ \\neg P(x)",
    "spoken_en": "not for all x, P of x, is equivalent to there exists x such that not P of x",
    "examples": [
      {
        "en": "When you negate \"for all,\" it turns into \"there exists.\"",
        "ja": "「すべての」の否定をとると「ある」に変わる。",
        "register": "spoken"
      },
      {
        "en": "If you negate \"p and q,\" you get \"not p or not q.\"",
        "ja": "「p かつ q」の否定をとると「p でない または q でない」になる。",
        "register": "spoken"
      },
      {
        "en": "Negate the statement: For every real number x, x² ≥ 0.",
        "ja": "命題「すべての実数 x について x² ≧ 0」の否定をつくれ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "negate は「否定をとる（否定の命題をつくる）」で、「否定する（正しくないと主張する）」ではない。P を negate しても P が偽だと言ったことにはならない。",
      "「x > 0」の否定は「x ≦ 0」（英語でも x ≤ 0）。x < 0 にしない。"
    ],
    "related": [
      "negation",
      "de-morgans-laws",
      "all",
      "exist",
      "proposition"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.10 Negative Statements・2.12 Converse Inverse and Contrapositive Statements"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.3 Rules of Logic: negate a conjunction ／ negate a disjunction ／ negate both parts（3 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 negate the statement | negate a statement | negate statements | negate complex statements | negate an atomic sentence | negate the original statement | negate both parts | negate a conjunction | negate a disjunction | negate for all | negate the hypothesis | negate p and q（CK-12 Geometry 2.10 Negative Statements・計 8 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "negation",
    "ja": {
      "term": "否定",
      "reading": "ひてい",
      "alt": []
    },
    "en": {
      "term": "negation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "命題や条件 p に対して「p でない」という命題・条件。p が真なら偽、p が偽なら真になる。",
    "definition_en": "The statement \"not p\" formed from a statement p; it is true exactly when p is false.",
    "latex": "\\neg p",
    "spoken_en": "not p",
    "examples": [
      {
        "en": "The negation of \"x > 3\" is \"x ≤ 3,\" not \"x < 3.\" Don't forget the equal case.",
        "ja": "「x > 3」の否定は「x ≦ 3」で、「x < 3」ではない。等号の場合を忘れないように。",
        "register": "spoken"
      },
      {
        "en": "Write the negation of the statement \"For every real number x, x² ≥ 0.\"",
        "ja": "命題「すべての実数 x について x² ≧ 0」の否定を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "センター試験（平成 27 年度 本試験 数学I 第1問）は条件 p の否定を p̄ と上線で書き、記号 ¬ は、高等学校学習指導要領解説では理数に関する学科の科目「理数数学I」の発展・拡充の内容（真理値表を用いて、命題「p → q」の否定が「p ∧ (¬q)」であること）にだけ出てくる。日本語版 Wikipedia「否定」は ¬P・P̄・!P を並べ、Levin は ¬p、CK-12 Geometry（2.8）は ∼p と書く。「〜を否定する」の動詞は negate で、別エントリ。",
      "「すべての x について p」の否定は「ある x について p でない」、「p かつ q」の否定は「p でないまたは q でない」（ド・モルガンの法則）。"
    ],
    "related": [
      "negate",
      "de-morgans-laws",
      "complement",
      "proposition",
      "contrapositive"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "否定",
        "en": "Negation"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.3 Rules of Logic"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.10 Negative Statements・2.8 Truth Tables・2.12 Converse Inverse and Contrapositive Statements"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "センター試験 平成 27 年度 数学I 第1問: 条件 p の否定を p̄ と書く"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学I (1) 数と式（発展・拡充）: 真理値表を用いて命題「p→q」の否定が「p∧（¬q）」であること（数学I の節には無い）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "negation": 3
      },
      "sources": [
        "mit-6.042",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 negation（CK-12 Geometry 2.10 Negative Statements・CK-12 Geometry 2.8 Truth Tables・CK-12 Geometry 2.12 Converse Inverse and Contrapositive Statements・計 25 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "negative-correlation",
    "ja": {
      "term": "負の相関",
      "reading": "ふのそうかん",
      "alt": []
    },
    "en": {
      "term": "negative association",
      "alt": [
        "negative correlation"
      ],
      "variants": [
        {
          "term": "negative linear relationship",
          "register": "spoken",
          "note": "話し言葉では negative association・negative correlation より多いが、すべて Khan Academy（AP Statistics と中学の講義）。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Integrated Math 1",
        "Intro Statistics"
      ]
    },
    "definition_ja": "一方の変量が増えると、他方の変量は減る傾向があること。散布図の点が右下がりに並ぶ。",
    "definition_en": "A tendency for one variable to decrease as the other increases; the points of the scatter plot fall from left to right.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The more you practice, the fewer mistakes you make, so that's a negative association.",
        "ja": "練習するほどミスが減るので、これは負の相関。",
        "register": "spoken"
      },
      {
        "en": "Give an example of two variables that have a negative correlation.",
        "ja": "負の相関がある 2 つの変量の例を挙げよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP Statistics の CED（topic 5.1）と IM Grade 8 は、散布図の右下がりの傾向を negative association と呼ぶ。OpenStax Introductory Statistics・Algebra and Trigonometry と IM Algebra 1 は negative correlation と書く（CED の negative correlation は相関係数の値の話の 1 件）。"
    ],
    "related": [
      "positive-correlation",
      "correlation",
      "scatter-plot"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 5.1（negative association）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "Unit 6 glossary: negative association"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "negative linear relationship": 12,
        "negative correlation": 7
      },
      "written": {
        "negative correlation": 7,
        "negative association": 1,
        "negative linear relationship": 1
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-algtrig",
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "negative-exponent",
    "ja": {
      "term": "負の指数",
      "reading": "ふのしすう",
      "alt": []
    },
    "en": {
      "term": "negative exponent",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1",
        "Algebra 1"
      ]
    },
    "definition_ja": "a ≠ 0 のとき a⁻ⁿ = 1/aⁿ と定めた、負の整数の指数。",
    "definition_en": "An exponent that is a negative integer, defined by a⁻ⁿ = 1/aⁿ for a ≠ 0.",
    "latex": "a^{-n} = \\frac{1}{a^n}",
    "spoken_en": "a to the negative n equals one over a to the n",
    "examples": [
      {
        "en": "A negative exponent just means one over — it doesn't make the number negative.",
        "ja": "負の指数は「分の 1」という意味。数が負になるわけではない。",
        "register": "spoken"
      },
      {
        "en": "Write (2x⁻³y²)⁻² without negative exponents.",
        "ja": "(2x⁻³y²)⁻² を負の指数を使わずに表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "x⁻² を −x² とする誤りに注意（The negative exponent does not make it negative.）。用例コーパスの話し言葉では Professor Leonard が最も多く、次が The Organic Chemistry Tutor。",
      "OpenStax Algebra and Trigonometry（1.2）は、この定め方を negative rule of exponents と呼ぶ。"
    ],
    "related": [
      "zero-exponent",
      "rational-exponent",
      "laws-of-exponents"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "1.2 Exponents and Scientific Notation（negative rule of exponents）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 7.5〜7.8（negative exponents）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "10.5 Integer Exponents and Scientific Notation"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "negative exponent": 78
      },
      "written": {
        "negative exponent": 119
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "negative-number",
    "ja": {
      "term": "負の数",
      "reading": "ふのかず",
      "alt": []
    },
    "en": {
      "term": "negative number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "0 より小さい数。−3 のように数の前に − を付けて表し、数直線では 0 より左にある。",
    "definition_en": "A number less than zero. It is written with a − sign in front, like −3, and is to the left of zero on a number line.",
    "latex": "-3",
    "spoken_en": "negative three",
    "examples": [
      {
        "en": "If you owe your friend 10 dollars, you can show that with a negative number: negative 10.",
        "ja": "友だちに 10 ドル借りているなら、負の数で −10 と表せる。",
        "register": "spoken"
      },
      {
        "en": "The temperature was 4°F at noon and dropped 9 degrees by midnight. Write the midnight temperature as a negative number.",
        "ja": "正午の気温は 4°F で、真夜中までに 9 度下がった。真夜中の気温を負の数で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "−3 は negative three と読む。minus は引き算の読み（5 − 3 は five minus three）。英語版 Wikipedia「Negative number」は −3 の読みに minus three も挙げる。",
      "0 は負の数ではない。「負でない数」（0 以上の数）は nonnegative number で、0 を含む。"
    ],
    "related": [
      "positive-number",
      "positive-and-negative-numbers",
      "sign",
      "opposite",
      "number-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正の数と負の数",
        "en": "Negative number"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Negative_number",
        "note": "記事「Negative number」の冒頭: −3 は minus three とも negative three とも読む（地域は書いていない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "negative number": 446
      },
      "written": {
        "negative number": 200
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "negative-reciprocal",
    "ja": {
      "term": "符号を変えた逆数",
      "reading": "ふごうをかえたぎゃくすう",
      "alt": [
        "逆数の負"
      ]
    },
    "en": {
      "term": "negative reciprocal",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「符号を変えた逆数」は学習指導要領解説に無い、本プロジェクトの訳語。日本では垂直条件を「傾きの積が −1」（m₁m₂ = −1）と言い、この数に名前を付けない。OpenStax Algebra and Trigonometry は、垂直な直線の一方の傾きは他方の negative reciprocal だと説明する。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "0 でない数 a に対する −1/a。逆数をとって符号を変えた数で、2/3 に対しては −3/2。",
    "definition_en": "For a nonzero number a, the number −1/a: flip the number and change its sign. For example, the negative reciprocal of 2/3 is −3/2.",
    "latex": "-\\frac{1}{a}",
    "spoken_en": "negative one over a",
    "examples": [
      {
        "en": "Perpendicular slopes are negative reciprocals, so if one line has a slope of 2/3, the other one has a slope of negative 3/2.",
        "ja": "垂直な直線の傾きは互いに符号を変えた逆数だから、一方の傾きが 2/3 なら、もう一方の傾きは −3/2。",
        "register": "spoken"
      },
      {
        "en": "The slope of a line perpendicular to y = −4x + 1 is the negative reciprocal of −4. Find this slope, and write an equation of the perpendicular line through (8, 3).",
        "ja": "直線 y = −4x + 1 に垂直な直線の傾きは、−4 の符号を変えた逆数である。この傾きを求め、点 (8, 3) を通り y = −4x + 1 に垂直な直線の式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "negative は「負の」ではなく「符号を変えた」の意味。負の数の negative reciprocal は正になる（−4 なら 1/4）。",
      "opposite reciprocal とも言う。opposite は符号を変えた数のこと（エントリ opposite）。",
      "傾きが negative reciprocal になるのは、垂直な 2 直線のどちらも垂直・水平でないときだけ。水平な直線（傾き 0）には逆数がなく、垂直な直線の傾きは定義されない（エントリ perpendicular-lines）。"
    ],
    "related": [
      "reciprocal",
      "perpendicular-lines",
      "slope",
      "opposite",
      "parallel-lines"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "negative reciprocal": 13
      },
      "written": {
        "negative reciprocal": 27
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-elemalg",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "nested-radical",
    "ja": {
      "term": "二重根号",
      "reading": "にじゅうこんごう",
      "alt": []
    },
    "en": {
      "term": "nested radical",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（nested radical は CED・OpenStax・IM・CK-12 とも 0 件）。用例コーパスでも話・書とも出てこない。見出しは英語版 Wikipedia の記事名「Nested radical」（冒頭で太字で定義。外す操作は節 Denesting）。授業で説明するときは a square root inside a square root と中身を言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": []
    },
    "definition_ja": "根号の中にさらに根号を含む式。√((a + b) + 2√(ab)) の形なら √a + √b に外せる（a, b > 0）。",
    "definition_en": "A radical expression that contains another radical inside it, such as √(5 + 2√6). When it has the form √((a + b) + 2√(ab)) with a, b > 0, it simplifies to √a + √b.",
    "latex": "\\sqrt{(a + b) + 2\\sqrt{ab}} = \\sqrt{a} + \\sqrt{b} \\quad (a, b > 0)",
    "spoken_en": "the square root of a plus b plus two root a b equals root a plus root b",
    "examples": [
      {
        "en": "This is a nested radical, a square root inside a square root, so let's try to write it as √a + √b.",
        "ja": "これは二重根号、つまり根号の中に根号があるので、√a + √b の形に直せないか考えよう。",
        "register": "spoken"
      },
      {
        "en": "Simplify the nested radical √(5 + 2√6).",
        "ja": "二重根号 √(5 + 2√6) を外せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "引き算の形は大きいほうから引く: a > b > 0 のとき √((a + b) − 2√(ab)) = √a − √b。√b − √a とすると負になって誤り。",
      "中の根号の前が 2 でないときは、先に 2 をくくり出す形に直す（√(7 − 4√3) = √(7 − 2√12) = 2 − √3）。"
    ],
    "related": [
      "real-number",
      "expansion",
      "symmetric-expression",
      "radical-expression",
      "simplify-radicals"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Nested_radical",
        "note": "記事名 Nested radical（en.term と同じ。冒頭で太字で定義。数学カテゴリから 3 段）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学I 数と式: 二重根号をはずす計算まで扱うことも考えられる（数学I の節には無い。2 件）"
      }
    ],
    "confidence": "verified",
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 nested radical（英語版「Nested radical」、en.term のリダイレクト先）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "net-change",
    "ja": {
      "term": "純変化量",
      "reading": "じゅんへんかりょう",
      "alt": []
    },
    "en": {
      "term": "net change",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学III）は、速度 v を時刻 t₁ から t₂ まで積分して点の「位置の変化」を、|v| を積分して「道のり」を求める例を挙げるが、定積分で求まる変化の合計一般の名前は出てこない。OpenStax Calculus Volume 1 と AP の CED（topic 8.3）は、増えた分と減った分を差し引いた変化を net change と呼び、OpenStax Calculus Volume 1 は ∫ₐᵇ F′(x)dx = F(b) − F(a) を Net Change Theorem として示す。AP の CED は Unit 6 を accumulation of change（変化の積み重ね）として組み立て、net change を topic 8.3 で扱う。見出しの「純変化量」と例文の和訳の「純変化定理」は、学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "ある量の、区間の始めから終わりまでの変化の合計。増えた分と減った分を差し引いたもので、変化率の定積分で求まる（∫ₐᵇ F′(x)dx = F(b) − F(a)。OpenStax Calculus はこの形を Net Change Theorem と呼ぶ）。",
    "definition_en": "The total change in a quantity over an interval, gains minus losses. It is the integral of the rate of change: ∫ₐᵇ F′(x) dx = F(b) − F(a) (the net change theorem).",
    "latex": "\\int_a^b F'(x)\\,dx = F(b) - F(a)",
    "spoken_en": "the integral from a to b of F prime of x dx equals F of b minus F of a",
    "examples": [
      {
        "en": "The net change in the amount of water is the integral of the rate it flows in.",
        "ja": "水の量の純変化量は、流れ込む割合の積分。",
        "register": "spoken"
      },
      {
        "en": "Think of the integral as accumulation: water flows in at this rate, and the integral tells you how much has built up.",
        "ja": "積分は累積だと考えて。この割合で水が流れ込むとき、積分はたまった量を表す。",
        "register": "spoken"
      },
      {
        "en": "By the Net Change Theorem, the change in position from t = 0 to t = 3 is ∫₀³ v(t) dt.",
        "ja": "純変化定理より、t = 0 から t = 3 までの位置の変化は ∫₀³ v(t) dt である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "net は「差し引きの」の意味。net change が 0 でも、途中で動かなかったとは限らない。",
      "積分を変化の積み重ねとして見るときは accumulation と言い（AP の CED の Unit 6 の題は Integration and Accumulation of Change）、動詞は accumulate（たまる・積み重なる）。"
    ],
    "related": [
      "displacement",
      "total-distance-traveled",
      "fundamental-theorem-of-calculus-part-2",
      "accumulation-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "5.4 Integration Formulas and the Net Change Theorem（mapping_note・definition_ja）"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 6（Integration and Accumulation of Change）、Topic 8.3（net change）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 積分法: 速度の定積分で「位置の変化」と「道のり」を求める例（定積分で求まる変化の合計の名前と「純変化量」は出てこない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "net change": 9
      },
      "written": {
        "net change": 23
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "openstax-calculus"
      ],
      "counted": "2026-10-09"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "net",
    "ja": {
      "term": "展開図",
      "reading": "てんかいず",
      "alt": []
    },
    "en": {
      "term": "net",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "立体の表面を切り開いて、平面の上に広げた図。組み立てるともとの立体になる。",
    "definition_en": "A flat figure you get by cutting open the surface of a solid and laying it out in a plane. Folding it back up gives the original solid.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Here's the net of a cube: six squares, and when you fold it up, you get a box.",
        "ja": "これが立方体の展開図。正方形が 6 つあって、折りたたむと箱になる。",
        "register": "spoken"
      },
      {
        "en": "The net of a square pyramid is made up of a square with a side length of 4 cm and four triangles, each with a height of 6 cm. Find the area of the net.",
        "ja": "正四角錐の展開図は、1 辺 4 cm の正方形 1 つと、高さ 6 cm の三角形 4 つでできている。この展開図の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "net は日常語の「網」でもあり、「正味の・差し引きの」の意味（net change、エントリ net-change）でも使う。展開図とは別の意味。",
      "1 つの立体の展開図は 1 通りとは限らない。立方体の展開図は 11 通りある。",
      "球の表面は平面に広げられないので、球には展開図がない。"
    ],
    "related": [
      "surface-area",
      "lateral-area",
      "polyhedron",
      "prism",
      "net-change"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "展開図",
        "en": "Net (polyhedron)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 6)",
        "url": "https://im.kendallhunt.com/MS/students/1/index.html",
        "note": "Grade 6 glossary・1.14 Nets and Surface Area・1.15 More Nets, More Surface Area"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "net of a | net of the | a net for | the net for": 1
      },
      "sources": [
        "khan-middle"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 net of a | net of the | a net for | the net for（Grade 6 glossary・Grade 6 1.18 Surface Area of a Cube・Grade 6 1.13 Polyhedra・計 21 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "newtons-law-of-cooling",
    "ja": {
      "term": "ニュートンの冷却法則",
      "reading": "にゅーとんのれいきゃくほうそく",
      "alt": [
        "ニュートンの冷却の法則"
      ]
    },
    "en": {
      "term": "Newton's law of cooling",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "物体の温度の変化率は、物体と周囲の温度差に比例するという法則。T(t) = Ae^(kt) + Tₛ で表す。",
    "definition_en": "The law that an object's rate of temperature change is proportional to the difference between its temperature and the surroundings', giving T(t) = Ae^(kt) + Tₛ.",
    "latex": "T(t) = A e^{kt} + T_s",
    "spoken_en": "T of t equals A e to the k t plus T sub s",
    "examples": [
      {
        "en": "By Newton's law of cooling, the coffee cools fast at first and then slowly.",
        "ja": "ニュートンの冷却法則から、コーヒーは最初は速く、やがてゆっくり冷める。",
        "register": "spoken"
      },
      {
        "en": "A cake at 165°F is placed in a 70°F room and cools to 120°F in 10 minutes. Use Newton's law of cooling to find its temperature after 20 minutes.",
        "ja": "165°F のケーキを 70°F の部屋に置くと、10 分で 120°F になった。ニュートンの冷却法則を用いて、20 分後の温度を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校の学習指導要領（数学）には含まれない。米国では Precalculus（OpenStax Algebra and Trigonometry）の指数関数の応用に出てくる。"
    ],
    "related": [
      "exponential-decay",
      "exponential-model",
      "separable-differential-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "Newton's law of cooling": 1
      },
      "written": {
        "Newton's law of cooling": 28
      },
      "sources": [
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "newtons-method",
    "ja": {
      "term": "ニュートン法",
      "reading": "にゅーとんほう",
      "alt": []
    },
    "en": {
      "term": "Newton's method",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "f(x) = 0 の解を、接線と x 軸の交点を繰り返し求めて近似する方法。",
    "definition_en": "A method for approximating a solution of f(x) = 0 by repeatedly following the tangent line down to the x-axis.",
    "latex": "x_{n+1} = x_n - \\frac{f(x_n)}{f'(x_n)}",
    "spoken_en": "x sub n plus one equals x sub n minus f of x sub n over f prime of x sub n",
    "examples": [
      {
        "en": "Newton's method: start with a guess and slide down the tangent line to the axis.",
        "ja": "ニュートン法は、最初の予想から接線をたどって x 軸に下りることを繰り返す。",
        "register": "spoken"
      },
      {
        "en": "Use Newton's method with x₀ = 1 to approximate √2 to four decimal places.",
        "ja": "x₀ = 1 としてニュートン法を用い、√2 を小数第 4 位まで求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "linear-approximation",
      "slope-of-the-tangent-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ニュートン法",
        "en": "Newton's method"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Newton's method": 18
      },
      "written": {
        "Newton's method": 66
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "no-solution",
    "ja": {
      "term": "解なし",
      "reading": "かいなし",
      "alt": [
        "矛盾する連立方程式"
      ]
    },
    "en": {
      "term": "no solution",
      "alt": [
        "inconsistent system"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "連立方程式を満たす x, y の値の組が 1 つもないこと。2 つの式のグラフが平行で交わらないときで、解いていくと 0 = 5 のような成り立たない式になる。",
    "definition_en": "A system has no solution when no pair of values makes both equations true. The two lines are parallel and never meet, and solving leads to a statement that is never true, such as 0 = 5.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The x's and y's are all gone and you're left with 0 = 5, which is never true, so there's no solution.",
        "ja": "x も y も消えて 0 = 5 が残る。これは絶対に成り立たないから、解なしだ。",
        "register": "spoken"
      },
      {
        "en": "Same slope, different y-intercepts, so the lines are parallel and it's an inconsistent system.",
        "ja": "傾きが同じで切片が違うから 2 直線は平行で、この連立方程式は解なしだ。",
        "register": "spoken"
      },
      {
        "en": "Show that the system y = 2x + 1, y = 2x − 4 has no solution.",
        "ja": "連立方程式 y = 2x + 1、y = 2x − 4 は解をもたないことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "教科書（OpenStax）は、解のない連立方程式そのものを inconsistent system と呼ぶ。「解なし」という答えは no solution と書く。",
      "解が無数にある場合（エントリ infinitely-many-solutions）と区別する。式を整理して x も y も消えたとき、残る式が 0 = 5 のように成り立たなければ解なし、0 = 0 のようにいつでも成り立てば解が無数にある。"
    ],
    "related": [
      "infinitely-many-solutions",
      "parallel-lines",
      "system-of-linear-equations",
      "solution",
      "point-of-intersection"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "no solution": 100,
        "inconsistent system": 9
      },
      "written": {
        "no solution": 257,
        "inconsistent system": 27
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "non-terminating-decimal",
    "ja": {
      "term": "無限小数",
      "reading": "むげんしょうすう",
      "alt": []
    },
    "en": {
      "term": "non-terminating decimal",
      "alt": [
        "infinite decimal expansion"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "小数点以下の数字が限りなく続く小数。同じ数字の並びがくり返す循環小数（有理数）と、くり返さない小数（√2 = 1.41421…、π などの無理数）がある。",
    "definition_en": "A decimal whose digits after the decimal point go on forever. The digits may repeat, as in 0.333…, or never repeat, as in π = 3.14159….",
    "latex": "\\sqrt{2} = 1.41421\\ldots",
    "spoken_en": "the square root of two equals one point four one four two one, and so on",
    "examples": [
      {
        "en": "Pi is a non-terminating decimal, and its digits never repeat, so it's irrational.",
        "ja": "π は無限小数で、しかも数字の並びがくり返さないから、無理数だ。",
        "register": "spoken"
      },
      {
        "en": "Write 5/11 as a decimal. Is it a terminating decimal or a non-terminating decimal?",
        "ja": "5/11 を小数で表せ。また、それは有限小数か無限小数か答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "無限小数には、循環する小数（repeating decimal、エントリ repeating-decimal）と循環しない小数がある。循環する無限小数は有理数（エントリ rational-number）、循環しない無限小数は無理数（エントリ irrational-number）。",
      "CK-12 Algebra は non-terminating decimal（ハイフンあり）と書き、OpenStax Algebra and Trigonometry（1.1）は無理数の説明で nonterminating（ハイフンなし）と書く。IM Grade 8 は infinite decimal expansion（無限小数展開）と言う（8.15 Infinite Decimal Expansions）。",
      "3.14 や 1.414 は無限小数を途中で切った近似値（エントリ approximate-value）で、π や √2 そのものではない。"
    ],
    "related": [
      "terminating-decimal",
      "repeating-decimal",
      "irrational-number",
      "rational-number",
      "real-number"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "1.1 Real Numbers"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.3 Use Diameter Radius and Pi"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "Grade 8 8.15 Infinite Decimal Expansions（infinite decimal expansion）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 non-terminating decimal（CK-12 Algebra 1.1 Real Numbers・CK-12 Geometry 6.3 Use Diameter Radius and Pi・計 6 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "nonlinear-system",
    "ja": {
      "term": "連立二次方程式",
      "reading": "れんりつにじほうていしき",
      "alt": []
    },
    "en": {
      "term": "system of nonlinear equations",
      "alt": [
        "nonlinear system of equations"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本では数I・数II で「連立方程式（2 次を含む）」として扱い、放物線と直線の共有点を求める問題として学ぶ。英語の nonlinear は二次に限らず、円や分数関数の式を含む連立方程式も指す。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "少なくとも 1 つの式が一次式でない連立方程式。代入法や加減法で解き、それぞれのグラフの交点の座標が解になる。",
    "definition_en": "A system of equations in which at least one equation is not linear. It can be solved by substitution or elimination, and its solutions are the points where the graphs intersect.",
    "latex": "\\begin{cases} y = x^2 \\\\ y = x + 2 \\end{cases}",
    "spoken_en": "the system y equals x squared and y equals x plus 2",
    "examples": [
      {
        "en": "In a system of nonlinear equations, the graphs can cross more than once, so this parabola and this line give you two solutions.",
        "ja": "連立二次方程式では、グラフが 2 回以上交わることもある。だからこの放物線と直線からは解が 2 つ出てくるよ。",
        "register": "spoken"
      },
      {
        "en": "Solve the system of nonlinear equations y = x² and y = x + 2.",
        "ja": "連立方程式 y = x²、y = x + 2 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "y = x² と y = x + 2 は、代入法（エントリ substitution-method）で x² = x + 2 から x = 2、−1。解は (2, 4) と (−1, 1) で、放物線と直線の交点になる。",
      "英語の nonlinear は二次に限らない。円や分数関数の式を含む連立方程式も system of nonlinear equations。",
      "nonlinear system は大学の微分方程式の講義では、非線形の連立微分方程式の意味で使う。"
    ],
    "related": [
      "substitution-method",
      "system-of-linear-equations",
      "elimination",
      "point-of-intersection",
      "parabola"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "system of nonlinear equations": 62,
        "nonlinear system of equations": 5
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "nonresponse",
    "ja": {
      "term": "無回答",
      "reading": "むかいとう",
      "alt": []
    },
    "en": {
      "term": "nonresponse",
      "alt": [
        "nonresponse bias"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "標本に選ばれた人が回答しないこと。答えない人の意見が結果に入らず、偏りの原因になる。",
    "definition_en": "Failure of people chosen for a sample to respond; their views are missing from the results, which can cause bias.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Half the people never mailed the survey back, so nonresponse is a real problem here.",
        "ja": "半分の人が調査票を返送しなかったので、ここでは無回答が大きな問題。",
        "register": "spoken"
      },
      {
        "en": "Why is nonresponse a problem in surveys?",
        "ja": "調査で無回答が問題になるのはなぜか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は non-response とハイフンで書く。"
    ],
    "related": [
      "voluntary-response-bias",
      "sample-survey",
      "is-biased"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "nonresponse": 6
      },
      "written": {
        "nonresponse": 3
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 6 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは AP Statistics の CED の呼び方 nonresponse（topic 1.12）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "normal-approximation-to-the-binomial",
    "ja": {
      "term": "二項分布の正規近似",
      "reading": "にこうぶんぷのせいききんじ",
      "alt": []
    },
    "en": {
      "term": "normal approximation to the binomial",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "n が大きいとき、二項分布 B(n, p) を平均 np、分散 np(1 − p) の正規分布で近似すること。",
    "definition_en": "For large n, approximating the binomial distribution B(n, p) by a normal distribution with mean np and variance np(1 − p).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Since np and n times one minus p are both at least 10, the normal approximation is fine here.",
        "ja": "np も n(1 − p) も 10 以上なので、ここでは正規近似を使ってよい。",
        "register": "spoken"
      },
      {
        "en": "Use the normal approximation to the binomial to estimate the probability that at least 60 of 100 voters support the measure.",
        "ja": "二項分布の正規近似を用いて、有権者 100 人のうち 60 人以上が賛成する確率を推定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "目安は資料で違う: OpenStax Introductory Statistics（7.3）は np > 5 かつ nq > 5、AP Statistics の CED（topic 3.2。標本比率の分布）は np ≥ 10 かつ n(1 − p) ≥ 10。",
      "the normal approximation だけだと、ほかの分布の正規近似も指す（OpenStax Introductory Statistics 10.2 は normal approximation to the Student's t と書く）。"
    ],
    "related": [
      "binomial-distribution",
      "normal-distribution",
      "continuity-correction",
      "sampling-distribution-of-a-proportion"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "7.3 Using the Central Limit Theorem・9.3（normal approximation to the binomial、np > 5 and nq > 5）・10.2（normal approximation to the Student's t）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.2（np ≥ 10、n(1 − p) ≥ 10）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "normal approximation to the binomial": 9
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 normal approximation to the binomial（本文 9 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "normal-distribution",
    "ja": {
      "term": "正規分布",
      "reading": "せいきぶんぷ",
      "alt": []
    },
    "en": {
      "term": "normal distribution",
      "alt": [
        "normal curve",
        "bell curve"
      ],
      "uk": null,
      "variants": [
        {
          "term": "normally distributed",
          "register": "written",
          "note": "形容詞の言い方（X is normally distributed）。話し言葉にも書き言葉にも出てくるが、書き言葉のほうが多い（ほとんど OpenStax Introductory Statistics）。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3",
        "Algebra 2"
      ]
    },
    "definition_ja": "平均 μ を中心に左右対称な釣り鐘形の確率密度関数をもつ連続型の確率分布。N(μ, σ²) と書く。",
    "definition_en": "A continuous probability distribution with a symmetric, bell-shaped density centered at the mean μ; written N(μ, σ²).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Heights are roughly normally distributed, so we can use the 68-95-99.7 rule.",
        "ja": "身長はほぼ正規分布に従うので、68–95–99.7 の法則が使える。",
        "register": "spoken"
      },
      {
        "en": "X follows a normal distribution with mean 50 and standard deviation 10. Find P(X > 65).",
        "ja": "X は平均 50、標準偏差 10 の正規分布に従う。P(X > 65) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（数学B）は N(μ, σ²) と分散を書くが、OpenStax Introductory Statistics（6.1）は X ~ N(μ, σ) と標準偏差を書く（同じ本の巻末の公式集は X ~ N(μ, σ²)）。記号を見たら 2 つ目が分散か標準偏差かを確かめる。",
      "「正規分布に従う」は follows a normal distribution か is normally distributed。",
      "平均から ±1σ・±2σ・±3σ の範囲に約 68%・95%・99.7% が入ることを empirical rule（68-95-99.7 rule）と言う（AP Statistics の CED topic 2.11）。"
    ],
    "related": [
      "standard-normal-distribution",
      "z-score",
      "standard-normal-table",
      "central-limit-theorem",
      "empirical-rule",
      "probability-density-function",
      "probability-distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正規分布",
        "en": "Normal distribution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "6.1 The Standard Normal Distribution（X ~ N(μ, σ)）、巻末の公式集（X ~ N(μ, σ²)）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B (2) 統計的な推測（正規分布 N(μ, σ²)）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.11（normal distribution・empirical rule）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 7.5 Normal Distributions（normal distribution）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "normal distribution": 174,
        "bell curve": 52,
        "normal curve": 7,
        "normally distributed": 32
      },
      "written": {
        "normal distribution": 248,
        "normally distributed": 87,
        "bell curve": 2,
        "normal curve": 10
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "normal-line",
    "ja": {
      "term": "法線",
      "reading": "ほうせん",
      "alt": []
    },
    "en": {
      "term": "normal line",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線上の点を通り、その点での接線に垂直な直線。",
    "definition_en": "The line through a point on a curve that is perpendicular to the tangent line there.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The normal line is perpendicular to the tangent line, so its slope is the negative reciprocal.",
        "ja": "法線は接線に垂直だから、傾きは接線の傾きの逆数の符号を変えたものになる。",
        "register": "spoken"
      },
      {
        "en": "Find an equation of the normal line to y = x² at the point (1, 1).",
        "ja": "点 (1, 1) における y = x² の法線の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「逆数の符号を変えたもの」は英語で negative reciprocal と言う。"
    ],
    "related": [
      "slope-of-the-tangent-line",
      "point-of-tangency",
      "find-the-equation-of-the-tangent-line"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "normal line": 52
      },
      "written": {
        "normal line": 6
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "normal-probability-plot",
    "ja": {
      "term": "正規確率プロット",
      "reading": "せいきかくりつぷろっと",
      "alt": []
    },
    "en": {
      "term": "normal probability plot",
      "alt": [
        "normal quantile plot"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「正規確率プロット」は学習指導要領解説に無い（日本語版 Wikipedia「Q-Qプロット」にはある）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "データの値を、正規分布なら予想される値（z スコア）に対して並べた散布図。点がほぼ直線に並べば、データは正規分布に近い。",
    "definition_en": "A plot of the data values against the values (z-scores) expected if the data were normal; points lying close to a line suggest the data are approximately normal.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If the normal probability plot is pretty straight, the normal model is reasonable.",
        "ja": "正規確率プロットがほぼまっすぐなら、正規分布のモデルは妥当。",
        "register": "spoken"
      },
      {
        "en": "Use a normal probability plot to decide whether the data come from an approximately normal population.",
        "ja": "正規確率プロットを使って、データがほぼ正規分布の母集団から来ているか判断せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパス・OpenStax Introductory Statistics・AP Statistics の CED（2026 年版）のどれにも出てこない。"
    ],
    "related": [
      "normal-distribution",
      "z-score"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Normal_probability_plot",
        "note": "記事名 Normal probability plot（en.term と同じ。冒頭で太字で定義。数学カテゴリから 3 段。alt の normal quantile plot は記事に出てこない）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Q-Qプロット」",
        "url": "https://ja.wikipedia.org/wiki/Q-Q%E3%83%97%E3%83%AD%E3%83%83%E3%83%88",
        "note": "正規確率プロット"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "normal-vector",
    "ja": {
      "term": "法線ベクトル",
      "reading": "ほうせんべくとる",
      "alt": []
    },
    "en": {
      "term": "normal vector",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "直線や平面に垂直な 0 でないベクトル。",
    "definition_en": "A nonzero vector perpendicular to a line or a plane.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The coefficients a, b, c in the plane's equation give you a normal vector.",
        "ja": "平面の方程式の係数 a, b, c がそのまま法線ベクトルになる。",
        "register": "spoken"
      },
      {
        "en": "Find a normal vector to the plane 2x − y + 3z = 6.",
        "ja": "平面 2x − y + 3z = 6 の法線ベクトルを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉は大半が MIT OCW 18.02（多変数の微積分）。"
    ],
    "related": [
      "equation-of-a-plane",
      "direction-vector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "法線ベクトル",
        "en": "Normal vector"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "normal vector": 43
      },
      "written": {
        "normal vector": 138
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "not-differentiable",
    "ja": {
      "term": "微分不可能",
      "reading": "びぶんふかのう",
      "alt": []
    },
    "en": {
      "term": "not differentiable",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "その点で微分係数が存在しないこと。",
    "definition_en": "Having no derivative at a point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "At a corner, the function is not differentiable.",
        "ja": "尖ったところでは、関数は微分可能でない。",
        "register": "spoken"
      },
      {
        "en": "Show that f(x) = |x| is not differentiable at x = 0.",
        "ja": "f(x) = |x| が x = 0 で微分可能でないことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「微分不可能な点」は a point where f is not differentiable と文で言う。1 語の nondifferentiable は用例コーパスには出てこない（AP の CED は Unit 1 の指導例で 1 回使う）。"
    ],
    "related": [
      "differentiable",
      "differentiability",
      "cusp",
      "corner-nondifferentiable",
      "vertical-tangent"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 1 の指導例に nondifferentiable が 1 件。topic の本文ではない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "not differentiable": 33
      },
      "written": {
        "not differentiable": 41
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "nth-derivative",
    "ja": {
      "term": "第 n 次導関数",
      "reading": "だいえぬじどうかんすう",
      "alt": []
    },
    "en": {
      "term": "nth derivative",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "関数を n 回微分してできる導関数 f⁽ⁿ⁾(x)。",
    "definition_en": "The derivative obtained by differentiating n times, f⁽ⁿ⁾(x).",
    "latex": "f^{(n)}(x)",
    "spoken_en": "the nth derivative of f of x",
    "examples": [
      {
        "en": "The nth derivative of e to the x is still e to the x.",
        "ja": "eˣ の第 n 次導関数は eˣ のまま。",
        "register": "spoken"
      },
      {
        "en": "Find the nth derivative of f(x) = 1/x.",
        "ja": "f(x) = 1/x の第 n 次導関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "f⁽ⁿ⁾(x) の (n) は累乗ではなく微分の回数。"
    ],
    "related": [
      "higher-order-derivative",
      "second-derivative"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "nth derivative": 34
      },
      "written": {
        "nth derivative": 7
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "nth-root",
    "ja": {
      "term": "累乗根",
      "reading": "るいじょうこん",
      "alt": [
        "n 乗根"
      ]
    },
    "en": {
      "term": "nth root",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II",
        "数C"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "n 乗すると a になる数。実数の範囲では a と同じ符号のもの（a > 0 で n が偶数なら正のもの）を ⁿ√a と書き、複素数の範囲では 0 でない a に n 個ある。",
    "definition_en": "A number whose nth power is a; among the complex numbers every nonzero a has exactly n of them.",
    "latex": "\\sqrt[n]{a}",
    "spoken_en": "the nth root of a",
    "examples": [
      {
        "en": "The cube root is just the nth root with n equal to three.",
        "ja": "立方根は、累乗根で n が 3 の場合。",
        "register": "spoken"
      },
      {
        "en": "Find all the fourth roots of −16 in the complex numbers.",
        "ja": "複素数の範囲で −16 の 4 乗根をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the nth roots of a complex number",
        "ja": "複素数の n 乗根"
      }
    ],
    "pitfalls": [
      "nth は「エンス」と読む（the enth root。Merriam-Webster: ˈen(t)th）。2 乗根は square root、3 乗根は cube root と特別な名前で呼ぶ。"
    ],
    "related": [
      "nth",
      "nth-roots-of-unity",
      "de-moivres-theorem",
      "rational-exponent"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "冪根",
        "en": "Nth root"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「nth」",
        "url": "https://www.merriam-webster.com/dictionary/nth",
        "note": "発音 ˈen(t)th（2026-09-27 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "nth root": 13
      },
      "written": {
        "nth root": 45
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "nth-roots-of-unity",
    "ja": {
      "term": "1 の n 乗根",
      "reading": "いちのえぬじょうこん",
      "alt": [
        "1 の 3 乗根",
        "1 の累乗根",
        "1 の冪根"
      ]
    },
    "en": {
      "term": "nth roots of unity",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II",
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "zⁿ = 1 を満たす n 個の複素数。複素数平面では単位円に内接する正 n 角形の頂点になる。",
    "definition_en": "The n complex numbers z with zⁿ = 1; in the complex plane they are the vertices of a regular n-gon inscribed in the unit circle.",
    "latex": "z_k = \\cos\\frac{2k\\pi}{n} + i\\sin\\frac{2k\\pi}{n} \\quad (k = 0, 1, \\ldots, n-1)",
    "spoken_en": null,
    "examples": [
      {
        "en": "The nth roots of unity are evenly spaced around the unit circle.",
        "ja": "1 の n 乗根は単位円上に等間隔に並ぶ。",
        "register": "spoken"
      },
      {
        "en": "Find all the sixth roots of unity and plot them in the complex plane.",
        "ja": "1 の 6 乗根をすべて求め、複素数平面上に図示せよ。",
        "register": "written"
      },
      {
        "en": "Let ω be one of the non-real cube roots of unity. Find the value of ω^{100} + ω^{50} + 1.",
        "ja": "ω を 1 の 3 乗根のうち虚数であるものの 1 つとする。ω¹⁰⁰ + ω⁵⁰ + 1 の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "参照では Nicholson（Appendix A、Theorem A.3）だけが nth roots of unity と書き、OpenStax・IM・CK-12 には roots of unity が出てこない（0 件）。OpenStax Algebra and Trigonometry（Polar Form of Complex Numbers）は、一般の複素数の n 乗根（nth roots of a complex number）として扱う。",
      "1 の 3 乗根 ω（ω² + ω + 1 = 0、ω³ = 1）は、英語でも cube roots of unity（Nicholson Appendix A の例）。英語版 Wikipedia「Root of unity」は、1 以外の 2 つを x² + x + 1 の根として書く。OpenStax・IM・CK-12 には cube roots of unity も出てこない（0 件）。"
    ],
    "related": [
      "nth-root",
      "de-moivres-theorem",
      "complex-plane"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "1の冪根",
        "en": "Root of unity"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Polar Form of Complex Numbers（nth roots of a complex number）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "Appendix A Complex Numbers（Theorem A.3 nth Roots of Unity、cube roots of unity）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Root_of_unity",
        "note": "記事「Root of unity」（1 の 3 乗根は x² + x + 1 の根。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「1の冪根」",
        "url": "https://ja.wikipedia.org/wiki/1%E3%81%AE%E5%86%AA%E6%A0%B9",
        "note": "1の冪根、1の累乗根"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-24。docs/DECISIONS.md「Phase 2 代数 2・Precalculus の単元の前の修正」の、残った ③ を人間が決めた行）。en.term は nth roots of unity のまま。register は主張しない。",
        "raised": "2026-09-24"
      }
    ],
    "evidence": {
      "written": {
        "nth roots of unity": 2
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "nth-term-test",
    "ja": {
      "term": "n 項判定法",
      "reading": "えぬこうはんていほう",
      "alt": []
    },
    "en": {
      "term": "divergence test",
      "alt": [
        "nth term test",
        "test for divergence"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「n 項判定法」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書では「級数が収束すれば aₙ → 0」の対偶として扱い、判定法の名前はつけない。",
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "aₙ が 0 に収束しなければ、級数 Σaₙ は発散するという判定法。",
    "definition_en": "The test that says a series Σaₙ diverges if its terms aₙ do not approach 0.",
    "latex": "\\lim_{n\\to\\infty} a_n \\ne 0 \\;\\Rightarrow\\; \\sum a_n \\text{ diverges}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Always try the divergence test first: if the terms don't go to zero, you're done.",
        "ja": "まず発散判定を試す。項が 0 に近づかなければ、それで終わり。",
        "register": "spoken"
      },
      {
        "en": "Use the divergence test to show that Σ n/(2n + 1) diverges.",
        "ja": "n 項判定法を用いて Σ n/(2n + 1) が発散することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は nth term test for divergence と呼ぶ（topic 10.3）。用例コーパスでは divergence test が話し言葉・書き言葉ともよく使われ、nth term test はほとんど出てこない。",
      "aₙ → 0 でも収束するとは限らない（調和級数）。この判定法で分かるのは発散だけ。"
    ],
    "related": [
      "diverge",
      "harmonic-series",
      "infinite-series",
      "integral-test"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.3（nth term test for divergence）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "divergence test": 33,
        "test for divergence": 1
      },
      "written": {
        "divergence test": 36,
        "nth term test": 1,
        "test for divergence": 1
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "nth-term",
    "ja": {
      "term": "第 n 項",
      "reading": "だいえぬこう",
      "alt": []
    },
    "en": {
      "term": "nth term",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus BC",
        "Integrated Math 1",
        "Calculus II"
      ]
    },
    "definition_ja": "数列の n 番目の項 aₙ。n の式で表したものを一般項という。",
    "definition_en": "The term aₙ in the nth position of a sequence; written as a formula in n, it gives every term.",
    "latex": "a_n",
    "spoken_en": "a sub n",
    "examples": [
      {
        "en": "Plug n = 10 into the nth term to get the tenth term.",
        "ja": "第 n 項に n = 10 を入れれば第 10 項になる。",
        "register": "spoken"
      },
      {
        "en": "Write a formula for the nth term of the sequence 5, 8, 11, 14, ….",
        "ja": "数列 5, 8, 11, 14, … の第 n 項を式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "nth は「エンス」と読む（Merriam-Webster: ˈen(t)th）。the nth term = a sub n。",
      "数列の一般項（センター試験 平成 26 年度 数学II・数学B の「数列 {aₙ} の一般項を求めよう」は aₙ を n の式で表すこと）は、英語でも general term（エントリ general-term）。OpenStax Algebra and Trigonometry（13.1）・Intermediate Algebra（12.1）は the general term (nth term) と書く。n の式そのものは、OpenStax Algebra and Trigonometry は explicit formula、Calculus Volume 2（5.1）は an explicit formula for the nth term と書く（find-the-nth-term を参照）。"
    ],
    "related": [
      "sequence",
      "first-term",
      "find-the-nth-term",
      "nth",
      "general-term"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「nth」",
        "url": "https://www.merriam-webster.com/dictionary/nth",
        "note": "発音 ˈen(t)th（2026-09-27 に確かめた）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成26年度 大学入試センター試験 本試験 数学Ⅱ・数学Ｂ（問題）",
        "url": "https://web.archive.org/web/20150601/https://www.dnc.ac.jp/albums/abm.php?f=abm00003015.pdf&n=26suugaku_2b.pdf",
        "note": "数列 {aₙ} の一般項を求めよう"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.1 Sequences（general term (nth term)）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "nth term": 19
      },
      "written": {
        "nth term": 98
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "nth",
    "ja": {
      "term": "n 番目",
      "reading": "えぬばんめ",
      "alt": []
    },
    "en": {
      "term": "nth",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "notation",
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus BC",
        "Calculus I",
        "Calculus II"
      ]
    },
    "definition_ja": "n 番目の。第 n 項（the nth term）のほか、n 乗根（the nth root。n は根の指数）・第 n 次導関数（the nth derivative。n は微分の回数）のように、位置でない n にも使う。",
    "definition_en": "The ordinal form of n: in position n (the nth term), or with n as the index or the order (the nth root, the nth derivative).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "What's the nth term? Just replace 5 with n.",
        "ja": "n 番目の項は？ 5 を n に置き換えるだけ。",
        "register": "spoken"
      },
      {
        "en": "Find the nth term of the sequence 1, 4, 9, 16, ….",
        "ja": "数列 1, 4, 9, 16, … の第 n 項を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「エンス」と読む（Merriam-Webster: ˈen(t)th）。n-th とハイフンを入れて書くこともある（IM Algebra 1 5.4、Levin 0.2）。"
    ],
    "related": [
      "nth-term",
      "nth-root",
      "nth-derivative",
      "raise-to-the-nth-power"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.1 Sequences and Their Notations（the nth term）・1.3 Radicals and Rational Exponents（the nth root）"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「nth」",
        "url": "https://www.merriam-webster.com/dictionary/nth",
        "note": "発音 ˈen(t)th（2026-09-27 に確かめた）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 1 5.4（n-th）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "0.2（n-th）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.1（nth partial sum）・Topic 10.3（nth term test）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "3.2 The Derivative as a Function（from the second derivative to the nth derivative）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences・6.3 Taylor and Maclaurin Series（the nth term、the nth Taylor polynomial）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "nth": 182
      },
      "written": {
        "nth": 234
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "null-hypothesis",
    "ja": {
      "term": "帰無仮説",
      "reading": "きむかせつ",
      "alt": []
    },
    "en": {
      "term": "null hypothesis",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "仮説検定で、差がない・効果がないとして立てる仮説。H₀ と書く。",
    "definition_en": "In a hypothesis test, the hypothesis of no difference or no effect, written H₀.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The null hypothesis is always the boring one: nothing's going on.",
        "ja": "帰無仮説はいつも「何も起きていない」という退屈なほうだ。",
        "register": "spoken"
      },
      {
        "en": "State the null and alternative hypotheses.",
        "ja": "帰無仮説と対立仮説を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "読み上げでは the null hypothesis と言うのが、用例コーパスの話し言葉では多い。記号を読むときの H sub zero は、用例コーパスでは Professor Leonard だけ（記号エントリ h-naught-null）。"
    ],
    "related": [
      "alternative-hypothesis",
      "hypothesis-testing",
      "reject"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "null hypothesis": 292
      },
      "written": {
        "null hypothesis": 261
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "null-space",
    "ja": {
      "term": "零空間",
      "reading": "れいくうかん",
      "alt": [
        "核",
        "カーネル"
      ]
    },
    "en": {
      "term": "null space",
      "alt": [
        "kernel",
        "nullspace"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "Ax = 0 を満たすベクトル x 全体。線形写像では 0 に移るベクトル全体で、核ともいう。",
    "definition_en": "The set of all vectors x with Ax = 0; for a linear map, the set of vectors sent to 0, also called the kernel.",
    "latex": "\\operatorname{Nul} A = \\{\\vec{x} : A\\vec{x} = \\vec{0}\\}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Anything in the null space gets squashed to zero.",
        "ja": "零空間のベクトルは、どれも 0 につぶされる。",
        "register": "spoken"
      },
      {
        "en": "Find a basis for the null space of the matrix.",
        "ja": "行列の零空間の基底を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "行列には null space、線形写像には kernel と言うことが多い（Nicholson は kernel を 7.2 で使う）。用例コーパスの話し言葉はほぼすべて MIT 18.06。"
    ],
    "related": [
      "column-space",
      "rank-nullity-theorem",
      "homogeneous-system",
      "subspace"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "零空間",
        "en": "Kernel (linear algebra)"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.4 Rank of a Matrix・7.2 Kernel and Image of a Linear Transformation"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "null space": 504,
        "kernel": 3
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "number-line",
    "ja": {
      "term": "数直線",
      "reading": "すうちょくせん",
      "alt": []
    },
    "en": {
      "term": "number line",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic",
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直線上に基準の点 0 と単位の長さを決め、各点に 1 つの実数を対応させたもの。右へ行くほど数が大きくなる。",
    "definition_en": "A straight line on which every point stands for a real number, measured from a point marked 0 with a fixed unit length; numbers increase to the right.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's put −3 on the number line: start at zero and go three units to the left.",
        "ja": "−3 を数直線上にとってみよう。0 から左へ 3 目盛り進む。",
        "register": "spoken"
      },
      {
        "en": "Graph the solution set of x ≥ −2 on a number line.",
        "ja": "x ≧ −2 の解を数直線上に図示せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "不等式の解を数直線に表すとき、端点を含むなら塗りつぶした丸（closed circle）、含まないなら白丸（open circle）で表す（IM・OpenStax）。",
      "絶対値 |a| は、数直線上で原点から a までの距離として説明する（the distance from 0 on the number line）。",
      "「数直線上に」は on the number line（書き言葉によく出てくる）。見出しの number line を中に含むので、候補表現には数えない。"
    ],
    "related": [
      "real-number",
      "linear-inequality",
      "solution-set",
      "absolute-value-inequality",
      "coordinates"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "closed circle ／ open circle（Grade 6〜8 で 9 件・10 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "number line": 461
      },
      "written": {
        "number line": 363
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "number-of-digits",
    "ja": {
      "term": "桁数",
      "reading": "けたすう",
      "alt": []
    },
    "en": {
      "term": "number of digits",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "整数を 10 進法で書いたときの数字の個数。常用対数を使うと、正の整数 N が n 桁 ⇔ n − 1 ≦ log₁₀ N < n。",
    "definition_en": "How many digits an integer has in base 10; with common logs, a positive integer N has n digits exactly when n − 1 ≤ log₁₀ N < n.",
    "latex": "n - 1 \\le \\log_{10} N < n",
    "spoken_en": null,
    "examples": [
      {
        "en": "Log base 10 of 2 to the 50th is about 15.05, so the number has 16 digits.",
        "ja": "log₁₀ 2⁵⁰ は約 15.05 なので、16 桁の数。",
        "register": "spoken"
      },
      {
        "en": "Using log₁₀ 2 = 0.3010, find the number of digits of 2^100.",
        "ja": "log₁₀ 2 = 0.3010 として、2¹⁰⁰ は何桁の数か求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "参照（CED・OpenStax・IM・CK-12）には、常用対数で桁数や最高位の数字を求める問題が出てこない。",
      "number of digits は、小数点以下の桁数（the number of digits after the decimal point）や、近似の精度（the number of digits of accuracy）にも使う。"
    ],
    "related": [
      "common-logarithm",
      "leading-digit",
      "digit",
      "place"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles（How many digits long …）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "number of digits": 9
      },
      "written": {
        "number of digits": 13
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "number-of-divisors",
    "ja": {
      "term": "約数の個数",
      "reading": "やくすうのこすう",
      "alt": []
    },
    "en": {
      "term": "number of divisors",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "約数の個数を返す関数は divisor function（d(n)。σ₀(n) とも書く）。英語版 Wikipedia の Divisor function（number of divisors のリダイレクト先）は約数の k 乗和 σₖ(n) の族の記事で、約数の個数はその k = 0 の場合。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "ある自然数の正の約数が全部でいくつあるかという数。n = pᵃ·qᵇ（p, q は異なる素数）なら (a + 1)(b + 1) 個になる。",
    "definition_en": "How many positive divisors a positive integer has. If n = pᵃ·qᵇ with distinct primes p and q, it has (a + 1)(b + 1) divisors.",
    "latex": "n = p^{a} q^{b} \\;\\Rightarrow\\; (a+1)(b+1)",
    "spoken_en": "if n is p to the a times q to the b, the number of divisors is the quantity a plus one times the quantity b plus one",
    "examples": [
      {
        "en": "72 is 2³ times 3², so the number of divisors is 4 times 3, which is 12.",
        "ja": "72 = 2³·3² だから、約数の個数は 4 × 3 = 12 個。",
        "register": "spoken"
      },
      {
        "en": "Find the number of positive divisors of 360.",
        "ja": "360 の正の約数の個数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "指数にそれぞれ 1 を足してかける。1 と n 自身も数に入っている。",
      "正の約数に限ることを英語の問題文ではっきりさせるときは positive divisors と書く。約数は divisor のほか factor とも言う（the factors of 12）。"
    ],
    "related": [
      "prime-factorization",
      "multiple",
      "greatest-common-divisor",
      "composite-number",
      "divisor"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Divisor_function",
        "note": "記事「Divisor function」の節 Definition（number of divisors、σ₀。数学カテゴリから 3 段）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.2 Combining Outcomes（練習問題: How many divisors does it have?）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成30年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20190601/https://www.dnc.ac.jp/albums/abm.php?f=abm00033141.pdf&n=h30+suugaku1+02suugaku1A.pdf",
        "note": "第4問: 144 の正の約数の個数"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元 3 の前の修正」の、残った ③ を人間が決めた行）。en.term は number of divisors のまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "number-of-elements",
    "ja": {
      "term": "要素の個数",
      "reading": "ようそのこすう",
      "alt": [
        "濃度"
      ]
    },
    "en": {
      "term": "cardinality",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "number of elements",
          "register": "written",
          "note": "高校・入門の書き方で、OpenStax はこちらだけを使う。話し言葉はすべて MIT OCW で、cardinality よりずっと少ない。書き言葉では MIT の講義ノートの cardinality と同じくらい使う。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "高等学校学習指導要領解説は、要素の個数の関係式 n(A ∪ B) = n(A) + n(B) − n(A ∩ B) を数学A（場合の数と確率）で扱う。英語の cardinality は無限集合の大きさ（日本語版 Wikipedia「濃度 (数学)」の濃度）まで含む語で、用例コーパスでは話し言葉はすべて MIT OCW（6.042）、書き言葉はすべて MIT の講義ノート。OpenStax（Algebra and Trigonometry・Precalculus）は number of elements と書き、cardinality は Levin（離散数学）に出る。記号は Levin が |A|、OpenStax Algebra and Trigonometry（13.7 Probability）が n(E)。",
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "集合に含まれる要素の数。有限集合 A の要素の個数を n(A) と書く。",
    "definition_en": "The number of elements in a set; for a finite set A it is written n(A) or |A|.",
    "latex": "n(A \\cup B) = n(A) + n(B) - n(A \\cap B)",
    "spoken_en": "n of A union B equals n of A plus n of B minus n of A intersect B",
    "examples": [
      {
        "en": "This set has five elements, so its cardinality is 5.",
        "ja": "この集合は要素を 5 つもつので、要素の個数は 5 だ。",
        "register": "spoken"
      },
      {
        "en": "Find the number of elements in A ∪ B.",
        "ja": "A ∪ B の要素の個数を求めよ。",
        "register": "written"
      },
      {
        "en": "If |A| = 12 and |B| = 8, what is the largest possible cardinality of A ∩ B?",
        "ja": "n(A) = 12, n(B) = 8 のとき、A ∩ B の要素の個数として考えられる最大の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "|A| は絶対値と同じ記号だが、集合に付くと要素の個数を表す。n(A) は n of A と読む。",
      "cardinality は無限集合にも使う（自然数全体と実数全体では cardinality が違う）。有限集合の要素の個数の意味だけなら number of elements でよい（OpenStax の言い方）。"
    ],
    "related": [
      "set",
      "element",
      "union",
      "venn-diagram",
      "universal-set"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "濃度 (数学)",
        "en": "Cardinality"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 場合の数と確率: 集合の要素の個数 n(A∪B) = n(A) + n(B) − n(A∩B)（数学I の節は数学A に回すと書く）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "Sets: |A| denotes the cardinality"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.7 Probability: n(E) ／ n(S)（number of elements）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "確率の節: number of elements（5 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "cardinality": 82,
        "number of elements": 10
      },
      "written": {
        "number of elements": 15,
        "cardinality": 36
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "number-of-possible-outcomes",
    "ja": {
      "term": "起こりうる場合の数",
      "reading": "おこりうるばあいのかず",
      "alt": [
        "すべての場合"
      ]
    },
    "en": {
      "term": "total number of outcomes",
      "alt": [
        "number of possible outcomes"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「場合の数」は、確率の分母の「起こりうる場合の数」にも、数え上げの単元の名前（エントリ counting）にも使う。英語では、起こりうるすべての結果（outcomes）の数を total number of outcomes と言い、OpenStax Algebra and Trigonometry もこの言い方で確率を求める。結果の集まりそのものは sample space（エントリ sample-space）と呼ぶ。",
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "ある試行で起こりうるすべての結果の個数。どの結果も同様に確からしいとき、確率は、あることがらの起こる場合の数をこの数でわって求める。",
    "definition_en": "How many outcomes an experiment can have in all. If every outcome is equally likely, an event's probability is its count of outcomes divided by this number.",
    "latex": "P(A) = \\frac{\\text{number of outcomes in } A}{\\text{total number of outcomes}}",
    "spoken_en": "P of A equals the number of outcomes in A over the total number of outcomes",
    "examples": [
      {
        "en": "Roll one die and the total number of outcomes is 6; three of them are even, so the probability is 3 over 6, or one half.",
        "ja": "さいころを 1 個投げると、起こりうる場合の数は 6 通り。そのうち偶数は 3 通りだから、確率は 6 分の 3、つまり 2 分の 1。",
        "register": "spoken"
      },
      {
        "en": "Two dice are rolled. Find the total number of outcomes, and then find the probability that the sum of the numbers is 7.",
        "ja": "2 個のさいころを投げる。起こりうる場合の数を求め、出た目の和が 7 になる確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「場合」を case と直訳しない。確率で数えるのは 1 つ 1 つの結果（outcome、エントリ elementary-event）で、その個数を言う。",
      "この数でわって確率を求めてよいのは、どの結果も同様に確からしい（equally likely、エントリ equally-likely）ときだけ。",
      "2 個のさいころの目の和は 2〜12 の 11 通りだが、これを起こりうる場合の数にしない。和の 11 通りは同様に確からしくないので、目の出方 6 × 6 = 36 通りで数える。"
    ],
    "related": [
      "sample-space",
      "counting",
      "probability",
      "equally-likely",
      "elementary-event"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "number of possible outcomes": 3
      },
      "written": {
        "number of possible outcomes": 6,
        "total number of outcomes": 13
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "number-of-real-solutions",
    "ja": {
      "term": "方程式の実数解の個数",
      "reading": "ほうていしきのじっすうかいのこすう",
      "alt": [
        "解の個数"
      ]
    },
    "en": {
      "term": "number of real solutions",
      "alt": [
        "how many real solutions"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "calculus"
    ],
    "level": {
      "jp": [
        "中1",
        "中3",
        "数II",
        "数III"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 2",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "方程式の解のうち実数であるものの個数。一次方程式・二次方程式では式の形から、微分を使うときは y = f(x) のグラフと直線 y = k の共有点の個数として数える。",
    "definition_en": "How many real solutions an equation has; with calculus, counted as the number of points where the graph of y = f(x) meets the line y = k.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The number of real solutions is the number of times the graph crosses the line y = k.",
        "ja": "実数解の個数は、グラフが直線 y = k と交わる回数に等しい。",
        "register": "spoken"
      },
      {
        "en": "Determine the number of real solutions of x³ − 3x − k = 0 for each value of k.",
        "ja": "k の値によって、方程式 x³ − 3x − k = 0 の実数解の個数を調べよ。",
        "register": "written"
      },
      {
        "en": "A quadratic can have two real solutions, one real solution, or no real solutions.",
        "ja": "二次方程式の実数解は、2 つのことも、1 つのことも、1 つもないこともある。",
        "register": "spoken"
      },
      {
        "en": "How many solutions does the equation 3x − 5 = 3x + 2 have?",
        "ja": "方程式 3x − 5 = 3x + 2 の解の個数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では solution（方程式の解）のほかに、root ／ zero（多項式の根・零点）とも言う。",
      "一次方程式でも、OpenStax Elementary Algebra は解が 1 つ（conditional equation）・解が無数（identity）・解が無い（contradiction）の 3 つに分けて答えさせる。3x − 5 = 3x + 2 は解が無い（no solution）。"
    ],
    "related": [
      "increasing-and-decreasing",
      "local-extremum"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "number of real solutions": 3
      },
      "sources": [
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-24。docs/DECISIONS.md「Phase 2 数列・級数の単元の前の修正」の、人間が見出しを決めた ③ の行（number-of-real-solutions・prime-notation・find-the-inflection-points））。register は主張しない。",
        "raised": "2026-09-24"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "numeral-system",
    "ja": {
      "term": "記数法",
      "reading": "きすうほう",
      "alt": []
    },
    "en": {
      "term": "numeral system",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（numeral system は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "definition_ja": "数を数字の並びで書き表す方法。特に、基数 n を決めて各位の数字に n の累乗の重みをつける位取り記数法を指す。",
    "definition_en": "A way of writing numbers with a string of digits; in particular, a place-value system in which each digit is weighted by a power of a fixed base.",
    "latex": "d_k n^k + d_{k-1} n^{k-1} + \\cdots + d_1 n + d_0",
    "spoken_en": null,
    "examples": [
      {
        "en": "Our usual numeral system is base ten, so each place is worth ten times the place to its right.",
        "ja": "ふだん使う記数法は 10 進法なので、各位は 1 つ右の位の 10 倍の重みをもつ。",
        "register": "spoken"
      },
      {
        "en": "In a positional numeral system with base n, the digits range from 0 to n − 1.",
        "ja": "n を基数とする位取り記数法では、使う数字は 0 から n − 1 までである。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語版 Wikipedia「Numeral system」は、記数法を number system とも呼ぶが、実数・複素数のような「数の体系」の意味と紛れると書く。位取り記数法は英語版 Wikipedia「Positional notation」の言い方。"
    ],
    "related": [
      "base-n",
      "binary",
      "hexadecimal",
      "digit",
      "place",
      "decimal-system"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Numeral_system",
        "note": "記事名 Numeral system（en.term のリダイレクト先）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Positional_notation",
        "note": "記事「Positional notation」（位取り記数法。数学カテゴリから 2 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 numeral system（英語版「Numeral system」、en.term のリダイレクト先）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "numerator",
    "ja": {
      "term": "分子",
      "reading": "ぶんし",
      "alt": []
    },
    "en": {
      "term": "numerator",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "小学校",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "分数の横線の上にある数・式。",
    "definition_en": "The number or expression above the fraction bar.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Factor the numerator: x² − 9 is (x + 3)(x − 3).",
        "ja": "分子を因数分解しよう。x² − 9 は (x + 3)(x − 3)。",
        "register": "spoken"
      },
      {
        "en": "Rationalize the numerator of (√(x + 1) − 1)/x.",
        "ja": "(√(x + 1) − 1)/x の分子を有理化せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では the top（と the bottom）とも言う。"
    ],
    "related": [
      "denominator",
      "reduce",
      "rational-expression",
      "fraction"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "numerator": 1474
      },
      "written": {
        "numerator": 857
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "numerical-integration",
    "ja": {
      "term": "数値積分",
      "reading": "すうちせきぶん",
      "alt": []
    },
    "en": {
      "term": "numerical integration",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "定積分の値を、区間を分けて計算した近似値で求める方法。台形公式やシンプソンの公式などがある。",
    "definition_en": "Approximating the value of a definite integral with a computed sum, as in the trapezoidal rule or Simpson's rule.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "There's no elementary antiderivative here, so we'll fall back on numerical integration.",
        "ja": "これは初等関数で原始関数が書けないから、数値積分に頼ろう。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説に「数値積分」の語は出てこない。理数に関する学科の科目「理数数学II」の積分法が、発展・拡充させる内容として、面積を台形公式やシンプソンの公式で求めることを挙げる（数学III には無い）。OpenStax Calculus Volume 2 は誤差の上限（error bound）まで扱う。"
    ],
    "related": [
      "trapezoidal-rule",
      "simpsons-rule",
      "midpoint-riemann-sum"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "数値積分",
        "en": "Numerical integration"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「数値積分」の語は無い。理数数学II (7) 積分法（発展・拡充）: 面積を台形公式やシンプソンの公式で求める（数学III には無い）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "numerical integration": 6
      },
      "written": {
        "numerical integration": 12
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "objective-function",
    "ja": {
      "term": "目的関数",
      "reading": "もくてきかんすう",
      "alt": []
    },
    "en": {
      "term": "objective function",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数II",
        "大学"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "最適化問題で、最大値・最小値を求めたい関数。線形計画法では 1 次式。",
    "definition_en": "In an optimization problem, the function to be maximized or minimized; linear in linear programming.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Plug each corner into the objective function and see which is biggest.",
        "ja": "各頂点を目的関数に代入して、どれが一番大きいか見る。",
        "register": "spoken"
      },
      {
        "en": "Maximize the objective function P = 3x + 2y subject to the given constraints.",
        "ja": "与えられた制約条件のもとで、目的関数 P = 3x + 2y を最大にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉はすべて MIT 18.02（ラグランジュの未定乗数法）。日本の高校の問題文は「x + y の最大値を求めよ」のように書く。"
    ],
    "related": [
      "linear-programming",
      "constraint",
      "feasible-region",
      "lagrange-multipliers"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "objective function": 10
      },
      "written": {
        "objective function": 5
      },
      "sources": [
        "mit-18.02",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "observational-study",
    "ja": {
      "term": "観察研究",
      "reading": "かんさつけんきゅう",
      "alt": []
    },
    "en": {
      "term": "observational study",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "研究者が処理を割り当てず、対象のありのままの様子を観察・記録して調べる研究。標本調査もその一種。",
    "definition_en": "A study in which researchers observe and record what happens without assigning any treatment to the individuals.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Nobody told people whether to drink coffee, so this is an observational study, and we can't say the coffee caused anything.",
        "ja": "コーヒーを飲むかどうかを誰も指示していないから、これは観察研究で、コーヒーが原因だとは言えない。",
        "register": "spoken"
      },
      {
        "en": "Researchers recorded the daily screen time and hours of sleep for a group of students. Explain why this is an observational study rather than an experiment.",
        "ja": "研究者は、ある生徒の集団について 1 日のスクリーンタイムと睡眠時間を記録した。これが実験ではなく観察研究である理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "観察研究からは因果関係（cause-and-effect）は言えない。2 つの変数に関係が見えても、交絡変数（エントリ confounding-variable）のせいかもしれない。AP Statistics の CED は実験（エントリ experiment）とこの点で区別する。",
      "実験との違いは、研究者が処理を割り当てるかどうか。標本調査（エントリ sample-survey）も処理を割り当てないので observational study の一種になる。"
    ],
    "related": [
      "experiment",
      "sample-survey",
      "causation",
      "confounding-variable",
      "treatment"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.5・3.6"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "observational study": 23
      },
      "written": {
        "observational study": 1
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "observed-count",
    "ja": {
      "term": "観測度数",
      "reading": "かんそくどすう",
      "alt": []
    },
    "en": {
      "term": "observed value",
      "alt": [
        "observed frequency",
        "observed count"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "実際のデータで各カテゴリ（分割表の各セル）に入った度数。",
    "definition_en": "The count actually found in each category (or cell of a two-way table) in the data.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Observed minus expected, squared, over expected: add that up over every cell.",
        "ja": "観測度数から期待度数を引いて 2 乗し、期待度数で割る。それを全部のセルで足す。",
        "register": "spoken"
      },
      {
        "en": "Record the observed values in the first list and the expected values in the second.",
        "ja": "観測度数を 1 つ目のリストに、期待度数を 2 つ目のリストに入力せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は O = observed values と書く（確かめた文脈はすべてカイ二乗検定の度数）。AP Statistics の CED（2026 年版）は observed count（2 件）。",
      "observed value は検定統計量の実現値など、度数でない値にも使う。"
    ],
    "related": [
      "expected-count",
      "chi-square-test"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.15"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "observed value": 20,
        "observed frequency": 4
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "obtuse-angle",
    "ja": {
      "term": "鈍角",
      "reading": "どんかく",
      "alt": []
    },
    "en": {
      "term": "obtuse angle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "大きさが 90° より大きく 180° より小さい角。直角よりも開きが大きい。",
    "definition_en": "An angle that measures more than 90° and less than 180°, so it is wider than a right angle.",
    "latex": "90^\\circ < \\theta < 180^\\circ",
    "spoken_en": "ninety degrees is less than theta, which is less than one hundred eighty degrees",
    "examples": [
      {
        "en": "This one opens wider than a right angle, so it's an obtuse angle.",
        "ja": "これは直角より大きく開いているから、鈍角だね。",
        "register": "spoken"
      },
      {
        "en": "One angle of an isosceles triangle is an obtuse angle that measures 110°. Find the measures of the other two angles.",
        "ja": "二等辺三角形の 1 つの角が 110° の鈍角である。残りの 2 つの角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "obtuse は日常語では「鈍い」の意味の単語。数学では 90° より大きく 180° より小さい角を表す。",
      "90° と 180° は含まない。180° の角は平角（エントリ straight-angle）、180° より大きい角は優角（エントリ reflex-angle）。",
      "obtuse は「オブトゥース」に近く、後半に強勢を置いて読む（Merriam-Webster: äb-ˈtüs）。"
    ],
    "related": [
      "acute-angle",
      "right-angle",
      "obtuse-triangle",
      "straight-angle",
      "reflex-angle"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.1 Classify Triangles・4.3 Classify Triangles by Side Measurement"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「obtuse」",
        "url": "https://www.merriam-webster.com/dictionary/obtuse",
        "note": "発音 äb-ˈtüs（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "obtuse angle": 2
      },
      "written": {
        "obtuse angle": 7
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは CK-12 の呼び方 obtuse angle（CK-12 Geometry 4.1 Classify Triangles・CK-12 Geometry 2.12 Converse Inverse and Contrapositive Statements・CK-12 Geometry 4.3 Classify Triangles by Side Measurement・計 16 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "obtuse-triangle",
    "ja": {
      "term": "鈍角三角形",
      "reading": "どんかくさんかくけい",
      "alt": []
    },
    "en": {
      "term": "obtuse triangle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "1 つの角が鈍角である三角形。鈍角は 1 つしかなく、残りの 2 つの角は鋭角になる。",
    "definition_en": "A triangle with one obtuse angle. A triangle can have only one obtuse angle, so its other two angles are acute.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This angle is 120 degrees, so it's an obtuse triangle, and the other two angles have to share the remaining 60.",
        "ja": "この角が 120° だから鈍角三角形。残りの 2 つの角で、あと 60° を分け合うことになる。",
        "register": "spoken"
      },
      {
        "en": "Explain why an obtuse triangle cannot have two obtuse angles.",
        "ja": "鈍角三角形が 2 つの鈍角をもつことはない。その理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "鈍角三角形の鈍角は必ず 1 つだけ。鈍角が 2 つあると、その 2 つだけで和が 180° を超えてしまう。",
      "鈍角をはさむ辺を底辺にすると、高さは三角形の外にはみ出る（底辺を延長した直線に垂線を下ろす）。面積を求めるときに注意する。",
      "3 辺の長さ a, b, c（c がいちばん長い）から見分けるときは、c² > a² + b² なら鈍角三角形。"
    ],
    "related": [
      "obtuse-angle",
      "acute-triangle",
      "right-triangle",
      "angle-sum-of-a-triangle",
      "triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "鈍角三角形",
        "en": "Obtuse triangle"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.2 Classify Triangles by Angle Measurement・4.29 Pythagorean Theorem to Classify Triangles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "obtuse triangle": 3
      },
      "written": {
        "obtuse triangle": 3
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは CK-12 の呼び方 obtuse triangle（CK-12 Geometry 4.29 Pythagorean Theorem to Classify Triangles・CK-12 Geometry 4.1 Classify Triangles・CK-12 Geometry 4.2 Classify Triangles by Angle Measurement・計 29 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "occur",
    "ja": {
      "term": "起こる",
      "reading": "おこる",
      "alt": []
    },
    "en": {
      "term": "occur",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "happen",
          "register": "both",
          "note": "事象について話すときは、話し言葉で occur と同じくらい使う（MIT OCW・Professor Leonard・Khan Academy）。書き言葉では occur のほうが多い。"
        }
      ],
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "試行の結果として、ある事象が実際に成り立つ。確率では「事象 A が起こる確率」のように使う。",
    "definition_en": "To take place as the result of a trial; in probability, an event occurs when the outcome belongs to it.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "What's the chance that at least one six happens in four rolls? Easier to look at the complement.",
        "ja": "4 回振って少なくとも 1 回 6 の目が出る確率は？ 余事象を考えたほうが楽だ。",
        "register": "spoken"
      },
      {
        "en": "A coin is tossed 5 times. Find the probability that heads occurs exactly 3 times.",
        "ja": "硬貨を 5 回投げる。表がちょうど 3 回出る確率を求めよ。",
        "register": "written"
      },
      {
        "en": "Events A and B cannot occur at the same time, so they are mutually exclusive.",
        "ja": "事象 A と B は同時には起こらないので、互いに排反である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "事象について言うとき、話し言葉では occur と happen を同じくらい使い、書き言葉では occur のほうが多い。occur は「〜で生じる」の意味でも使う（the maximum occurs at x = 2）。",
      "「起こりうるすべての場合」は all possible outcomes と言う。"
    ],
    "related": [
      "trial",
      "elementary-event",
      "mutually-exclusive-events",
      "repeated-trials",
      "probability"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.5・2.6（the probability that event A will occur）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "event … happen | event happen | probability that … happen | probability of … happening | happen at the same time | happen together": 62,
        "event … occur | event occur | probability that … occur | probability of … occurring | occur at the same time | occur together": 68
      },
      "written": {
        "event … occur | event occur | probability that … occur | probability of … occurring | occur at the same time | occur together": 82,
        "event … happen | event happen | probability that … happen | probability of … happening | happen at the same time | happen together": 24
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "odd-function",
    "ja": {
      "term": "奇関数",
      "reading": "きかんすう",
      "alt": []
    },
    "en": {
      "term": "odd function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry",
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "f(−x) = −f(x) がすべての x で成り立つ関数。グラフは原点に関して対称。",
    "definition_en": "A function with f(−x) = −f(x) for every x; its graph is symmetric about the origin.",
    "latex": "f(-x) = -f(x)",
    "spoken_en": "f of negative x equals negative f of x",
    "examples": [
      {
        "en": "Sine is odd, so sin(−θ) is negative sin θ.",
        "ja": "sin は奇関数なので、sin(−θ) は −sin θ。",
        "register": "spoken"
      },
      {
        "en": "Show that f(x) = x³ − 5x is an odd function.",
        "ja": "f(x) = x³ − 5x が奇関数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "even-function",
      "symmetric",
      "integrals-of-even-and-odd-functions",
      "point-symmetry"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "偶関数と奇関数",
        "en": "Even and odd functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "odd function": 91
      },
      "written": {
        "odd function": 56
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "odd-number",
    "ja": {
      "term": "奇数",
      "reading": "きすう",
      "alt": []
    },
    "en": {
      "term": "odd number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 で割り切れない整数。n を整数として 2n + 1 と表せる。",
    "definition_en": "An integer that cannot be divided evenly by 2, such as −3, 1, 7, or 15. Every odd number can be written as 2n + 1, where n is an integer.",
    "latex": "2n + 1",
    "spoken_en": "two n plus one",
    "examples": [
      {
        "en": "Take any even number, add 1, and you get an odd number.",
        "ja": "どんな偶数でも、1 を足せば奇数になるよ。",
        "register": "spoken"
      },
      {
        "en": "Show that the sum of two odd numbers is an even number.",
        "ja": "2 つの奇数の和は偶数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "odd は日常語では「変な、奇妙な」の意味にもなる（an odd question）。数学の文では odd number や 7 is odd の形で「奇数」。",
      "奇数を 2n − 1 と表してもよい。n を整数全体で動かせば、2n + 1 と同じ数の集まりになる。",
      "奇関数（odd function、エントリ odd-function）の odd と同じ語。"
    ],
    "related": [
      "even-number",
      "parity",
      "integer",
      "remainder",
      "odd-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "奇数",
        "en": "Odd number"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "odd number": 62
      },
      "written": {
        "odd number": 46
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "one-sample-t-test",
    "ja": {
      "term": "1 標本 t 検定",
      "reading": "いちひょうほんてぃーけんてい",
      "alt": []
    },
    "en": {
      "term": "one-sample t-test",
      "alt": [
        "t-test for a mean"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "母標準偏差が分からないとき、1 つの標本の平均をもとに母平均についての仮説を t 分布で検定する方法。",
    "definition_en": "A test of a claim about one population mean when the population standard deviation is unknown, based on one sample mean and the t-distribution.",
    "latex": "t = \\frac{\\bar{x} - \\mu_0}{s / \\sqrt{n}}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Since sigma is unknown, this calls for a one-sample t-test.",
        "ja": "σ が分からないので、1 標本 t 検定を使う場面だ。",
        "register": "spoken"
      },
      {
        "en": "Carry out a one-sample t-test of H₀: μ = 50 at the 5% significance level.",
        "ja": "有意水準 5% で、H₀: μ = 50 の 1 標本 t 検定を行え。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパス・OpenStax Introductory Statistics のどちらにも出てこない。OpenStax は a single population mean using the Student t distribution（節の名前）として扱う。"
    ],
    "related": [
      "t-distribution",
      "hypothesis-testing",
      "test-statistic",
      "two-sample-t-test",
      "paired-t-test"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 4.4・4.5"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは AP Statistics の CED の呼び方 one-sample t-test（topic 4.4・topic 4.5）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "one-sided-limit",
    "ja": {
      "term": "片側極限",
      "reading": "かたがわきょくげん",
      "alt": []
    },
    "en": {
      "term": "one-sided limit",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "右側極限と左側極限をまとめた呼び方。",
    "definition_en": "A limit taken from only one side: a limit from the left or a limit from the right.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For a piecewise function, check both one-sided limits at the break point.",
        "ja": "区分的に定義された関数では、つなぎ目で両側の片側極限を確かめる。",
        "register": "spoken"
      },
      {
        "en": "The limit exists if and only if both one-sided limits exist and are equal.",
        "ja": "極限が存在するのは、両側の片側極限が存在して等しいとき、かつそのときに限る。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "right-hand-limit",
      "left-hand-limit",
      "limit"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "片側極限",
        "en": "One-sided limit"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.5（one-sided limits）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "one-sided limit": 26
      },
      "written": {
        "one-sided limit": 26
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "one-sixth-formula",
    "ja": {
      "term": "1/6 公式",
      "reading": "ろくぶんのいちこうしき",
      "alt": []
    },
    "en": {
      "term": "one-sixth formula",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "日本語版 Wikipedia「1/6公式」は、途中の計算を書かないマークシート式の試験で積分の計算に使われることがあると書く。OpenStax Calculus にも AP の CED にも無く、用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。one-sixth formula も用例コーパスと参照に出てこないので、答案で使うなら公式そのもの（∫_α^β (x − α)(x − β) dx = −(β − α)³/6）を書くか、ふつうに積分して見せる。公式そのものは AP の CED に無い。放物線と直線で囲まれた面積の問題は CED topic 8.4（Finding the Area Between Curves Expressed as Functions of x）で扱う。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "放物線と直線で囲まれた部分の面積を、x² の係数と、交点の x 座標 α、β から求める公式。∫_α^β (x − α)(x − β) dx = −(β − α)³/6 を使う。",
    "definition_en": "A shortcut used on some Japanese exams: the integral of (x − α)(x − β) from α to β is −(β − α)³/6, so the area between a parabola and a line follows from the leading coefficient and the intersection points.",
    "latex": "\\int_{\\alpha}^{\\beta} (x-\\alpha)(x-\\beta)\\,dx = -\\frac{(\\beta-\\alpha)^3}{6}",
    "spoken_en": "the integral from alpha to beta of the quantity x minus alpha times the quantity x minus beta, dx, equals negative the quantity beta minus alpha, cubed, over six",
    "examples": [
      {
        "en": "I used a shortcut I learned in Japan: the integral of (x − α)(x − β) from α to β is −(β − α)³ over 6. Is it OK to use that on the exam?",
        "ja": "日本で習った近道を使いました。(x − α)(x − β) を α から β まで積分すると −(β − α)³/6 になります。試験で使ってもいいですか。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は数学的実践（Mathematical Practices）に justification と communication and notation を挙げ、記述問題では答えを支える途中の式（supporting work）を求める。公式だけで答えを出すなら、公式を示してから使う。"
    ],
    "related": [
      "area-between-two-curves"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Mathematical Practices（justification、communication and notation）と記述問題の説明。Topic 8.4（Finding the Area Between Curves Expressed as Functions of x）: 放物線と直線で囲まれた面積の問題が当たる場面（1/6 公式そのものは CED に無い）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「1/6公式」",
        "url": "https://ja.wikipedia.org/wiki/1/6%E5%85%AC%E5%BC%8F",
        "note": "マークシート式の試験で積分計算に使われることがある"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "one-tailed-test",
    "ja": {
      "term": "片側検定",
      "reading": "かたがわけんてい",
      "alt": []
    },
    "en": {
      "term": "one-tailed test",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "OpenStax Introductory Statistics は片側検定をまとめて呼ぶより、右側（right-tailed test）・左側（left-tailed test）と分けて呼ぶことが多い（right-tailed 38 件・left-tailed 26 件に対し one-tailed 4 件）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "対立仮説が「μ > μ₀」や「μ < μ₀」のように一方向の違いだけを考え、棄却域を分布の片側にとる検定。",
    "definition_en": "A test whose alternative hypothesis allows a difference in one direction only (such as μ > μ₀), with the rejection region in one tail.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Since the claim is that the mean increased, this is a right-tailed test.",
        "ja": "平均が増えたという主張なので、これは右側検定。",
        "register": "spoken"
      },
      {
        "en": "State the hypotheses for a left-tailed test of the claim that μ < 30.",
        "ja": "μ < 30 という主張の左側検定の仮説を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "two-tailed-test",
      "hypothesis-testing",
      "alternative-hypothesis"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "one-tailed test | right-tailed test | left-tailed test": 8
      },
      "written": {
        "one-tailed test | right-tailed test | left-tailed test": 16
      },
      "sources": [
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "one-to-one-property",
    "ja": {
      "term": "一対一の性質",
      "reading": "いちたいいちのせいしつ",
      "alt": []
    },
    "en": {
      "term": "one-to-one property",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「一対一の性質」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書では、指数関数・対数関数が単調であることから直接 aˣ = aʸ ⇔ x = y を使い、名前をつけない。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "a > 0, a ≠ 1 のとき、aˣ = aʸ ⇔ x = y、log_a M = log_a N ⇔ M = N（M, N > 0）であること。",
    "definition_en": "For a > 0, a ≠ 1: aˣ = aʸ exactly when x = y, and log_a M = log_a N exactly when M = N (M, N > 0).",
    "latex": "a^x = a^y \\iff x = y",
    "spoken_en": "a to the x equals a to the y if and only if x equals y",
    "examples": [
      {
        "en": "Both sides are powers of 3, so by the one-to-one property the exponents are equal.",
        "ja": "両辺とも 3 の累乗なので、一対一の性質から指数が等しい。",
        "register": "spoken"
      },
      {
        "en": "Use the one-to-one property to solve log₅(3x − 1) = log₅(x + 7).",
        "ja": "一対一の性質を用いて log₅(3x − 1) = log₅(x + 7) を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "書き言葉は大半が OpenStax Algebra and Trigonometry で、残りは OpenStax Intermediate Algebra。話し言葉には出てこない。"
    ],
    "related": [
      "one-to-one",
      "exponential-equation",
      "logarithmic-equation",
      "write-with-the-same-base"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "one-to-one property": 51
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "one-to-one",
    "ja": {
      "term": "1 対 1",
      "reading": "いちたいいち",
      "alt": [
        "単射"
      ]
    },
    "en": {
      "term": "one-to-one",
      "alt": [],
      "variants": [
        {
          "term": "injective",
          "register": "spoken",
          "note": "大学（MIT OCW）の言い方。話し言葉はすべて MIT OCW、書き言葉はすべて MIT の講義ノート。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Linear Algebra",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "異なる x には異なる値が対応する（x₁ ≠ x₂ なら f(x₁) ≠ f(x₂)）こと。",
    "definition_en": "Of a function: sending different inputs to different outputs.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's one-to-one, so it passes the horizontal line test and has an inverse.",
        "ja": "1 対 1 なので、水平線テストに通り、逆関数をもつ。",
        "register": "spoken"
      },
      {
        "en": "Show that f(x) = x³ is one-to-one.",
        "ja": "f(x) = x³ が 1 対 1 であることを示せ。",
        "register": "written"
      },
      {
        "en": "A linear transformation is one-to-one exactly when its null space contains only the zero vector.",
        "ja": "線形変換が 1 対 1（単射）になるのは、零空間が 0 ベクトルだけのときに限る。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "米国の教科書（OpenStax Algebra and Trigonometry・Calculus Volume 1）は、グラフと水平な直線の交点が 1 つ以下かどうかで調べる horizontal line test を使う。",
      "線形代数では、線形写像 T が 1 対 1 であることは T(x) = 0 となる x が 0 だけ（ker T = {0}）であることと同じ。Nicholson（7.2・7.3）は one-to-one と onto で単射・全射を言う。"
    ],
    "related": [
      "find-the-inverse",
      "composite-function",
      "one-to-one-property",
      "onto",
      "null-space",
      "isomorphism"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "単射",
        "en": "Injective function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "7.2 Kernel and Image of a Linear Transformation"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "one-to-one": 29,
        "injective": 13
      },
      "written": {
        "one-to-one": 283,
        "injective": 12
      },
      "sources": [
        "khan-algebra",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "one-variable-data",
    "ja": {
      "term": "一変量データ",
      "reading": "いちへんりょうでーた",
      "alt": []
    },
    "en": {
      "term": "one-variable data",
      "alt": [
        "univariate data"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Integrated Math 1"
      ]
    },
    "definition_ja": "1 つの変数だけを記録したデータ。クラス全員の身長のように、対象 1 つにつき値が 1 つある。",
    "definition_en": "Data that record just one variable for each individual, such as the heights of all the students in a class.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This is one-variable data, just heights, so a histogram or a box plot works well.",
        "ja": "これは身長だけの一変量データだから、ヒストグラムか箱ひげ図がいい。",
        "register": "spoken"
      },
      {
        "en": "Make a dot plot of the one-variable data below and describe its center and spread.",
        "ja": "次の一変量データのドットプロットをかき、中心と散らばりを述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「一変量」は変数（variable）が 1 つという意味で、データの値が 1 個という意味ではない。univariate data とも言う。",
      "1 つの対象について 2 つの変数を組にしたデータは bivariate data（エントリ two-variable-data）。一変量データはヒストグラム・箱ひげ図・ドットプロットで表し、散布図は使わない。"
    ],
    "related": [
      "two-variable-data",
      "histogram",
      "box-plot",
      "dot-plot",
      "statistical-variable"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Unit 1・Topic 1.1〜1.3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "univariate data": 5
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは AP Statistics の CED の呼び方 one-variable data（topic 1.1・topic 1.2・topic 1.3・topic 1.4・topic 1.5・topic 1.6・topic 1.7・topic 1.8・topic 1.9・topic 1.10・topic 1.11・topic 1.12・topic 1.13）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "onto",
    "ja": {
      "term": "全射",
      "reading": "ぜんしゃ",
      "alt": []
    },
    "en": {
      "term": "surjective",
      "alt": [
        "onto"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "写像の値域が行き先の空間全体になること。どの b についても T(x) = b となる x がある。",
    "definition_en": "Of a map: its range is the whole target space, so every b has some x with T(x) = b.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The map is surjective, or onto, if every vector in the target gets hit.",
        "ja": "行き先のどのベクトルにも届くなら、その写像は全射だ。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the linear transformation is one-to-one, onto, both, or neither.",
        "ja": "線形写像が単射か、全射か、その両方か、どちらでもないか判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "onto はふつうの前置詞（project onto the line）と同じ語。",
      "Nicholson（7.2・7.3）は onto と書く。"
    ],
    "related": [
      "one-to-one",
      "isomorphism",
      "linear-transformation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "全射",
        "en": "Surjective function"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "7.2 Kernel and Image of a Linear Transformation"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "surjective": 8
      },
      "written": {
        "surjective": 16
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "open-interval",
    "ja": {
      "term": "開区間",
      "reading": "かいくかん",
      "alt": []
    },
    "en": {
      "term": "open interval",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "両端の数を含まない区間。a < x < b を満たす実数 x 全体で、(a, b) と書く。",
    "definition_en": "An interval that does not include its endpoints: all real numbers x with a < x < b, written (a, b).",
    "latex": "(a, b) = \\{\\, x \\mid a < x < b \\,\\}",
    "spoken_en": "the open interval from a to b",
    "examples": [
      {
        "en": "The function is increasing on the open interval from 1 to 4, so we don't include the endpoints.",
        "ja": "この関数は開区間 1 < x < 4 で増加するから、端の点は含めない。",
        "register": "spoken"
      },
      {
        "en": "Show that f(x) = x³ − 3x is decreasing on the open interval (−1, 1).",
        "ja": "f(x) = x³ − 3x が開区間 −1 < x < 1 で減少することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "開区間 (a, b) は点の座標 (a, b) と同じ見た目になる。どちらの意味か文脈で判断し、口頭では the open interval from a to b と言えば紛れない。",
      "∞ を端にもつ区間 (a, ∞) は丸括弧で書く。∞ は数ではないので角括弧 [ ] にはしない。"
    ],
    "related": [
      "closed-interval",
      "interval",
      "endpoint",
      "increasing-and-decreasing",
      "interval-notation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.1・5.6・10.13（open interval）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Interval_(mathematics)",
        "note": "節 Open and closed intervals: open interval を太字で定義（日本語版「区間 (数学)」も同じ）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "開区間の語は出てこない。数学II (5) 微分・積分の考え: 区間を制限した関数の最大値・最小値。数学III (3) 積分法: 閉区間 [a, b]（level.jp）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "open interval": 121
      },
      "written": {
        "open interval": 64
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "opens-downward",
    "ja": {
      "term": "上に凸",
      "reading": "うえにとつ",
      "alt": []
    },
    "en": {
      "term": "opens downward",
      "alt": [
        "opens down"
      ],
      "uk": null,
      "variants": [
        {
          "term": "concave down",
          "register": "both",
          "note": "微積分での言い方（凹凸）で、書き言葉では最も多い（ほとんど OpenStax Calculus）。話し言葉でも最も多く、MIT OCW・Professor Leonard・Khan Academy が中心。"
        }
      ],
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「上に凸」は放物線にも一般の曲線にも使うが、英語では放物線には opens downward ／ opens down（開く向き）、一般の曲線には concave down（凹凸）と言い分ける。用例コーパスでは話し言葉・書き言葉とも opens downward のほうが opens down より多い。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2",
        "AP Calculus AB"
      ]
    },
    "definition_ja": "グラフが上にふくらんだ形であること。二次関数 y = ax² + bx + c では a < 0 のとき放物線が上に凸で、頂点が最も高い点になる。",
    "definition_en": "Said of a parabola y = ax² + bx + c with a < 0: it is shaped like an upside-down U and its vertex is the highest point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The leading coefficient is negative, so this parabola opens downward and the vertex is the max.",
        "ja": "最高次の係数が負なので、この放物線は上に凸で、頂点で最大になる。",
        "register": "spoken"
      },
      {
        "en": "Since a = −3 < 0, the graph opens downward, and f has a maximum value at its vertex.",
        "ja": "a = −3 < 0 よりグラフは上に凸で、f は頂点で最大値をとる。",
        "register": "written"
      },
      {
        "en": "Because f″(x) < 0 for x < 0, the graph of f is concave down on (−∞, 0).",
        "ja": "x < 0 で f″(x) < 0 だから、f のグラフは x < 0 で上に凸である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「上に凸」を convex upward と直訳しない（用例コーパスには出てこない）。放物線は opens downward ／ opens down、一般の曲線は concave down と言う。",
      "日本語は「上」、英語は down（下に開く）で、上下の単語が逆になる。"
    ],
    "related": [
      "concavity",
      "opens-upward",
      "vertex",
      "maximum",
      "check-the-concavity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "concave down": 37,
        "opens downward": 16,
        "opens down": 3
      },
      "written": {
        "opens downward": 28,
        "concave down": 59,
        "opens down": 9
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "opens-upward",
    "ja": {
      "term": "下に凸",
      "reading": "したにとつ",
      "alt": []
    },
    "en": {
      "term": "opens upward",
      "alt": [
        "opens up"
      ],
      "uk": null,
      "variants": [
        {
          "term": "concave up",
          "register": "both",
          "note": "微積分での言い方（凹凸）。書き言葉はほとんど OpenStax Calculus。話し言葉は Khan Academy・Professor Leonard・MIT OCW が中心。"
        }
      ],
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「下に凸」は放物線にも一般の曲線にも使うが、英語では放物線には opens upward ／ opens up（開く向き）、一般の曲線には concave up（凹凸。CED topic 5.6）と言い分ける。用例コーパスでは opens upward と opens up を話し言葉・書き言葉とも同じくらい使う。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2",
        "AP Calculus AB"
      ]
    },
    "definition_ja": "グラフが下にふくらんだ形であること。二次関数 y = ax² + bx + c では a > 0 のとき放物線が下に凸で、頂点が最も低い点になる。",
    "definition_en": "Said of a parabola y = ax² + bx + c with a > 0: it is U-shaped and its vertex is the lowest point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Since a is positive, the parabola opens up, so the vertex gives the minimum.",
        "ja": "a が正なので放物線は下に凸で、頂点で最小になる。",
        "register": "spoken"
      },
      {
        "en": "The graph of y = 2x² − 8x + 3 opens upward, so the function has a minimum value.",
        "ja": "y = 2x² − 8x + 3 のグラフは下に凸なので、この関数は最小値をもつ。",
        "register": "written"
      },
      {
        "en": "Since f″(x) > 0 on (1, ∞), f is concave up on that interval.",
        "ja": "区間 x > 1 で f″(x) > 0 なので、f はこの区間で下に凸である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「下に凸」を convex downward と直訳しない。放物線は opens upward ／ opens up、一般の曲線は concave up と言う。",
      "日本語は「下」、英語は up（上に開く）で、上下の単語が逆になる。上に凸・下に凸を取り違えやすい。"
    ],
    "related": [
      "concavity",
      "opens-downward",
      "vertex",
      "minimum",
      "check-the-concavity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.6（concave up）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "concave up": 48,
        "opens up": 27,
        "opens upward": 21
      },
      "written": {
        "opens upward": 47,
        "concave up": 60,
        "opens up": 31
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "opposite-angle",
    "ja": {
      "term": "対角",
      "reading": "たいかく",
      "alt": []
    },
    "en": {
      "term": "opposite angle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "四角形で、向かい合う角。三角形では、ある辺の向かいにある角をその辺の対角という。",
    "definition_en": "In a quadrilateral, the angle across from a given angle (their vertices are not next to each other). In a triangle, the angle across from a given side.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Opposite angles in a parallelogram are equal, so if this angle is 70 degrees, the opposite angle is 70 degrees too.",
        "ja": "平行四辺形の対角は等しいから、この角が 70° なら、向かいの角も 70° だよ。",
        "register": "spoken"
      },
      {
        "en": "In quadrilateral ABCD, both pairs of opposite angles are equal. Prove that ABCD is a parallelogram.",
        "ja": "四角形 ABCD で、2 組の対角がそれぞれ等しい。四角形 ABCD は平行四辺形であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「対角」と「対角線」を取り違えない。opposite angle は角、対角線は diagonal（エントリ diagonal）で線分。",
      "対頂角（vertical angles、エントリ vertical-angles）とは別。対頂角は 2 直線が交わってできる向かい合う角で、四角形の対角とは違う。ただし CK-12 Geometry（1.17）は、2 直線が交わってできる対頂角を a pair of opposite angles と説明するので、聞くときは文脈で見分ける。",
      "円に内接する四角形では、対角の和が 180° になる（エントリ cyclic-quadrilateral）。平行四辺形の対角は等しいので、区別する。"
    ],
    "related": [
      "opposite-side",
      "parallelogram",
      "vertical-angles",
      "cyclic-quadrilateral",
      "diagonal"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.17 Vertical Angles（a pair of opposite angles created by intersecting lines）・5.9 Parallelograms（Opposite Angles Theorem）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "2.12 Proofs about Quadrilaterals（opposite angles in a parallelogram）・7.4 Quadrilaterals in Circles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "opposite angles of | opposite angles in | pairs of opposite angles | its opposite angle": 1
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-10-09"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 opposite angles of | opposite angles in | pairs of opposite angles | its opposite angle（Geometry 7.4 Quadrilaterals in Circles・Grade 6 1.4 Parallelograms・Geometry 2.10 Practicing Proofs・計 7 件）。register は主張しない。",
        "raised": "2026-10-09"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "opposite-side",
    "ja": {
      "term": "対辺",
      "reading": "たいへん",
      "alt": []
    },
    "en": {
      "term": "opposite side",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "四角形で、向かい合う辺。三角形では、ある角の向かいにある辺をその角の対辺という。",
    "definition_en": "In a quadrilateral, the side across from a given side (the two sides share no vertex). In a triangle, the side across from a given angle.",
    "latex": "\\sin\\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}",
    "spoken_en": "sine theta equals opposite over hypotenuse",
    "examples": [
      {
        "en": "Look at angle A. Its opposite side is the one it's facing, side BC.",
        "ja": "∠A を見て。対辺は ∠A が向いている辺、つまり辺 BC だよ。",
        "register": "spoken"
      },
      {
        "en": "Prove that the opposite sides of parallelogram ABCD are equal, that is, AB = DC and AD = BC.",
        "ja": "平行四辺形 ABCD の 2 組の対辺はそれぞれ等しい、すなわち AB = DC、AD = BC であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "三角比では、角 θ の向かいの辺を単に opposite と呼ぶ（sin θ = opposite/hypotenuse、エントリ sohcahtoa）。どちらの鋭角から見るかで、対辺と隣辺（エントリ adjacent-side）が入れ替わる。",
      "数の opposite は「反数」（エントリ opposite）。the opposite of 5 は −5 のことで、図形の対辺とは関係がない。",
      "「対辺」「対角」の「対」は opposite だが、「対角線」は diagonal（エントリ diagonal）という別の語。"
    ],
    "related": [
      "side",
      "opposite-angle",
      "parallelogram",
      "adjacent-side",
      "sohcahtoa"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "opposite side": 104
      },
      "written": {
        "opposite side": 87
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "opposite-vector",
    "ja": {
      "term": "逆ベクトル",
      "reading": "ぎゃくべくとる",
      "alt": []
    },
    "en": {
      "term": "opposite vector",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスには opposite vector ／ negative of a vector とも出てこない）。−v は negative v と読み、名前をつけずに the vector in the opposite direction と説明する。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトル a と大きさが同じで向きが反対のベクトル。−a と書く。",
    "definition_en": "The vector with the same magnitude as a but the opposite direction, written −a.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Negative v points the other way but has the same length.",
        "ja": "−v は反対向きで、長さは同じ。",
        "register": "spoken"
      },
      {
        "en": "Draw −a and a − b for the vectors shown.",
        "ja": "図のベクトルについて −a と a − b をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector",
      "vector-subtraction",
      "zero-vector"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "opposite",
    "ja": {
      "term": "反数",
      "reading": "はんすう",
      "alt": []
    },
    "en": {
      "term": "opposite",
      "alt": [
        "additive inverse"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある数に足すと 0 になる数。a の反数は −a で、数直線では 0 をはさんで反対側の、0 から同じ距離のところにある。",
    "definition_en": "For a number a, the number −a: it lies on the other side of zero at the same distance, and a + (−a) = 0.",
    "latex": "a + (-a) = 0",
    "spoken_en": "a plus the opposite of a equals zero",
    "examples": [
      {
        "en": "The opposite of 5 is negative 5, and the opposite of negative 5 is 5.",
        "ja": "5 の反数は −5 で、−5 の反数は 5 だ。",
        "register": "spoken"
      },
      {
        "en": "Find the opposite of each number: 8, −2.4, 0.",
        "ja": "次の数の反数を求めよ。8、−2.4、0",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "opposite は日常語（反対の・反対側）でもある。数の反数は the opposite of 5 の形で言う。",
      "反数と逆数（エントリ reciprocal）を取り違えない。5 の反数は −5、逆数は 1/5。反数は足して 0、逆数はかけて 1 になる数。",
      "−a はいつも負の数とは限らない。a = −3 なら −a = 3 で正の数。"
    ],
    "related": [
      "negative-number",
      "absolute-value",
      "reciprocal",
      "number-line",
      "sign"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "反数",
        "en": "Additive inverse"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the opposite of | its opposite": 133,
        "additive inverse": 7
      },
      "written": {
        "the opposite of | its opposite": 180,
        "additive inverse": 47
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "optimization-problem",
    "ja": {
      "term": "最適化問題",
      "reading": "さいてきかもんだい",
      "alt": [
        "最適化"
      ]
    },
    "en": {
      "term": "optimization problem",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学II・III）は、文章題の最大・最小を「最適化」とは呼ばない（解説の「最適化」は IoT の話だけ）。AP の CED（topic 5.10・5.11）と OpenStax Calculus は optimization problems と呼ぶ。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "条件のもとで、ある量を最大または最小にする値を求める問題。微分を使うほか、不等式の表す領域を使うもの（線形計画法。学習指導要領解説の数学II の例）もある。",
    "definition_en": "A problem of finding the values that make a quantity as large or as small as possible under given conditions, solved with derivatives or with regions defined by inequalities (linear programming, an example in 数学II).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In an optimization problem, write the thing you're maximizing as a function of one variable first.",
        "ja": "最適化問題では、最大にしたい量をまず 1 変数の関数で表す。",
        "register": "spoken"
      },
      {
        "en": "A farmer has 200 m of fencing to enclose a rectangular field along a river, with no fence on the river side. What dimensions maximize the area?",
        "ja": "川沿いの長方形の畑を 200 m の柵で囲む（川側には柵を置かない）。面積を最大にする縦と横の長さを求めよ。",
        "register": "written"
      },
      {
        "en": "A factory makes two products with limited labor and materials. How many of each should it make to maximize profit?",
        "ja": "ある工場は、限られた労働力と原料で 2 種類の製品を作る。利益を最大にするにはそれぞれいくつ作ればよいか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "absolute-extrema",
      "critical-point",
      "candidates-test",
      "lagrange-multipliers",
      "linear-programming",
      "feasible-region",
      "objective-function",
      "constraint",
      "mathematical-model"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "最適化問題",
        "en": "Optimization problem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.10・5.11（optimization problems）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 図形と方程式: 不等式の表す領域・線形計画法の問題"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "optimization problem": 15
      },
      "written": {
        "optimization problem": 33
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "order-matters",
    "ja": {
      "term": "順序を考える",
      "reading": "じゅんじょをかんがえる",
      "alt": []
    },
    "en": {
      "term": "order matters",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "取り出したものの並ぶ順番が違えば別の場合として数えること。順序を考えるなら順列、考えないなら組合せになる。",
    "definition_en": "Treating different orders of the same objects as different outcomes; when order matters you count permutations, and when it does not you count combinations.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Order matters here — first place and second place are different prizes — so it's a permutation.",
        "ja": "ここは順序を考える。1 位と 2 位は違う賞だから順列だ。",
        "register": "spoken"
      },
      {
        "en": "If order doesn't matter, divide by 3! to get rid of the repeats.",
        "ja": "順序を考えないなら、重なりを除くために 3! で割る。",
        "register": "spoken"
      },
      {
        "en": "Since the order does not matter, the number of ways to choose 3 of the 10 books is C(10, 3) = 120.",
        "ja": "順序を考えないので、10 冊から 3 冊を選ぶ方法は ₁₀C₃ = 120 通り。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は「（人が）順序を考える」だが、英語は order を主語にして order matters ／ order doesn't matter と言う。consider the order は直訳。order doesn't matter は、足し算・掛け算の交換法則の説明（OpenStax Prealgebra）や行列の積・積分の順序の話にも使う。",
      "順序を考えた組は ordered（ordered pair）、考えない組は unordered と形容詞でも言う。"
    ],
    "related": [
      "permutation",
      "combination",
      "arrange",
      "choose"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "交換法則の説明の order doesn't matter"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "order matters | order doesn't matter | order does not matter": 20
      },
      "written": {
        "order matters | order doesn't matter | order does not matter": 10
      },
      "sources": [
        "khan-middle",
        "mit-18.02",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "order-of-operations",
    "ja": {
      "term": "演算の順序",
      "reading": "えんざんのじゅんじょ",
      "alt": []
    },
    "en": {
      "term": "order of operations",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "式を計算するときの順序のきまり。かっこの中、累乗、乗法・除法、加法・減法の順に計算し、同じ順位の計算は左から順に行う。",
    "definition_en": "The agreed order for working out an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In 8 minus 2 times 3, the order of operations says multiply first, so it's 8 minus 6, which is 2.",
        "ja": "8 − 2 × 3 では、演算の順序でかけ算が先だから、8 − 6 で 2 になる。",
        "register": "spoken"
      },
      {
        "en": "Use the order of operations to evaluate 8 − 2 × 3 + 4².",
        "ja": "演算の順序にしたがって、8 − 2 × 3 + 4² を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "乗法と除法は同じ順位で、左から順に計算する。加法と減法も同じ。12 ÷ 3 × 2 は左から計算して 8 で、12 ÷ 6 = 2 ではない。",
      "米国の教材（CK-12 Algebra）は、演算の順序をよく PEMDAS（エントリ pemdas）と呼ぶと書く。PEMDAS の M が D より先という意味ではない。"
    ],
    "related": [
      "pemdas",
      "parentheses",
      "exponent",
      "the-four-operations",
      "evaluate"
    ],
    "sources": [
      {
        "type": "wikidata",
        "qid": "Q845118"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "order of operations": 68
      },
      "written": {
        "order of operations": 136
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "orientation",
    "ja": {
      "term": "向き",
      "reading": "むき",
      "alt": [
        "曲線の向き"
      ]
    },
    "en": {
      "term": "orientation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「向き」は一般語で、媒介変数表示の曲線が t の増加とともに進む向きにも、ベクトルの向きにも使う。英語の orientation は曲線（や曲面）の進む向きに限って使い、ベクトルの向きは direction。",
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II",
        "Calculus III"
      ]
    },
    "definition_ja": "媒介変数表示の曲線で、媒介変数 t が増えるときに点が進む向き。",
    "definition_en": "For a parametric curve, the direction in which the point moves as the parameter t increases.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Put arrows on the curve to show the orientation — which way it goes as t increases.",
        "ja": "曲線に矢印をかいて向きを示そう。t が増えるとどちらへ進むか。",
        "register": "spoken"
      },
      {
        "en": "Sketch the curve x = cos t, y = 2 sin t (0 ≤ t ≤ 2π) and indicate its orientation.",
        "ja": "曲線 x = cos t, y = 2 sin t（0 ≦ t ≦ 2π）をかき、向きを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "parametric-equations",
      "parameter"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "向き",
        "en": "Orientation (space)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "orientation of the curve": 16
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "origin",
    "ja": {
      "term": "原点",
      "reading": "げんてん",
      "alt": []
    },
    "en": {
      "term": "origin",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "数直線で 0 を表す点。座標平面では x 軸と y 軸が交わる点で、座標は (0, 0) である。",
    "definition_en": "The point 0 on a number line; on the coordinate plane, the point (0, 0) where the two axes cross.",
    "latex": "(0, 0)",
    "spoken_en": "zero comma zero",
    "examples": [
      {
        "en": "Start at the origin, go right 3, then go up 2.",
        "ja": "原点から右に 3、上に 2 進んで。",
        "register": "spoken"
      },
      {
        "en": "On a number line, point A is 7 units to the left of the origin. What number does point A represent?",
        "ja": "数直線上で、点 A は原点から左へ 7 の距離にある。点 A が表す数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "origin は日常語の「起源・出身」と同じ単語。数学では 1 つに決まった点なので、the origin と the を付けて言う。",
      "日本の問題文（共通テスト・センター試験）や学習指導要領解説は原点を文字 O（オー）で表す。数字の 0 ではない。"
    ],
    "related": [
      "number-line",
      "coordinates",
      "rectangular-coordinates",
      "distance",
      "absolute-value"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "原点 (数学)",
        "en": "Origin (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "問題文の「原点 O」"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "原点 O"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "origin": 498
      },
      "written": {
        "origin": 617
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "orthocenter",
    "ja": {
      "term": "垂心",
      "reading": "すいしん",
      "alt": []
    },
    "en": {
      "term": "orthocenter",
      "alt": [],
      "uk": "orthocentre",
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "三角形の 3 つの頂点から対辺（またはその延長）に下ろした垂線が交わる点。",
    "definition_en": "The point where the three altitudes of a triangle, or the lines containing them, meet.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "You only need two of the altitudes; where they cross is the orthocenter.",
        "ja": "高さ（垂線）は 2 本引けば十分。その交点が垂心。",
        "register": "spoken"
      },
      {
        "en": "Find the orthocenter of the triangle with vertices A(0, 0), B(6, 0), and C(2, 4).",
        "ja": "3 点 A(0, 0), B(6, 0), C(2, 4) を頂点とする三角形の垂心の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "頂点から対辺に下ろした垂線は英語で altitude（高さ）。垂心は鋭角三角形では内部、直角三角形では直角の頂点、鈍角三角形では外部にある。",
      "用例コーパスの orthocenter は話し言葉ではすべて The Organic Chemistry Tutor。書き言葉には出てこない。",
      "英国式のつづりは orthocentre。"
    ],
    "related": [
      "the-five-centers-of-a-triangle",
      "circumcenter",
      "centroid",
      "foot-of-the-perpendicular",
      "height"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "垂心",
        "en": "Orthocenter"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences#re_er",
        "note": "記事「American and British English spelling differences」の節 -re, -er（英国の綴りは centre、米国は center）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "orthocenter": 11
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "orthogonal-complement",
    "ja": {
      "term": "直交補空間",
      "reading": "ちょっこうほくうかん",
      "alt": []
    },
    "en": {
      "term": "orthogonal complement",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（orthogonal complement は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "部分空間 W のすべてのベクトルと直交するベクトル全体。W⊥ と書く。",
    "definition_en": "The set of all vectors orthogonal to every vector in a subspace W, written W⊥.",
    "latex": "W^{\\perp}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The null space is the orthogonal complement of the row space.",
        "ja": "零空間は行空間の直交補空間だ。",
        "register": "spoken"
      },
      {
        "en": "Find a basis for the orthogonal complement of W.",
        "ja": "W の直交補空間の基底を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "W⊥ は W perp と読む。",
      "用例コーパスでは話し言葉にわずかに出てくるだけ。"
    ],
    "related": [
      "orthogonal",
      "null-space",
      "row-space",
      "subspace"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "直交補空間",
        "en": "Orthogonal complement"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.1 Orthogonal Complements and Projections"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "orthogonal complement": 5
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 orthogonal complement（8.1 Orthogonal Complements and Projections・8.6 The Singular Value Decomposition・10.2 Orthogonal Sets of Vectors）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "orthogonal-projection",
    "ja": {
      "term": "正射影",
      "reading": "せいしゃえい",
      "alt": [
        "直交射影"
      ]
    },
    "en": {
      "term": "projection",
      "alt": [
        "orthogonal projection"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C",
        "大学"
      ],
      "us": [
        "Precalculus",
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "definition_ja": "点や図形から直線（平面・部分空間）に垂線を下ろし、その足に移すこと。またその像。",
    "definition_en": "Mapping a point or figure to the foot of the perpendicular on a line (plane, subspace); also the resulting image.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The projection of b onto a is the closest point to b on the line through a.",
        "ja": "b の a への正射影は、a を通る直線上で b に一番近い点。",
        "register": "spoken"
      },
      {
        "en": "Find the orthogonal projection of the point (4, 1) onto the line y = x.",
        "ja": "点 (4, 1) の直線 y = x への正射影を求めよ。",
        "register": "written"
      },
      {
        "en": "Project b onto the column space: that's the closest vector to b in the subspace.",
        "ja": "b を列空間へ正射影すると、部分空間の中で b にいちばん近いベクトルになる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの projection の話し言葉は大半が MIT 18.06（部分空間への射影）。orthogonal projection とはっきり言う例は無く、授業では単に projection と言う。",
      "ベクトルの正射影は vector-projection を参照。",
      "線形代数では、直線だけでなく部分空間 W への正射影 proj_W を考える。W の正規直交基底があれば、各基底ベクトルへの射影の和になる（Nicholson 8.1）。"
    ],
    "related": [
      "vector-projection",
      "dot-product",
      "perpendicular-lines",
      "orthogonal",
      "orthogonal-complement",
      "orthonormal-basis",
      "gram-schmidt-process"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.1 Orthogonal Complements and Projections"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "projection": 265
      },
      "written": {
        "projection": 86
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "orthogonal",
    "ja": {
      "term": "直交",
      "reading": "ちょっこう",
      "alt": [
        "直交する",
        "垂直"
      ]
    },
    "en": {
      "term": "orthogonal",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C",
        "大学"
      ],
      "us": [
        "Precalculus",
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "definition_ja": "2 つのベクトルの内積が 0 であること。0 でないベクトルなら互いに垂直になる。",
    "definition_en": "Of two vectors: having dot product zero; nonzero orthogonal vectors are perpendicular.",
    "latex": "\\vec{a} \\cdot \\vec{b} = 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "Their dot product is zero, so the vectors are orthogonal.",
        "ja": "内積が 0 なので、この 2 つのベクトルは直交している。",
        "register": "spoken"
      },
      {
        "en": "Find a value of k for which the vectors ⟨2, k⟩ and ⟨3, −1⟩ are orthogonal.",
        "ja": "ベクトル ⟨2, k⟩ と ⟨3, −1⟩ が直交するような k の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ベクトルの垂直は英語で orthogonal（線分・直線は perpendicular、エントリ perpendicular）。0 ベクトルはどのベクトルとも orthogonal だが、perpendicular とは言わない。",
      "用例コーパスの書き言葉はすべて OpenStax Calculus。"
    ],
    "related": [
      "perpendicular",
      "dot-product",
      "perpendicular-lines",
      "orthonormal-basis",
      "orthogonal-complement",
      "orthogonal-projection"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "直交",
        "en": "Orthogonality"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.3 Orthogonality"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "orthogonal": 301
      },
      "written": {
        "orthogonal": 96
      },
      "sources": [
        "khan-algebra",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "orthographic-projection",
    "ja": {
      "term": "投影図",
      "reading": "とうえいず",
      "alt": []
    },
    "en": {
      "term": "orthographic projection",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。中学校学習指導要領解説の投影図は、平面図（上から見た図）と立面図（前から見た図）で空間図形を表す図で、英語では IM Grade 7 が front view・top view と、見る向きで図を呼ぶ。見出しの orthographic projection はこの図法の英語の名前（英語版 Wikipedia の記事名。数学カテゴリから 5 段で、見出しを決める記事名の 4 段には当たらない）で、米国の中学・高校の教材（OpenStax・IM・CK-12）には出てこない。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "立体を正面から見た図（立面図）と真上から見た図（平面図）を組にして、その立体を表した図。",
    "definition_en": "A way to show a solid with flat drawings of what you see when you look at it straight on from different directions, such as a front view and a top view.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This orthographic projection shows a triangle from the front and a circle from the top, so the solid is a cone.",
        "ja": "この投影図は立面図が三角形で平面図が円だから、立体は円錐だね。",
        "register": "spoken"
      },
      {
        "en": "In the orthographic projection of a solid, the front view is a rectangle and the top view is a circle. Name the solid.",
        "ja": "ある立体の投影図で、立面図は長方形、平面図は円である。この立体の名前を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "立面図は front view、平面図は top view、横から見た図は side view。平面図の「平面」は plane（エントリ plane）の意味ではなく、真上から見た図のこと。",
      "綴りの似た orthogonal projection（正射影、エントリ orthogonal-projection）は、点や図形から直線・平面に垂線を下ろして写すことの名前で、図の名前の投影図とは別の語。"
    ],
    "related": [
      "solid",
      "net",
      "orthogonal-projection",
      "cross-section",
      "cone"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 B 図形: 投影図（上から見た図（平面図）や前から見た図（立面図））"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "1.7 Scale Drawings・1.12 Units in Scale Drawings（front view）、7.16 Applying Volume and Surface Area（top view）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Orthographic_projection",
        "note": "記事「Orthographic projection」（数学カテゴリから 5 段。見出しを決める記事名の 4 段には当たらない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "orthonormal-basis",
    "ja": {
      "term": "正規直交基底",
      "reading": "せいきちょっこうきてい",
      "alt": []
    },
    "en": {
      "term": "orthonormal basis",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "互いに直交し、どれも長さ 1 のベクトルからなる基底。",
    "definition_en": "A basis of mutually orthogonal vectors, each of length 1.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "With an orthonormal basis, the coordinates are just dot products.",
        "ja": "正規直交基底なら、座標は内積をとるだけで求まる。",
        "register": "spoken"
      },
      {
        "en": "Find an orthonormal basis for the subspace spanned by the given vectors.",
        "ja": "与えられたベクトルが張る部分空間の正規直交基底を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はほぼすべて MIT 18.06。"
    ],
    "related": [
      "basis",
      "orthogonal",
      "gram-schmidt-process",
      "qr-factorization"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正規直交基底",
        "en": "Orthonormal basis"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.1 Orthogonal Complements and Projections"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "orthonormal basis": 25
      },
      "sources": [
        "mit-18.03",
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "oscillate",
    "ja": {
      "term": "振動する",
      "reading": "しんどうする",
      "alt": []
    },
    "en": {
      "term": "oscillate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "数列や関数が、収束もせず、正・負の無限大にも発散しない。",
    "definition_en": "Of a sequence or function: to neither converge nor go to plus or minus infinity, typically bouncing between values.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Sine of one over x oscillates wildly near zero, so there's no limit.",
        "ja": "sin(1/x) は 0 の近くで激しく振動するので、極限はない。",
        "register": "spoken"
      },
      {
        "en": "The sequence (−1)ⁿ oscillates and therefore diverges.",
        "ja": "数列 (−1)ⁿ は振動するので発散する。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の oscillate は物理の振動も指すが、数列・極限の文脈では「収束しないで行ったり来たりする」の意味。"
    ],
    "related": [
      "divergence",
      "convergence"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.3（oscillate）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "oscillate": 110
      },
      "written": {
        "oscillate": 64
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "outer-function",
    "ja": {
      "term": "外側の関数",
      "reading": "そとがわのかんすう",
      "alt": []
    },
    "en": {
      "term": "outer function",
      "alt": [],
      "variants": [
        {
          "term": "outside function",
          "register": "spoken",
          "note": "話し言葉ではこちらが最も多い。話し言葉は NancyPi・Khan Academy・MIT OCW が中心で、書き言葉には少ない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "合成関数 f(g(x)) で、あとから施す外側の関数 f。",
    "definition_en": "In a composite function f(g(x)), the function f that is applied last.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The outside function is the fifth power, and the inside is 3x plus 1.",
        "ja": "外側の関数は 5 乗、内側は 3x + 1。",
        "register": "spoken"
      },
      {
        "en": "For y = (3x + 1)⁵, the outer function is u⁵ and the inner function is u = 3x + 1.",
        "ja": "y = (3x + 1)⁵ では、外側の関数は u⁵、内側の関数は u = 3x + 1 である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "inner-function",
      "chain-rule",
      "composite-function"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "outside function": 52,
        "outer function": 35
      },
      "written": {
        "outer function": 13,
        "outside function": 4
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "outlier",
    "ja": {
      "term": "外れ値",
      "reading": "はずれち",
      "alt": []
    },
    "en": {
      "term": "outlier",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Intro Statistics",
        "AP Statistics",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "ほかのデータから大きく離れた値。Q1 − 1.5 × IQR より小さいか Q3 + 1.5 × IQR より大きい値を外れ値とする決め方が多い。",
    "definition_en": "A data value that lies well apart from the others; a common rule calls a value an outlier when it is below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "That one point way out there is an outlier, and it pulls the mean up.",
        "ja": "あの 1 つだけ離れた点は外れ値で、平均値を引き上げている。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the data contain any outliers. Justify your answer.",
        "ja": "データに外れ値があるかどうかを判定せよ。理由も示せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "remove the outlier",
        "ja": "外れ値を除く"
      }
    ],
    "pitfalls": [
      "回帰直線を大きく動かす点は influential point（影響の大きい点）と言い、外れ値と区別する。"
    ],
    "related": [
      "interquartile-range",
      "box-plot",
      "median"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "外れ値",
        "en": "Outlier"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "outlier": 144
      },
      "written": {
        "outlier": 168
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.06",
        "mit-notes",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "overestimate-and-underestimate",
    "ja": {
      "term": "過大評価・過小評価",
      "reading": "かだいひょうか・かしょうひょうか",
      "alt": []
    },
    "en": {
      "term": "overestimate and underestimate",
      "alt": [],
      "variants": [
        {
          "term": "underestimate",
          "register": "both",
          "note": "話し言葉は Khan Academy が中心。話し言葉・書き言葉とも overestimate と同じくらい使い、書き言葉はほとんど OpenStax Calculus。"
        },
        {
          "term": "overestimate",
          "register": "both",
          "note": "話し言葉は Khan Academy が中心。話し言葉・書き言葉とも underestimate と同じくらい使い、書き言葉はほとんど OpenStax Calculus。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB"
      ]
    },
    "definition_ja": "近似値が真の値より大きいこと（過大評価）と小さいこと（過小評価）。リーマン和などの誤差の向きを言う。",
    "definition_en": "Whether an approximation comes out above the true value (overestimate) or below it (underestimate), as with Riemann sums.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "f is decreasing, so the left sum is an overestimate and the right sum is an underestimate.",
        "ja": "f は減少関数だから、左の和は過大評価、右の和は過小評価になる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では動詞としても使う（The left sum overestimates the integral. ＝ 左の和は積分を多めに見積もる）。「評価」と訳さず「多めの見積もり」と考えると分かりやすい。"
    ],
    "related": [
      "left-riemann-sum",
      "right-riemann-sum",
      "riemann-sum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "underestimate": 65,
        "overestimate": 51
      },
      "written": {
        "underestimate": 17,
        "overestimate": 11
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "p-series",
    "ja": {
      "term": "p 級数",
      "reading": "ぴーきゅうすう",
      "alt": []
    },
    "en": {
      "term": "p-series",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "級数 Σ1/nᵖ。p > 1 のとき収束し、p ≤ 1 のとき発散する。",
    "definition_en": "The series Σ1/nᵖ, which converges when p > 1 and diverges when p ≤ 1.",
    "latex": "\\sum_{n=1}^{\\infty} \\frac{1}{n^p}",
    "spoken_en": "the sum of one over n to the p",
    "examples": [
      {
        "en": "It's a p-series with p equal to 3/2, so it converges.",
        "ja": "p = 3/2 の p 級数だから収束する。",
        "register": "spoken"
      },
      {
        "en": "Determine whether Σ 1/n^0.9 is a convergent p-series.",
        "ja": "Σ 1/n^0.9 が収束する p 級数かどうか答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "p = 1 は調和級数（harmonic series）。"
    ],
    "related": [
      "harmonic-series",
      "integral-test",
      "comparison-test"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.5（p-series）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "p-series": 71
      },
      "written": {
        "p-series": 40
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "p-value",
    "ja": {
      "term": "p 値",
      "reading": "ぴーち",
      "alt": []
    },
    "en": {
      "term": "p-value",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "帰無仮説が正しいとしたとき、観測したデータと同じかそれ以上に極端な結果が起こる確率。",
    "definition_en": "The probability, assuming the null hypothesis is true, of a result at least as extreme as the one observed.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A small p-value means our data would be really unusual if the null hypothesis were true.",
        "ja": "p 値が小さいとは、帰無仮説が正しければ、このデータはとても珍しいということ。",
        "register": "spoken"
      },
      {
        "en": "The p-value is 0.03. What is your conclusion at the 5% significance level?",
        "ja": "p 値は 0.03 である。有意水準 5% で結論を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "p 値は「帰無仮説が正しい確率」ではない。",
      "OpenStax Introductory Statistics は p 値と有意水準を比べて判断する（p-value 475 件）。"
    ],
    "related": [
      "significance-level",
      "test-statistic",
      "hypothesis-testing",
      "reject"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "p-value": 79
      },
      "written": {
        "p-value": 475
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "openstax-introstats",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "paired-t-test",
    "ja": {
      "term": "対応のある t 検定",
      "reading": "たいおうのあるてぃーけんてい",
      "alt": [
        "対応のある標本"
      ]
    },
    "en": {
      "term": "paired samples",
      "alt": [
        "paired t-test",
        "matched pairs t-test"
      ],
      "uk": null,
      "variants": [
        {
          "term": "matched pairs",
          "register": "written",
          "note": "OpenStax Introductory Statistics の言い方（matched or paired samples）。実験計画の matched pairs design は除いて数えた。話し言葉にはほとんど出てこず、書き言葉は大半が OpenStax Introductory Statistics。"
        }
      ],
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語は検定の名前。OpenStax Introductory Statistics は検定に名前を付けず、標本の種類（Matched or Paired Samples。節の名前）で呼ぶ。AP Statistics の CED（2026 年版 topic 4.4）は matched pairs design の差に対する one-sample t-test として扱う。paired t-test は用例コーパスには出てこない。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "同じ対象を 2 回測るなど、2 つの標本の値が組になっているとき、組ごとの差の平均が 0 かどうかを t 分布で検定する方法。",
    "definition_en": "A t-test for data that come in pairs (such as before-and-after measurements on the same subjects): it tests whether the mean of the paired differences is zero.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These are the same students before and after, so work with the differences: it's paired data.",
        "ja": "前後で同じ生徒なので、差で考える。対応のあるデータだ。",
        "register": "spoken"
      },
      {
        "en": "Test whether the mean difference between the paired samples is zero.",
        "ja": "対応のある標本の差の平均が 0 かどうかを検定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 つの独立な標本の平均の差を比べる 2 標本 t 検定（two-sample t-test）と取り違えない。差を 1 つの標本とみなすので、計算は 1 標本 t 検定と同じ。",
      "matched pairs design（対応のある実験計画）は同じ語で実験の組み方を指す。"
    ],
    "related": [
      "one-sample-t-test",
      "two-sample-t-test",
      "t-distribution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 4.2・4.4（matched pairs design）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "matched pairs !design": 1
      },
      "written": {
        "paired samples": 16,
        "matched pairs !design": 12
      },
      "sources": [
        "khan-ap-stats",
        "mit-notes",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "parabola",
    "ja": {
      "term": "放物線",
      "reading": "ほうぶつせん",
      "alt": []
    },
    "en": {
      "term": "parabola",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "二次関数 y = ax² のグラフのような形の曲線。軸について線対称で、軸との交点を頂点という。",
    "definition_en": "A U-shaped curve like the graph of y = ax². It is symmetric about its axis of symmetry, and the point where it turns around is its vertex.",
    "latex": "y = ax^2",
    "spoken_en": "y equals a x squared",
    "examples": [
      {
        "en": "This parabola opens upward, so the vertex is the lowest point on the graph.",
        "ja": "この放物線は下に凸だから、頂点がグラフのいちばん低い点だね。",
        "register": "spoken"
      },
      {
        "en": "The parabola y = ax² passes through the point (−2, 8). Find the value of a.",
        "ja": "放物線 y = ax² が点 (−2, 8) を通るとき、a の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "発音は日本語の「パラボラ」と違い、2 つ目の音節 ra を強く読む（Merriam-Webster: pə-ˈra-bə-lə）。",
      "a > 0 なら opens upward（日本語の「下に凸」）、a < 0 なら opens downward（「上に凸」）。上下の語が日本語と逆になる（エントリ opens-upward、opens-downward）。",
      "放物線の軸は axis of symmetry、頂点は vertex（エントリ axis-of-symmetry、vertex）。axis だけでは x 軸・y 軸と紛れる。"
    ],
    "related": [
      "quadratic-function",
      "vertex",
      "axis-of-symmetry",
      "opens-upward",
      "opens-downward"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "放物線",
        "en": "Parabola"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「parabola」",
        "url": "https://www.merriam-webster.com/dictionary/parabola",
        "note": "発音 pə-ˈra-bə-lə（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "parabola": 334
      },
      "written": {
        "parabola": 728
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "paragraph-proof",
    "ja": {
      "term": "段落証明",
      "reading": "だんらくしょうめい",
      "alt": []
    },
    "en": {
      "term": "paragraph proof",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「段落証明」は学習指導要領解説に無い、本プロジェクトの訳語。日本の証明はふつうこの形（文章で書く）なので、わざわざ名前を付けない。CK-12 Geometry は paragraph proof と呼ぶ（2.15 Proofs- Angle Pairs and Segments、4 件）。",
    "domains": [
      "proof",
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "各段階の主張とその理由を、表や図にせず文章でつなげて書く証明の書き方。二段組みの証明やフローチャート証明と区別するときの呼び名。",
    "definition_en": "A proof written in ordinary sentences, giving each step and its reason in prose rather than in a table or diagram.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "You don't need the table for this one. Just write a paragraph proof and explain each step in a sentence.",
        "ja": "この問題は表にしなくていい。段落証明で、各ステップを 1 文ずつ説明すればいい。",
        "register": "spoken"
      },
      {
        "en": "Write a paragraph proof that the base angles of an isosceles triangle are congruent.",
        "ja": "二等辺三角形の 2 つの底角が等しいことを、文章で証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の答案の証明はほぼこの形。英語で書くときは、理由を because … や by the … theorem の形で文の中に入れる。",
      "用例コーパス・IM とも paragraph proof は出てこない。"
    ],
    "related": [
      "two-column-proof",
      "flowchart-proof",
      "proof"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.15 Proofs- Angle Pairs and Segments"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 paragraph proof（CK-12 Geometry 2.15 Proofs- Angle Pairs and Segments・計 4 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "parallel-lines",
    "ja": {
      "term": "平行な直線",
      "reading": "へいこうなちょくせん",
      "alt": [
        "2 直線の平行"
      ]
    },
    "en": {
      "term": "parallel lines",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "geometry"
    ],
    "level": {
      "jp": [
        "中2",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "交わらない 2 直線。座標平面では傾きが等しい（y 軸に平行な場合を除く）。",
    "definition_en": "Two lines that never meet; on the coordinate plane, lines with equal slopes (apart from vertical lines).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Parallel lines have the same slope but different y-intercepts.",
        "ja": "平行な 2 直線は、傾きが同じで y 切片が違う。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of the line through (2, 1) parallel to 3x − y + 4 = 0.",
        "ja": "点 (2, 1) を通り、直線 3x − y + 4 = 0 に平行な直線の方程式を求めよ。",
        "register": "written"
      },
      {
        "en": "When a transversal cuts two parallel lines, the alternate interior angles are congruent.",
        "ja": "平行な 2 直線に 1 本の直線が交わるとき、錯角は等しい。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 直線 a₁x + b₁y + c₁ = 0, a₂x + b₂y + c₂ = 0 の平行条件 a₁b₂ − a₂b₁ = 0 は、米国では傾きが等しいことで説明することが多い（OpenStax Algebra and Trigonometry も傾きで扱う）。"
    ],
    "related": [
      "equation-of-a-line",
      "intersect",
      "perpendicular-lines",
      "transversal",
      "alternate-interior-angles-theorem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "parallel lines": 86
      },
      "written": {
        "parallel lines": 58
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "parallel-vectors",
    "ja": {
      "term": "平行なベクトル",
      "reading": "へいこうなべくとる",
      "alt": []
    },
    "en": {
      "term": "parallel vectors",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "向きが同じか反対の 2 つの 0 でないベクトル。b = ka（k は 0 でない実数）と書ける。",
    "definition_en": "Two nonzero vectors that point in the same or opposite directions, so that b = ka for some nonzero real number k.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Check if one is a multiple of the other; if it is, the vectors are parallel.",
        "ja": "一方が他方の実数倍か調べる。そうなら平行。",
        "register": "spoken"
      },
      {
        "en": "Find k so that ⟨2, k⟩ and ⟨6, −9⟩ are parallel vectors.",
        "ja": "⟨2, k⟩ と ⟨6, −9⟩ が平行になるような k を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の教科書の「平行条件」（a ∥ b ⇔ b = ka）は、英語では条件に名前をつけず、one vector is a scalar multiple of the other のように言う（scalar multiple of は用例コーパスに少し出てくる）。"
    ],
    "related": [
      "scalar-multiplication",
      "vector",
      "orthogonal-projection"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "parallel vectors": 5
      },
      "written": {
        "parallel vectors": 8
      },
      "sources": [
        "mit-18.02",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは OpenStax の呼び方 parallel vectors（本文 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "parallel",
    "ja": {
      "term": "平行",
      "reading": "へいこう",
      "alt": [
        "平行である"
      ]
    },
    "en": {
      "term": "parallel",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "同じ平面上の 2 直線が、どこまでのばしても交わらないこと。直線 AB と CD が平行であることを、中学校学習指導要領の〔用語・記号〕どおり AB // CD と書く（英語の教材の記号は ∥。エントリ parallel-sign）。",
    "definition_en": "Two lines in the same plane are parallel if they never meet, no matter how far you extend them.",
    "latex": "AB \\parallel CD",
    "spoken_en": "A B is parallel to C D",
    "examples": [
      {
        "en": "Opposite sides of a rectangle are parallel, so they never meet.",
        "ja": "長方形の向かい合う辺は平行だから、交わることはないよ。",
        "register": "spoken"
      },
      {
        "en": "Show that the line y = 3x + 2 is parallel to the line y = 3x − 5.",
        "ja": "直線 y = 3x + 2 と直線 y = 3x − 5 が平行であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "前置詞は to。AB ∥ CD は AB is parallel to CD。平行な直線は parallel lines（エントリ parallel-lines）。",
      "parallelogram（平行四辺形）の parallel は同じ語。向かい合う 2 組の辺がそれぞれ parallel な四角形という意味。"
    ],
    "related": [
      "parallel-lines",
      "perpendicular",
      "line",
      "quadrilateral",
      "alternate-interior-angles-theorem"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）〔用語・記号〕",
        "note": "中1 B 図形: //"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "parallel !lines !line": 278
      },
      "written": {
        "parallel !lines !line": 424
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "parallelogram",
    "ja": {
      "term": "平行四辺形",
      "reading": "へいこうしへんけい",
      "alt": []
    },
    "en": {
      "term": "parallelogram",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "2 組の向かい合う辺がそれぞれ平行な四角形。向かい合う辺の長さ、向かい合う角の大きさはそれぞれ等しく、2 本の対角線はそれぞれの中点で交わる。",
    "definition_en": "A quadrilateral in which each pair of opposite sides is parallel. Its opposite sides are equal, its opposite angles are equal, and its diagonals bisect each other.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a parallelogram, opposite sides are parallel and the same length, so if this side is 7, the side across from it is 7 too.",
        "ja": "平行四辺形では向かい合う辺は平行で長さも同じだから、この辺が 7 なら、向かいの辺も 7 だね。",
        "register": "spoken"
      },
      {
        "en": "In parallelogram ABCD, m∠A = 65°. Find m∠B, m∠C, and m∠D.",
        "ja": "平行四辺形 ABCD で ∠A = 65° のとき、∠B、∠C、∠D の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "長方形・ひし形・正方形はどれも平行四辺形の特別な場合（エントリ rectangle、rhombus、square-shape）。",
      "台形の定義によっては、平行四辺形も台形に入る（エントリ trapezoid）。IM Geometry は台形を平行な辺が「少なくとも 1 組」の四角形とし、CK-12 Geometry は「ちょうど 1 組」として平行四辺形を除く。",
      "「対角線がそれぞれの中点で交わる」は英語で the diagonals bisect each other（エントリ bisect）。"
    ],
    "related": [
      "quadrilateral",
      "rectangle",
      "rhombus",
      "trapezoid",
      "conditions-for-a-parallelogram"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "平行四辺形",
        "en": "Parallelogram"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 6.17 Lines in Triangles の練習問題（Unit 6 Lesson 14 から）: 台形は a quadrilateral with at least one pair of parallel sides"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.7 Quadrilateral Classification・5.13 Trapezoids: 台形は a quadrilateral with exactly one pair of parallel sides"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "parallelogram": 137
      },
      "written": {
        "parallelogram": 78
      },
      "sources": [
        "khan-middle",
        "mit-18.02",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "parameter",
    "ja": {
      "term": "媒介変数",
      "reading": "ばいかいへんすう",
      "alt": [
        "パラメータ"
      ]
    },
    "en": {
      "term": "parameter",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions",
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III",
        "数C"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "x と y をそれぞれ別の変数（t など）の式で表すときの、その変数。",
    "definition_en": "An extra variable, often t, in terms of which both x and y are written.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Here t is the parameter. As t goes from 0 to 2π, the point goes once around the circle.",
        "ja": "ここでは t が媒介変数。t が 0 から 2π まで動くと、点は円を 1 周する。",
        "register": "spoken"
      },
      {
        "en": "Eliminate the parameter to find an equation of the curve in x and y.",
        "ja": "媒介変数を消去して、曲線の方程式を x と y で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の parameter は、統計の母数（population parameter）や、式に含まれる定数（y = ax² の a）も指す。「媒介変数」の意味のときは parametric equations の文脈で使う。"
    ],
    "related": [
      "point-of-tangency"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "媒介変数",
        "en": "Parameter"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "parameter t": 15
      },
      "written": {
        "parameter t": 31
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "parametric-equations",
    "ja": {
      "term": "媒介変数表示",
      "reading": "ばいかいへんすうひょうじ",
      "alt": [
        "媒介変数方程式"
      ]
    },
    "en": {
      "term": "parametric equations",
      "alt": [],
      "variants": [
        {
          "term": "parametric curve",
          "register": "both",
          "note": "曲線のほうを指す言い方で、件数は少ない。話し言葉・書き言葉とも parametric equations よりずっと少なく、書き言葉はほとんど OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "ja.alt の「媒介変数方程式」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "functions",
      "calculus"
    ],
    "level": {
      "jp": [
        "数III",
        "数C"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "曲線上の点の座標 x, y を、媒介変数 t の関数 x = f(t), y = g(t) で表したもの。",
    "definition_en": "A pair of equations x = f(t), y = g(t) that describe a curve in terms of a parameter t.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "With parametric equations, x and y both depend on t, like x = cos t, y = sin t.",
        "ja": "媒介変数表示では、x = cos t, y = sin t のように x も y も t で決まる。",
        "register": "spoken"
      },
      {
        "en": "Sketch the curve given by the parametric equations x = t², y = t − 1.",
        "ja": "媒介変数表示 x = t², y = t − 1 で表される曲線をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語は 2 つの式の組として複数形 parametric equations で言う。"
    ],
    "related": [
      "parameter",
      "derivative-of-a-parametric-curve",
      "cycloid",
      "vector-valued-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 9.1（parametric equations）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「媒介変数方程式」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "parametric curve": 11,
        "parametric equations": 43
      },
      "written": {
        "parametric equations": 258,
        "parametric curve": 31
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "parent-function",
    "ja": {
      "term": "親関数",
      "reading": "おやかんすう",
      "alt": []
    },
    "en": {
      "term": "parent function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「親関数」は学習指導要領解説に無い、本プロジェクトの訳語。日本では「y = x² のグラフを平行移動したもの」と言い、もとになる関数に名前を付けない（エントリ transformations-of-functions）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "平行移動・拡大・対称移動をする前の、いちばん基本の形の関数。y = x²、y = |x|、y = √x、y = x³、y = 1/x など。",
    "definition_en": "The simplest function in a family of functions, before any shifts, stretches, or reflections, such as y = x², y = |x|, or y = √x.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The parent function here is y equals x squared, and this graph is just that shifted up 3.",
        "ja": "ここでの親関数は y = x² で、このグラフはそれを上に 3 ずらしただけ。",
        "register": "spoken"
      },
      {
        "en": "Identify the parent function of g(x) = |x − 2| + 5 and describe the transformations.",
        "ja": "g(x) = |x − 2| + 5 の親関数を答え、どのように移動・変形したかを述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "parent function は基本の形の 1 つの関数。そこから変換してできる関数の集まりは family of functions と言う。",
      "OpenStax Algebra and Trigonometry は、基本の関数の一覧を toolkit functions とも呼ぶ。",
      "合成関数の外側の関数（outer function、エントリ outer-function）とは別の語。"
    ],
    "related": [
      "transformations-of-functions",
      "translation",
      "stretch-vertically",
      "reflection",
      "absolute-value-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "parent function": 19
      },
      "written": {
        "parent function": 45
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "parentheses",
    "ja": {
      "term": "かっこ",
      "reading": "かっこ",
      "alt": [
        "括弧",
        "丸かっこ"
      ]
    },
    "en": {
      "term": "parentheses",
      "alt": [],
      "uk": "brackets",
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「かっこ」は ( )・[ ]・{ } のどれにも使うが、米国の教科書（OpenStax・IM）は ( ) を parentheses（単数 parenthesis）、[ ] を brackets、{ } を braces と呼び分ける。英国では ( ) も brackets と呼ぶ。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "式の一部を囲み、その部分をひとまとまりとして先に計算することを表す記号。英語の parentheses は丸かっこ ( ) を指す。",
    "definition_en": "The curved symbols ( ) placed around part of an expression to group it, showing that the part inside is worked out first.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In 4(2 + 3), do what's inside the parentheses first, so that's 4 times 5, or 20.",
        "ja": "4(2 + 3) では、かっこの中を先に計算するから、4 × 5 = 20 だね。",
        "register": "spoken"
      },
      {
        "en": "Insert one pair of parentheses to make the statement true: 8 − 3 × 2 + 1 = 11.",
        "ja": "次の等式が成り立つように、かっこを 1 組つけよ。8 − 3 × 2 + 1 = 11",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "parentheses は複数形で、( ) の 1 組を指す。片側だけなら単数の parenthesis。",
      "[ ] は brackets、{ } は braces と呼び分ける。英国では ( ) も brackets と呼ぶので、英国の教材の brackets は丸かっこのことがある。",
      "区間の記法では (a, b) は端を含まず、[a, b] は含む（エントリ interval）。(a, b) は座標にも使うので、文脈で見分ける。"
    ],
    "related": [
      "the-four-operations",
      "distributive-property",
      "expression",
      "interval",
      "set"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Bracket",
        "note": "記事「Bracket」の冒頭: brackets だけで言うと英国英語では ( )、米国英語では [ ] を指す"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "parentheses ／ brackets ／ braces の呼び分け"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "parentheses": 895
      },
      "written": {
        "parentheses": 240
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "parity",
    "ja": {
      "term": "偶奇",
      "reading": "ぐうき",
      "alt": [
        "偶奇性"
      ]
    },
    "en": {
      "term": "parity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "整数が偶数であるか奇数であるかという性質。2 で割った余りが 0 か 1 かで決まる。",
    "definition_en": "Whether an integer is even or odd, that is, whether it leaves remainder 0 or 1 when divided by 2.",
    "latex": "n \\equiv 0 \\text{ or } 1 \\pmod{2}",
    "spoken_en": "n is congruent to zero or one mod two",
    "examples": [
      {
        "en": "Look at the parity here: x and y have to be both even or both odd.",
        "ja": "ここで偶奇に注目しよう。x と y はどちらも偶数か、どちらも奇数でなければならない。",
        "register": "spoken"
      },
      {
        "en": "Since m + n and m − n have the same parity, their product is either odd or a multiple of 4.",
        "ja": "m + n と m − n の偶奇は一致するので、その積は奇数か 4 の倍数である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "説明的に whether n is even or odd と言っても同じ意味になる。",
      "「偶奇が異なる」は have opposite parity（英語版 Wikipedia「Parity (mathematics)」の冒頭は、連続する 2 つの整数は opposite parity と書く）。「偶奇が一致する」は have the same parity と言う。"
    ],
    "related": [
      "classification-by-remainder",
      "modulus",
      "remainder",
      "integer-solution"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Parity_%28mathematics%29",
        "note": "記事「Parity (mathematics)」の冒頭（整数の偶奇。opposite parity。same parity は節 Higher dimensions のチェス盤のマスについてだけ。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「偶奇性」",
        "url": "https://ja.wikipedia.org/wiki/%E5%81%B6%E5%A5%87%E6%80%A7",
        "note": "英: parity"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "!power parity": 21
      },
      "written": {
        "!power parity": 14
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "partial-derivative",
    "ja": {
      "term": "偏導関数",
      "reading": "へんどうかんすう",
      "alt": [
        "偏微分係数"
      ]
    },
    "en": {
      "term": "partial derivative",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "多変数関数を 1 つの変数だけについて微分し、ほかの変数は定数とみなしたもの。",
    "definition_en": "The derivative of a function of several variables with respect to one variable, holding the others constant.",
    "latex": "\\frac{\\partial f}{\\partial x}",
    "spoken_en": "partial f partial x",
    "examples": [
      {
        "en": "For the partial derivative with respect to x, treat y like a constant.",
        "ja": "x についての偏導関数では、y を定数とみなす。",
        "register": "spoken"
      },
      {
        "en": "Find the partial derivatives f_x and f_y of f(x, y) = x³y² − 2xy.",
        "ja": "f(x, y) = x³y² − 2xy の偏導関数 f_x と f_y を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "記号 ∂ は partial と読む（∂f/∂x は partial f partial x）。del と読むこともある（MIT 18.02、3Blue1Brown）。",
      "f_x は f sub x と読む。"
    ],
    "related": [
      "take-the-partial-derivative",
      "function-of-several-variables",
      "gradient",
      "tangent-plane",
      "total-differential"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "偏微分",
        "en": "Partial derivative"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "partial derivative": 265
      },
      "written": {
        "partial derivative": 146
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "partial-fraction-decomposition",
    "ja": {
      "term": "部分分数分解",
      "reading": "ぶぶんぶんすうぶんかい",
      "alt": [
        "部分分数",
        "部分分数による積分"
      ]
    },
    "en": {
      "term": "partial fraction decomposition",
      "alt": [
        "integration by partial fractions"
      ],
      "variants": [
        {
          "term": "partial fractions",
          "register": "both",
          "note": "do partial fractions のように手法の名前として使う。話し言葉では最も多く（MIT OCW が中心）、書き言葉では partial fraction decomposition に次ぐ（OpenStax Algebra and Trigonometry・OpenStax Calculus が中心）。"
        },
        {
          "term": "partial fraction decomposition",
          "register": "both",
          "note": "話し言葉では partial fractions のほうが多く、書き言葉ではこちらが最も多い。話し言葉は MIT OCW が中心で、書き言葉は OpenStax Algebra and Trigonometry・OpenStax Calculus。"
        },
        {
          "term": "method of partial fractions",
          "register": "written",
          "note": "積分の技法の名前として書く。話し言葉には出てこず、書き言葉は OpenStax Calculus と MIT の講義ノート。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数B",
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "分数式を、分母の因数ごとの簡単な分数式の和に分けること。分数関数の積分では、分けた各項を積分する（部分分数による積分）。",
    "definition_en": "Splitting a rational expression into a sum of simpler fractions, one for each factor of the denominator; integrating a rational function this way is integration by partial fractions.",
    "latex": "\\frac{1}{x(x+1)} = \\frac{1}{x} - \\frac{1}{x+1}",
    "spoken_en": "one over x times x plus one equals one over x minus one over x plus one",
    "examples": [
      {
        "en": "The denominator factors, so let's do partial fractions.",
        "ja": "分母が因数分解できるから、部分分数に分けよう。",
        "register": "spoken"
      },
      {
        "en": "The partial fraction decomposition of 1/(x² − 1) is (1/2)/(x − 1) − (1/2)/(x + 1).",
        "ja": "1/(x² − 1) を部分分数分解すると (1/2)/(x − 1) − (1/2)/(x + 1) となる。",
        "register": "written"
      },
      {
        "en": "After partial fractions, each piece is just a natural log.",
        "ja": "部分分数に分けたら、各項はただの自然対数になる。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "do partial fractions",
        "ja": "部分分数分解する"
      }
    ],
    "pitfalls": [
      "数列の和で使う 1/k(k + 1) = 1/k − 1/(k + 1) の分解も英語では同じく partial fractions。和が打ち消し合って求まるものは telescoping sum と呼ぶ。",
      "log|x| − log|x + 1| + C を log|x/(x + 1)| + C とまとめるかどうかは自由。OpenStax Calculus は ln を使う。",
      "学習指導要領解説（数学III）は「分数関数」について「簡単な分数関数」のグラフを扱う。OpenStax の rational function は多項式 ÷ 多項式の関数全般を指す。",
      "分子の次数が分母以上なら、先に多項式の割り算（long division）で次数を下げる。",
      "∫(1/x)dx は、日本語では log|x| + C、OpenStax Calculus では ln|x| + C と書く。"
    ],
    "related": [
      "decompose-into-partial-fractions",
      "integration-formulas",
      "indefinite-integral",
      "integration-by-long-division",
      "substituting-values",
      "compare-coefficients"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "部分分数分解",
        "en": "Partial fraction decomposition"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.12（linear partial fractions、BC のみ）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III: 簡単な分数関数のグラフ"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "partial fractions": 63,
        "do partial fractions": 6,
        "partial fraction decomposition": 42
      },
      "written": {
        "partial fraction decomposition": 53,
        "partial fractions": 40,
        "method of partial fractions": 11,
        "do partial fractions": 2
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "partial-order",
    "ja": {
      "term": "半順序",
      "reading": "はんじゅんじょ",
      "alt": [
        "半順序関係"
      ]
    },
    "en": {
      "term": "partial order",
      "alt": [
        "partial ordering"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete",
      "set-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "反射律・反対称律・推移律をみたす二項関係のこと。どの 2 つの元も比べられるとは限らない点が、全順序と違う。",
    "definition_en": "A relation that is reflexive, antisymmetric, and transitive. Unlike a total order, it can have pairs of elements that are not comparable.",
    "latex": "(a \\preceq b \\land b \\preceq a) \\Rightarrow a = b",
    "spoken_en": "if a precedes b and b precedes a, then a equals b",
    "examples": [
      {
        "en": "Divisibility is a partial order on the positive integers, but 2 and 3 aren't comparable, since neither one divides the other.",
        "ja": "割り切る関係は正の整数上の半順序だが、2 と 3 はどちらも他方を割り切らないので比べられない。",
        "register": "spoken"
      },
      {
        "en": "Show that ⊆ is a partial order on the power set of {a, b, c}, and find two elements that are not comparable.",
        "ja": "{a, b, c} のべき集合上の関係 ⊆ が半順序であることを示し、比較できない 2 つの元を挙げよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "反対称律は antisymmetric。対称律（symmetric）の否定ではなく、a ⪯ b かつ b ⪯ a なら a = b という条件。",
      "本によっては、反射律の代わりに a ⪯ a が成り立たない狭義の順序（< 型、strict partial order）を partial order と呼ぶ。定義を確かめる。",
      "半順序が定められた集合は partially ordered set（英語版 Wikipedia の記事名）。用例コーパスでは話し言葉はすべて MIT OCW、書き言葉はすべて MIT の講義ノート。Levin の本では partial order と partial ordering が 3 件ずつ。"
    ],
    "related": [
      "relation",
      "equivalence-relation",
      "power-set",
      "subset"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "partial order": 30
      },
      "written": {
        "partial order": 54
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "partial-sum",
    "ja": {
      "term": "部分和",
      "reading": "ぶぶんわ",
      "alt": [
        "初項から第 n 項までの和"
      ]
    },
    "en": {
      "term": "partial sum",
      "alt": [],
      "variants": [
        {
          "term": "partial sum",
          "register": "both",
          "note": "大学（OpenStax Calculus・MIT）の言い方。書き言葉は大半が OpenStax Calculus。話し言葉は Khan Academy と MIT OCW が中心で、sum of the first n terms よりずっと多い。"
        },
        {
          "term": "sum of the first n terms",
          "register": "written",
          "note": "高校レベル（OpenStax Algebra and Trigonometry・Intermediate Algebra）の言い方。話し言葉にはほとんど出てこず（すべて Khan Academy）、書き言葉は OpenStax Algebra and Trigonometry と Intermediate Algebra が中心。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B",
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "数列の初項から第 n 項までの和 Sₙ。無限級数の収束は、部分和の数列の極限で決める。",
    "definition_en": "Sₙ = a₁ + a₂ + ⋯ + aₙ, the total of a sequence's terms up through the nth; an infinite series converges when its sequence of partial sums has a limit.",
    "latex": "S_n = \\sum_{k=1}^{n} a_k",
    "spoken_en": "S sub n equals the sum from k equals one to n of a sub k",
    "examples": [
      {
        "en": "Write out the partial sums and see where they're heading.",
        "ja": "部分和を書き出して、どこに近づくか見よう。",
        "register": "spoken"
      },
      {
        "en": "Find a formula for the nth partial sum Sₙ of the series Σ 1/(k(k + 1)).",
        "ja": "級数 Σ 1/(k(k + 1)) の第 n 部分和 Sₙ を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the sequence of partial sums",
        "ja": "部分和の数列"
      }
    ],
    "pitfalls": [
      "共通テスト・センター試験（数学II・数学B）の問題文は「初項から第 n 項までの和」と書き、学習指導要領解説（数学III）は無限級数で「部分和」と書く。OpenStax Algebra and Trigonometry（13.4）は sum of the first n terms を nth partial sum と呼ぶ。"
    ],
    "related": [
      "infinite-series",
      "sum-of-the-series",
      "summation-notation",
      "find-the-sum",
      "sum-of-a-geometric-sequence",
      "sum-of-an-arithmetic-sequence",
      "telescoping-series",
      "write-out-the-first-few-terms"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.2 Infinite Series（partial sum）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.4 Series and Their Notations（nth partial sum）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.1（partial sum）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成30年度 大学入試センター試験 本試験 数学Ⅱ・数学Ｂ（問題）",
        "url": "https://web.archive.org/web/20190601/https://www.dnc.ac.jp/albums/abm.php?f=abm00033144.pdf&n=h30+suugaku2+02suugaku2B.pdf",
        "note": "第3問: 初項から第 n 項までの和"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III (1) 極限 無限級数（部分和）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "partial sum": 66,
        "sum of the first n terms": 7
      },
      "written": {
        "partial sum": 170,
        "sum of the first n terms": 55
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "particular-solution",
    "ja": {
      "term": "特殊解",
      "reading": "とくしゅかい",
      "alt": []
    },
    "en": {
      "term": "particular solution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II"
      ]
    },
    "definition_ja": "微分方程式の一般解のうち、初期条件などで任意定数の値を決めた 1 つの解。",
    "definition_en": "The one solution of a differential equation picked out of the general solution by fixing the constant, for example with an initial condition.",
    "latex": "y = 3e^{x^2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Once you plug in the initial condition, you get a particular solution.",
        "ja": "初期条件を代入すれば特殊解が得られる。",
        "register": "spoken"
      },
      {
        "en": "Find the particular solution of dy/dx = 6x² that passes through (1, 5).",
        "ja": "点 (1, 5) を通る dy/dx = 6x² の特殊解を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "任意定数を含む解は general solution（一般解）。"
    ],
    "related": [
      "initial-condition",
      "differential-equation",
      "separation-of-variables"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 7.7・7.8（particular solution）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "particular solution": 191
      },
      "written": {
        "particular solution": 98
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "partitioning-into-groups",
    "ja": {
      "term": "組分け",
      "reading": "くみわけ",
      "alt": []
    },
    "en": {
      "term": "split … into groups",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスで split … into groups は話し言葉（Khan Academy）にわずかにあるだけで、書き言葉には出てこない）。参照（CED・OpenStax・IM・CK-12・Levin）にも組分けの問題の名前は出てこず、Levin（3.9）は split up 11 kids into five named teams のように、何をするかを文で問う。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "いくつかのものを、決まった人数（個数）ずつのいくつかの組に分ける方法を数える問題。組に区別がないときは、同じ大きさの組の並べ替えの分だけ割る。",
    "definition_en": "Counting the ways to split a set of people or objects into groups of given sizes; when groups of the same size are not labeled, divide by the number of ways to reorder those groups.",
    "latex": "\\frac{6!}{2!\\,2!\\,2!\\,3!} = 15",
    "spoken_en": "six factorial over the quantity two factorial cubed times three factorial equals fifteen",
    "examples": [
      {
        "en": "How many ways can we split six people into three groups of two? Careful, the groups don't have names.",
        "ja": "6 人を 2 人ずつ 3 組に組分けする方法は何通りか。組には名前がないので注意。",
        "register": "spoken"
      },
      {
        "en": "In how many ways can 9 students be divided into three groups of 3 students each?",
        "ja": "9 人の生徒を 3 人ずつ 3 組に分ける方法は何通りあるか。",
        "register": "written"
      },
      {
        "en": "In how many ways can 9 students be split into groups named A, B, and C with 3 students in each?",
        "ja": "9 人の生徒を A, B, C の 3 組に 3 人ずつ分ける方法は何通りあるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "組に名前（A, B, C や Team 1, 2, 3）があるかないかで答えが変わる。9 人を 3 人ずつ分けるとき、名前ありは 9!/(3!3!3!) = 1680 通り、名前なしはそれを 3! で割って 280 通り。",
      "大きさの違う組（2 人と 3 人と 4 人など）は、名前がなくても大きさで区別できるので割らない。"
    ],
    "related": [
      "combination",
      "distinguish",
      "permutation-of-a-multiset",
      "counting",
      "factorial"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.9（split up 11 kids into five named teams）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "split … into groups": 1
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 1 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "pascals-triangle",
    "ja": {
      "term": "パスカルの三角形",
      "reading": "ぱすかるのさんかくけい",
      "alt": []
    },
    "en": {
      "term": "Pascal's triangle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "両端を 1 とし、各数をすぐ上の段の隣り合う 2 数の和として三角形に並べた数の表。n 段目には (a + b)ⁿ の二項係数が並ぶ。",
    "definition_en": "A triangular array of numbers with 1s on the edges in which each entry is the sum of the two entries just above it; row n lists the binomial coefficients of (a + b)ⁿ.",
    "latex": "\\binom{n}{k} = \\binom{n-1}{k-1} + \\binom{n-1}{k}",
    "spoken_en": "n choose k equals n minus one choose k minus one plus n minus one choose k",
    "examples": [
      {
        "en": "Go down to the fourth row of Pascal's triangle and you get 1, 4, 6, 4, 1.",
        "ja": "パスカルの三角形の 4 段目まで下りると 1, 4, 6, 4, 1 が出てくる。",
        "register": "spoken"
      },
      {
        "en": "Use Pascal's triangle to expand (x + y)⁵.",
        "ja": "パスカルの三角形を用いて (x + y)⁵ を展開せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "一番上の 1 を row 0（0 段目）と数えると、n 段目が (a + b)ⁿ の係数になる。一番上を 1 段目と数えると n と 1 ずれる。",
      "Pascal's は人名の所有格で、P は大文字、アポストロフィを落とさない。"
    ],
    "related": [
      "binomial-coefficient",
      "binomial-theorem",
      "combination",
      "binomial-identities"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "パスカルの三角形",
        "en": "Pascal's triangle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "Pascal's triangle": 18
      },
      "written": {
        "Pascal's triangle": 38
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pass-through",
    "ja": {
      "term": "通る",
      "reading": "とおる",
      "alt": [
        "点を通る"
      ]
    },
    "en": {
      "term": "pass through",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2",
        "中1",
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "点が直線や曲線の上にあるとき、その直線（曲線）はその点を通るという。グラフが点 (a, b) を通るとは、x = a のとき y = b になること。",
    "definition_en": "A line or curve passes through a point when the point lies on it. A graph passes through (a, b) when y = b for x = a.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If the line passes through (2, 5), then when x is 2, y has to be 5.",
        "ja": "直線が (2, 5) を通るなら、x が 2 のとき y は 5 になるはずだね。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of the line that passes through the origin and the point (3, −6).",
        "ja": "原点と点 (3, −6) を通る直線の式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「点を通る」は pass through と through を付けて言う。主語は直線やグラフで、The line passes through (2, 5). の形になる。",
      "点を主語にするときは The point (2, 5) is on the line.（点が直線上にある）と言う。",
      "「原点を通る」は passes through the origin（エントリ origin）。origin には the を付ける。"
    ],
    "related": [
      "line",
      "origin",
      "graph",
      "coordinates",
      "satisfy"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "pass through": 72
      },
      "written": {
        "pass through": 367
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "path-in-a-graph",
    "ja": {
      "term": "経路（グラフ）",
      "reading": "けいろ",
      "alt": [
        "道",
        "パス",
        "経路"
      ]
    },
    "en": {
      "term": "path",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフで、辺でつながった頂点を順にたどる道筋のうち、同じ頂点を 2 度通らないもの。",
    "definition_en": "In a graph, a route along edges from vertex to vertex that does not visit any vertex twice.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Is there a path from A to F? Follow the edges and see.",
        "ja": "A から F への経路はあるか。辺をたどって確かめよう。",
        "register": "spoken"
      },
      {
        "en": "Find the shortest path from vertex A to vertex F.",
        "ja": "頂点 A から頂点 F への最短経路を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "同じ頂点を何度通ってもよいものは walk（歩道）と呼び分ける（Levin）。",
      "Nicholson（2.3 Matrix Multiplication）は、頂点をくり返してよい道も path（長さ r の r-path）と呼び、隣接行列の累乗で数える（Levin の walk に当たる）。"
    ],
    "related": [
      "edge-of-a-graph",
      "vertex-graph",
      "graph-network",
      "eulerian-path"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.1 Problems and Definitions"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "2.3 Matrix Multiplication（paths of length r、r-path）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「道 (グラフ理論)」",
        "url": "https://ja.wikipedia.org/wiki/%E9%81%93_%28%E3%82%B0%E3%83%A9%E3%83%95%E7%90%86%E8%AB%96%29",
        "note": "道（パス）、歩道（walk）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学C 数学的な表現の工夫: 離散グラフ（経路、最短経路の探索）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "path in the graph | path in a graph | path from u to v | path of length | hamilton path | hamiltonian path | shortest path | simple path | walk or path": 22
      },
      "written": {
        "path in the graph | path in a graph | path from u to v | path of length | hamilton path | hamiltonian path | shortest path | simple path | walk or path": 36
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pemdas",
    "ja": {
      "term": "PEMDAS",
      "reading": "ぺむだす",
      "alt": []
    },
    "en": {
      "term": "PEMDAS",
      "alt": [],
      "uk": "BODMAS"
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "日本には演算の順序の覚え方の決まった名前がない。M と D、A と S はそれぞれ同じ順位で、左から順に計算する（M が D より先ではない）。英国では同じ覚え方を BODMAS ／ BIDMAS と言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "演算の順序を覚えるための英語の頭文字語。Parentheses（かっこ）、Exponents（累乗）、Multiplication and Division（乗法・除法）、Addition and Subtraction（加法・減法）の頭文字を並べたもの。",
    "definition_en": "An acronym for remembering the order of operations: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "PEMDAS puts exponents before multiplication, but multiplication and division are a tie, so you just go left to right.",
        "ja": "PEMDAS では累乗がかけ算より先だけど、かけ算とわり算は同じ順位だから、左から順に計算すればいい。",
        "register": "spoken"
      },
      {
        "en": "Use PEMDAS to simplify 20 ÷ 4 × (3 + 2).",
        "ja": "PEMDAS を使って、20 ÷ 4 × (3 + 2) を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "PEMDAS は M が D より先、A が S より先という意味ではない。20 ÷ 4 × 5 は左から計算して 25 で、20 ÷ 20 = 1 ではない。",
      "OpenStax Elementary Algebra は語呂合わせ Please Excuse My Dear Aunt Sally を紹介する。英国では BODMAS ／ BIDMAS と言う（B はかっこ brackets）。",
      "演算の順序そのものは order of operations（エントリ order-of-operations）。CK-12 Algebra は、演算の順序を PEMDAS と呼ぶことが多いと書く。"
    ],
    "related": [
      "order-of-operations",
      "parentheses",
      "exponent",
      "the-four-operations",
      "evaluate"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "1.4.2 PEMDAS in Numerical Expressions"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Order_of_operations#Mnemonics",
        "note": "記事「Order of operations」の節 Mnemonics: 英国などの英連邦の国は BODMAS、I（Indices）を使う BIDMAS もある"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "PEMDAS": 4
      },
      "written": {
        "PEMDAS": 1
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CK-12 の呼び方 PEMDAS（CK-12 Algebra 1.4.2 PEMDAS in Numerical Expressions・CK-12 Algebra 1.4.1 Order of Operations・CK-12 Algebra 1.4.5 Order of Operations and Variable Substitution・計 19 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "percent-change",
    "ja": {
      "term": "パーセントの増減",
      "reading": "ぱーせんとのぞうげん",
      "alt": []
    },
    "en": {
      "term": "percent increase",
      "alt": [],
      "variants": [
        {
          "term": "percent decrease",
          "register": "written",
          "note": "減るときの言い方。percent increase と対にして使う。"
        },
        {
          "term": "percent change",
          "register": "written",
          "note": "増減をまとめた言い方。教科書（OpenStax）では percent increase ／ percent decrease よりずっと少ない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本では「〜% 増える」「〜% 減る」「増加率」と言い、増減をまとめた名前はない。英語は増えるとき percent increase、減るとき percent decrease と分けて言う。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "もとの量に対して、増えた量（減った量）が何パーセントにあたるかを表したもの。増えた量 ÷ もとの量 × 100 で求める。",
    "definition_en": "The amount of increase (or decrease) written as a percent of the original amount. The percent increase is the amount of increase divided by the original amount, times 100.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It went from 20 to 23, so that's 3 out of the original 20, which is a 15 percent increase.",
        "ja": "20 から 23 になったから、もとの 20 に対して 3 増えた。15 % の増加だね。",
        "register": "spoken"
      },
      {
        "en": "The price of a ticket rose from $12 to $15. Find the percent increase.",
        "ja": "チケットの値段が 12 ドルから 15 ドルに上がった。何パーセント増えたかを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "割る数はもとの量で、新しい量ではない。12 ドルから 15 ドルは 3 ÷ 12 で 25 % の増加で、3 ÷ 15 = 20 % ではない。",
      "減るときは percent decrease と言う。OpenStax Prealgebra は percent increase と percent decrease をそれぞれ定義する（増えた量・減った量が、もとの量の何パーセントか）。",
      "増減をまとめて percent change とも言う（OpenStax Algebra and Trigonometry）。"
    ],
    "related": [
      "percent",
      "growth-rate",
      "simple-interest",
      "rate-of-change",
      "ratio"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "percent increase": 4,
        "percent decrease": 1
      },
      "written": {
        "percent change": 7,
        "percent increase": 51,
        "percent decrease": 45
      },
      "sources": [
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "percent",
    "ja": {
      "term": "百分率",
      "reading": "ひゃくぶんりつ",
      "alt": [
        "パーセント"
      ]
    },
    "en": {
      "term": "percent",
      "alt": [],
      "variants": [
        {
          "term": "percentage",
          "register": "spoken",
          "note": "話し言葉では percent に次いで多い（ほとんどが Khan Academy）。書き言葉では percent のほうがずっと多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "もとにする量を 100 としたとき、比べる量がいくつに当たるかで表した割合。記号 % を使い、0.25 を 25% と表す。",
    "definition_en": "A way of writing a number as a part of 100, using the symbol %. For example, 25% means 25 out of every 100, or 0.25.",
    "latex": "x\\% = \\frac{x}{100}",
    "spoken_en": "x percent equals x over one hundred",
    "examples": [
      {
        "en": "Percent means out of 100, so 30 percent of 50 is 0.3 times 50, which is 15.",
        "ja": "パーセントは「100 のうちいくつ」ということ。だから 50 の 30% は 0.3 × 50 で 15 だね。",
        "register": "spoken"
      },
      {
        "en": "What percent of 40 is 14?",
        "ja": "14 は 40 の何 % にあたるかを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "percent は数に付けて 25 percent のように言う。percentage は「割合（の大きさ）」を表す名詞で、a large percentage of … のように使う。",
      "歩合も英語では percent で言う。2 割は 20 percent、3 割 5 分は 35 percent。",
      "「割合」（比べる量 ÷ もとにする量）に当たる 1 語は英語に無く、場面で ratio ／ rate ／ fraction ／ percent を使い分ける（エントリ rate）。"
    ],
    "related": [
      "rate",
      "ratio",
      "fraction",
      "decimal",
      "value-of-a-ratio"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "percent": 332,
        "percentage": 199
      },
      "written": {
        "percent": 914,
        "percentage": 224
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "percentile",
    "ja": {
      "term": "パーセンタイル",
      "reading": "ぱーせんたいる",
      "alt": [
        "百分位数"
      ]
    },
    "en": {
      "term": "percentile",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "データを小さい順に並べたとき、その値以下のデータが全体の p% になる値を第 p パーセンタイルという。",
    "definition_en": "The pth percentile is the value at or below which p percent of the ordered data lie.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Scoring in the 90th percentile means you did better than about 90 percent of people.",
        "ja": "第 90 パーセンタイルとは、約 90% の人より上だったということ。",
        "register": "spoken"
      },
      {
        "en": "Find the 30th percentile of the data.",
        "ja": "データの第 30 パーセンタイルを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "第 1 四分位数は第 25 パーセンタイル、中央値は第 50 パーセンタイル。"
    ],
    "related": [
      "quartile",
      "median",
      "cumulative-relative-frequency",
      "standard-score"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "percentile": 96
      },
      "written": {
        "percentile": 282
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "perfect-square-trinomial",
    "ja": {
      "term": "完全平方式",
      "reading": "かんぜんへいほうしき",
      "alt": []
    },
    "en": {
      "term": "perfect square trinomial",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Geometry",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "(a + b)² や (a − b)² を展開した形の、3 つの項からなる式。たとえば x² + 6x + 9 = (x + 3)² である。",
    "definition_en": "A trinomial that is the square of a binomial, such as x² + 6x + 9 = (x + 3)². It has the form a² + 2ab + b² or a² − 2ab + b².",
    "latex": "a^2 + 2ab + b^2 = (a + b)^2",
    "spoken_en": "a squared plus two a b plus b squared equals the quantity a plus b, squared",
    "examples": [
      {
        "en": "x² + 10x + 25 is a perfect square trinomial: half of 10 is 5, and 5 squared is 25, so it's (x + 5)².",
        "ja": "x² + 10x + 25 は完全平方式。10 の半分が 5 で、5 の 2 乗が 25 だから、(x + 5)² になる。",
        "register": "spoken"
      },
      {
        "en": "Find the value of c that makes x² − 14x + c a perfect square trinomial.",
        "ja": "x² − 14x + c が完全平方式になるような c の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "perfect square だけだと、16 や 25 のような平方数の意味にもなる。3 つの項の式であることを表すのが trinomial。IM Algebra 1（7.11 What are Perfect Squares?）は x² + 6x + 9 のような式そのものも perfect square と呼ぶ。",
      "まん中の項が 2ab になっているかを確かめる。x² + 6x + 16 は、x² と 16 = 4² がどちらも 2 乗でも、2 × x × 4 = 8x にならないので完全平方式ではない。",
      "x² + bx に (b/2)² を足して完全平方式を作るのが平方完成（エントリ completing-the-square）。"
    ],
    "related": [
      "square-of-a-binomial",
      "completing-the-square",
      "trinomial",
      "factoring",
      "special-products"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 7.11 What are Perfect Squares?（式も perfect square）・Geometry 6.5〜6.14（perfect square trinomial）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "perfect square trinomial": 11
      },
      "written": {
        "perfect square trinomial": 87
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "perimeter",
    "ja": {
      "term": "周の長さ",
      "reading": "しゅうのながさ",
      "alt": []
    },
    "en": {
      "term": "perimeter",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "図形のまわりの長さ。多角形では、すべての辺の長さの和になる。",
    "definition_en": "The distance around a flat figure. For a polygon, add up the lengths of all the sides.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Perimeter is the distance around, so just add up all the sides.",
        "ja": "周の長さはまわりの長さだから、辺を全部たせばいい。",
        "register": "spoken"
      },
      {
        "en": "A rectangle is 8 cm long and 5 cm wide. Find its perimeter and its area.",
        "ja": "縦 5 cm、横 8 cm の長方形の周の長さと面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "円のまわりの長さは circumference（エントリ circumference）。perimeter は多角形などに使う。",
      "面積は area（エントリ area）。perimeter の単位は cm・m などの長さの単位で、area は cm² などの square units（エントリ square-units）。",
      "発音は pe-RIM-e-ter で、2 つ目の音節に強勢がある（Merriam-Webster: pə-ˈri-mə-tər）。"
    ],
    "related": [
      "circumference",
      "area",
      "polygon",
      "side",
      "square-units"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「perimeter」",
        "url": "https://www.merriam-webster.com/dictionary/perimeter",
        "note": "発音 pə-ˈri-mə-tər（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "perimeter": 177
      },
      "written": {
        "perimeter": 300
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "period",
    "ja": {
      "term": "周期",
      "reading": "しゅうき",
      "alt": []
    },
    "en": {
      "term": "period",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "関数の値が同じ変化を繰り返す最小の正の間隔 p（すべての x で f(x + p) = f(x)）。",
    "definition_en": "The smallest positive p with f(x + p) = f(x) for all x: the length after which a function repeats.",
    "latex": "f(x + p) = f(x)",
    "spoken_en": null,
    "examples": [
      {
        "en": "The period of sine is 2π, but sin 3x has period 2π over 3.",
        "ja": "sin の周期は 2π だが、sin 3x の周期は 2π/3。",
        "register": "spoken"
      },
      {
        "en": "Find the period of y = tan(2θ − π/3).",
        "ja": "y = tan(2θ − π/3) の周期を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "period は a period of time（期間）の意味にも使う。関数の周期は the period of the function ／ the period of f のように言う（IM Algebra 2 6.9・6.11 ほか）。"
    ],
    "related": [
      "periodic-function",
      "amplitude",
      "trigonometric-function",
      "midline"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 6.9・6.11（period、glossary）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "period of !time": 37
      },
      "written": {
        "period of !time": 109
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "periodic-function",
    "ja": {
      "term": "周期関数",
      "reading": "しゅうきかんすう",
      "alt": []
    },
    "en": {
      "term": "periodic function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "ある正の数 p について f(x + p) = f(x) がすべての x で成り立つ関数。",
    "definition_en": "A function for which some positive p gives f(x + p) = f(x) for every x.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Sine and cosine are periodic functions with period 2π.",
        "ja": "sin と cos は周期 2π の周期関数。",
        "register": "spoken"
      },
      {
        "en": "Show that f(x) = sin x + cos 2x is a periodic function and find its period.",
        "ja": "f(x) = sin x + cos 2x が周期関数であることを示し、その周期を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉は大半が MIT 18.03（フーリエ級数）。"
    ],
    "related": [
      "period",
      "trigonometric-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "周期関数",
        "en": "Periodic function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "periodic function": 24
      },
      "written": {
        "periodic function": 32
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "permutation-of-a-multiset",
    "ja": {
      "term": "同じものを含む順列",
      "reading": "おなじものをふくむじゅんれつ",
      "alt": []
    },
    "en": {
      "term": "permutation of n non-distinct objects",
      "alt": [
        "multiset permutation"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "n 個のうち p 個、q 個、r 個がそれぞれ同じものであるとき、それら全部を 1 列に並べる並べ方。総数は n!/(p! q! r!) 通り（p + q + r = n）。",
    "definition_en": "An arrangement in a row of n objects of which p, q, and r are alike; there are n!/(p! q! r!) of them, where p + q + r = n.",
    "latex": "\\frac{n!}{p!\\,q!\\,r!}",
    "spoken_en": "n factorial over p factorial q factorial r factorial",
    "examples": [
      {
        "en": "Since the two S's look the same, divide by 2 factorial: it's a multiset permutation.",
        "ja": "2 つの S は区別できないので 2! で割る。これは同じものを含む順列だ。",
        "register": "spoken"
      },
      {
        "en": "Use the formula for permutations of n non-distinct objects to count the arrangements of the letters in BANANA.",
        "ja": "同じものを含む順列の公式を使って、BANANA の 6 文字の並べ方の数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry（13.5 Counting Principles）は permutations of n non-distinct objects と呼び、n!/(r₁! r₂! ⋯ r_k!) で数える。英語版 Wikipedia「Permutation」の節 Permutations of multisets は multiset permutation と呼ぶ。",
      "「重複順列」（同じものを何度でも選べる、nʳ）とは別物。こちらは並べるものの中に同じものが決まった個数だけある。",
      "割るのは同じものの個数の階乗。BANANA なら A が 3 個、N が 2 個なので 6!/(3! 2!) = 60。"
    ],
    "related": [
      "permutation",
      "factorial",
      "permutation-with-repetition",
      "shortest-path",
      "distinguish",
      "arrange"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles（Finding the Number of Permutations of n Non-Distinct Objects）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Permutation",
        "note": "記事「Permutation」の節 Permutations of multisets（multiset permutation。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.4・3.6（anagrams、STATISTICS の並べ方）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "permutation of n non-distinct objects": 4
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは OpenStax の呼び方 permutation of n non-distinct objects（本文 4 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "permutation-with-repetition",
    "ja": {
      "term": "重複順列",
      "reading": "ちょうふくじゅんれつ",
      "alt": []
    },
    "en": {
      "term": "permutation with repetition",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスでは permutation with repetition は話し言葉には出てこず、書き言葉も MIT の講義ノートにわずかにあるだけ）。英語版 Wikipedia「Permutation」の節 Permutations with repetition は、重複を許して並べたものを k-tuples（words、strings）と呼び、permutations with repetition とも呼ばれると書く（日本語版 Wikipedia には記事「重複順列」がある）。OpenStax Algebra and Trigonometry（13.5 の問題）と Levin（3.3）は名前を出さず、can be repeated と条件を書いて積の法則で 10³・26³ を求める。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "異なる n 個のものから、同じものを何度でも選んでよいとして r 個を並べる並べ方。総数は nʳ 通り。",
    "definition_en": "An arrangement of r objects chosen from n kinds when the same kind may be used again and again; there are nʳ of them.",
    "latex": "n^r",
    "spoken_en": "n to the r",
    "examples": [
      {
        "en": "Each of the four digits can be any of 10 numbers, so this is a permutation with repetition: 10 to the fourth.",
        "ja": "4 桁のどれにも 10 通りの数字が入るので、これは重複順列で 10⁴ 通りだ。",
        "register": "spoken"
      },
      {
        "en": "Find the number of permutations with repetition of length 5 formed from the letters a, b, and c.",
        "ja": "a, b, c の 3 文字から重複を許して 5 個並べる重複順列の総数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「同じものを含む順列」（並べるものの中に同じものがある）と「重複順列」（同じものを何度でも選べる）は別の数え方。前者は n!/(p! q! r!)、後者は nʳ。",
      "r 個のものを n 人に配る問題では、何を「並べる位置」、何を「選択肢」とするかで nʳ と rⁿ を取り違えやすい。"
    ],
    "related": [
      "permutation",
      "permutation-of-a-multiset",
      "combination-with-repetition",
      "counting",
      "circular-permutation"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Permutation",
        "note": "記事「Permutation」の節 Permutations with repetition（k-tuples。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「重複順列」",
        "url": "https://ja.wikipedia.org/wiki/%E9%87%8D%E8%A4%87%E9%A0%86%E5%88%97",
        "note": "総数 nᵏ"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles（a string of 3 digits if numbers can be repeated → 10³）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.3（letters can be repeated → 26³）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "permutation with repetition": 1
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 1 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "permutation",
    "ja": {
      "term": "順列",
      "reading": "じゅんれつ",
      "alt": []
    },
    "en": {
      "term": "permutation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "いくつかのものから何個かを取り出し、順序を付けて 1 列に並べたもの。異なる n 個から r 個を取る順列の総数は ₙPᵣ = n!/(n − r)!。",
    "definition_en": "An arrangement of some or all of a set of objects in a definite order; r of n distinct objects can be arranged in order in n!/(n − r)! ways.",
    "latex": "{}_n P_r = \\frac{n!}{(n-r)!}",
    "spoken_en": "n P r equals n factorial over n minus r factorial",
    "examples": [
      {
        "en": "Order matters here, so it's a permutation, not a combination.",
        "ja": "ここでは順序が関係するので、組合せではなく順列だ。",
        "register": "spoken"
      },
      {
        "en": "How many permutations of the letters A, B, C, D, E are there?",
        "ja": "A, B, C, D, E の 5 文字の順列は何通りあるか。",
        "register": "written"
      },
      {
        "en": "A permutation of a finite set is an ordering of all of its elements.",
        "ja": "有限集合の順列とは、そのすべての要素を並べる 1 つの順序のことである。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高等学校学習指導要領〔用語・記号〕（数学A）は nPr と書く。OpenStax Algebra and Trigonometry（13.5 Counting Principles）は P(n, r) と書き、nPr は計算機の表記として出てくる。",
      "大学の数学では permutation は集合からそれ自身への全単射（置換）の意味にもなる。線形代数の permutation matrix は置換行列。"
    ],
    "related": [
      "combination",
      "factorial",
      "circular-permutation",
      "permutation-with-repetition",
      "order-matters"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles（P(n, r)）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Permutation",
        "note": "記事「Permutation」の節 k-permutations of n（ₙPₖ・P(n, k)。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「順列」",
        "url": "https://ja.wikipedia.org/wiki/%E9%A0%86%E5%88%97",
        "note": "定義（n 個から k 個を取る順列）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）〔用語・記号〕",
        "note": "数学A 場合の数と確率: nPr"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.4 Combinations and Permutations"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "permutation": 196
      },
      "written": {
        "permutation": 106
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "perpendicular-bisector",
    "ja": {
      "term": "垂直二等分線",
      "reading": "すいちょくにとうぶんせん",
      "alt": []
    },
    "en": {
      "term": "perpendicular bisector",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "線分の中点を通り、その線分に垂直な直線。線分の両端から等しい距離にある点の集まりになる。",
    "definition_en": "The line through the midpoint of a segment and perpendicular to it; every point on it is equally far from the two endpoints.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Any point on the perpendicular bisector is the same distance from A and from B.",
        "ja": "垂直二等分線上の点は、A からも B からも同じ距離にある。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of the perpendicular bisector of the segment joining A(1, 2) and B(5, 4).",
        "ja": "2 点 A(1, 2), B(5, 4) を結ぶ線分の垂直二等分線の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉・書き言葉とも少ない（書き言葉は OpenStax Algebra and Trigonometry だけ）。IM Geometry では glossary の見出しで、レッスン（作図の 1 章、2.8 の垂直二等分線の定理、7.5 ほか）に 123 件。"
    ],
    "related": [
      "midpoint",
      "equation-of-a-line",
      "perpendicular-lines"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二等分線",
        "en": "Bisection"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary・Geometry 2.8 The Perpendicular Bisector Theorem・Geometry 7.5 Triangles in Circles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "perpendicular bisector": 5
      },
      "written": {
        "perpendicular bisector": 1
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 perpendicular bisector（Geometry glossary・Geometry 2.8 The Perpendicular Bisector Theorem・Geometry 7.5 Triangles in Circles・計 123 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "perpendicular-lines",
    "ja": {
      "term": "2 直線の垂直",
      "reading": "にちょくせんのすいちょく",
      "alt": [
        "垂直な 2 直線",
        "垂線"
      ]
    },
    "en": {
      "term": "perpendicular lines",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直角に交わる 2 直線で、一方を他方の垂線ともいう。座標平面では傾きの積が −1 になる（座標軸に平行な場合を除く）。",
    "definition_en": "Two lines that meet at a right angle; on the coordinate plane, their slopes multiply to −1 (apart from lines parallel to the axes).",
    "latex": "m_1 m_2 = -1",
    "spoken_en": "m one times m two equals negative one",
    "examples": [
      {
        "en": "Perpendicular lines have slopes that are negative reciprocals, like 2 and −1/2.",
        "ja": "垂直な 2 直線の傾きは、2 と −1/2 のように「逆数の符号を変えたもの」になる。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of the line through (1, 4) perpendicular to y = 3x − 2.",
        "ja": "点 (1, 4) を通り、直線 y = 3x − 2 に垂直な直線の方程式を求めよ。",
        "register": "written"
      },
      {
        "en": "Construct the line through point P that is perpendicular to line ℓ.",
        "ja": "点 P を通り、直線 ℓ に垂直な直線（垂線）を作図せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax（Elementary Algebra・Intermediate Algebra・Algebra and Trigonometry）は傾きの条件を negative reciprocals（負の逆数）と言う。日本の「傾きの積が −1」と同じ内容。",
      "「垂線」に当たる 1 語の名詞はなく、a line perpendicular to ℓ や the perpendicular from P to ℓ と言う。垂線の足は foot of the perpendicular（エントリ foot-of-the-perpendicular）。"
    ],
    "related": [
      "parallel-lines",
      "perpendicular-bisector",
      "equation-of-a-line",
      "distance-from-a-point-to-a-line",
      "foot-of-the-perpendicular"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "negative reciprocals"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "negative reciprocals"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "perpendicular lines": 37
      },
      "written": {
        "perpendicular lines": 61
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "perpendicular-postulate",
    "ja": {
      "term": "垂線の公準",
      "reading": "すいせんのこうじゅん",
      "alt": []
    },
    "en": {
      "term": "perpendicular postulate",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「垂線の公準」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書は「直線外の 1 点を通る垂線はただ 1 本」を公準として立てない（作図で扱う）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "直線 ℓ 上にない点 P を通り、ℓ に垂直な直線はちょうど 1 本ある、という公準。",
    "definition_en": "The postulate that through a point not on a line, there is exactly one line perpendicular to the given line.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "From P there's exactly one perpendicular to line ℓ — that's the perpendicular postulate — and its length is the distance from P to the line.",
        "ja": "P から直線 ℓ に引ける垂線はちょうど 1 本。これが垂線の公準で、その長さが P と直線の距離になる。",
        "register": "spoken"
      },
      {
        "en": "Point P is not on line ℓ. Explain why there cannot be two different lines through P that are both perpendicular to ℓ.",
        "ja": "点 P は直線 ℓ 上にない。P を通り ℓ に垂直な直線が 2 本あり得ない理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "平行線の公準（parallel postulate）は「直線外の 1 点を通り、その直線に平行な直線はただ 1 本」で、別の公準。",
      "用例コーパスにも IM にも perpendicular postulate は出てこない。"
    ],
    "related": [
      "perpendicular-lines",
      "postulate",
      "foot-of-the-perpendicular"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。そのまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "perpendicular",
    "ja": {
      "term": "垂直",
      "reading": "すいちょく",
      "alt": [
        "垂直である"
      ]
    },
    "en": {
      "term": "perpendicular",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの直線（線分）が直角に交わっていること。直線 AB と CD が垂直であることを AB ⊥ CD と書く。",
    "definition_en": "Meeting at a right angle. Two lines are perpendicular if they cross to form a 90° angle.",
    "latex": "AB \\perp CD",
    "spoken_en": "A B is perpendicular to C D",
    "examples": [
      {
        "en": "These two lines meet at right angles, so they're perpendicular.",
        "ja": "この 2 本の直線は直角に交わっているから、垂直だね。",
        "register": "spoken"
      },
      {
        "en": "Construct a line through point P that is perpendicular to line ℓ.",
        "ja": "点 P を通り、直線 ℓ に垂直な直線を作図せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "前置詞は to。AB ⊥ CD は AB is perpendicular to CD。「垂直に交わる」は meet at right angles。",
      "垂直な 2 直線は perpendicular lines（エントリ perpendicular-lines）で、垂線もこの語で言う。ベクトルの直交は orthogonal（エントリ orthogonal）。",
      "日常語の「垂直」（垂直跳びなど）は上下の向きのことで、英語では vertical。座標平面の vertical line は x = 3 のような縦の直線で、2 本が直角に交わる関係を表す perpendicular とは別の語。"
    ],
    "related": [
      "perpendicular-lines",
      "parallel",
      "orthogonal",
      "perpendicular-bisector",
      "foot-of-the-perpendicular"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "perpendicular !bisector !bisectors !lines !line": 353
      },
      "written": {
        "perpendicular !bisector !bisectors !lines !line": 276
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "phase-shift",
    "ja": {
      "term": "位相のずれ",
      "reading": "いそうのずれ",
      "alt": []
    },
    "en": {
      "term": "phase shift",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「位相のずれ」は学習指導要領解説に無い、本プロジェクトの訳語（日本語版 Wikipedia「位相のずれ」は量子力学の散乱理論の位相差の記事で、別の概念）。学習指導要領解説（数学II 三角関数）は、この内容を三角関数の式の定数とグラフの関係として扱い、「位相」の語は使わない。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "y = sin(b(x − c)) のグラフが y = sin bx のグラフから x 軸方向に c だけ平行移動していること。その量 c。",
    "definition_en": "The horizontal shift c of the graph of y = sin(b(x − c)) from that of y = sin bx.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Factor out the 2 first: sin(2x − π) is sin 2(x − π/2), so the phase shift is π/2 to the right.",
        "ja": "まず 2 をくくり出す。sin(2x − π) = sin 2(x − π/2) なので、右に π/2 ずれる。",
        "register": "spoken"
      },
      {
        "en": "State the amplitude, period, and phase shift of y = 3 cos(2x + π/3).",
        "ja": "y = 3 cos(2x + π/3) の振幅・周期・位相のずれを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "horizontal shift（x 軸方向の平行移動）は三角関数に限らないグラフの平行移動の言い方で、OpenStax Algebra and Trigonometry は正弦曲線の説明にも使う。IM Algebra 2（6.15・6.17〜6.19）は phase shift と言わず horizontal translation と書く。",
      "学習指導要領解説（数学I）と共通テスト（令和7年度 本試験 数学I 第3問）は、グラフの横のずれを「x 軸方向に p だけ平行移動」と書く（二次関数のグラフ）。",
      "b をくくり出さずに c を読むのはよくある間違い（sin(2x − π) のずれは π ではなく π/2）。",
      "微分方程式（OpenStax Calculus Volume 3 の 7.3 Applications、Volume 2 の 4.5 First-order Linear Equations）では、A sin(ωt + ϕ) の角 ϕ そのものを phase shift と呼ぶ（グラフは左に ϕ/ω ずれる）。Algebra and Trigonometry の phase shift C/B はずれの量そのもので、数の意味が違う。"
    ],
    "related": [
      "phase",
      "amplitude",
      "period",
      "translation",
      "sinusoid",
      "auxiliary-angle-form"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II (4) 三角関数: 三角関数の式における定数とグラフとの関係（「位相」の語は無い）。数学I 二次関数: x 軸方向に p だけ平行移動"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和7年度 大学入学共通テスト 本試験 数学Ⅰ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=771&f=abm00005954.pdf&n=2025_op_27_sugaku1.pdf",
        "note": "第3問: x 軸方向に s … だけ平行移動した放物線"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「位相のずれ」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BD%8D%E7%9B%B8%E3%81%AE%E3%81%9A%E3%82%8C",
        "note": "量子力学の散乱理論の位相差（別の概念）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 6.15・6.17・6.18・6.19（horizontal translation）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "7.3 Applications（the constant ϕ is called a phase shift）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "4.5 First-order Linear Equations（the phase shift φ）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "phase shift": 10
      },
      "written": {
        "phase shift": 56
      },
      "sources": [
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "phase",
    "ja": {
      "term": "位相",
      "reading": "いそう",
      "alt": []
    },
    "en": {
      "term": "phase",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "正弦波 y = A sin(ωt + φ) の ωt + φ（または φ）。波がどの位置にあるかを表す。",
    "definition_en": "In a sinusoid y = A sin(ωt + φ), the quantity ωt + φ (or φ alone), telling where in its cycle the wave is.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These two waves have the same frequency but they're out of phase.",
        "ja": "この 2 つの波は振動数が同じだが、位相がずれている。",
        "register": "spoken"
      },
      {
        "en": "Write y = sin t + √3 cos t in the form A sin(t + φ), and give the amplitude and the phase.",
        "ja": "y = sin t + √3 cos t を A sin(t + φ) の形に表し、振幅と位相を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数学の「位相」（topology、位相空間の位相）とは別の意味。",
      "用例コーパスの話し言葉は大半が MIT OCW 18.03（振動の微分方程式）。"
    ],
    "related": [
      "phase-shift",
      "amplitude",
      "auxiliary-angle-form"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "位相",
        "en": "Phase (waves)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "phase !shift !shifts !shifted !line !lines !portrait !portraits !space !spaces !plane !planes !diagram !diagrams": 74
      },
      "written": {
        "phase !shift !shifts !shifted !line !lines !portrait !portraits !space !spaces !plane !planes !diagram !diagrams": 41
      },
      "sources": [
        "khan-middle",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-precalculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pi",
    "ja": {
      "term": "円周率",
      "reading": "えんしゅうりつ",
      "alt": []
    },
    "en": {
      "term": "pi",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円周の長さを直径でわった値で、どんな大きさの円でも同じになる。3.14159… と限りなく続く数で、記号 π で表す。",
    "definition_en": "The ratio of a circle's circumference to its diameter, written with the Greek letter π. Its decimal form never ends or repeats, and it is about 3.14.",
    "latex": "\\pi \\approx 3.14",
    "spoken_en": "pi is approximately three point one four",
    "examples": [
      {
        "en": "Leave your answer in terms of pi, so just write 16 pi. Don't multiply it out.",
        "ja": "答えは π を使って表して。16π と書けばよくて、かけ算して小数にしなくていい。",
        "register": "spoken"
      },
      {
        "en": "The radius of a circle is 4 cm. Find its area. Use 3.14 for pi.",
        "ja": "半径 4 cm の円の面積を求めよ。円周率は 3.14 とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "問題文の leave your answer in terms of pi は「π を使って表せ」で、π を 3.14 に置き換えない。use 3.14 for pi は「円周率は 3.14 とする」。",
      "pi と、円グラフ pie chart（エントリ pie-chart）の pie は同じ発音で（Merriam-Webster はどちらも ˈpī）、綴りが違う。"
    ],
    "related": [
      "circumference",
      "diameter",
      "circle",
      "approximate-value",
      "area"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円周率",
        "en": "Pi"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「pi」",
        "url": "https://www.merriam-webster.com/dictionary/pi",
        "note": "発音 ˈpī。pie も ˈpī（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "pi": 4447
      },
      "written": {
        "pi": 206
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pie-chart",
    "ja": {
      "term": "円グラフ",
      "reading": "えんぐらふ",
      "alt": []
    },
    "en": {
      "term": "pie chart",
      "alt": [
        "pie graph"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Pre-Algebra",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "円を全体とし、各項目の割合をおうぎ形の大きさで表したグラフ。",
    "definition_en": "A circle divided into sectors whose sizes show the share of each category in the whole.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A pie chart only makes sense when the pieces add up to the whole.",
        "ja": "円グラフは、部分を足すと全体になるときだけ意味がある。",
        "register": "spoken"
      },
      {
        "en": "Make a pie chart of the survey results.",
        "ja": "調査結果の円グラフをかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "bar-chart",
      "stacked-bar-chart",
      "statistical-graph"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "円グラフ",
        "en": "Pie chart"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "pie chart": 3,
        "pie graph": 3
      },
      "written": {
        "pie chart": 14
      },
      "sources": [
        "khan-middle",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "piecewise-function",
    "ja": {
      "term": "区分的関数",
      "reading": "くぶんてきかんすう",
      "alt": [
        "区分的定義"
      ]
    },
    "en": {
      "term": "piecewise function",
      "alt": [
        "piecewise-defined function"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の高校では「場合分けして定義された関数」として扱い、名前を付けないことが多い。英語では piecewise function という名前がある。OpenStax Algebra and Trigonometry も piecewise function として扱う。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "definition_ja": "x の範囲ごとに違う式で定義される関数。f(x) = { x + 1（x < 0）、x²（x ≧ 0）} のように、波かっこで式と範囲を並べて書く。",
    "definition_en": "A function defined by different formulas on different parts of its domain. It is written with a brace that lists each formula next to the interval where it applies.",
    "latex": "f(x) = \\begin{cases} x + 1 & x < 0 \\\\ x^2 & x \\ge 0 \\end{cases}",
    "spoken_en": "f of x equals x plus one when x is less than zero, and x squared when x is greater than or equal to zero",
    "examples": [
      {
        "en": "This is a piecewise function, so before you plug in, check which piece your x-value falls in.",
        "ja": "これは区分的関数だから、代入する前に、その x の値がどの式の範囲に入るかを確かめよう。",
        "register": "spoken"
      },
      {
        "en": "For the piecewise function f(x) = {x + 1 if x < 0, x² if x ≥ 0}, find f(−3) and f(2).",
        "ja": "関数 f(x) = { x + 1（x < 0）、x²（x ≧ 0）} について、f(−3) と f(2) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "グラフでは、範囲の端の点を含むか（●）含まないか（○）をかき分ける。x < 0 の式の端 (0, 1) は ○、x ≧ 0 の式の端 (0, 0) は ●。",
      "絶対値関数 |x| も、|x| = { −x（x < 0）、x（x ≧ 0）} と piecewise function の形で書ける（エントリ absolute-value-function）。",
      "piece は「部分」という意味で、式と範囲の 1 組 1 組を piece と呼ぶ（the first piece、the second piece）。"
    ],
    "related": [
      "absolute-value-function",
      "step-function",
      "function-notation",
      "domain",
      "function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "piecewise function": 28,
        "piecewise-defined function": 2
      },
      "written": {
        "piecewise-defined function": 7,
        "piecewise function": 48
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pigeonhole-principle",
    "ja": {
      "term": "鳩の巣原理",
      "reading": "はとのすげんり",
      "alt": [
        "部屋割り論法",
        "ディリクレの箱入れ原理"
      ]
    },
    "en": {
      "term": "pigeonhole principle",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics",
      "proof"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "n 個の箱に n より多くの物を入れると、2 個以上入る箱が少なくとも 1 つあるという原理。一般に n 個の物を k 個の箱に入れると、⌈n/k⌉ 個以上入る箱がある。",
    "definition_en": "The principle that if more than n objects are put into n boxes, some box gets at least two of them. More generally, if n objects go into k boxes, some box holds at least ⌈n/k⌉ objects.",
    "latex": "\\left\\lceil \\frac{n}{k} \\right\\rceil",
    "spoken_en": "the ceiling of n over k",
    "examples": [
      {
        "en": "There are 13 people in this room, so by the pigeonhole principle, at least two of them were born in the same month.",
        "ja": "この部屋には 13 人いるので、鳩の巣原理から、少なくとも 2 人は同じ月に生まれている。",
        "register": "spoken"
      },
      {
        "en": "Use the pigeonhole principle to show that among any 5 integers, there are two that leave the same remainder when divided by 4.",
        "ja": "鳩の巣原理を用いて、任意の 5 つの整数の中に、4 で割った余りが等しいものが 2 つあることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "pigeonhole は「鳩の巣」ではなく、鳩小屋の仕切りや書類の仕分け棚を指す語。英語では pigeon nest とは言わず、pigeonhole principle で 1 つの名前。",
      "日本の入試問題の解説では「部屋割り論法」と呼ぶことがある。証明の中では by the pigeonhole principle と理由を添える。",
      "用例コーパスでは話し言葉には出てこず、書き言葉はすべて MIT の講義ノート。Levin の本では 5 件（1.4 Proofs ほか）。"
    ],
    "related": [
      "counting",
      "proof-by-contradiction",
      "ceiling-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "鳩の巣原理",
        "en": "Pigeonhole principle"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "pigeonhole principle": 18
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pivot",
    "ja": {
      "term": "ピボット",
      "reading": "ぴぼっと",
      "alt": []
    },
    "en": {
      "term": "pivot",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "行階段形で、各行の先頭にある 0 でない成分。その位置の列をピボット列という。",
    "definition_en": "The first nonzero entry in a row of a row-echelon form; its column is a pivot column.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The second pivot is zero, so we have to swap rows.",
        "ja": "2 つ目のピボットが 0 なので、行を入れ替えないといけない。",
        "register": "spoken"
      },
      {
        "en": "Circle the pivot in each row and identify the pivot columns.",
        "ja": "各行のピボットに丸をつけ、ピボット列を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Nicholson は pivot と言わず leading 1（先頭の 1）と書く。"
    ],
    "related": [
      "reduced-row-echelon-form",
      "row-echelon-form",
      "basic-variable",
      "free-variable"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Pivot_element",
        "note": "記事「Pivot element」の冒頭で pivot（pivot element）を太字で定義（行列の消去で最初に選ぶ 0 でない成分）。数学カテゴリから 5 段（本文の定義を出典に使える 5 段以内。見出しを記事名で決める 4 段の外）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "pivot column | pivot position | pivot entry | first pivot | second pivot | third pivot | product of the pivots": 103
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "place",
    "ja": {
      "term": "位",
      "reading": "くらい",
      "alt": [
        "十の位",
        "一の位"
      ]
    },
    "en": {
      "term": "place",
      "alt": [],
      "uk": null,
      "register": "both",
      "variants": [
        {
          "term": "place value",
          "register": "written",
          "note": "位取りの考え方の名前として。書き言葉では OpenStax Prealgebra・Elementary Algebra に多い。話し言葉はほとんどが Khan Academy の中学の講義。"
        },
        {
          "term": "place",
          "register": "both",
          "note": "ones place ／ tens place ／ hundreds place の形で数えた。話し言葉・書き言葉ともよく使う。話し言葉は大半が Khan Academy の中学の講義、書き言葉は大半が OpenStax Prealgebra。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「位」は単独でも言えるが、英語の place はふつう ones place ／ tens place ／ hundreds place のように位の名前と組で使う。place value は「位取り」、つまり数字の値が置かれた位で決まるという考え方を指す。小数の位は tenths place（小数第 1 位）、hundredths place（小数第 2 位）と言う。",
    "domains": [
      "algebra",
      "number-theory"
    ],
    "level": {
      "jp": [
        "中2",
        "数A"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Discrete Math"
      ]
    },
    "definition_ja": "位取り記数法で、数字が置かれる位置。一の位・十の位・百の位のように、位置によって数字の重みが決まる。",
    "definition_en": "A position in a number written in a place-value system; the ones, tens, and hundreds places give the digit in them a weight of 1, 10, and 100.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "What digit is in the tens place of 4,735?",
        "ja": "4735 の十の位の数字は何？",
        "register": "spoken"
      },
      {
        "en": "Carry the one over to the hundreds place.",
        "ja": "繰り上がった 1 を百の位に足してね。",
        "register": "spoken"
      },
      {
        "en": "Let a be the digit in the hundreds place, b the digit in the tens place, and c the digit in the ones place.",
        "ja": "百の位の数を a、十の位の数を b、一の位の数を c とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "書き言葉では OpenStax Prealgebra が位取りの章で place value を多く使い、ones place の形と並ぶ。",
      "「一の位の数」は the digit in the ones place（the ones digit とも言える）。the one place と単数にしない。"
    ],
    "related": [
      "digit",
      "numeral-system",
      "number-of-digits",
      "base-n"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "ones place | tens place | hundreds place": 250,
        "place value": 55
      },
      "written": {
        "ones place | tens place | hundreds place": 120,
        "place value": 143
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "placebo",
    "ja": {
      "term": "プラセボ",
      "reading": "ぷらせぼ",
      "alt": [
        "偽薬"
      ]
    },
    "en": {
      "term": "placebo",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "有効成分を含まない、見かけだけ本物と同じ薬や処理。",
    "definition_en": "A fake treatment, such as a pill with no active ingredient, that looks like the real one.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Some people feel better just from the placebo.",
        "ja": "プラセボだけで具合がよくなる人もいる。",
        "register": "spoken"
      },
      {
        "en": "Why did the researchers give the control group a placebo?",
        "ja": "研究者が対照群にプラセボを与えたのはなぜか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "プラセボで効果が出ることを placebo effect（プラセボ効果）と言う。"
    ],
    "related": [
      "control-group",
      "blinding",
      "treatment"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "placebo": 32
      },
      "written": {
        "placebo": 15
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "planar-graph",
    "ja": {
      "term": "平面グラフ",
      "reading": "へいめんぐらふ",
      "alt": [
        "平面的グラフ"
      ]
    },
    "en": {
      "term": "planar graph",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "辺どうしが交わらないように平面上にかくことのできるグラフのこと。交差してかかれていても、かき直して交差をなくせれば平面グラフである。",
    "definition_en": "A graph that can be drawn in the plane so that no two edges cross. A drawing with crossings does not rule it out, as long as some other drawing has none.",
    "latex": "v - e + f = 2",
    "spoken_en": "v minus e plus f equals two",
    "examples": [
      {
        "en": "It looks like these edges cross, but you can redraw it with no crossings at all, so it's a planar graph.",
        "ja": "辺が交差しているように見えるけれど、交差がないようにかき直せるから、これは平面グラフだ。",
        "register": "spoken"
      },
      {
        "en": "Explain why K₅, the complete graph on 5 vertices, is not a planar graph.",
        "ja": "5 頂点の完全グラフ K₅ が平面グラフでない理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の本によっては、交差なくかけるグラフを「平面的グラフ」、交差なくかいた図そのものを「平面グラフ」と呼び分ける。英語では前者が planar graph、後者が plane graph。",
      "連結な平面グラフでは v − e + f = 2（頂点・辺・面の数。外側の領域も 1 つの面と数える）が成り立つ。K₅ と K₃,₃ は平面グラフでない。",
      "用例コーパスでは書き言葉はすべて MIT の講義ノートで、話し言葉には出てこない。Levin の本では 53 件（2.3 Planar Graphs 43 ほか）。"
    ],
    "related": [
      "eulers-formula-for-polyhedra",
      "graph-coloring",
      "graph-network"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "平面グラフ",
        "en": "Planar graph"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "planar graph": 54
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "plane",
    "ja": {
      "term": "平面",
      "reading": "へいめん",
      "alt": []
    },
    "en": {
      "term": "plane",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "どの方向にも限りなく広がっている、平らな面。厚みはない。",
    "definition_en": "A flat surface with no thickness that extends forever in every direction.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Think of a plane as a sheet of paper that goes on forever in every direction.",
        "ja": "平面は、どこまでも広がっている紙のようなものだと思って。",
        "register": "spoken"
      },
      {
        "en": "Lines ℓ and m lie in plane P and do not intersect. What is the relationship between ℓ and m?",
        "ja": "直線 ℓ と m は平面 P 上にあり、交わらない。ℓ と m はどのような位置関係にあるか答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "座標平面は coordinate plane（エントリ coordinate-plane）、複素平面は complex plane、接平面は tangent plane、平面図形は plane figure。",
      "日常語の plane（飛行機）と同じ綴り。plain（平原、ふつうの）とは発音が同じで（Merriam-Webster はどちらも ˈplān）綴りが違う。"
    ],
    "related": [
      "coordinate-plane",
      "complex-plane",
      "tangent-plane",
      "equation-of-a-plane",
      "solid"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "平面",
        "en": "Plane (mathematics)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「plane」",
        "url": "https://www.merriam-webster.com/dictionary/plane",
        "note": "発音 ˈplān。plain も ˈplān（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!coordinate !complex !cartesian !tangent !projective !osculating plane": 798
      },
      "written": {
        "!coordinate !complex !cartesian !tangent !projective !osculating plane": 1066
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "plot-a-point",
    "ja": {
      "term": "点をとる",
      "reading": "てんをとる",
      "alt": [
        "点をとる操作"
      ]
    },
    "en": {
      "term": "plot a point",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "座標をもとに、座標平面上のその位置に点をかき入れること。原点から x 座標の分だけ左右に、y 座標の分だけ上下に進んだところに点を打つ。",
    "definition_en": "To mark a point at its location on the coordinate plane, using its coordinates. Start at the origin, move left or right by the x-coordinate, then up or down by the y-coordinate.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's plot a point at (3, −2): start at the origin, go right 3, then down 2.",
        "ja": "(3, −2) に点をとろう。原点から右に 3、下に 2 進んだところだよ。",
        "register": "spoken"
      },
      {
        "en": "Plot a point at (−4, 1) and label it A.",
        "ja": "点 A(−4, 1) を座標平面上にとれ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「点をとる」の「とる」を take と訳さない。take a point は「（条件を満たす）点を 1 つ選んで考える」（take a point P on the circle）の意味で、座標平面に点を打つことではない。",
      "plot は名詞で統計の図の名前にもなる（scatter plot ／ box plot。エントリ scatter-plot、box-plot）。"
    ],
    "related": [
      "coordinate-plane",
      "x-coordinate",
      "y-coordinate",
      "draw-a-graph",
      "scatter-plot"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "plot … point | plot point": 93
      },
      "written": {
        "plot … point | plot point": 167
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "point-estimate",
    "ja": {
      "term": "点推定",
      "reading": "てんすいてい",
      "alt": [
        "点推定値"
      ]
    },
    "en": {
      "term": "point estimate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3",
        "Algebra 2"
      ]
    },
    "definition_ja": "母数を 1 つの値で推定すること。また、その値。母平均の点推定には標本平均を使う。",
    "definition_en": "An estimate of a population parameter by a single value, or that value; the sample mean is the point estimate of the population mean.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The sample mean, 72.4, is our point estimate of the population mean.",
        "ja": "標本平均 72.4 が母平均の点推定値。",
        "register": "spoken"
      },
      {
        "en": "Give a point estimate for the proportion of all voters who support the proposal.",
        "ja": "提案を支持する有権者全体の割合の点推定値を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて Professor Leonard。"
    ],
    "related": [
      "estimation",
      "interval-estimation",
      "confidence-interval",
      "error-bound"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "点推定",
        "en": "Point estimation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 7.10 Estimating Proportions from Samples・7.12 Estimating a Population Mean"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "point estimate": 14
      },
      "written": {
        "point estimate": 40
      },
      "sources": [
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "point-of-external-division",
    "ja": {
      "term": "外分点",
      "reading": "がいぶんてん",
      "alt": [
        "外分"
      ]
    },
    "en": {
      "term": "point that divides the segment externally",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスには divides … externally は出てこず、OpenStax にもない）。the point that divides the segment AB externally in the ratio m:n と説明する。英語版 Wikipedia「Section formula」は外分を節 External divisions で扱い、divided externally と書く。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A",
        "数II"
      ],
      "us": []
    },
    "definition_ja": "線分 AB の延長上にあって、AP : PB = m : n（m ≠ n）となる点 P。m > n なら B の外側、m < n なら A の外側にある。",
    "definition_en": "The point P on line AB but outside segment AB with AP : PB = m : n (m ≠ n); it lies beyond B when m > n and beyond A when m < n.",
    "latex": "\\left(\\frac{-n x_1 + m x_2}{m - n}, \\frac{-n y_1 + m y_2}{m - n}\\right)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Dividing externally in the ratio 3 to 1 means P is outside the segment, past B, and AP is three times PB.",
        "ja": "3 : 1 に外分するというのは、P が線分の外、B の先にあって AP が PB の 3 倍ということ。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinates of the point that divides the segment externally in the ratio 3 : 1, where A = (1, 2) and B = (5, 4).",
        "ja": "2 点 A(1, 2), B(5, 4) を結ぶ線分を 3 : 1 に外分する点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "外分点の座標は、内分点の公式で n を −n に置き換えたもの。m = n のときは外分点は存在しない。",
      "線分 AB を m : n に内分する点と外分する点を直径の両端とする円が、アポロニウスの円になる。"
    ],
    "related": [
      "point-of-internal-division",
      "midpoint",
      "directed-segment",
      "apollonian-circle",
      "exterior-angle-bisector",
      "angle-bisector-theorem"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Section_formula",
        "note": "記事「Section formula」の節 External divisions（数学カテゴリから 5 段）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 図形と方程式: 線分を内分する点、外分する点"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "point-of-internal-division",
    "ja": {
      "term": "内分点",
      "reading": "ないぶんてん",
      "alt": [
        "内分する点",
        "内分"
      ]
    },
    "en": {
      "term": "point that divides the segment internally",
      "alt": [
        "point of internal division"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。the point that divides the segment AB internally in the ratio m:n と説明する。内分点の座標の公式は、CK-12 Geometry（1.6 Points that Partition Line Segments）が section formula と呼ぶ（10 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A",
        "数II"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "線分 AB 上にあって、AP : PB = m : n となる点 P。座標は両端の座標を n : m の重みで平均したもの。",
    "definition_en": "The point P on segment AB with AP : PB = m : n; its coordinates are the weighted average of the endpoints with weights n and m.",
    "latex": "\\left(\\frac{n x_1 + m x_2}{m + n}, \\frac{n y_1 + m y_2}{m + n}\\right)",
    "spoken_en": null,
    "examples": [
      {
        "en": "The point that divides the segment internally in the ratio 2 to 1 is two-thirds of the way from A to B.",
        "ja": "AB を 2 : 1 に内分する点は、A から B への 3 分の 2 のところにある。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinates of the point that divides the segment joining A(1, 2) and B(7, 5) internally in the ratio 2 : 1.",
        "ja": "2 点 A(1, 2), B(7, 5) を結ぶ線分を 2 : 1 に内分する点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "外分点は point-of-external-division（外分点）。英語版 Wikipedia「Section formula」は内分を節 Internal divisions で扱い、divided internally と書く。",
      "IM Geometry（6.15〜6.17）と CK-12 Geometry（1.6）は、内分点を partition（分ける）で言う（the point that partitions segment AB …）。"
    ],
    "related": [
      "midpoint",
      "centroid",
      "point-of-external-division",
      "angle-bisector-theorem",
      "apollonian-circle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.6 Points that Partition Line Segments（section formula、10 件）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 6.15〜6.17（the point that partitions segment AB）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Section_formula",
        "note": "記事「Section formula」の節 Internal divisions（数学カテゴリから 5 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "point-of-intersection",
    "ja": {
      "term": "交点",
      "reading": "こうてん",
      "alt": [
        "交点の座標",
        "共有点"
      ]
    },
    "en": {
      "term": "point of intersection",
      "alt": [],
      "variants": [
        {
          "term": "point of intersection",
          "register": "both",
          "note": "話し言葉・書き言葉とも intersection point より多い。話し言葉は Khan Academy が半分ほど、書き言葉は OpenStax Algebra and Trigonometry が半分ほど。"
        },
        {
          "term": "intersection point",
          "register": "written",
          "note": "話し言葉・書き言葉とも point of intersection より少ない。話し言葉は 3Blue1Brown が半分以上、書き言葉は大半が OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「交点」は交わる点で、接する点も含めて言うときは「共有点」と言う（高等学校学習指導要領解説（数学I）は二次方程式の解をグラフと x 軸との共有点の x 座標と捉え、共通テスト（令和 3 年度 第 1 日程 数学II 第 3 問）は「2 点で交わる」「接する」「共有点をもたない」を分けて言う）。英語の point of intersection は intersect（エントリ intersect）する点で、接する場合の点も含む（IM Geometry の用語集（7.3 Tangent Lines）は接線を円と exactly one point で intersect する直線と定義し、OpenStax Algebra and Trigonometry 11.3 も接線は円と intersects … at exactly one point と書き、放物線と直線の Points of Intersection の場合の 1 つに直線が放物線に接する場合を挙げる）。接する点だとはっきり言うときは point of tangency（エントリ point-of-tangency）か、the graph touches the x-axis と言う。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中2",
        "数I",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Geometry",
        "Precalculus",
        "Algebra 2",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 つの線（直線・曲線）が交わる点。交点の座標は 2 つの方程式を連立して求める。",
    "definition_en": "The point where two lines or curves meet; its coordinates come from solving the two equations together.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Set the two equations equal to each other to find the point of intersection.",
        "ja": "2 つの式を等しいとおいて交点を求める。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinates of the points of intersection of the line y = x + 1 and the circle x² + y² = 5.",
        "ja": "直線 y = x + 1 と円 x² + y² = 5 の交点の座標を求めよ。",
        "register": "written"
      },
      {
        "en": "How many points of intersection does the graph of y = x² − 4x + k have with the x-axis?",
        "ja": "y = x² − 4x + k のグラフと x 軸の共有点の個数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数の交点は points of intersection（intersection points）。",
      "接することをはっきり言うときは、the graph touches the x-axis（x 軸に接する）や point of tangency（エントリ point-of-tangency）と言う。"
    ],
    "related": [
      "intersect",
      "circle-through-the-intersections-of-two-circles",
      "point-of-tangency"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.3 Systems of Nonlinear Equations and Inequalities: Two Variables（intersects the circle at exactly one point・Points of Intersection of a Parabola and a Line。mapping_note）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 二次関数: 二次方程式の解をグラフと x 軸との共有点の x 座標と捉える（mapping_note）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和3年度 大学入学共通テスト 第1日程 数学Ⅱ（問題）",
        "url": "https://web.archive.org/web/20220601/https://www.dnc.ac.jp/albums/abm.php?d=88&f=abm00001261.pdf&n=R3_%E6%95%B0%E5%AD%A6%E2%85%A1%EF%BC%88%EF%BC%91%E6%9C%8816%E6%97%A5%E3%83%BB17%E6%97%A5%EF%BC%89%E3%81%AE%E5%95%8F%E9%A1%8C.pdf",
        "note": "第 3 問: 2 点で交わる・接する・共有点をもたない を分ける解答群（mapping_note）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.3 Tangent Lines・glossary（tangent (line): intersects the circle at exactly one point。mapping_note）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 C(1) 一次関数: 連立二元一次方程式の解を 2 直線の交点の座標として求める（level.jp 中2）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "intersection point": 13,
        "point of intersection": 53
      },
      "written": {
        "point of intersection": 64,
        "intersection point": 16
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "point-of-tangency",
    "ja": {
      "term": "接点",
      "reading": "せってん",
      "alt": [
        "接点の座標"
      ]
    },
    "en": {
      "term": "point of tangency",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Geometry",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "曲線や円に接線が接している点。",
    "definition_en": "The point at which a curve or a circle meets its tangent line.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Call the point of tangency (a, a²). The slope there is 2a.",
        "ja": "接点を (a, a²) とおく。そこでの傾きは 2a。",
        "register": "spoken"
      },
      {
        "en": "Find the point of tangency of the line y = 2x − 1 and the parabola y = x².",
        "ja": "直線 y = 2x − 1 と放物線 y = x² の接点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「接点の x 座標を t とおく」は、英語では Let the point of tangency be (t, f(t)). のように書ける（本プロジェクトの書き方の例。用例コーパスと参照には出てこない）。"
    ],
    "related": [
      "slope-of-the-tangent-line",
      "normal-line",
      "find-the-equation-of-the-tangent-line"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 4.6（point of tangency）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "point of tangency": 5
      },
      "written": {
        "point of tangency": 3
      },
      "sources": [
        "mit-18.01",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは CED の呼び方 point of tangency（topic 4.6）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "point-slope-form",
    "ja": {
      "term": "点傾き形",
      "reading": "てんかたむきけい",
      "alt": []
    },
    "en": {
      "term": "point-slope form",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「点傾き形」は学習指導要領解説に無い、本プロジェクトの訳語。日本では数II で「点 (x₁, y₁) を通り傾き m の直線の方程式」として同じ式を学ぶが、形の名前はない。中 2 では y = ax + b に点の座標を代入して b を求めることが多い。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "点 (x₁, y₁) を通り傾き m の直線の式を、y − y₁ = m(x − x₁) の形に書いたもの。",
    "definition_en": "The form y − y₁ = m(x − x₁) of the equation of the line with slope m that passes through the point (x₁, y₁).",
    "latex": "y - y_1 = m(x - x_1)",
    "spoken_en": "y minus y sub one equals m times the quantity x minus x sub one",
    "examples": [
      {
        "en": "We know the slope and one point, so point-slope form is the fastest way to write the equation.",
        "ja": "傾きと 1 点がわかっているから、点傾き形を使うのがいちばん早く式を書ける。",
        "register": "spoken"
      },
      {
        "en": "Write an equation in point-slope form for the line through (2, −1) with slope 3.",
        "ja": "点 (2, −1) を通り、傾きが 3 の直線の式を点傾き形で書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "点の座標が負のときは符号に注意する。点 (2, −1) なら y − (−1) で、y + 1 = 3(x − 2) になる。",
      "x₁ は x sub one と読む。",
      "slope-intercept form（エントリ slope-intercept-form-of-a-line）に直すと、y + 1 = 3(x − 2) は y = 3x − 7。"
    ],
    "related": [
      "slope",
      "slope-intercept-form-of-a-line",
      "standard-form-of-a-line",
      "equation-of-a-line",
      "slope-formula"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "point-slope form": 25
      },
      "written": {
        "point-slope form": 44
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "point-symmetry",
    "ja": {
      "term": "点対称",
      "reading": "てんたいしょう",
      "alt": []
    },
    "en": {
      "term": "point symmetry",
      "alt": [
        "symmetric about a point"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。日本語の「点対称」は、180° の回転で重なる回転対称（rotational symmetry、エントリ rotational-symmetry）と同じ性質で、IM Geometry は 180 degree rotational symmetry と言い、CK-12 Geometry は rotational symmetry（8.4 Rotation Symmetry）と呼ぶ。関数のグラフが原点について対称（symmetric about the origin）なのも点対称の 1 つで、グラフがそうなる関数が奇関数（エントリ odd-function）。英語版 Wikipedia「Point reflection」（ja 記事「点対称」の langlink 先）は冒頭で point symmetry を太字で定義する。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "図形をある点のまわりに 180° 回転させると、もとの図形にぴったり重なる性質。その点を対称の中心という。",
    "definition_en": "The property a figure has when a 180° turn about a point makes it land exactly on itself. It is the same as having rotational symmetry of 180°.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The letter S has point symmetry: turn it upside down, 180 degrees, and it looks exactly the same.",
        "ja": "S という文字は点対称。180° 回してさかさまにしても、まったく同じに見えるよね。",
        "register": "spoken"
      },
      {
        "en": "List all the letters among H, N, S, and T that have point symmetry.",
        "ja": "H、N、S、T のうち、点対称な文字をすべて答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "rotational symmetry（エントリ rotational-symmetry）は 120° や 90° などの回転で重なる場合も含む。点対称は 180° の回転で重なる場合で、正三角形は 120° の回転対称をもつが点対称ではない。",
      "線対称（エントリ line-symmetry）と区別する。平行四辺形は点対称だが、長方形でもひし形でもない平行四辺形は線対称ではない。",
      "グラフが原点に関して点対称であることは symmetric about the origin と言う（エントリ symmetric）。y = 2x や y = 6/x のグラフがその例。"
    ],
    "related": [
      "rotational-symmetry",
      "line-symmetry",
      "symmetric",
      "odd-function",
      "rotation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "点対称",
        "en": "Point reflection"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "180 degree rotational symmetry（2 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.4 Rotation Symmetry（rotational symmetry 16 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "poisson-distribution",
    "ja": {
      "term": "ポアソン分布",
      "reading": "ぽあそんぶんぷ",
      "alt": []
    },
    "en": {
      "term": "Poisson distribution",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "一定の時間や範囲の中で、まれな出来事が起こる回数が従う確率分布。平均と分散がどちらも λ。",
    "definition_en": "The distribution of the number of times a rare event happens in a fixed interval of time or space; its mean and variance are both λ.",
    "latex": "P(X = k) = \\frac{\\lambda^k e^{-\\lambda}}{k!}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Calls coming in at an average of four an hour: that's a Poisson distribution with lambda equals four.",
        "ja": "1 時間に平均 4 本かかってくる電話は、λ = 4 のポアソン分布だ。",
        "register": "spoken"
      },
      {
        "en": "The number of typos per page follows a Poisson distribution with mean 1.5. Find the probability of no typos on a page.",
        "ja": "1 ページの誤植の数は平均 1.5 のポアソン分布に従う。あるページに誤植がない確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Poisson は人名（フランスの数学者）。英語の発音は「ポワソン」に近い（Merriam-Webster「Poisson distribution」: pwä-ˈsōⁿ）。用例コーパスの書き言葉はすべて OpenStax Introductory Statistics。"
    ],
    "related": [
      "binomial-distribution",
      "exponential-distribution",
      "probability-distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ポアソン分布",
        "en": "Poisson distribution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「Poisson distribution」",
        "url": "https://www.merriam-webster.com/dictionary/Poisson%20distribution",
        "note": "発音 pwä-ˈsōⁿ-（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Poisson distribution": 2
      },
      "written": {
        "Poisson distribution": 38
      },
      "sources": [
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "polar-angle",
    "ja": {
      "term": "偏角",
      "reading": "へんかく",
      "alt": []
    },
    "en": {
      "term": "polar angle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスでは polar angle は話し言葉・書き言葉ともほとんど出てこない）。極座標の θ は、英語ではふつう名前をつけずに the angle θ と呼ぶ。複素数の偏角は argument（argument-of-a-complex-number を参照）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "極座標 (r, θ) の θ。始線から反時計回りに測った角。",
    "definition_en": "The θ of polar coordinates (r, θ): the angle measured counterclockwise from the polar axis.",
    "latex": "\\theta",
    "spoken_en": null,
    "examples": [
      {
        "en": "θ is the angle you sweep out from the positive x-axis.",
        "ja": "θ は x 軸の正の向きから回した角。",
        "register": "spoken"
      },
      {
        "en": "Find r and θ for the point (1, −1), with 0 ≤ θ < 2π.",
        "ja": "点 (1, −1) の r と θ を 0 ≤ θ < 2π の範囲で求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "polar-coordinates",
      "pole",
      "argument-of-a-complex-number"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "polar angle": 1
      },
      "written": {
        "polar angle": 1
      },
      "sources": [
        "mit-18.03",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 1 件 ／ 書 1 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "polar-coordinates",
    "ja": {
      "term": "極座標",
      "reading": "きょくざひょう",
      "alt": []
    },
    "en": {
      "term": "polar coordinates",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "極からの距離 r と、始線から測った角 θ の組 (r, θ) で平面上の点を表す座標。",
    "definition_en": "Coordinates (r, θ) that locate a point by its distance r from the pole and its angle θ from the polar axis.",
    "latex": "x = r\\cos\\theta, \\quad y = r\\sin\\theta",
    "spoken_en": "x equals r cosine theta, y equals r sine theta",
    "examples": [
      {
        "en": "In polar coordinates, a circle centered at the origin is just r equals a constant.",
        "ja": "極座標では、原点中心の円は r = 定数 になるだけ。",
        "register": "spoken"
      },
      {
        "en": "Convert the point (−1, √3) to polar coordinates.",
        "ja": "点 (−1, √3) を極座標で表せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "write … in polar coordinates",
        "ja": "極座標で表す"
      }
    ],
    "pitfalls": [
      "始線は polar axis（用例コーパスの話し言葉ではすべて Professor Leonard）。",
      "同じ点に (r, θ) の表し方が何通りもある（θ に 2π を足しても同じ点）。"
    ],
    "related": [
      "pole",
      "polar-equation",
      "rectangular-coordinates",
      "polar-angle",
      "area-in-polar-coordinates"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "極座標系",
        "en": "Polar coordinate system"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 9.7–9.9（polar coordinates）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "polar coordinates": 127
      },
      "written": {
        "polar coordinates": 223
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "polar-equation",
    "ja": {
      "term": "極方程式",
      "reading": "きょくほうていしき",
      "alt": []
    },
    "en": {
      "term": "polar equation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "曲線を r と θ の関係 r = f(θ) などで表した方程式。",
    "definition_en": "An equation of a curve written as a relation between r and θ, such as r = f(θ).",
    "latex": "r = \\frac{ed}{1 + e\\cos\\theta}",
    "spoken_en": "r equals e d over one plus e cosine theta",
    "examples": [
      {
        "en": "A polar equation like r = 2 sin θ is actually a circle.",
        "ja": "r = 2 sin θ のような極方程式は、実は円。",
        "register": "spoken"
      },
      {
        "en": "Identify the conic with polar equation r = 4/(1 − cos θ).",
        "ja": "極方程式 r = 4/(1 − cos θ) で表される二次曲線を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "二次曲線は、焦点を極とした極方程式 r = ed/(1 ± e cos θ) で一まとめに書ける（離心率 e）。"
    ],
    "related": [
      "polar-coordinates",
      "conic-section",
      "eccentricity",
      "rectangular-coordinates"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 9.7（polar equation）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "polar equation": 45
      },
      "written": {
        "polar equation": 123
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "polar-form",
    "ja": {
      "term": "極形式",
      "reading": "きょくけいしき",
      "alt": []
    },
    "en": {
      "term": "polar form",
      "alt": [
        "trigonometric form"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "複素数を絶対値 r と偏角 θ を使って r(cos θ + i sin θ) と表した形。",
    "definition_en": "The form r(cos θ + i sin θ) of a complex number, written with its modulus r and argument θ.",
    "latex": "z = r(\\cos\\theta + i\\sin\\theta)",
    "spoken_en": "z equals r times cosine theta plus i sine theta",
    "examples": [
      {
        "en": "In polar form, multiplying is easy: multiply the r's and add the angles.",
        "ja": "極形式なら掛け算は簡単。r どうしを掛けて、角を足す。",
        "register": "spoken"
      },
      {
        "en": "Write z = 1 + √3 i in polar form.",
        "ja": "z = 1 + √3 i を極形式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "write … in polar form",
        "ja": "極形式で表す"
      }
    ],
    "pitfalls": [
      "trigonometric form とも言うが少ない（用例コーパスには話し言葉・書き言葉ともほとんど出てこない）。",
      "積と商: 絶対値は掛け算・割り算、偏角は足し算・引き算。英語でも multiply the moduli and add the arguments と言える。",
      "polar form は極方程式（曲線を r と θ で表した形）の意味でも使う。"
    ],
    "related": [
      "de-moivres-theorem",
      "complex-plane",
      "eulers-formula",
      "modulus-of-a-complex-number",
      "argument-of-a-complex-number"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "polar form": 42,
        "trigonometric form": 1
      },
      "written": {
        "polar form": 97,
        "trigonometric form": 2
      },
      "sources": [
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pole",
    "ja": {
      "term": "極",
      "reading": "きょく",
      "alt": []
    },
    "en": {
      "term": "pole",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "極座標の基準となる点。直交座標の原点にあたる。",
    "definition_en": "The reference point of a polar coordinate system, playing the role of the origin.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "r is just the distance from the pole to the point.",
        "ja": "r は極から点までの距離にすぎない。",
        "register": "spoken"
      },
      {
        "en": "The point with r = 0 is the pole, whatever the value of θ.",
        "ja": "r = 0 の点は、θ の値によらず極である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "pole は複素関数の極（MIT 18.03 の pole diagram など）や「棒」の意味でも使う。",
      "始線（極から出る半直線）は polar axis。"
    ],
    "related": [
      "polar-coordinates",
      "polar-angle",
      "polar-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "from the pole": 14
      },
      "written": {
        "from the pole": 22
      },
      "sources": [
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "polygon",
    "ja": {
      "term": "多角形",
      "reading": "たかくけい",
      "alt": []
    },
    "en": {
      "term": "polygon",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "3 本以上の線分で囲まれた平面図形。三角形・四角形・五角形などをまとめて多角形という。",
    "definition_en": "A closed flat figure made of three or more straight sides that meet only at their endpoints. Triangles, quadrilaterals, and pentagons are all polygons.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A circle isn't a polygon — a polygon has to be made of straight sides, and a circle doesn't have any.",
        "ja": "円は多角形じゃないよ。多角形はまっすぐな辺でできていないといけないけど、円にはまっすぐな辺がないからね。",
        "register": "spoken"
      },
      {
        "en": "Find the sum of the interior angles of a polygon with 9 sides.",
        "ja": "九角形の内角の和を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "n 角形は英語で n-gon と言う（9-gon は nine-gon と読む）。五角形 pentagon、六角形 hexagon、八角形 octagon のように決まった名前をもつものもある。",
      "polygon は平面図形。立体の多面体は polyhedron（エントリ polyhedron）で、poly-（多い）が共通なので取り違えない。",
      "辺はすべて線分なので、円や、曲線の辺をもつおうぎ形（エントリ sector）は polygon ではない。"
    ],
    "related": [
      "quadrilateral",
      "triangle",
      "regular-polygon",
      "polyhedron",
      "side"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "多角形",
        "en": "Polygon"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.18 Classify Polygons"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "polygon": 46
      },
      "written": {
        "polygon": 35
      },
      "sources": [
        "khan-middle",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "polyhedron",
    "ja": {
      "term": "多面体",
      "reading": "ためんたい",
      "alt": []
    },
    "en": {
      "term": "polyhedron",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "平面の多角形だけで囲まれた立体。囲んでいる多角形を面、面と面が交わる線分を辺、辺が集まる点を頂点という。",
    "definition_en": "A solid whose surface is made up entirely of flat polygons. The polygons are its faces, the segments where two faces meet are its edges, and the points where edges meet are its vertices.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A cylinder isn't a polyhedron, because it has a curved surface.",
        "ja": "円柱は多面体じゃないよ。曲がった面があるからね。",
        "register": "spoken"
      },
      {
        "en": "A polyhedron has 8 faces and 12 vertices. How many edges does it have?",
        "ja": "面の数が 8、頂点の数が 12 の多面体がある。この多面体の辺の数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形は polyhedra（polyhedrons とも言う）。",
      "円柱・円錐・球は曲面をもつので、多面体ではない。",
      "多面体の面・辺・頂点は face・edge・vertex（複数形 vertices）。多角形の辺は side（エントリ side）だが、多面体の辺は edge と言う。"
    ],
    "related": [
      "polygon",
      "regular-polyhedron",
      "prism",
      "pyramid",
      "eulers-formula-for-polyhedra"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "多面体",
        "en": "Polyhedron"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "9.1 Polyhedrons"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "polyhedron": 13
      },
      "written": {
        "polyhedron": 9
      },
      "sources": [
        "khan-middle",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "polynomial-division",
    "ja": {
      "term": "整式の除法",
      "reading": "せいしきのじょほう",
      "alt": [
        "多項式の除法",
        "筆算除法"
      ]
    },
    "en": {
      "term": "long division",
      "alt": [],
      "uk": null,
      "register": "both",
      "variants": [
        {
          "term": "long division",
          "register": "both",
          "note": "話し言葉ではほとんどがこの言い方で、Khan Academy・Professor Leonard・MIT OCW に多い。書き言葉でも最も多く、OpenStax Algebra and Trigonometry に多い。"
        },
        {
          "term": "dividing polynomials",
          "register": "written",
          "note": "節の名前や指示の言い方（Dividing Polynomials）。書き言葉（OpenStax Algebra and Trigonometry・Intermediate Algebra が中心）で多く、話し言葉では少ない。"
        },
        {
          "term": "polynomial division",
          "register": "written",
          "note": "書き言葉では少なく（大半が OpenStax Algebra and Trigonometry）、話し言葉には出てこない。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語は割り算の方法の名前 long division（polynomial long division）で呼ぶ。数の筆算の割り算も long division。OpenStax Algebra and Trigonometry は節の名前を Dividing Polynomials とし、本文で polynomial division とも書く。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "整式 A を整式 B で割って、A = BQ + R（R の次数は B の次数より小さい）となる商 Q と余り R を求める計算。",
    "definition_en": "Dividing a polynomial A by a polynomial B to get a quotient Q and remainder R with A = BQ + R, where R has lower degree than B.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's do long division: how many times does x go into x³?",
        "ja": "筆算で割ろう。x³ の中に x は何回入る？",
        "register": "spoken"
      },
      {
        "en": "Use long division to divide 2x³ − 3x² + 4 by x² + 1.",
        "ja": "2x³ − 3x² + 4 を x² + 1 で割れ（筆算を使う）。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "long division は数の筆算の割り算にも、積分の前に分子の次数を下げる割り算（エントリ integration-by-long-division。CED topic 6.10）にも使う。",
      "1 次式で割るときの簡便法は synthetic division（組立除法）。"
    ],
    "related": [
      "synthetic-division",
      "division-algorithm",
      "remainder",
      "integration-by-long-division",
      "slant-asymptote"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式: 多項式の除法（158 ÷ 12 の筆算と関連付ける）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「組立除法」",
        "url": "https://ja.wikipedia.org/wiki/%E7%B5%84%E7%AB%8B%E9%99%A4%E6%B3%95",
        "note": "多項式の筆算除法"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Polynomial_long_division",
        "note": "記事「Polynomial long division」の冒頭の太字（polynomial long division、long division。数学カテゴリから 4 段）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.10（long division）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "long division": 130,
        "dividing polynomials": 5
      },
      "written": {
        "long division": 68,
        "dividing polynomials": 38,
        "polynomial division": 10
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "polynomial-equation-of-higher-degree",
    "ja": {
      "term": "高次方程式",
      "reading": "こうじほうていしき",
      "alt": [
        "n 次方程式"
      ]
    },
    "en": {
      "term": "polynomial equation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「3 次以上」をまとめる決まった言い方は、用例コーパスと参照に出てこない（higher-degree equation も出てこない）。OpenStax Intermediate Algebra は 1 次・2 次も polynomial equation に含め、次数は polynomial equation of degree n と言う。OpenStax Algebra and Trigonometry は、2 次より高い次数の多項式を含む方程式を因数分解で解く。個別には cubic equation、fourth-degree equation（OpenStax Algebra and Trigonometry）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "3 次以上の整式 = 0 の形の方程式。",
    "definition_en": "An equation of the form polynomial = 0 of degree three or more.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For a cubic, guess a root first, then divide it out to get a quadratic.",
        "ja": "3 次方程式は、まず解を 1 つ見つけて割り、2 次式にする。",
        "register": "spoken"
      },
      {
        "en": "Solve the polynomial equation x⁴ − 5x² + 4 = 0.",
        "ja": "方程式 x⁴ − 5x² + 4 = 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "n 次方程式は a polynomial equation of degree n。",
      "学習指導要領解説（数学II）は、簡単な高次方程式を因数定理などを用いて解くことを扱う。"
    ],
    "related": [
      "cubic-equation",
      "quartic-equation",
      "factor-theorem",
      "fundamental-theorem-of-algebra",
      "rational-root-theorem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Polynomial Equations（polynomial equation of degree n）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式: 因数定理について理解し，簡単な高次方程式について因数定理などを用いてその解を求める"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "polynomial equation": 3
      },
      "written": {
        "polynomial equation": 41
      },
      "sources": [
        "mit-18.01",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "polynomial-function",
    "ja": {
      "term": "多項式関数",
      "reading": "たこうしきかんすう",
      "alt": [
        "多項式函数"
      ]
    },
    "en": {
      "term": "polynomial function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "f(x) = aₙ xⁿ + … + a₁x + a₀ のように、x の多項式で表される関数。一次関数や二次関数もその仲間である。",
    "definition_en": "A function whose rule is a polynomial in x, such as f(x) = 2x³ − x + 5. Linear and quadratic functions are polynomial functions too.",
    "latex": "f(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0",
    "spoken_en": "f of x equals a sub n x to the n, plus a sub n minus one x to the n minus one, plus dot dot dot, plus a sub one x, plus a sub zero",
    "examples": [
      {
        "en": "Linear and quadratic functions? Those are both polynomial functions, just with degree 1 and degree 2.",
        "ja": "一次関数と二次関数？ どちらも多項式関数で、次数がそれぞれ 1 と 2 なだけだよ。",
        "register": "spoken"
      },
      {
        "en": "Find the degree and the leading coefficient of the polynomial function f(x) = −3x⁴ + 2x² − 7.",
        "ja": "多項式関数 f(x) = −3x⁴ + 2x² − 7 の次数と最高次の係数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "多項式（エントリ polynomial）は式、polynomial function はその式で x の値から f(x) の値を決める関数。",
      "次数（degree）は最高次の項の次数。英語の degree は角度の「度」にも使う。",
      "グラフはなめらかで切れ目がない。分母に x を含む関数（エントリ rational-function）は polynomial function ではない。"
    ],
    "related": [
      "polynomial",
      "degree",
      "leading-coefficient",
      "zeros-of-a-polynomial",
      "end-behavior"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "polynomial function": 11
      },
      "written": {
        "polynomial function": 323
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "polynomial-inequality",
    "ja": {
      "term": "多項式不等式",
      "reading": "たこうしきふとうしき",
      "alt": [
        "高次不等式"
      ]
    },
    "en": {
      "term": "polynomial inequality",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「多項式不等式」は学習指導要領解説に無い、本プロジェクトの訳語（日本では 3 次以上のものを高次不等式と呼ぶ）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "多項式を含む不等式 P(x) > 0 など。零点で数直線を区切り、区間ごとの符号を調べて解く。",
    "definition_en": "An inequality such as P(x) > 0 for a polynomial P, solved by splitting the number line at the zeros and checking the sign on each interval.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Find the zeros, put them on a number line, and test a point in each interval.",
        "ja": "零点を求めて数直線に並べ、各区間で 1 点ずつ試す。",
        "register": "spoken"
      },
      {
        "en": "Solve the polynomial inequality (x + 2)(x − 1)(x − 4) < 0.",
        "ja": "不等式 (x + 2)(x − 1)(x − 4) < 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパス（OpenStax の教科書を含む）には出てこない。区間ごとに符号を調べる手順は test-point と sign-chart-inequality を参照。"
    ],
    "related": [
      "sign-chart-inequality",
      "test-point",
      "zeros-of-a-polynomial"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "polynomial",
    "ja": {
      "term": "多項式",
      "reading": "たこうしき",
      "alt": [
        "整式"
      ]
    },
    "en": {
      "term": "polynomial",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Algebra 2",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "定数と、変数の 0 以上の整数乗との積（単項式）を有限個たし合わせた式。",
    "definition_en": "An expression made by adding finitely many terms, each a constant times variables raised to whole-number powers.",
    "latex": "a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_1 x + a_0",
    "spoken_en": "a sub n x to the n, plus a sub n minus one x to the n minus one, and so on down to a sub zero",
    "examples": [
      {
        "en": "This is a polynomial of degree three, so it has at most three real roots.",
        "ja": "これは 3 次の多項式だから、実数解は多くても 3 個だ。",
        "register": "spoken"
      },
      {
        "en": "Write the polynomial 3x − 5 + x³ − 2x² in descending order of degree.",
        "ja": "整式 3x − 5 + x³ − 2x² を降べきの順に整理せよ。",
        "register": "written"
      },
      {
        "en": "Divide the polynomial x³ − 2x + 4 by x + 2.",
        "ja": "整式 x³ − 2x + 4 を x + 2 で割れ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "degree of a polynomial",
        "ja": "多項式の次数"
      },
      {
        "en": "factor the polynomial",
        "ja": "多項式を因数分解する"
      }
    ],
    "pitfalls": [
      "高校の「整式」を integral expression と直訳しない。整式も多項式も英語では polynomial。",
      "中学校学習指導要領解説は「単項式と多項式の意味」と対比で挙げ、日本語版 Wikipedia「多項式」は「多項式」を単項式でない整式の意味で用いる流儀があると書く。英語の polynomial は単項式（monomial）も含む。範囲は高校の「整式」と同じ。"
    ],
    "related": [
      "polynomial-division",
      "zeros-of-a-polynomial",
      "write-in-descending-order",
      "common-factor",
      "arrange-by-the-variable-of-lowest-degree"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "多項式",
        "en": "Polynomial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 A 数と式: 単項式と多項式の意味"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「多項式」",
        "url": "https://ja.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F",
        "note": "注意（多項式と整式）: 「多項式」を単項式でない整式の意味で用いる流儀"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "polynomial": 859
      },
      "written": {
        "polynomial": 1585
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "population-mean",
    "ja": {
      "term": "母平均",
      "reading": "ぼへいきん",
      "alt": [
        "母集団の平均"
      ]
    },
    "en": {
      "term": "population mean",
      "alt": [
        "true mean"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3",
        "数B"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "母集団全体の平均値。",
    "definition_en": "The mean of the whole population, written μ.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We almost never know the population mean, so we estimate it with the sample mean.",
        "ja": "母平均はほとんどわからないので、標本平均で推定する。",
        "register": "spoken"
      },
      {
        "en": "Test the claim that the population mean is 50 at the 5% significance level.",
        "ja": "母平均が 50 であるという主張を有意水準 5% で検定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "μ は mu と読む。",
      "学習指導要領解説（数学B）と共通テストの問題文（令和 6 年度 数学II・数学B ほか）は母平均を m と書く。AP Statistics の CED と OpenStax Introductory Statistics は μ と書く。"
    ],
    "related": [
      "sample-mean",
      "population"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B 統計的な推測: 母平均 m"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和6年度 大学入学共通テスト 本試験 数学Ⅱ・数学Ｂ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=666&f=abm00004690.pdf&n=2024_or_21_sugaku2B.pdf",
        "note": "母平均 m と書く問題文"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Unit 4（population mean）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "true mean": 23,
        "population mean": 86
      },
      "written": {
        "population mean": 186,
        "true mean": 24
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.03",
        "mit-notes",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "population-proportion",
    "ja": {
      "term": "母比率",
      "reading": "ぼひりつ",
      "alt": []
    },
    "en": {
      "term": "population proportion",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "母集団の中で、ある性質をもつものの割合。p と書く。",
    "definition_en": "The fraction of the population that has a given characteristic, written p.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We don't know the population proportion p, so we estimate it with p-hat.",
        "ja": "母比率 p はわからないので、p̂ で推定する。",
        "register": "spoken"
      },
      {
        "en": "Construct a 95% confidence interval for the population proportion of students who own a car.",
        "ja": "車をもっている生徒の母比率の 95% 信頼区間を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "sample-proportion",
      "population-mean"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "population proportion": 72
      },
      "written": {
        "population proportion": 86
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "population-standard-deviation",
    "ja": {
      "term": "母標準偏差",
      "reading": "ぼひょうじゅんへんさ",
      "alt": []
    },
    "en": {
      "term": "population standard deviation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "母集団全体の標準偏差。σ と書く。",
    "definition_en": "The standard deviation of the whole population, written σ.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When we know the population standard deviation, we use z; when we don't, we use t.",
        "ja": "母標準偏差がわかっていれば z を、わからなければ t を使う。",
        "register": "spoken"
      },
      {
        "en": "Assume the population standard deviation is 15. Construct a 95% confidence interval for the mean.",
        "ja": "母標準偏差を 15 とする。平均の 95% 信頼区間を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "σ は sigma と読む。"
    ],
    "related": [
      "population-variance",
      "standard-deviation",
      "population-mean"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "population standard deviation": 26
      },
      "written": {
        "population standard deviation": 128
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "population-variance",
    "ja": {
      "term": "母分散",
      "reading": "ぼぶんさん",
      "alt": []
    },
    "en": {
      "term": "population variance",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "母集団全体の分散。σ² と書く。",
    "definition_en": "The variance of the whole population, written σ².",
    "latex": "\\sigma^2",
    "spoken_en": "sigma squared",
    "examples": [
      {
        "en": "The population variance divides by N, but the sample variance divides by n minus 1.",
        "ja": "母分散は N で割るが、標本分散は n − 1 で割る。",
        "register": "spoken"
      },
      {
        "en": "The population variance is 16. Find the population standard deviation.",
        "ja": "母分散は 16 である。母標準偏差を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "σ² は sigma squared と読む。"
    ],
    "related": [
      "population-standard-deviation",
      "variance",
      "population-mean",
      "unbiased-estimator"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "2.7 Measures of the Spread of the Data（population variance）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Variance",
        "note": "節 Population variance（太字・節の見出し。数学カテゴリから 4 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "population variance": 56
      },
      "written": {
        "population variance": 22
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "population",
    "ja": {
      "term": "母集団",
      "reading": "ぼしゅうだん",
      "alt": []
    },
    "en": {
      "term": "population",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Pre-Algebra",
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "調査の対象となる集団全体。標本調査では、その一部を標本として取り出す。",
    "definition_en": "The whole group being studied; in a sample survey, a sample is taken from it.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The population is all the students at the school, and the sample is the 50 we actually asked.",
        "ja": "母集団は学校の生徒全員で、標本は実際に聞いた 50 人。",
        "register": "spoken"
      },
      {
        "en": "Identify the population and the sample in the study.",
        "ja": "この調査の母集団と標本を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "population は「人口」「（生物の）個体数」の意味でもよく使う（population growth は人口・個体数の増加）。用例コーパスでも、この意味で使っている文脈がかなりあった。",
      "母集団の特徴を表す数（母平均など）を parameter、標本から計算する数を statistic と呼び分ける。"
    ],
    "related": [
      "sample",
      "census",
      "statistic"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "母集団",
        "en": "Statistical population"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "population": 997
      },
      "written": {
        "population": 1658
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "position-of-a-line-relative-to-a-circle",
    "ja": {
      "term": "円と直線の位置関係",
      "reading": "えんとちょくせんのいちかんけい",
      "alt": []
    },
    "en": {
      "term": "position of a line relative to a circle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。OpenStax Algebra and Trigonometry は、グラフが共有点をもたないことを they have no points in common と言う（3 つの平面の位置関係の説明）。本プロジェクトの例文は、円と直線の位置関係を how many points the line and the circle have in common と共有点の個数で問う形にした（本プロジェクトの書き方の例）。判別式か、中心と直線の距離 d と半径 r の比較で判定する手順は、英語でも文で説明する。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Geometry",
        "Precalculus"
      ]
    },
    "definition_ja": "円と直線が 2 点で交わる・接する・共有点をもたない、の 3 通りの関係。判別式か、中心と直線の距離と半径の大小で判定する。",
    "definition_en": "Whether a line meets a circle in two points, touches it, or misses it, decided by a discriminant or by comparing the center-to-line distance with the radius.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If the distance from the center to the line is less than the radius, the line cuts the circle in two points.",
        "ja": "中心と直線の距離が半径より小さければ、直線は円と 2 点で交わる。",
        "register": "spoken"
      },
      {
        "en": "Determine, depending on k, how many points the line y = x + k and the circle x² + y² = 2 have in common.",
        "ja": "直線 y = x + k と円 x² + y² = 2 の共有点の個数を、k の値によって調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "tangent-line",
      "distance-from-a-point-to-a-line",
      "discriminant",
      "equation-of-a-circle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "no points in common（3 元 1 次方程式の系の解の説明）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "position-vector",
    "ja": {
      "term": "位置ベクトル",
      "reading": "いちべくとる",
      "alt": []
    },
    "en": {
      "term": "position vector",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "基準の点 O を決めたとき、点 P の位置を表すベクトル OP。",
    "definition_en": "For a fixed origin O, the vector OP that gives the position of a point P.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The position vector of a point just has the point's coordinates as its components.",
        "ja": "点の位置ベクトルは、その点の座標を成分にもつ。",
        "register": "spoken"
      },
      {
        "en": "Let a and b be the position vectors of A and B. Find the position vector of the point that divides AB internally in the ratio 2 : 1.",
        "ja": "A, B の位置ベクトルを a, b とする。AB を 2 : 1 に内分する点の位置ベクトルを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "position vector of the dividing point",
        "ja": "内分点の位置ベクトル"
      }
    ],
    "pitfalls": [
      "運動では、時刻 t における点の位置を表すベクトル関数 r(t) も position vector と言う。"
    ],
    "related": [
      "vector",
      "point-of-internal-division",
      "vector-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "position vector": 80
      },
      "written": {
        "position vector": 62
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "positive-and-negative-numbers",
    "ja": {
      "term": "正負の数",
      "reading": "せいふのかず",
      "alt": []
    },
    "en": {
      "term": "positive and negative numbers",
      "alt": [],
      "variants": [
        {
          "term": "positive and negative numbers",
          "register": "both",
          "note": "話し言葉ではこちらだけを使う（ほとんどが Khan Academy の中学の講義）。書き言葉では signed numbers と同じくらい。"
        },
        {
          "term": "signed numbers",
          "register": "written",
          "note": "教科書（OpenStax Prealgebra・Elementary Algebra）の言い方で、話し言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「正負の数」は正の数と負の数をまとめた呼び名で、英語では positive and negative numbers と 2 つを並べて言うか、signed numbers（符号の付いた数）と言う。どちらも数の集合の名前ではなく、整数（integer）や有理数（rational number）とは別の語。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "正の数と負の数をまとめた呼び名。+ や − の符号を付けて表した数として扱う。",
    "definition_en": "Positive numbers and negative numbers talked about together, as numbers that carry a sign (+ or −). They are also called signed numbers.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Today we're adding positive and negative numbers, so pay close attention to the signs.",
        "ja": "今日は正負の数の足し算をやるから、符号によく注意してね。",
        "register": "spoken"
      },
      {
        "en": "Use positive and negative numbers to describe each change: a rise of 6 degrees and a drop of 4 degrees.",
        "ja": "次の変化を、正負の数を使って表せ。6 度上がる、4 度下がる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「正負の数」を integers と言い換えない。integer は小数や分数の部分をもたない数だけなので、2.5 や −1/2 のような正負の数が抜ける（エントリ integer）。",
      "signed numbers の signed は「符号（sign）の付いた」の意味で、日常語の sign（署名する）とは別。"
    ],
    "related": [
      "positive-number",
      "negative-number",
      "sign",
      "integer",
      "absolute-value"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "positive and negative numbers": 12
      },
      "written": {
        "positive and negative numbers": 27,
        "signed numbers": 26
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "positive-correlation",
    "ja": {
      "term": "正の相関",
      "reading": "せいのそうかん",
      "alt": []
    },
    "en": {
      "term": "positive association",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "positive correlation",
          "register": "spoken",
          "note": "話し言葉の首位だが、すべて Khan Academy（AP Statistics）。OpenStax Introductory Statistics も positive correlation と書く。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Integrated Math 1",
        "Intro Statistics"
      ]
    },
    "definition_ja": "一方の変量が増えると、他方の変量も増える傾向があること。散布図の点が右上がりに並ぶ。",
    "definition_en": "A tendency for one variable to increase as the other increases; the points of the scatter plot rise from left to right.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Height and weight have a positive association: taller people tend to weigh more.",
        "ja": "身長と体重には正の相関がある。背が高い人ほど体重が重い傾向がある。",
        "register": "spoken"
      },
      {
        "en": "Does the scatter plot show a positive correlation, a negative correlation, or no correlation?",
        "ja": "散布図は正の相関、負の相関、相関なしのどれを示しているか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP Statistics の CED（topic 5.1）と IM Grade 8 は、散布図の右上がりの傾向を positive association と呼ぶ。Khan Academy は positive correlation のほか positive linear relationship とも言う。CED の positive correlation は相関係数の値の話に 1 件。"
    ],
    "related": [
      "negative-correlation",
      "correlation",
      "scatter-plot"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 5.1（positive association）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "Unit 6 glossary: positive association"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "positive correlation": 14
      },
      "written": {
        "positive association": 1,
        "positive correlation": 4
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-algtrig",
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "positive-number",
    "ja": {
      "term": "正の数",
      "reading": "せいのかず",
      "alt": []
    },
    "en": {
      "term": "positive number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "0 より大きい数。数直線では 0 より右にある。",
    "definition_en": "A number greater than zero; it sits to the right of 0 on the number line.",
    "latex": "+3",
    "spoken_en": "positive three",
    "examples": [
      {
        "en": "Every number to the right of zero on the number line is a positive number.",
        "ja": "数直線で 0 より右にある数は、どれも正の数だよ。",
        "register": "spoken"
      },
      {
        "en": "Circle every positive number in the list: −4, 0, 3, −1.5, 7/2.",
        "ja": "次の数のうち、正の数をすべて選べ。−4、0、3、−1.5、7/2",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "0 は正の数でも負の数でもない。英語の positive number も 0 を含まない。0 も含めて「0 以上の数」と言うときは nonnegative number。",
      "正の整数（positive integer）は 1, 2, 3, … で、日本の自然数と同じ範囲（エントリ natural-number）。"
    ],
    "related": [
      "negative-number",
      "positive-and-negative-numbers",
      "sign",
      "number-line",
      "natural-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正の数と負の数",
        "en": "Negative number"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "positive number": 258
      },
      "written": {
        "positive number": 98
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "postulate",
    "ja": {
      "term": "公準",
      "reading": "こうじゅん",
      "alt": []
    },
    "en": {
      "term": "postulate",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の高校数学では「公理」と言い、「公準」はユークリッド原論の訳語として出てくる程度。米国の Geometry では証明の前提を postulate と呼び、名前付きで使う（segment addition postulate、parallel postulate など）。axiom はほぼ同じ意味だが別のエントリ（axiom）。",
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "証明せずに正しいと認め、定理を導く出発点にする事柄。",
    "definition_en": "A statement that is accepted as true without proof and used as a starting point for proving theorems.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We don't prove this one. It's a postulate, so we just accept it and use it.",
        "ja": "これは証明しない。公準だから、正しいと認めてそのまま使う。",
        "register": "spoken"
      },
      {
        "en": "Which postulate justifies the statement AB + BC = AC?",
        "ja": "AB + BC = AC が成り立つ根拠となる公準（公理）はどれか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスで postulate は話し言葉ではすべて The Organic Chemistry Tutor で、書き言葉にはほとんど出てこない（OpenStax Algebra and Trigonometry・Precalculus）。IM には出てこない。",
      "定理（theorem）は証明によって導いた事柄で、証明せずに認める postulate とは区別する。"
    ],
    "related": [
      "axiom",
      "segment-addition-postulate",
      "angle-addition-postulate",
      "ruler-postulate",
      "perpendicular-postulate",
      "corresponding-angles-postulate"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "postulate": 10
      },
      "written": {
        "postulate": 2
      },
      "sources": [
        "openstax-algtrig",
        "openstax-precalculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "potential-function",
    "ja": {
      "term": "ポテンシャル関数",
      "reading": "ぽてんしゃるかんすう",
      "alt": []
    },
    "en": {
      "term": "potential function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "保存場 F に対して ∇f = F を満たす関数 f。",
    "definition_en": "A function f with ∇f = F for a conservative vector field F.",
    "latex": "\\nabla f = \\mathbf{F}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Once you have a potential function, the line integral is just f at the end minus f at the start.",
        "ja": "ポテンシャル関数が見つかれば、線積分は終点での f から始点での f を引くだけ。",
        "register": "spoken"
      },
      {
        "en": "Find a potential function for F(x, y) = ⟨y cos x, sin x⟩.",
        "ja": "F(x, y) = ⟨y cos x, sin x⟩ のポテンシャル関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "物理のポテンシャル（位置エネルギー）とは符号の約束が逆になることがある（F = −∇U）。"
    ],
    "related": [
      "conservative-vector-field",
      "gradient",
      "line-integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "potential function": 47
      },
      "written": {
        "potential function": 109
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "power-of-a-point",
    "ja": {
      "term": "方べきの定理",
      "reading": "ほうべきのていり",
      "alt": [
        "方冪の定理"
      ]
    },
    "en": {
      "term": "power of a point",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "power of a point という名前は、米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（CED・OpenStax・IM・CK-12・Nicholson・Levin とも 0 件）。用例コーパスにも出てこない。CK-12 Geometry は方べきの定理を 1 つの名前でまとめず、3 つの場合を別の定理として扱う: 2 本の弦は Intersecting Chords Theorem（6.13 Segments from Chords。本辞典の intersecting-chords-theorem）、2 本の割線は Two Secants Segments Theorem（6.19 Intersecting Secants Theorem）、接線と割線は Tangent Secant Segment Theorem（6.20 Tangent Secant Theorem。本辞典の secant-tangent-theorem）。見出しは英語版 Wikipedia の記事名「Power of a point」（ja 記事「方べきの定理」の langlink 先）で、記事が冒頭で太字で定義する power of a point は、点 P と円に対して決まる量（中心との距離 d と半径 r で d² − r²。P が円の外なら PA·PB に等しく、内なら −PA·PB）の名前。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "点 P を通る 2 直線が円とそれぞれ A, B と C, D で交わるとき PA·PB = PC·PD が成り立ち、一方が点 T で接するときは PA·PB = PT² が成り立つという定理。",
    "definition_en": "For a point P and a circle, if two lines through P meet the circle at A, B and at C, D, then PA·PB = PC·PD; if one line is tangent at T, then PA·PB = PT².",
    "latex": "PA \\cdot PB = PC \\cdot PD = PT^2",
    "spoken_en": "PA times PB equals PC times PD, which equals PT squared",
    "examples": [
      {
        "en": "The two chords cross at P, so by the power of a point, PA times PB has to equal PC times PD.",
        "ja": "2 本の弦が P で交わっているので、方べきの定理より PA·PB = PC·PD になる。",
        "register": "spoken"
      },
      {
        "en": "Chords AB and CD intersect at P. If PA = 4, PB = 6, and PC = 3, use the power of a point to find PD.",
        "ja": "弦 AB と CD が点 P で交わっている。PA = 4, PB = 6, PC = 3 のとき、方べきの定理を用いて PD を求めよ。",
        "register": "written"
      },
      {
        "en": "A secant from P meets the circle at A and B, and PT is tangent to the circle at T. Show that PA·PB = PT².",
        "ja": "点 P から引いた直線が円と A, B で交わり、PT は点 T で円に接している。PA·PB = PT² を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の power of a point は定理の名前というより「点の方べき」という量を指す。日本語版 Wikipedia「方べきの定理」は定理の英語名を power of a point theorem と書く（英語版「Power of a point」では外部リンクの題にだけ出てくる）。式 PA·PB = PC·PD を示して言ってもよい。",
      "「方べき」の「べき」は冪（power）。power を「力」と訳すと意味が通らない。"
    ],
    "related": [
      "chord",
      "tangent-line",
      "similar-triangles",
      "tangent-chord-theorem",
      "tangent-segments-are-equal",
      "intersecting-chords-theorem",
      "secant-tangent-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "方べきの定理",
        "en": "Power of a point"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.13 Segments from Chords（Intersecting Chords Theorem）・6.19 Intersecting Secants Theorem（Two Secants Segments Theorem）・6.20 Tangent Secant Theorem（Tangent Secant Segment Theorem）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「方べきの定理」",
        "url": "https://ja.wikipedia.org/wiki/%E6%96%B9%E3%81%B9%E3%81%8D%E3%81%AE%E5%AE%9A%E7%90%86",
        "note": "冒頭: 方冪の定理、英: power of a point theorem"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 power of a point（英語版「Power of a point」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "power-rule",
    "ja": {
      "term": "xⁿ の微分",
      "reading": "えっくすのえぬじょうのびぶん",
      "alt": [
        "べき乗則"
      ]
    },
    "en": {
      "term": "power rule",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説には (xⁿ)′ = nxⁿ⁻¹ の公式の名前が出てこない。AP の CED（topic 2.5）と OpenStax Calculus は power rule と呼び、名前で指して使う。ja.alt の「べき乗則」は学習指導要領解説に無い。日本語版 Wikipedia の「冪乗則」（べき乗則）は統計の power law の名前で、微分のこの公式とは別の概念。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "(xⁿ)′ = nxⁿ⁻¹ という微分の公式。",
    "definition_en": "The rule that the derivative of xⁿ is nxⁿ⁻¹.",
    "latex": "\\frac{d}{dx}x^n = nx^{n-1}",
    "spoken_en": "the derivative of x to the n is n x to the n minus one",
    "examples": [
      {
        "en": "Power rule: bring the exponent down in front and subtract one from it.",
        "ja": "xⁿ の微分は、指数を前に出して指数から 1 を引く。",
        "register": "spoken"
      },
      {
        "en": "By the power rule, the derivative of x⁵ is 5x⁴.",
        "ja": "xⁿ の微分の公式より、x⁵ の導関数は 5x⁴ である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "対数の性質（power rule for logarithms）や積分（power rule for integrals）にも同じ名前がある。"
    ],
    "related": [
      "derivative",
      "derivative-of-a-sum",
      "properties-of-logarithms"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 2.5（power rule）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「冪乗則」",
        "url": "https://ja.wikipedia.org/wiki/%E5%86%AA%E4%B9%97%E5%89%87",
        "note": "統計の power law（冪乗則）。微分の power rule とは別の概念"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "(xⁿ)′ = nxⁿ⁻¹ の公式の名前が出てこない・「べき乗則」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "power rule": 245
      },
      "written": {
        "power rule": 109
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "power-series",
    "ja": {
      "term": "べき級数",
      "reading": "べききゅうすう",
      "alt": [
        "冪級数",
        "整級数"
      ]
    },
    "en": {
      "term": "power series",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "Σcₙ(x − a)ⁿ の形の級数。x の値によって収束したり発散したりする。",
    "definition_en": "A series of the form Σcₙ(x − a)ⁿ, which converges for some values of x and diverges for others.",
    "latex": "\\sum_{n=0}^{\\infty} c_n (x-a)^n",
    "spoken_en": "the sum from n equals zero to infinity of c sub n times x minus a to the n",
    "examples": [
      {
        "en": "A power series is like a polynomial that never ends.",
        "ja": "べき級数は終わらない多項式のようなもの。",
        "register": "spoken"
      },
      {
        "en": "Find a power series representation for f(x) = 1/(1 + x²) and state its interval of convergence.",
        "ja": "f(x) = 1/(1 + x²) をべき級数で表し、収束区間を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "power series representation",
        "ja": "べき級数表示"
      }
    ],
    "pitfalls": [
      "収束する範囲内では項別に微分・積分できる（term-by-term-differentiation を参照）。"
    ],
    "related": [
      "radius-of-convergence",
      "interval-of-convergence",
      "taylor-series",
      "term-by-term-differentiation",
      "ratio-test"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "冪級数",
        "en": "Power series"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.13–10.15（power series）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "power series": 196
      },
      "written": {
        "power series": 266
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "power-set",
    "ja": {
      "term": "べき集合",
      "reading": "べきしゅうごう",
      "alt": [
        "冪集合"
      ]
    },
    "en": {
      "term": "power set",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（power set は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "set-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "集合 A のすべての部分集合を要素とする集合のこと。空集合と A 自身も含み、A の要素が n 個ならべき集合の要素は 2ⁿ 個ある。",
    "definition_en": "The set of all subsets of a set A, including the empty set and A itself. If A has n elements, its power set has 2ⁿ elements.",
    "latex": "|\\mathcal{P}(A)| = 2^{|A|}",
    "spoken_en": "the size of P of A equals two to the size of A",
    "examples": [
      {
        "en": "The power set always contains the empty set and the whole set, so don't forget those two.",
        "ja": "べき集合には空集合と集合全体が必ず入るので、その 2 つを忘れないように。",
        "register": "spoken"
      },
      {
        "en": "Let A = {1, 2, 3}. List the elements of the power set of A. How many elements does it have?",
        "ja": "A = {1, 2, 3} とする。A のべき集合の要素をすべて書き出せ。要素は何個あるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "べき集合の要素は集合（A の部分集合）で、A の要素そのものではない。{1} ∈ 𝒫(A) だが、1 ∈ 𝒫(A) ではない。",
      "記号は 𝒫(A) や 2^A など、本によって違う。",
      "用例コーパスでは書き言葉に少しあるだけ（すべて MIT の講義ノート）で、話し言葉には出てこない。Levin の本では 5.1 Sets に 4 件。"
    ],
    "related": [
      "subset",
      "set",
      "number-of-elements",
      "empty-set"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "冪集合",
        "en": "Power set"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "power set": 5
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは Levin の呼び方 power set（5.1 Sets）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "power",
    "ja": {
      "term": "累乗",
      "reading": "るいじょう",
      "alt": [
        "べき",
        "冪"
      ]
    },
    "en": {
      "term": "power",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "同じ数や文字をいくつか掛け合わせたもの。a を n 個掛け合わせたものを a の n 乗といい、aⁿ と書く。",
    "definition_en": "A product in which the same number is used as a factor again and again, written with an exponent. For example, 2 · 2 · 2 = 2³ is the third power of 2.",
    "latex": "a^n",
    "spoken_en": "a to the nth power",
    "examples": [
      {
        "en": "Two to the third power means 2 times 2 times 2, so it's 8, not 6.",
        "ja": "2 の 3 乗は 2 × 2 × 2 のことだから 8。6 じゃないよ。",
        "register": "spoken"
      },
      {
        "en": "Write 5 · 5 · 5 · 5 as a power, and then find its value.",
        "ja": "5 × 5 × 5 × 5 を累乗の形で表し、その値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "power は累乗 aⁿ 全体を指すほか、「何乗か」を指すこともある。two to the third power の third power は「3 乗」の意味。",
      "2³ は two to the third power または two cubed と読む。",
      "微分の power rule や power series（べき級数）など、power を含む別の語がある（エントリ power-rule、power-series）。"
    ],
    "related": [
      "exponent",
      "base-of-a-power",
      "square",
      "cube",
      "laws-of-exponents"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "to the … power | to the power of | raised to a power | raised to the power | to a power | powers of": 3015
      },
      "written": {
        "to the … power | to the power of | raised to a power | raised to the power | to a power | powers of": 623
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "predicate",
    "ja": {
      "term": "述語",
      "reading": "じゅつご",
      "alt": []
    },
    "en": {
      "term": "predicate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "変数を含み、変数に値を入れると真か偽かが決まる文や式のこと。P(x) のように書く。",
    "definition_en": "A statement that contains one or more variables and becomes true or false once values are put in for the variables. It is often written as P(x).",
    "latex": "P(x):\\ x^2 > 4",
    "spoken_en": "P of x: x squared is greater than four",
    "examples": [
      {
        "en": "P of x isn't true or false by itself. It's a predicate, and it only gets a truth value once you plug in a number.",
        "ja": "P(x) はそれだけでは真とも偽とも言えない。述語だから、数を代入して初めて真偽が決まるんだ。",
        "register": "spoken"
      },
      {
        "en": "Let P(x) be the predicate \"x is divisible by 3\" on the integers. Find the truth values of P(6) and P(7).",
        "ja": "整数全体で考え、P(x) を述語「x は 3 で割り切れる」とする。P(6), P(7) の真偽を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数I では、x > 3 のように変数を含み値を決めると真偽が決まる文を「条件」と呼び、「述語」とは言わない。",
      "英語の predicate は文法用語の「述部」の意味でも使う。",
      "用例コーパスの predicate は話し言葉（MIT OCW）・書き言葉（MIT の講義ノート）とも MIT のものだけ。Levin の本では 39 件。"
    ],
    "related": [
      "quantifier",
      "proposition",
      "truth-value",
      "all",
      "exist"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "predicate": 31
      },
      "written": {
        "predicate": 52
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "predict",
    "ja": {
      "term": "予測する",
      "reading": "よそくする",
      "alt": []
    },
    "en": {
      "term": "predict",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "モデルや回帰直線などを使って、まだ観測していない値を見積もる。",
    "definition_en": "To estimate a value that has not been observed, using a model such as a regression line.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Plug 12 into the regression equation to predict the score.",
        "ja": "回帰式に 12 を代入して得点を予測する。",
        "register": "spoken"
      },
      {
        "en": "Use the model to predict the population in 2030.",
        "ja": "モデルを使って 2030 年の人口を予測せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "予測した値（ŷ）を predicted value と言う（書き言葉では大半が OpenStax Introductory Statistics）。"
    ],
    "related": [
      "regression-line",
      "extrapolation",
      "interpolation",
      "mathematical-model",
      "residual"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 5.1・5.3（predict、predicted response value）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 1 3.4 Linear Models・5.21 Predicting Populations（predict）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "predict": 99
      },
      "written": {
        "predict": 220
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "preimage",
    "ja": {
      "term": "原像",
      "reading": "げんぞう",
      "alt": [
        "もとの図形"
      ]
    },
    "en": {
      "term": "preimage",
      "alt": [
        "pre-image"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の中学では「もとの図形」と言い、原像という言葉は写像（大学）で使う。英語の Geometry では変換する前の点や図形を preimage と呼ぶ。CK-12 Geometry は preimage と呼ぶ（8.17 Composite Transformations ほか、57 件）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "変換で移す前の、もとの点や図形。A′ が像のとき、A がその原像である。",
    "definition_en": "The original point or figure before a transformation is applied. If A′ is the image, then A is its preimage.",
    "latex": "A(1, 4) \\to A'(4, 1)",
    "spoken_en": "A at 1, 4 maps to A prime at 4, 1",
    "examples": [
      {
        "en": "The preimage is the figure you start with, and the image is what you get after the transformation.",
        "ja": "もとの図形は移動する前の図形で、移した図形は移動した後に得られる図形だ。",
        "register": "spoken"
      },
      {
        "en": "Point A′(−3, 5) is the image of point A under a reflection across the y-axis. Find the coordinates of the preimage A.",
        "ja": "点 A を y 軸に関して対称移動すると点 A′(−3, 5) に移る。点 A の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の問題文では「もとの図形」を preimage と書く。ハイフンを付けた pre-image もあり、CK-12 Geometry では preimage 57 件、pre-image 1 件。",
      "大学では関数 f と集合 B について、f(x) が B に入る x 全体 f⁻¹(B) を B の preimage（逆像）と呼ぶ。f に逆関数が無くても使う。",
      "用例コーパスには話・書とも出てこない。Nicholson では 7.1 Examples and Elementary Properties に 1 件。"
    ],
    "related": [
      "image",
      "rigid-motion",
      "reflection",
      "translation"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.17 Composite Transformations・8.11 Rotation Rules・8.13 Defining Reflection"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 preimage（CK-12 Geometry 8.17 Composite Transformations・CK-12 Geometry 8.11 Rotation Rules・CK-12 Geometry 8.13 Defining Reflection・計 57 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "prime-factor",
    "ja": {
      "term": "素因数",
      "reading": "そいんすう",
      "alt": []
    },
    "en": {
      "term": "prime factor",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある自然数の約数のうち、素数であるもの。たとえば 12 = 2 × 2 × 3 なので、12 の素因数は 2 と 3。",
    "definition_en": "A factor of a whole number that is also a prime number. For example, 12 = 2 × 2 × 3, so the prime factors of 12 are 2 and 3.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "So what are the prime factors of 30? Well, 30 is 2 times 3 times 5, so they're 2, 3, and 5.",
        "ja": "30 の素因数は何かな？ 30 = 2 × 3 × 5 だから、2 と 3 と 5 だね。",
        "register": "spoken"
      },
      {
        "en": "Find all the prime factors of 84.",
        "ja": "84 の素因数をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "素因数分解（エントリ prime-factorization）は数を素因数の積で表すこと。素因数はその積に出てくる素数のことで、「12 の素因数」と言えば異なる素数 2 と 3 を挙げるが、「素因数 2 を 2 個もつ」のように重なりを数える言い方もある（センター試験 平成 26 年度 本試験 数学II・数学B 第6問）。英語でも OpenStax Elementary Algebra 1.1 は 12 の prime factors を 2 · 2 · 3 と重ねて書く。",
      "factor だけでは「因数・約数」の意味で、12 の factor には素数でない 4 や 6 も入る。factor は「因数分解する」という動詞にもなる。",
      "1 は素数ではないので、素因数にならない。"
    ],
    "related": [
      "prime-factorization",
      "prime-number",
      "composite-number",
      "common-factor",
      "multiple"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "素因数",
        "en": "Prime factor"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.1 Introduction to Whole Numbers（the prime factors of 12（2 · 2 · 3）。pitfalls[0]）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "センター試験 平成 26 年度 数学II・数学B 第6問: 6! は素因数 2 を…個もつ（pitfalls[0]）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "prime factor": 31
      },
      "written": {
        "prime factor": 46
      },
      "sources": [
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "prime-factorization",
    "ja": {
      "term": "素因数分解",
      "reading": "そいんすうぶんかい",
      "alt": []
    },
    "en": {
      "term": "prime factorization",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic",
      "number-theory"
    ],
    "level": {
      "jp": [
        "中1",
        "数A"
      ],
      "us": [
        "Pre-Algebra",
        "Discrete Math"
      ]
    },
    "definition_ja": "自然数を素数だけの積の形に表すこと。また、その積の形。1 より大きい自然数では、積の順番を除いてただ 1 通りに決まる。",
    "definition_en": "Writing a whole number as a product of prime numbers, or the product itself; for any whole number greater than 1 it is unique apart from the order of the factors.",
    "latex": "n = p_1^{e_1} p_2^{e_2} \\cdots p_k^{e_k}",
    "spoken_en": "n equals p one to the e one, times p two to the e two, and so on, up to p k to the e k",
    "examples": [
      {
        "en": "Let's factor 60 into primes: that's 2 times 2 times 3 times 5, so the prime factorization is 2 squared times 3 times 5.",
        "ja": "60 を素因数分解しよう。2 × 2 × 3 × 5 だから、素因数分解は 2² × 3 × 5 だ。",
        "register": "spoken"
      },
      {
        "en": "Write the prime factorization of 360.",
        "ja": "360 を素因数分解せよ。",
        "register": "written"
      },
      {
        "en": "Since the prime factorization of 72 is 2³ · 3², the number of positive divisors of 72 is (3 + 1)(2 + 1) = 12.",
        "ja": "72 を素因数分解すると 2³ · 3² なので、72 の正の約数の個数は (3 + 1)(2 + 1) = 12 である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "prime factorization of 60",
        "ja": "60 の素因数分解"
      }
    ],
    "pitfalls": [
      "「素因数分解する」という動詞は factor 60 into primes や write 60 as a product of primes と言う。名詞の prime factorization を動詞に言い換えない。",
      "最大公約数・最小公倍数を素因数分解から求めるときは、共通する素因数の指数の小さい方・大きい方をとる、と手順を英語で説明する（take the smaller / larger exponent of each prime）。",
      "用例コーパスでは prime factorization は話し言葉・書き言葉とも同じくらい出てくる。話し言葉はほとんどが Khan Academy の中学の講義、書き言葉は大半が OpenStax Prealgebra と Elementary Algebra。"
    ],
    "related": [
      "multiple",
      "common-factor",
      "division-algorithm"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "素因数分解",
        "en": "Integer factorization"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "prime factorization": 102
      },
      "written": {
        "prime factorization": 79
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-notes",
        "openstax-elemalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "prime-notation",
    "ja": {
      "term": "ラグランジュの記法",
      "reading": "らぐらんじゅのきほう",
      "alt": []
    },
    "en": {
      "term": "prime notation",
      "alt": [
        "Lagrange notation",
        "Lagrange's notation"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "notation"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "導関数を f′(x)、f″(x) のようにプライム記号 ′ で書く記法。",
    "definition_en": "Writing derivatives with prime marks, as in f′(x) and f″(x).",
    "latex": "f'(x)",
    "spoken_en": "f prime of x",
    "examples": [
      {
        "en": "In prime notation, the derivative of f is just f prime.",
        "ja": "ラグランジュの記法では、f の導関数は f′ と書く。",
        "register": "spoken"
      },
      {
        "en": "Rewrite dy/dx = 3x² in prime notation.",
        "ja": "dy/dx = 3x² をラグランジュの記法で書き直せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語では ′ を「ダッシュ」と読むが、英語では prime。f′ は f prime、f″ は f double prime。"
    ],
    "related": [
      "leibniz-notation",
      "derivative"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "微分の記法",
        "en": "Notation for differentiation"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "prime notation": 5,
        "Lagrange notation": 4
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-24。docs/DECISIONS.md「Phase 2 数列・級数の単元の前の修正」の、人間が見出しを決めた ③ の行（number-of-real-solutions・prime-notation・find-the-inflection-points））。register は主張しない。",
        "raised": "2026-09-24"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "prime-number",
    "ja": {
      "term": "素数",
      "reading": "そすう",
      "alt": []
    },
    "en": {
      "term": "prime number",
      "alt": [],
      "variants": [
        {
          "term": "prime number",
          "register": "both",
          "note": "話し言葉では形容詞の prime より多い。"
        },
        {
          "term": "prime",
          "register": "both",
          "note": "形容詞で 7 is prime、are prime のように言う形。書き言葉では prime number と同じくらい使う。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "2 以上の自然数のうち、1 とその数自身のほかに正の約数をもたないもの。1 は素数ではない。",
    "definition_en": "A whole number greater than 1 whose only positive factors are 1 and itself. The number 1 is not prime.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Is 51 a prime number? Nope, it's 3 times 17.",
        "ja": "51 は素数かな？ 違うね、3 × 17 だ。",
        "register": "spoken"
      },
      {
        "en": "List all the prime numbers between 20 and 40.",
        "ja": "20 と 40 の間にある素数をすべて答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "1 は素数ではなく、合成数でもない（1 is neither prime nor composite。エントリ composite-number）。",
      "形容詞で 7 is prime とも言う。prime 1 語は f′（f prime）の prime（エントリ prime-notation）と同じ語なので、数の話か記号の話かを前後で見分ける。",
      "2 は偶数の素数でただ 1 つ（2 is the only even prime）。"
    ],
    "related": [
      "prime-factor",
      "prime-factorization",
      "composite-number",
      "natural-number",
      "relatively-prime"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "素数",
        "en": "Prime number"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "prime number": 101,
        "is prime | is a prime | are prime": 67
      },
      "written": {
        "is prime | is a prime | are prime": 64,
        "prime number": 52
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "prime-polynomial",
    "ja": {
      "term": "既約多項式",
      "reading": "きやくたこうしき",
      "alt": [
        "素多項式",
        "これ以上因数分解できない式"
      ]
    },
    "en": {
      "term": "prime polynomial",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の中学・高校では「これ以上因数分解できない」と言い、名前を付けない。「既約多項式」は大学の用語。英語では OpenStax Elementary Algebra が、因数分解できない式を prime と書く。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "整数の範囲では、これ以上因数分解できない多項式。x² + 1 や x² + x + 1 など。",
    "definition_en": "A polynomial that cannot be factored using integers. For example, x² + 1 and x² + x + 1 are prime polynomials.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Nothing multiplies to 3 and adds to 1, so x squared plus x plus 3 is a prime polynomial.",
        "ja": "掛けて 3、足して 1 になる整数の組はないから、x² + x + 3 はこれ以上因数分解できない。",
        "register": "spoken"
      },
      {
        "en": "Factor x² + 4x + 6 completely, or state that it is a prime polynomial.",
        "ja": "x² + 4x + 6 を因数分解せよ。因数分解できないときは、そのことを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "因数分解できるかどうかは、どの数の範囲で考えるかで変わる。x² − 2 は整数の範囲では prime だが、実数の範囲では (x − √2)(x + √2) と因数分解できる。",
      "OpenStax Elementary Algebra は、因数分解できない三項式を prime と形容詞で書く（The polynomial is prime.）。素数（prime number、エントリ prime-number）と同じ prime。",
      "大学の代数では irreducible polynomial（既約多項式）と言う。"
    ],
    "related": [
      "factoring",
      "polynomial",
      "trinomial",
      "prime-number",
      "difference-of-squares"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "prime polynomial | polynomial is prime | trinomial is prime | binomial is prime": 5
      },
      "sources": [
        "openstax-elemalg"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 prime polynomial | polynomial is prime | trinomial is prime | binomial is prime（本文 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "principal-square-root",
    "ja": {
      "term": "正の平方根",
      "reading": "せいのへいほうこん",
      "alt": [
        "主平方根"
      ]
    },
    "en": {
      "term": "principal square root",
      "alt": [],
      "variants": [
        {
          "term": "principal root",
          "register": "spoken",
          "note": "話し言葉で最も多い言い方（すべて Khan Academy。n 乗根にも使う）。書き言葉では少ない。"
        },
        {
          "term": "positive square root",
          "register": "both",
          "note": "話・書とも principal square root と並んで使う。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語の principal square root は、OpenStax Algebra and Trigonometry では負でない平方根（0 の平方根 0 も含む）、OpenStax Prealgebra・Elementary Algebra では正の平方根と定義され、記号 √a はこれを表す。日本語版 Wikipedia「平方根」は √a を正の（あるいは非負の）平方根、主平方根と書く。0 を含めるかどうかは日米どちらの資料でも割れる。英語では positive square root とも言い、話し言葉では principal root とも言う。",
    "domains": [
      "arithmetic",
      "functions"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2",
        "Algebra 2"
      ]
    },
    "definition_ja": "正の数 a の 2 つの平方根のうち、正のほう。記号 √a はこの数を表す。",
    "definition_en": "The nonnegative square root of a number, written with the radical sign as √a. For a positive number a, it is the positive one of its two square roots, and the principal square root of 0 is 0.",
    "latex": "\\sqrt{25} = 5",
    "spoken_en": "the square root of twenty-five equals five",
    "examples": [
      {
        "en": "25 has two square roots, 5 and negative 5, but the radical sign means the principal square root, so √25 is just 5.",
        "ja": "25 の平方根は 5 と −5 の 2 つあるけど、根号が表すのは正の平方根だから、√25 は 5 だけ。",
        "register": "spoken"
      },
      {
        "en": "Find the principal square root of 0.49.",
        "ja": "0.49 の正の平方根を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry の principal square root は 0 の平方根 0 も含む「負でない平方根」。日本語の「正の平方根」は字のとおりなら 0 を含まない（日本語版 Wikipedia「平方根」は「正の（あるいは非負の）平方根」と書く）。",
      "負のほうの平方根は −√a で、the negative square root と言う（エントリ square-root-of-a-number）。",
      "「25 の平方根」は ±5 の 2 つだが、√25 は 5 だけを表す。√25 = ±5 とは書かない。"
    ],
    "related": [
      "square-root-of-a-number",
      "radical-sign",
      "nth-root",
      "irrational-number",
      "square"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「平方根」",
        "url": "https://ja.wikipedia.org/wiki/%E5%B9%B3%E6%96%B9%E6%A0%B9",
        "note": "√a は正の（あるいは非負の）平方根、主平方根"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "positive square root": 57,
        "principal root": 70,
        "principal square root": 15
      },
      "written": {
        "positive square root": 15,
        "principal square root": 24,
        "principal root": 3
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "prism",
    "ja": {
      "term": "角柱",
      "reading": "かくちゅう",
      "alt": []
    },
    "en": {
      "term": "prism",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "合同な 2 つの多角形を底面とし、長方形の側面で囲まれた立体。底面の形によって三角柱、四角柱などと呼ぶ。",
    "definition_en": "A solid with two congruent, parallel polygon bases joined by rectangular faces. It is named by the shape of its bases, as in a triangular prism.",
    "latex": "V = Bh",
    "spoken_en": "V equals B times h",
    "examples": [
      {
        "en": "For any prism, find the area of the base first, then multiply by the height.",
        "ja": "角柱ならどれでも、まず底面積を求めて、それに高さをかければいい。",
        "register": "spoken"
      },
      {
        "en": "The base of a triangular prism is a right triangle with legs of 6 cm and 8 cm, and the height of the prism is 10 cm. Find the volume of the prism.",
        "ja": "底面が、直角をはさむ 2 辺の長さが 6 cm、8 cm の直角三角形で、高さが 10 cm の三角柱の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "V = Bh の B（大文字）は底面積（area of the base）。三角形の面積の公式 A = ½bh の b（底辺の長さ）とは別。",
      "角柱の名前は底面の形を前に付けて言う。三角柱は triangular prism、直方体は rectangular prism。",
      "prism は光を分ける「プリズム」と同じ語。"
    ],
    "related": [
      "pyramid",
      "cylinder",
      "polyhedron",
      "volume",
      "area-of-the-base"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "角柱",
        "en": "Prism (geometry)"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 5.12 Prisms and Pyramids ほか"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "prism": 76
      },
      "written": {
        "prism": 13
      },
      "sources": [
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "probability-density-function",
    "ja": {
      "term": "確率密度関数",
      "reading": "かくりつみつどかんすう",
      "alt": []
    },
    "en": {
      "term": "probability density function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics",
      "calculus"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Calculus II"
      ]
    },
    "definition_ja": "連続型の確率変数の分布を表す関数 f(x)。X が区間 a ≦ X ≦ b に入る確率は、f(x) をその区間で積分した値になり、全体での積分は 1。",
    "definition_en": "The function f(x) that describes how a continuous random variable X is distributed: P(a ≤ X ≤ b) = ∫ₐᵇ f(x) dx, and f integrates to 1 over the whole line.",
    "latex": "P(a \\le X \\le b) = \\int_a^b f(x)\\,dx",
    "spoken_en": "the probability that X is between a and b equals the integral from a to b of f of x dx",
    "examples": [
      {
        "en": "A probability density can be bigger than one; it's the area under it that has to be one.",
        "ja": "確率密度は 1 より大きくてもいい。1 にならないといけないのは、その下の面積のほう。",
        "register": "spoken"
      },
      {
        "en": "If the probability density function of X is f(x) = 2x on [0, 1], find P(0 ≤ X ≤ 1/2).",
        "ja": "X の確率密度関数が区間 [0, 1] で f(x) = 2x のとき、P(0 ≦ X ≦ 1/2) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "関数そのものは probability density function（pdf）、その値は probability density と言う。density function と略すことも多い。",
      "日本の共通テスト（数学B）は正規分布表を添えて確率を求めさせる。OpenStax Calculus にも、積分の応用として確率密度関数の例（P(a ≤ X ≤ b) = ∫ₐᵇ f(x)dx）がある。",
      "OpenStax Introductory Statistics は離散型の確率分布を probability distribution function（PDF）とも呼ぶ。略称が同じなので、PDF が連続型の確率密度関数か離散型の確率分布かを文脈で見分ける。"
    ],
    "related": [
      "definite-integral",
      "improper-integral",
      "continuous-random-variable",
      "normal-distribution",
      "probability-distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "確率密度関数",
        "en": "Probability density function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和5年度 大学入学共通テスト 本試験 数学Ⅱ・数学Ｂ（問題）",
        "url": "https://web.archive.org/web/20240408/https://www.dnc.ac.jp/albums/abm.php?d=432&f=abm00003626.pdf&n=2023_ot_21_sugaku2B.pdf",
        "note": "正規分布表"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "3.7 Improper Integrals（probability density function）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "probability density function": 15
      },
      "written": {
        "probability density function": 52
      },
      "sources": [
        "khan-ap-stats",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "probability-distribution",
    "ja": {
      "term": "確率分布",
      "reading": "かくりつぶんぷ",
      "alt": []
    },
    "en": {
      "term": "probability distribution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "確率変数のとる値（連続型では値の範囲）と、その確率との対応。表や式、グラフで表す。",
    "definition_en": "The correspondence between the values (for a continuous variable, intervals of values) of a random variable and their probabilities, shown as a table, a formula, or a graph.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a probability distribution, all the probabilities have to add up to one.",
        "ja": "確率分布では、確率を全部足すと 1 になる。",
        "register": "spoken"
      },
      {
        "en": "Construct the probability distribution of X and verify that it is valid.",
        "ja": "X の確率分布を作り、条件を満たしていることを確かめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は離散型の確率分布を probability distribution function（PDF）とも呼ぶ。連続型の確率密度関数（probability density function）と略称が同じなので注意。"
    ],
    "related": [
      "random-variable",
      "expected-value",
      "binomial-distribution",
      "normal-distribution",
      "probability-density-function",
      "discrete-random-variable"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "確率分布",
        "en": "Probability distribution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "4.1 Probability Distribution Function (PDF) for a Discrete Random Variable・5.1 Continuous Probability Functions（pdf）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.8（probability distribution）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "probability distribution": 55
      },
      "written": {
        "probability distribution": 112
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "probability",
    "ja": {
      "term": "確率",
      "reading": "かくりつ",
      "alt": []
    },
    "en": {
      "term": "probability",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "ある事柄の起こりやすさを 0 以上 1 以下の数で表したもの。起こりうる結果がどれも同様に確からしいときは、その事柄が起こる場合の数を、起こりうるすべての場合の数で割って求める。",
    "definition_en": "A number from 0 to 1 that measures how likely an event is. When all outcomes are equally likely, it is the number of outcomes in the event divided by the total number of outcomes.",
    "latex": "P(A) = \\frac{n(A)}{n(S)}",
    "spoken_en": "P of A equals n of A over n of S",
    "examples": [
      {
        "en": "So the probability of getting a sum of seven with two dice is six out of thirty-six, which is one sixth.",
        "ja": "だから、2 つのさいころの目の和が 7 になる確率は 36 分の 6、つまり 6 分の 1。",
        "register": "spoken"
      },
      {
        "en": "A bag contains 3 red balls and 5 white balls. Two balls are drawn at random. Find the probability that both are red.",
        "ja": "袋の中に赤玉 3 個と白玉 5 個が入っている。この中から 2 個を取り出すとき、2 個とも赤玉である確率を求めよ。",
        "register": "written"
      },
      {
        "en": "When the sample space is finite, you get P(A) by adding up the probabilities of the individual outcomes in A.",
        "ja": "標本空間が有限のとき、P(A) は A に含まれる 1 つ 1 つの結果（根元事象）の確率を足して求める。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "find the probability",
        "ja": "確率を求める"
      },
      {
        "en": "probability of an event",
        "ja": "事象の確率"
      }
    ],
    "pitfalls": [
      "odds（オッズ）は確率ではなく、起こる場合と起こらない場合の数の比。odds of 1 to 5 は確率 1/6 に当たる。",
      "教科書の「同様に確からしい」は equally likely（outcomes are equally likely）。",
      "高等学校学習指導要領解説（数学A）は全事象を U で表す（P(U) = 1）が、上の式では標本空間を S とした（sample-space を参照）。"
    ],
    "related": [
      "sample-space",
      "experimental-probability",
      "basic-properties-of-probability",
      "independence-of-events",
      "at-least-one"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "確率",
        "en": "Probability"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 場合の数と確率: 全事象 U の確率 P(U) = 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "probability": 2363
      },
      "written": {
        "probability": 2326
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "product-rule",
    "ja": {
      "term": "積の微分",
      "reading": "せきのびぶん",
      "alt": [
        "積の微分法"
      ]
    },
    "en": {
      "term": "product rule",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "積の導関数 {f(x)g(x)}′ = f′(x)g(x) + f(x)g′(x) を与える公式。",
    "definition_en": "The rule (fg)′ = f′g + fg′ for the derivative of a product.",
    "latex": "(fg)' = f'g + fg'",
    "spoken_en": "the derivative of f g is f prime g plus f g prime",
    "examples": [
      {
        "en": "Use the product rule: take the derivative of the first factor times the second, then add the first factor times the derivative of the second.",
        "ja": "積の微分を使う。前を微分して後ろはそのまま、足すことの、前はそのままで後ろを微分。",
        "register": "spoken"
      },
      {
        "en": "Use the product rule to differentiate y = x² sin x.",
        "ja": "積の微分を用いて y = x² sin x を微分せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "use the product rule",
        "ja": "積の微分を使う"
      }
    ],
    "pitfalls": [
      "対数の性質にも product rule（log MN = log M + log N）がある。"
    ],
    "related": [
      "quotient-rule",
      "chain-rule",
      "derivative-of-a-sum"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "積の微分法則",
        "en": "Product rule"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 2.8（product rule）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "product rule": 346
      },
      "written": {
        "product rule": 165
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "product-to-sum-formulas",
    "ja": {
      "term": "積和公式",
      "reading": "せきわこうしき",
      "alt": []
    },
    "en": {
      "term": "product-to-sum formulas",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "sin α cos β などの積を、三角関数の和・差で表す公式。",
    "definition_en": "The formulas that rewrite products such as sin α cos β as sums or differences of trigonometric functions.",
    "latex": "\\sin\\alpha\\cos\\beta = \\frac{1}{2}\\{\\sin(\\alpha + \\beta) + \\sin(\\alpha - \\beta)\\}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The product-to-sum formulas come straight from adding or subtracting two addition formulas.",
        "ja": "積和公式は、加法定理の 2 式を足したり引いたりすればすぐ出る。",
        "register": "spoken"
      },
      {
        "en": "Use a product-to-sum formula to write sin 5x cos 3x as a sum.",
        "ja": "積和公式を用いて sin 5x cos 3x を和の形で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はすべて OpenStax Algebra and Trigonometry（節 Sum-to-Product and Product-to-Sum Formulas）。話し言葉には出てこない。"
    ],
    "related": [
      "sum-to-product-formulas",
      "angle-addition-formulas",
      "trigonometric-integrals"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Sum-to-Product and Product-to-Sum Formulas（product-to-sum formulas）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角関数の公式の一覧」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0%E3%81%AE%E5%85%AC%E5%BC%8F%E3%81%AE%E4%B8%80%E8%A6%A7",
        "note": "節「和積公式と積和公式」"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "product-to-sum formulas": 18
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "product",
    "ja": {
      "term": "積",
      "reading": "せき",
      "alt": []
    },
    "en": {
      "term": "product",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "掛け算の答え。たとえば 3 と 4 の積は 12 である。",
    "definition_en": "What you get when you multiply two or more numbers or expressions together. For example, the product of 3 and 4 is 12.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Product means multiply, so the product of 6 and 7 is 42.",
        "ja": "積はかけ算の答えだから、6 と 7 の積は 42。",
        "register": "spoken"
      },
      {
        "en": "Write “the product of 9 and a number n” as an algebraic expression.",
        "ja": "「9 と数 n の積」を文字式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日常語の product（製品）と同じ語。数学の文では掛け算の答え。",
      "内積・外積は dot product・cross product（エントリ dot-product・cross-product）。微分の積の公式は product rule（エントリ product-rule）。",
      "場合の数の「積の法則」は multiplication principle（エントリ multiplication-principle）で、微分の product rule とは別。"
    ],
    "related": [
      "multiplication",
      "multiply",
      "sum",
      "quotient",
      "dot-product"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "積",
        "en": "Product (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the product of | a product of": 402
      },
      "written": {
        "the product of | a product of": 658
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "projectile-motion",
    "ja": {
      "term": "放物運動",
      "reading": "ほうぶつうんどう",
      "alt": [
        "投射運動"
      ]
    },
    "en": {
      "term": "projectile motion",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「放物運動」は数学の学習指導要領解説には無い（高等学校学習指導要領の理科「物理」の内容（解説 数学編 理数編の付録）の項目「放物運動」と日本語版 Wikipedia「放物運動」にはある）。中学校学習指導要領解説（第 3 学年 関数 y = ax²）は、y = ax² で捉える事象の例に斜面をころがる物の運動・車の制動距離・噴水の水が作る形を挙げ、落下の式は挙げていない。本プロジェクトは落ちる距離 y m と時間 x 秒の関係 y = 4.9x² を例にする。米国の教科書 OpenStax Algebra and Trigonometry は、二次関数の応用として projectile motion の問題を扱い、高さを h(t) = −16t² + v₀t + h₀ のような式で表す（長さをフィートで書く）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中3",
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2",
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "投げ上げたり落としたりした物体の運動。空気の抵抗を考えなければ、物体の高さや落ちる距離は時間の二次関数で表される。",
    "definition_en": "The motion of an object that is thrown, launched, or dropped and then moves only under the pull of gravity. Its height, or the distance it has fallen, is a quadratic function of time.",
    "latex": "h(t) = -16t^2 + v_0 t + h_0",
    "spoken_en": "h of t equals negative sixteen t squared plus v naught t plus h naught",
    "examples": [
      {
        "en": "In projectile motion, the height is a quadratic function of time, so its graph is a parabola: the ball goes up, slows down, and comes back down.",
        "ja": "放物運動では高さが時間の二次関数なので、そのグラフは放物線になる。ボールは上がって、だんだん遅くなり、また落ちてくる。",
        "register": "spoken"
      },
      {
        "en": "An object in free fall travels y meters in x seconds, where y = 4.9x². Find the distance it falls in the first 3 seconds.",
        "ja": "物体を落とすとき、落ち始めてから x 秒間に落ちる距離を y m とすると、y = 4.9x² である。最初の 3 秒間に落ちる距離を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "落ちた距離を表す y = 4.9x²（下向きに測る）は x² の係数が正。高さ（上向きに測る）を表す h(t) = −16t² + … は t² の係数が負になる。",
      "係数の 4.9 はメートル（9.8 m/s² の半分）、16 はフィート（32 ft/s² の半分）で、長さの単位が違うと係数も変わる。",
      "最高点の高さ（maximum height）を求める問題は、放物線の頂点を求める問題になる（エントリ vertex）。",
      "投げ上げずに落とすだけの運動は free fall（自由落下）と言う（OpenStax Calculus Volume 1）。projectile motion は投げ上げ・打ち上げも含む。"
    ],
    "related": [
      "quadratic-function",
      "parabola",
      "vertex",
      "mathematical-model",
      "word-problem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "中学校 第 3 学年 関数: 関数 y = ax² を具体的な事象と関連付けて扱う（落下の式 y = 4.9x² は 0 件）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「放物運動」",
        "url": "https://ja.wikipedia.org/wiki/%E6%94%BE%E7%89%A9%E9%81%8B%E5%8B%95",
        "note": "見出しの日本語の記事"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "付録 学習指導要領 理科「物理」: 放物運動（水平投射及び斜方投射された物体の運動）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "projectile motion": 3
      },
      "written": {
        "projectile motion": 18
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "proof-by-cases",
    "ja": {
      "term": "場合分けによる証明",
      "reading": "ばあいわけによるしょうめい",
      "alt": []
    },
    "en": {
      "term": "proof by cases",
      "alt": [
        "proof by exhaustion",
        "case analysis"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（proof by cases ／ proof by exhaustion ／ case analysis は CED・OpenStax・IM・CK-12 とも 0 件）。英語版 Wikipedia の記事名は Proof by exhaustion（proof by cases はそのリダイレクト）。",
    "domains": [
      "proof"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": []
    },
    "definition_ja": "起こりうる場合を漏れなくいくつかに分け、それぞれの場合で結論が成り立つことを示して全体を証明する方法のこと。",
    "definition_en": "A method of proof that splits the situation into a finite number of cases covering every possibility and proves the statement in each case separately.",
    "latex": "\\big((p \\lor q) \\land (p \\to r) \\land (q \\to r)\\big) \\to r",
    "spoken_en": "if p or q, and p implies r, and q implies r, then r",
    "examples": [
      {
        "en": "We'll do a proof by cases here: case one, n is even, and case two, n is odd.",
        "ja": "ここは場合分けで証明しよう。n が偶数の場合と、n が奇数の場合だ。",
        "register": "spoken"
      },
      {
        "en": "Use a proof by cases to show that n² + n is even for every integer n. (Consider the cases n even and n odd.)",
        "ja": "すべての整数 n について n² + n が偶数であることを、場合分けによって証明せよ（n が偶数の場合と奇数の場合を考えよ）。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "proof by exhaustion は、有限個の場合をすべて調べ尽くす総当たりの証明を指すこともある。面積を求める method of exhaustion（取り尽くし法）とは別物。",
      "日本語の「場合分け」は証明だけでなく、絶対値を外す計算や不等式の解法にも使うが、この見出しは証明の方法の名前。",
      "用例コーパスでは proof by cases と case analysis が書き言葉に少しあるだけ（どちらも MIT の講義ノート）で、proof by exhaustion は出てこない。"
    ],
    "related": [
      "prove",
      "direct-proof",
      "proof",
      "parity"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "proof by cases": 3,
        "case analysis": 1
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 2）。en.term は proof by cases、proof by exhaustion は en.alt。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "proof-by-contradiction",
    "ja": {
      "term": "背理法",
      "reading": "はいりほう",
      "alt": [
        "背理法による証明"
      ]
    },
    "en": {
      "term": "proof by contradiction",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "number-theory"
    ],
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "示したい命題の否定を仮定し、そこから矛盾を導くことで、元の命題が正しいと結論する証明法。",
    "definition_en": "A method of proof in which you assume the negation of the statement you want to prove and derive a contradiction, which shows the statement must be true.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Use proof by contradiction to show that √2 is irrational.",
        "ja": "背理法を用いて √2 が無理数であることを証明せよ。",
        "register": "written"
      },
      {
        "en": "We prove this by contradiction: suppose there are only finitely many primes p₁, p₂, …, pₙ.",
        "ja": "背理法で示す。素数が有限個 p₁, p₂, …, pₙ しかないと仮定する。",
        "register": "written"
      },
      {
        "en": "This is a classic proof by contradiction. Assume it's rational, and watch everything fall apart.",
        "ja": "これは典型的な背理法だね。有理数だと仮定すると、全部つじつまが合わなくなる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「背理法で示す」は prove … by contradiction。Levin（1.4 Proofs）は Suppose, for the sake of contradiction, that … と書き出し（4 件）、OpenStax Calculus Volume 1（平均値の定理）は By way of contradiction, suppose … と書く。矛盾を示して終える。",
      "対偶を利用した証明（proof by contrapositive）とは別物。背理法は結論の否定から矛盾を導き、対偶の証明は結論の否定から仮定の否定を導く。",
      "用例コーパスでは話し言葉は MIT OCW だけ、書き言葉は MIT の講義ノートだけ。",
      "英語版 Wikipedia「Proof by contradiction」は indirect proof・reductio ad absurdum とも呼ぶが、参照と用例コーパスにはどちらも出てこない。"
    ],
    "related": [
      "contradiction",
      "proof-by-contrapositive",
      "assume",
      "negation",
      "proof",
      "infinitude-of-primes"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "背理法",
        "en": "Proof by contradiction"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.4 Proofs: proof by contradiction（28 件）、Suppose, for the sake of contradiction（4 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "平均値の定理の証明: By way of contradiction, suppose"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "proof by contradiction": 8
      },
      "written": {
        "proof by contradiction": 15
      },
      "sources": [
        "mit-18.02",
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "proof-by-contrapositive",
    "ja": {
      "term": "対偶を利用した証明",
      "reading": "たいぐうをりようしたしょうめい",
      "alt": [
        "対偶による証明"
      ]
    },
    "en": {
      "term": "proof by contrapositive",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（proof by contrapositive ／ proof by contraposition は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "「p ならば q」を示す代わりに、真偽が一致する対偶「q でないならば p でない」を証明する方法。",
    "definition_en": "A method of proving \"if p, then q\" by proving its contrapositive, \"if not q, then not p,\" which has the same truth value.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Use a proof by contrapositive to show that if n² is even, then n is even.",
        "ja": "対偶を利用して、n² が偶数ならば n は偶数であることを証明せよ。",
        "register": "written"
      },
      {
        "en": "Going directly is messy, so let's prove the contrapositive: if n is odd, then n² is odd.",
        "ja": "直接示すのは面倒だから対偶を証明しよう。n が奇数ならば n² は奇数。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "prove the contrapositive",
        "ja": "対偶を証明する"
      }
    ],
    "pitfalls": [
      "背理法（proof by contradiction）と名前が似ているので聞き間違えない。contrapositive は対偶、contradiction は矛盾。"
    ],
    "related": [
      "contrapositive",
      "proof-by-contradiction",
      "direct-proof",
      "proof"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.4 Proofs"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "prove the contrapositive": 4
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは Levin の呼び方 proof by contrapositive（1.4 Proofs・2.2 Trees・6.2 Introduction to Number Theory）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "proof-by-induction",
    "ja": {
      "term": "数学的帰納法",
      "reading": "すうがくてききのうほう",
      "alt": []
    },
    "en": {
      "term": "proof by induction",
      "alt": [
        "mathematical induction"
      ],
      "uk": null,
      "register": "spoken",
      "variants": [
        {
          "term": "induction proof",
          "register": "written",
          "note": "MIT 6.042（離散数学）のテキストの言い方。Levin（4.5）も Induction Proof Structure と書く。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof",
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "n = 1 で成り立つことと、n = k で成り立てば n = k + 1 でも成り立つことを示して、すべての自然数 n で成り立つことを証明する方法。",
    "definition_en": "A method of proving a statement for every positive integer n: show it for n = 1, then show that if it holds for n = k it holds for n = k + 1.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Proof by induction has two parts: the base case and the inductive step.",
        "ja": "数学的帰納法は 2 段階。基底段階と帰納段階。",
        "register": "spoken"
      },
      {
        "en": "Use mathematical induction to prove that 1 + 2 + ⋯ + n = n(n + 1)/2 for every positive integer n.",
        "ja": "すべての正の整数 n について 1 + 2 + ⋯ + n = n(n + 1)/2 が成り立つことを、数学的帰納法で証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "by induction on n",
        "ja": "n についての数学的帰納法により"
      }
    ],
    "pitfalls": [
      "話し言葉はほとんどが MIT OCW 6.042（離散数学）。単に by induction と言うことが多い（用例コーパスでは話し言葉・書き言葉とも、ほとんどが MIT 6.042 のもの）。",
      "n = k + 1 を示す段は inductive step、仮定は induction hypothesis（inductive-hypothesis を参照）。"
    ],
    "related": [
      "inductive-hypothesis",
      "base-case",
      "prove-by-induction",
      "recursive-definition",
      "inductive-step",
      "strong-induction"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "数学的帰納法",
        "en": "Mathematical induction"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.5 Proof by Induction（proof by induction、mathematical induction、Induction Proof Structure）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "mathematical induction": 3,
        "by induction on n": 2,
        "proof by induction": 10,
        "induction proof": 2
      },
      "written": {
        "proof by induction": 4,
        "induction proof": 19,
        "by induction on n": 7
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "proof",
    "ja": {
      "term": "証明",
      "reading": "しょうめい",
      "alt": []
    },
    "en": {
      "term": "proof",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "logic"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1",
        "Discrete Math"
      ]
    },
    "definition_ja": "仮定と、すでに正しいとわかっている定義・定理から、筋道を立てて結論が正しいことを示す論証。",
    "definition_en": "A chain of reasoning that shows a statement must be true, starting from its hypotheses and using definitions and results already established.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's start the proof by writing down what we're given.",
        "ja": "まず仮定を書き出すところから証明を始めよう。",
        "register": "spoken"
      },
      {
        "en": "Complete the proof by giving a reason for each step.",
        "ja": "各段階の根拠を書いて、証明を完成させよ。",
        "register": "written"
      },
      {
        "en": "In this proof, the hypothesis is that n is even and the conclusion is that n² is even.",
        "ja": "この証明では、仮定は「n は偶数」、結論は「n² は偶数」である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "write a proof",
        "ja": "証明を書く"
      },
      {
        "en": "proof by contradiction",
        "ja": "背理法"
      }
    ],
    "pitfalls": [
      "名詞の「証明」は proof、動詞の「証明する」は prove。発音（Merriam-Webster: ˈprüf と ˈprüv）も綴りも近いので取り違えない。",
      "証明を書くときの「仮定」は hypothesis（given とも）、「結論」は conclusion。証明の終わりの印には ∎ や Q.E.D. を使う。"
    ],
    "related": [
      "prove",
      "counterexample",
      "proof-by-induction",
      "congruent"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "証明 (数学)",
        "en": "Mathematical proof"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「proof」",
        "url": "https://www.merriam-webster.com/dictionary/proof",
        "note": "発音 ˈprüf。prove は ˈprüv（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "proof": 696
      },
      "written": {
        "proof": 670
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "properties-of-equality",
    "ja": {
      "term": "等式の性質",
      "reading": "とうしきのせいしつ",
      "alt": []
    },
    "en": {
      "term": "properties of equality",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "等式の両辺に同じ数を足しても、両辺から同じ数を引いても、両辺に同じ数を掛けても、両辺を 0 でない同じ数で割っても、等式は成り立つという性質。方程式を解くときの変形の根拠になる。",
    "definition_en": "The rules that an equation stays true if you add the same number to both sides, subtract the same number from both sides, multiply both sides by the same number, or divide both sides by the same nonzero number. They justify each step when you solve an equation.",
    "latex": "A = B \\Rightarrow A + C = B + C,\\; A - C = B - C,\\; AC = BC,\\; \\frac{A}{C} = \\frac{B}{C} \\ (C \\neq 0)",
    "spoken_en": "if A equals B, then A plus C equals B plus C, A minus C equals B minus C, A C equals B C, and A over C equals B over C, where C is not zero",
    "examples": [
      {
        "en": "All four properties of equality boil down to one idea: both sides of the equation get the same treatment.",
        "ja": "4 つの等式の性質は、結局 1 つの考えにまとまる。片方の辺にしたことは、もう片方の辺にもする。",
        "register": "spoken"
      },
      {
        "en": "Solve 5x − 7 = 18, and state which of the properties of equality justifies each step.",
        "ja": "方程式 5x − 7 = 18 を解け。また、それぞれの変形にどの等式の性質を使ったかを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では 1 つずつ名前がある: addition property of equality（両辺に足す）、subtraction property of equality（両辺から引く）、multiplication property of equality（両辺に掛ける）、division property of equality（両辺を 0 でない数で割る）。",
      "解く手順は add 7 to both sides ／ divide both sides by 5 のように、両辺への操作で言う（エントリ both-sides）。",
      "「A = B ならば B = A」（両辺を入れかえても成り立つ）は、英語では symmetric property と呼ぶ（エントリ symmetric-property）。"
    ],
    "related": [
      "both-sides",
      "equation",
      "move-term-to-other-side",
      "linear-equation",
      "symmetric-property"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "properties of equality": 239
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "properties-of-inequalities",
    "ja": {
      "term": "不等式の性質",
      "reading": "ふとうしきのせいしつ",
      "alt": []
    },
    "en": {
      "term": "properties of inequalities",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "不等式の両辺に同じ数を足しても引いても、正の数を掛けても割っても不等号の向きは変わらないが、負の数を掛けたり割ったりすると向きが逆になるという性質。",
    "definition_en": "The rules that say what you may do to both sides of an inequality: add or subtract the same number, or multiply or divide by the same positive number, and the inequality sign stays as it is; multiply or divide by the same negative number, and the sign turns around.",
    "latex": "a < b,\\ c < 0 \\implies ac > bc",
    "spoken_en": "if a is less than b and c is negative, then a c is greater than b c",
    "examples": [
      {
        "en": "We're dividing both sides by negative two, so we have to flip the inequality.",
        "ja": "両辺を −2 で割るので、不等号の向きが変わる。",
        "register": "spoken"
      },
      {
        "en": "Dividing both sides of −4x ≥ 12 by −4 reverses the inequality, so x ≤ −3.",
        "ja": "−4x ≧ 12 の両辺を −4 で割ると不等号の向きが変わり、x ≦ −3。",
        "register": "written"
      },
      {
        "en": "Use the properties of inequalities to show that if a > b > 0, then 1/a < 1/b.",
        "ja": "不等式の性質を用いて、a > b > 0 ならば 1/a < 1/b であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "use the properties of inequalities",
        "ja": "不等式の性質を使う"
      }
    ],
    "pitfalls": [
      "「不等号の向きが変わる」は、用例コーパスでは話し言葉で flip the inequality がわずかに出てきて、書き言葉では reverse the inequality を使う（OpenStax Elementary Algebra ほか）。",
      "「両辺を負の数で割る」は divide both sides by a negative number。負の数を掛けたり割ったりしたのに向きを変え忘れやすい。",
      "文字 a で割るときは a の符号が分からないので、a > 0 と a < 0 で場合分けする。a = 0 のときは割れないので別に扱う。",
      "OpenStax Elementary Algebra（Solve Linear Inequalities）は Subtraction Property of Inequality と Addition Property of Inequality を別々に名付けて Properties of Inequality の枠にまとめ、割る・掛けるは Division and Multiplication Properties of Inequality と 1 つにする。Intermediate Algebra（Solve Linear Inequalities）は Addition and Subtraction Property of Inequality と Multiplication and Division Property of Inequality の 2 つにまとめる。中身はこのエントリの定義と同じ（両辺に同じ数を足す・引く、掛ける・割る）。"
    ],
    "related": [
      "linear-inequality",
      "system-of-inequalities",
      "absolute-value-inequality",
      "case-analysis"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Solve Linear Inequalities: Subtraction and Addition Properties of Inequality ／ Division and Multiplication Properties of Inequality"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Solve Linear Inequalities: Addition and Subtraction Property of Inequality ／ Multiplication and Division Property of Inequality"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "written": {
        "properties of inequalities": 20
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "properties-of-integrals",
    "ja": {
      "term": "積分の性質",
      "reading": "せきぶんのせいしつ",
      "alt": []
    },
    "en": {
      "term": "properties of the definite integral",
      "alt": [],
      "variants": [
        {
          "term": "properties of the definite integral",
          "register": "written",
          "note": "OpenStax Calculus の書き方。話し言葉には出てこず、書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "properties of integrals",
          "register": "both",
          "note": "話し言葉ではこちら（すべて MIT OCW）。書き言葉では少ない（OpenStax Calculus・MIT の講義ノート）。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "定数倍や和を分けて積分できること、区間を分けて足せること、上端と下端を入れ替えると符号が変わること、区間で f(x) ≦ g(x) なら ∫ₐᵇ f(x)dx ≦ ∫ₐᵇ g(x)dx となること（比較の性質）など、定積分の基本的な性質。",
    "definition_en": "The basic rules for definite integrals: constants and sums can be pulled apart, intervals can be split and added, swapping the limits changes the sign, and f(x) ≤ g(x) on [a, b] gives ∫ₐᵇ f(x) dx ≤ ∫ₐᵇ g(x) dx (comparison).",
    "latex": "\\int_a^b f(x)\\,dx = -\\int_b^a f(x)\\,dx",
    "spoken_en": "the integral from a to b of f of x dx equals negative the integral from b to a of f of x dx",
    "examples": [
      {
        "en": "One property you'll use all the time: if you flip the limits, you pick up a negative sign.",
        "ja": "よく使う性質：上端と下端を入れ替えるとマイナスが付く。",
        "register": "spoken"
      },
      {
        "en": "By the properties of the definite integral, ∫₀² (3x + 1) dx = 3∫₀² x dx + ∫₀² 1 dx.",
        "ja": "積分の性質より ∫₀² (3x + 1) dx = 3∫₀² x dx + ∫₀² 1 dx。",
        "register": "written"
      },
      {
        "en": "Since one over x is decreasing, each rectangle sticks out above the curve, so the sum is bigger than the integral.",
        "ja": "1/x は減少関数だから、長方形はどれも曲線より上にはみ出す。だから和は積分より大きい。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "比較の性質は、定積分を使って不等式を証明するときに使う。OpenStax Calculus Volume 1 はこの性質を comparison properties として定積分の性質に並べる。",
      "等号を含まない不等式は英語で strict inequality と言う。"
    ],
    "related": [
      "definite-integral",
      "integrals-of-even-and-odd-functions",
      "riemann-sum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.6（properties of definite integrals）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "properties of integrals": 6
      },
      "written": {
        "properties of the definite integral": 8,
        "properties of integrals": 3
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 6 件（10 未満） ／ 書: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満）。見出しは OpenStax の呼び方 properties of the definite integral（本文 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "properties-of-logarithms",
    "ja": {
      "term": "対数の性質",
      "reading": "たいすうのせいしつ",
      "alt": []
    },
    "en": {
      "term": "properties of logarithms",
      "alt": [
        "logarithmic properties"
      ],
      "variants": [
        {
          "term": "rules of logarithms",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry の書き方。話し言葉では少なく（MIT OCW・3Blue1Brown・patrickJMT）、書き言葉はすべて OpenStax Algebra and Trigonometry。"
        },
        {
          "term": "log rules",
          "register": "spoken",
          "note": "口頭の略した言い方で、件数は少ない。話し言葉だけに出てきて（大半が 3Blue1Brown）、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "log_a MN = log_a M + log_a N、log_a Mᵏ = k log_a M などの、対数の計算規則。",
    "definition_en": "The rules for working with logarithms, such as log(MN) = log M + log N and log(Mᵏ) = k log M.",
    "latex": "\\log_a MN = \\log_a M + \\log_a N",
    "spoken_en": "log base a of M N equals log base a of M plus log base a of N",
    "examples": [
      {
        "en": "By the properties of logarithms, log of x squared is two log x.",
        "ja": "対数の性質より、log x² は 2 log x。",
        "register": "spoken"
      },
      {
        "en": "Use the properties of logarithms to expand log₂(8x³).",
        "ja": "対数の性質を使って log₂(8x³) を展開せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "個々の性質にも名前があり、OpenStax Algebra and Trigonometry は product rule ／ quotient rule ／ power rule for logarithms と呼ぶ。微分の product rule ／ quotient rule ／ power rule と同じ名前なので、どちらの話か文脈で聞き分ける。"
    ],
    "related": [
      "power-rule",
      "expanding-and-condensing-logs",
      "logarithm",
      "laws-of-exponents"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.5 Logarithmic Properties（product ／ quotient ／ power rule for logarithms）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.5 Exponential and Logarithmic Functions（Properties of Logarithms）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "rules of logarithms": 6,
        "log rules": 4,
        "properties of logarithms": 17
      },
      "written": {
        "properties of logarithms": 143,
        "logarithmic properties": 2,
        "rules of logarithms": 19
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "proportion",
    "ja": {
      "term": "比例式",
      "reading": "ひれいしき",
      "alt": []
    },
    "en": {
      "term": "proportion",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 つの比が等しいことを表す式。a : b = c : d の形で、a/b = c/d と同じ意味になる。",
    "definition_en": "An equation stating that two ratios are equal, such as a : b = c : d, which means the same as a/b = c/d.",
    "latex": "a : b = c : d \\quad \\left(\\frac{a}{b} = \\frac{c}{d}\\right)",
    "spoken_en": "a to b equals c to d, that is, a over b equals c over d",
    "examples": [
      {
        "en": "This is a proportion, 3 to 5 equals x to 20, so cross-multiply: 5x equals 60, and x is 12.",
        "ja": "これは比例式で、3 : 5 = x : 20。外項の積と内項の積が等しいから 5x = 60 で、x = 12 だね。",
        "register": "spoken"
      },
      {
        "en": "Solve the proportion x : 12 = 5 : 4.",
        "ja": "比例式 x : 12 = 5 : 4 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "関数の「比例」（y = ax）は別のエントリ（direct-proportion）。比例式の proportion と取り違えない。",
      "統計の proportion は「割合」の意味で、別の用語（標本比率 sample proportion、エントリ sample-proportion）。",
      "比例式を解くのに使う「外項の積 = 内項の積」（a : b = c : d ならば ad = bc）は、英語では cross-multiply と動詞で言う（エントリ cross-multiply）。"
    ],
    "related": [
      "ratio",
      "value-of-a-ratio",
      "cross-multiply",
      "direct-proportion",
      "similar-triangles"
    ],
    "sources": [
      {
        "type": "wikidata",
        "qid": "Q2631509"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "set up a proportion | set up the proportion | solve the proportion | solve a proportion | solve proportions | write a proportion | using a proportion | use a proportion | the proportion is true": 11
      },
      "written": {
        "set up a proportion | set up the proportion | solve the proportion | solve a proportion | solve proportions | write a proportion | using a proportion | use a proportion | the proportion is true": 47
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "proposition",
    "ja": {
      "term": "命題",
      "reading": "めいだい",
      "alt": []
    },
    "en": {
      "term": "statement",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "proposition",
          "register": "written",
          "note": "MIT 6.042 の講義とその講義ノートの言い方。Levin の離散数学は statement が主で、proposition は propositional logic などに少数（37 件）。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "正しいか正しくないか（真か偽か）がはっきり決まる文や式。",
    "definition_en": "A sentence or formula that is definitely either true or false.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Is this statement true or false? If it's false, give me a counterexample.",
        "ja": "この命題は真かな、偽かな。偽なら反例を挙げてごらん。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the following statement is true or false: if x² = 4, then x = 2.",
        "ja": "次の命題の真偽を調べよ。「x² = 4 ならば x = 2」",
        "register": "written"
      },
      {
        "en": "In logic, a proposition is a sentence that has exactly one truth value: it is true or it is false, never both.",
        "ja": "命題とは、真か偽かのどちらか一方に定まる文である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "statement は「命題」のほかに、一般の「文・主張」（problem statement ＝問題文など）の意味でも使う。",
      "「x > 3」のように変数の値で真偽が変わるものは、センター試験（平成 30 年度 数学I・A）は「実数 x に関する次の条件 p, q, r, s」、共通テスト（令和 3 年度 第1日程 数学I）は「条件 p, q を次のように定める」と「条件」と呼んで命題と区別する（エントリ condition）。",
      "日本語版 Wikipedia「命題」は命題を真偽が確定した言明と定義し、高等学校学習指導要領解説（数学I 集合と命題）は命題の真偽と「条件や結論」を扱う。Levin は statement を「真か偽かが定まる平叙文」と定義し、番号付きの結果の名前にも Proposition を使う（Proposition 2.2.2）。"
    ],
    "related": [
      "true",
      "false",
      "truth-value",
      "condition",
      "counterexample"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "命題",
        "en": "Proposition"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 集合と命題: 命題の条件や結論に着目（命題 34 件）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "平成 30 年度 数学I・A: 実数 x に関する次の条件 p, q, r, s。令和 3 年度 第1日程 数学I 第1問〔2〕: 条件 p, q を次のように定める（数学Ⅰ・数学Ａ には無い）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1 Mathematical Statements: Definition 1.1.5（a statement is a declarative sentence that is either true or false）、Proposition 2.2.2"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 図形: 命題の逆、推測や命題が常に成り立つとは限らない（命題 24 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "statement": 505,
        "proposition": 36
      },
      "written": {
        "statement": 351,
        "proposition": 152
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "propositional-logic",
    "ja": {
      "term": "命題論理",
      "reading": "めいだいろんり",
      "alt": []
    },
    "en": {
      "term": "propositional logic",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（propositional logic は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "真か偽かが決まる命題を「かつ」「または」「でない」「ならば」などでつないだときの真偽を調べる論理。命題の中の主語と述語の構造には立ち入らない。",
    "definition_en": "The part of logic that studies how true-or-false statements combine with connectives such as \"and,\" \"or,\" \"not,\" and \"if … then.\" It does not look inside a statement at its subject or predicate.",
    "latex": "\\lnot(p \\land q) \\equiv \\lnot p \\lor \\lnot q",
    "spoken_en": "not the quantity p and q is equivalent to not p or not q",
    "examples": [
      {
        "en": "In propositional logic, we don't care what p and q actually say, only whether each one is true or false.",
        "ja": "命題論理では、p や q が何を言っているかは気にしない。それぞれが真か偽かだけを見る。",
        "register": "spoken"
      },
      {
        "en": "In propositional logic, show that p → q is logically equivalent to ¬q → ¬p.",
        "ja": "命題論理において、p → q と ¬q → ¬p が同値であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「すべての x について」「ある x が存在して」のような量化子（∀、∃）を使うのは述語論理（predicate logic）で、命題論理には入らない。",
      "数I「集合と命題」では否定を p̄ と上に線を引いて書き、「かつ」「または」は言葉で書く。英語の本では ¬p、p ∧ q、p ∨ q と書く。",
      "用例コーパスでは書き言葉に少しあるだけ（MIT の講義ノート）で、話し言葉には出てこない。Levin では 6 件（1.3 Rules of Logic 5 ほか）。"
    ],
    "related": [
      "proposition",
      "truth-table",
      "logical-connective",
      "tautology",
      "negation",
      "quantifier"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "命題論理",
        "en": "Propositional logic"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.3 Rules of Logic・1.1 Mathematical Statements"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "propositional logic": 3
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは Levin の呼び方 propositional logic（1.3 Rules of Logic・1.1 Mathematical Statements）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "prove-by-induction",
    "ja": {
      "term": "数学的帰納法で証明する",
      "reading": "すうがくてききのうほうでしょうめいする",
      "alt": []
    },
    "en": {
      "term": "use induction",
      "alt": [
        "prove by induction"
      ],
      "variants": [
        {
          "term": "prove … by induction",
          "register": "both",
          "note": "Levin（2.2・4.7。4.5 は prove … by mathematical induction）と Nicholson（Appendix C）も prove … by induction と書く。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "数学的帰納法を使って、すべての自然数 n について命題が成り立つことを示す。",
    "definition_en": "To show, using mathematical induction, that a statement holds for every positive integer n.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We'll use induction on n.",
        "ja": "n についての数学的帰納法で証明しよう。",
        "register": "spoken"
      },
      {
        "en": "Use induction to prove that 2ⁿ > n for every positive integer n.",
        "ja": "すべての正の整数 n について 2ⁿ > n が成り立つことを、数学的帰納法で証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉・書き言葉とも MIT 6.042（離散数学）の講義とテキストが中心。Levin（4.5）の問題文は Use induction to prove … と書く。"
    ],
    "related": [
      "proof-by-induction",
      "inductive-hypothesis",
      "base-case"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.5 Proof by Induction（Use induction to prove、prove … by mathematical induction）・2.2・4.7（prove … by induction）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "Appendix C Mathematical Induction（prove … by induction）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "use induction !hypothesis": 33,
        "prove … by induction": 10
      },
      "written": {
        "use induction !hypothesis": 22,
        "prove by induction": 4
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "prove",
    "ja": {
      "term": "証明する",
      "reading": "しょうめいする",
      "alt": [
        "証明せよ",
        "示す"
      ]
    },
    "en": {
      "term": "prove",
      "alt": [],
      "variants": [
        {
          "term": "prove",
          "register": "both",
          "note": "話し言葉では show that より多く（MIT OCW が半分以上）、書き言葉では show that よりやや少ない（MIT の講義ノートと OpenStax Calculus が中心）。"
        },
        {
          "term": "show that",
          "register": "both",
          "note": "問題文の「〜を示せ」。話し言葉では prove より少なく（MIT OCW が半分ほど）、書き言葉では prove よりやや多い（半分以上が OpenStax Calculus）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "proof"
    ],
    "level": {
      "jp": [
        "中2",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Geometry",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "ある命題が正しいことを、定義・定理・仮定から筋道を立てて示す。",
    "definition_en": "To establish that a statement is true by reasoning from definitions, theorems, and assumptions.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To prove this inequality, just look at the left side minus the right side.",
        "ja": "この不等式を証明するには、左辺 − 右辺を考えればいい。",
        "register": "spoken"
      },
      {
        "en": "Prove that a² + b² ≥ 2ab for all real numbers a and b.",
        "ja": "すべての実数 a, b について a² + b² ≧ 2ab であることを証明せよ。",
        "register": "written"
      },
      {
        "en": "Given that M is the midpoint of AB and ∠1 ≅ ∠2, prove that △AMC ≅ △BMD.",
        "ja": "M が AB の中点で ∠1 = ∠2 のとき、△AMC ≡ △BMD を証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "prove the inequality",
        "ja": "不等式を証明する"
      }
    ],
    "pitfalls": [
      "問題文の「〜を示せ」は Show that …。用例コーパスでは話し言葉は prove、書き言葉は show that が多い（書き言葉の show that は半分以上が OpenStax Calculus）。",
      "不等式の証明の定番（差をとる、平方の和にする、相加平均と相乗平均の関係を使う）は、英語でも手順として説明する（take the difference ／ write it as a sum of squares ／ use AM-GM）。"
    ],
    "related": [
      "proving-an-identity",
      "am-gm-inequality",
      "proof-by-induction",
      "two-column-proof",
      "hypothesis",
      "sss-congruence"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "prove": 1018,
        "show that": 429
      },
      "written": {
        "show that": 635,
        "prove": 496
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "proving-an-identity",
    "ja": {
      "term": "等式を証明する",
      "reading": "とうしきをしょうめいする",
      "alt": [
        "等式の証明"
      ]
    },
    "en": {
      "term": "verify the identity",
      "alt": [
        "prove the identity"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "等式が成り立つことを示す。一方の辺を変形して他方に一致させる、両辺をそれぞれ同じ式に変形する、差が 0 になることを示す、などの方法がある。",
    "definition_en": "To show that an equation is an identity, for example by transforming one side into the other or by showing that the difference of the two sides is zero.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Start with the more complicated side and work toward the other side.",
        "ja": "複雑なほうの辺から始めて、もう一方の辺に近づけていく。",
        "register": "spoken"
      },
      {
        "en": "Verify the identity (sin θ + cos θ)² = 1 + 2 sin θ cos θ.",
        "ja": "等式 (sin θ + cos θ)² = 1 + 2 sin θ cos θ を証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は三角関数の恒等式の証明を verify the identity と書くことが多い。prove the identity も使う。用例コーパスの話し言葉にはどちらも出てこない。",
      "OpenStax Algebra and Trigonometry（Verifying Trigonometric Identities… の節）は、片方の辺（ふつうは複雑なほう）を変形して他方の辺に一致させる手順を示し、恒等式を確かめる手順は方程式を解く手順とは違うと書く（Sum-to-Product and Product-to-Sum Formulas）。両辺に同じ操作をして 1 = 1 を導く書き方は、この手順に当たらない。"
    ],
    "related": [
      "identity",
      "prove",
      "trigonometric-identities"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions・Sum-to-Product and Product-to-Sum Formulas（verify the identity）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式: 等式や不等式が成り立つことを証明する（等式の証明）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "verify the identity": 17,
        "prove the identity": 7
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pure-imaginary-number",
    "ja": {
      "term": "純虚数",
      "reading": "じゅんきょすう",
      "alt": []
    },
    "en": {
      "term": "pure imaginary number",
      "alt": [],
      "variants": [
        {
          "term": "purely imaginary",
          "register": "spoken",
          "note": "形容詞で「z は純虚数」と言う形。話し言葉は半分以上が 3Blue1Brown（ほかは MIT OCW）。書き言葉はすべて MIT の講義ノート。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II",
        "数C"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "実部が 0 で、虚部が 0 でない複素数 bi。",
    "definition_en": "A complex number bi with b real and b ≠ 0: its real part is zero and its imaginary part is not.",
    "latex": "z = bi \\quad (b \\ne 0)",
    "spoken_en": null,
    "examples": [
      {
        "en": "So z squared is purely imaginary: its real part is zero.",
        "ja": "だから z² は純虚数。実部が 0 だ。",
        "register": "spoken"
      },
      {
        "en": "Find all real numbers a for which (a + i)² is a pure imaginary number.",
        "ja": "(a + i)² が純虚数となる実数 a をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "講義では形容詞の purely imaginary（z is purely imaginary）で言う。OpenStax Intermediate Algebra・Algebra and Trigonometry と Nicholson は名詞の pure imaginary number と書く。共役複素数を使えば、z ≠ 0 かつ z̄ = −z と書ける。"
    ],
    "related": [
      "complex-conjugate",
      "complex-plane",
      "imaginary-axis",
      "imaginary-number",
      "imaginary-part"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Use the Complex Number System（pure imaginary number）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "Appendix A Complex Numbers（pure imaginary）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "purely imaginary": 11
      },
      "written": {
        "pure imaginary number": 4,
        "purely imaginary": 4
      },
      "sources": [
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pyramid",
    "ja": {
      "term": "角錐",
      "reading": "かくすい",
      "alt": []
    },
    "en": {
      "term": "pyramid",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "多角形の底面と、1 つの頂点に集まる三角形の側面で囲まれた立体。底面の形によって三角錐、四角錐などと呼ぶ。",
    "definition_en": "A solid with one polygon base and triangular faces that meet at a single point called the apex. It is named by the shape of its base, as in a square pyramid.",
    "latex": "V = \\frac{1}{3}Bh",
    "spoken_en": "V equals one third B times h",
    "examples": [
      {
        "en": "A pyramid holds one third of what the prism with the same base and height holds, so don't forget to divide by 3.",
        "ja": "角錐の体積は、底面と高さが同じ角柱の 3 分の 1。だから 3 で割るのを忘れないで。",
        "register": "spoken"
      },
      {
        "en": "A pyramid has a square base with sides of 6 m and a height of 5 m. Find its volume.",
        "ja": "底面が 1 辺 6 m の正方形で、高さが 5 m の角錐の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "側面の三角形が集まる頂点は apex と言う。底面の角の点も vertex（複数形 vertices）なので、どちらの頂点かを区別する。",
      "高さ（height）は apex から底面に下ろした垂線の長さで、側面の三角形の高さ（slant height、エントリ slant-height）とは別。",
      "角錐の名前は底面の形を前に付けて言う（square pyramid は底面が正方形の角錐）。"
    ],
    "related": [
      "prism",
      "cone",
      "polyhedron",
      "slant-height",
      "volume"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "角錐",
        "en": "Pyramid (geometry)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "pyramid": 46
      },
      "written": {
        "pyramid": 49
      },
      "sources": [
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pythagorean-identity",
    "ja": {
      "term": "ピタゴラスの恒等式",
      "reading": "ぴたごらすのこうとうしき",
      "alt": []
    },
    "en": {
      "term": "Pythagorean identity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「ピタゴラスの恒等式」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書では sin²θ + cos²θ = 1 を「三角関数の相互関係」の 1 つとして扱い、個別の名前をつけない。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "sin²θ + cos²θ = 1。単位円上の点の座標が三平方の定理を満たすことから出る。",
    "definition_en": "The identity sin²θ + cos²θ = 1, which comes from the Pythagorean theorem on the unit circle.",
    "latex": "\\sin^2\\theta + \\cos^2\\theta = 1",
    "spoken_en": "sine squared theta plus cosine squared theta equals one",
    "examples": [
      {
        "en": "Replace sine squared with one minus cosine squared — that's the Pythagorean identity.",
        "ja": "sin² を 1 − cos² に置きかえる。ピタゴラスの恒等式だ。",
        "register": "spoken"
      },
      {
        "en": "Use the Pythagorean identity to simplify (1 − cos θ)(1 + cos θ).",
        "ja": "(1 − cos θ)(1 + cos θ) を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "米国の教科書は 1 + tan²θ = sec²θ、1 + cot²θ = csc²θ も合わせて Pythagorean identities（複数形）と呼ぶ。"
    ],
    "related": [
      "trigonometric-identities",
      "trigonometric-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Pythagorean identity": 54
      },
      "written": {
        "Pythagorean identity": 43
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pythagorean-theorem",
    "ja": {
      "term": "三平方の定理",
      "reading": "さんへいほうのていり",
      "alt": [
        "ピタゴラスの定理"
      ]
    },
    "en": {
      "term": "Pythagorean theorem",
      "alt": [
        "Pythagoras' theorem"
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直角三角形で、直角をはさむ 2 辺の長さを a、b、斜辺の長さを c とすると、a² + b² = c² が成り立つという定理。",
    "definition_en": "For a right triangle, squaring the two shorter sides (the legs) and adding gives the square of the longest side (the hypotenuse): a² + b² = c².",
    "latex": "a^2 + b^2 = c^2",
    "spoken_en": "a squared plus b squared equals c squared",
    "examples": [
      {
        "en": "It's a right triangle, so we can use the Pythagorean theorem: 6 squared plus 8 squared equals c squared, so c is 10.",
        "ja": "直角三角形だから三平方の定理が使える。6 の 2 乗 + 8 の 2 乗 = c の 2 乗だから、c は 10 だ。",
        "register": "spoken"
      },
      {
        "en": "A right triangle has legs of length 5 and 12. Use the Pythagorean theorem to find the length of the hypotenuse.",
        "ja": "直角をはさむ 2 辺の長さが 5 と 12 の直角三角形がある。三平方の定理を用いて、斜辺の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Elementary Algebra・Intermediate Algebra は a と b を legs、c を hypotenuse と呼んで公式を書く。「直角をはさむ辺」には、英語では leg という 1 語の名前がある（エントリ legs、hypotenuse）。",
      "c は必ず斜辺（直角の向かいの辺）。直角をはさむ辺を c に入れると式が成り立たない。",
      "英語版 Wikipedia「Pythagorean theorem」は Pythagoras's theorem という別名も挙げ、Nicholson は Pythagoras' theorem と書く（19 件）。Pythagorean は人名 Pythagoras から来た語なので大文字で始める。"
    ],
    "related": [
      "legs",
      "hypotenuse",
      "right-triangle",
      "converse-of-the-pythagorean-theorem",
      "distance-formula"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ピタゴラスの定理",
        "en": "Pythagorean theorem"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Pythagorean_theorem",
        "note": "記事「Pythagorean theorem」の冒頭: Pythagoras's theorem とも呼ぶ（地域は書いていない）"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "4.1 Vectors and Lines ほか: Pythagoras' theorem（19 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Pythagorean theorem": 167
      },
      "written": {
        "Pythagorean theorem": 135
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "pythagorean-triple",
    "ja": {
      "term": "ピタゴラス数",
      "reading": "ぴたごらすすう",
      "alt": []
    },
    "en": {
      "term": "Pythagorean triple",
      "alt": [
        "Pythagorean number triple"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "a² + b² = c² を満たす 3 つの自然数の組 (a, b, c)。(3, 4, 5)、(5, 12, 13)、(8, 15, 17) など。",
    "definition_en": "Three positive integers a, b, c satisfying a² + b² = c², such as 3, 4, 5; a right triangle can have these numbers as its side lengths.",
    "latex": "3^2 + 4^2 = 5^2",
    "spoken_en": "three squared plus four squared equals five squared",
    "examples": [
      {
        "en": "3, 4, 5 is a Pythagorean triple, so 6, 8, 10 works too; just double everything.",
        "ja": "3、4、5 はピタゴラス数だから、6、8、10 もピタゴラス数になる。全部 2 倍すればいい。",
        "register": "spoken"
      },
      {
        "en": "Determine whether 8, 15, and 17 form a Pythagorean triple.",
        "ja": "8、15、17 がピタゴラス数であるかどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の triple は「3 つ組」。1 つの数の名前ではなく、3 つの数の組の名前。",
      "自然数（正の整数）の組に限る。1、1、√2 は直角三角形の辺の長さになるが、ピタゴラス数ではない（エントリ 45-45-90-triangle）。",
      "CK-12 Geometry は Pythagorean Triples の節で扱う。"
    ],
    "related": [
      "pythagorean-theorem",
      "converse-of-the-pythagorean-theorem",
      "natural-number",
      "right-triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ピタゴラス数",
        "en": "Pythagorean triple"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.30 Pythagorean Triples"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 Pythagorean triple（CK-12 Geometry 4.30 Pythagorean Triples・CK-12 Geometry 4.27 The Pythagorean Theorem・CK-12 Geometry 4.28 Basics of Pythagorean Theorem・計 61 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "qr-factorization",
    "ja": {
      "term": "QR 分解",
      "reading": "きゅーあーるぶんかい",
      "alt": []
    },
    "en": {
      "term": "QR factorization",
      "alt": [
        "QR decomposition"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（QR factorization ／ QR decomposition は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "列が 1 次独立な行列を、列が正規直交な行列 Q と上三角行列 R の積 A = QR に分けること。グラム・シュミットの直交化法で得られる。",
    "definition_en": "Writing a matrix with independent columns as A = QR, with Q having orthonormal columns and R upper triangular; the Gram-Schmidt process produces it.",
    "latex": "A = QR",
    "spoken_en": null,
    "examples": [
      {
        "en": "QR factorization is just Gram-Schmidt written as a matrix product.",
        "ja": "QR 分解は、グラム・シュミット法を行列の積で書いたものだ。",
        "register": "spoken"
      },
      {
        "en": "Find a QR factorization of the matrix.",
        "ja": "行列の QR 分解を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Nicholson は QR-factorization（ハイフン付き）と書く。用例コーパスには出てこない。"
    ],
    "related": [
      "gram-schmidt-process",
      "orthonormal-basis",
      "lu-decomposition"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.4 QR-Factorization"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 QR factorization（8.4 QR-Factorization・8.5 Computing Eigenvalues）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadrant",
    "ja": {
      "term": "象限",
      "reading": "しょうげん",
      "alt": []
    },
    "en": {
      "term": "quadrant",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "座標平面を x 軸と y 軸で 4 つに分けた部分のそれぞれ。x 軸・y 軸の上の点は、どの象限にも入らない。",
    "definition_en": "One of the four regions of the coordinate plane cut out by the two axes. A point on an axis is in no quadrant.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If x is negative and y is negative, the point is down here in Quadrant III.",
        "ja": "x が負で y も負なら、点は左下の第 3 象限にある。",
        "register": "spoken"
      },
      {
        "en": "In which quadrant does the point (5, −2) lie?",
        "ja": "点 (5, −2) は第何象限にあるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の「第 1 象限」〜「第 4 象限」（センター試験の問題文・学習指導要領解説も第 1 象限と書く）に対し、米国の教科書（OpenStax）はローマ数字で Quadrant I〜IV と書く（Quadrant III は quadrant three と読む）。授業では the first quadrant のように序数で言うことも、quadrant one と数で言うこともある（Khan Academy は序数、YouTube の解説は数が多い）。番号の付け方は同じで、第 1 象限から反時計回り。",
      "x 軸・y 軸の上の点（(4, 0)、(0, −3)、原点など）はどの象限にも入らない。",
      "quadrantal angle（エントリ quadrantal-angle）は「座標軸上の角」で、象限の中にある角のことではない。"
    ],
    "related": [
      "coordinate-plane",
      "coordinate-axes",
      "origin",
      "x-coordinate",
      "quadrantal-angle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "象限",
        "en": "Orthant"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "大学入試センター試験 平成29年度 本試験 数学（問題）",
        "note": "問題文の「第 1 象限」"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 第1象限"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "quadrant": 475
      },
      "written": {
        "quadrant": 306
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadrantal-angle",
    "ja": {
      "term": "座標軸上の角",
      "reading": "ざひょうじくじょうのかく",
      "alt": []
    },
    "en": {
      "term": "quadrantal angle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「座標軸上の角」は学習指導要領解説に無い、本プロジェクトの訳語。学習指導要領解説・共通テスト／センター試験・日本語版 Wikipedia にも、動径が座標軸に重なる角をまとめて呼ぶ名前は出てこない。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "動径が座標軸に重なる角（0°, 90°, 180°, 270° など、90° の整数倍）。",
    "definition_en": "An angle whose terminal side lies on a coordinate axis: an integer multiple of 90°.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "At quadrantal angles, one of sine or cosine is zero, and tangent may be undefined.",
        "ja": "座標軸上の角では sin か cos が 0 になり、tan は定義されないことがある。",
        "register": "spoken"
      },
      {
        "en": "Find sin θ, cos θ, and tan θ for the quadrantal angle θ = 3π/2.",
        "ja": "θ = 3π/2 のときの sin θ, cos θ, tan θ を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは書き言葉（OpenStax Algebra and Trigonometry）に少しあり、話し言葉にはほとんど出てこない。"
    ],
    "related": [
      "reference-angle",
      "terminal-side",
      "quadrant"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles（quadrantal angle）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「座標軸上の角」ほか、動径が座標軸に重なる角をまとめて呼ぶ名前は出てこない"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "「座標軸上の角」ほか、動径が座標軸に重なる角をまとめて呼ぶ名前は 0 件"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "quadrantal angle": 1
      },
      "written": {
        "quadrantal angle": 6
      },
      "sources": [
        "openstax-algtrig",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 quadrantal angle（本文 6 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadratic-equation",
    "ja": {
      "term": "二次方程式",
      "reading": "にじほうていしき",
      "alt": []
    },
    "en": {
      "term": "quadratic equation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "移項して整理すると ax² + bx + c = 0（a ≠ 0）の形になる方程式。x の 2 乗の項をふくみ、3 乗以上の項をふくまない。",
    "definition_en": "Any equation that rearranges to ax² + bx + c = 0 with a ≠ 0; the variable's highest power is 2.",
    "latex": "ax^2 + bx + c = 0 \\quad (a \\neq 0)",
    "spoken_en": "a x squared plus b x plus c equals zero, where a is not equal to zero",
    "examples": [
      {
        "en": "Since this is a quadratic equation, get everything on one side so it's equal to zero before you factor.",
        "ja": "これは二次方程式だから、因数分解する前に、全部を片側に集めて = 0 の形にしよう。",
        "register": "spoken"
      },
      {
        "en": "Solve the quadratic equation x² + 2x = 15.",
        "ja": "二次方程式 x² + 2x = 15 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "quadratic は 2 次の意味で、4 次は quartic（エントリ quartic-equation）。英語版 Wikipedia「Quadratic equation」は語源をラテン語 quadratus（正方形）と書く。",
      "二次式（エントリ quadratic-expression）や二次関数（エントリ quadratic-function）とは別の語。二次方程式は等式で、成り立つ x の値（解）を求めるもの。",
      "解き方は、因数分解（エントリ solving-by-factoring）、平方根の考え（エントリ solving-by-taking-square-roots）、平方完成（エントリ completing-the-square）、解の公式（エントリ quadratic-formula）。どれも、まず式を整理してから使う。"
    ],
    "related": [
      "quadratic-formula",
      "solving-by-factoring",
      "completing-the-square",
      "solving-by-taking-square-roots",
      "quadratic-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二次方程式",
        "en": "Quadratic equation"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Quadratic_equation",
        "note": "記事「Quadratic equation」（数学カテゴリから 4 段）の冒頭: 語源はラテン語 quadratus（square）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quadratic equation": 95
      },
      "written": {
        "quadratic equation": 542
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadratic-expression",
    "ja": {
      "term": "二次式",
      "reading": "にじしき",
      "alt": []
    },
    "en": {
      "term": "quadratic expression",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "次数が 2 の式。x² + 3x − 4 のように、2 次の項をふくみ、3 次以上の項をふくまない式。",
    "definition_en": "A polynomial expression of degree 2, such as x² + 3x − 4. In one variable, it can be written as ax² + bx + c with a ≠ 0.",
    "latex": "ax^2 + bx + c \\quad (a \\neq 0)",
    "spoken_en": "a x squared plus b x plus c, where a is not equal to zero",
    "examples": [
      {
        "en": "The highest power of x is 2, so it's a quadratic expression; if there were an x cubed, it wouldn't be.",
        "ja": "x の最高次が 2 だから、これは二次式。x³ があったら二次式じゃない。",
        "register": "spoken"
      },
      {
        "en": "Factor the quadratic expression x² + 3x − 4.",
        "ja": "二次式 x² + 3x − 4 を因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "quadratic は「2 次の」。quad- から「4」を連想しやすいが、ここでは「正方形」（x² は正方形の面積）から来ている。4 次は quartic（エントリ quartic-equation）。",
      "二次方程式は quadratic equation、二次関数は quadratic function と、「二次」はどれも quadratic。一次は linear（エントリ linear-expression）、三次は cubic。",
      "ax² + bx + c で a = 0 なら 1 次式になる。二次式であるためには a ≠ 0 が必要。"
    ],
    "related": [
      "degree",
      "polynomial",
      "linear-expression",
      "factored-form",
      "quadratic-formula"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quadratic expression": 20
      },
      "written": {
        "quadratic expression": 14
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadratic-form",
    "ja": {
      "term": "二次形式",
      "reading": "にじけいしき",
      "alt": []
    },
    "en": {
      "term": "quadratic form",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "変数について 2 次の項だけからなる式。対称行列 A を使って xᵀ A x と書ける。",
    "definition_en": "An expression in which every term has degree two in the variables; it can be written xᵀ A x with a symmetric matrix A.",
    "latex": "q(\\vec{x}) = \\vec{x}^{T}A\\vec{x}",
    "spoken_en": null,
    "examples": [
      {
        "en": "x squared plus four xy plus y squared is a quadratic form, and the matrix goes with it.",
        "ja": "x² + 4xy + y² は二次形式で、それに対応する行列がある。",
        "register": "spoken"
      },
      {
        "en": "Write the quadratic form as xᵀ A x with a symmetric matrix A.",
        "ja": "二次形式を対称行列 A を用いて xᵀ A x の形で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry や多くの講義の equation in quadratic form は、x⁴ − 5x² + 4 = 0 のように置き換えると 2 次方程式になる式のこと（別の意味）。"
    ],
    "related": [
      "orthogonal",
      "diagonalization",
      "transpose"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二次形式",
        "en": "Quadratic form"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.9 An Application to Quadratic Forms"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!in !the !standard !undoing quadratic form": 5
      },
      "written": {
        "!in !the !standard !undoing quadratic form": 3
      },
      "sources": [
        "mit-18.06",
        "openstax-algtrig",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは OpenStax の呼び方 !in !the !standard !undoing quadratic form（本文 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadratic-formula",
    "ja": {
      "term": "解の公式",
      "reading": "かいのこうしき",
      "alt": [
        "二次方程式の解の公式"
      ]
    },
    "en": {
      "term": "quadratic formula",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "二次方程式 ax²+bx+c=0 の解を係数から直接求める公式。",
    "definition_en": "A formula that gives the solutions of ax²+bx+c=0 directly from the coefficients.",
    "latex": "x = \\frac{-b \\pm \\sqrt{b^2-4ac}}{2a}",
    "spoken_en": "x equals negative b, plus or minus the square root of b squared minus four a c, all over two a",
    "examples": [
      {
        "en": "If it doesn't factor, just use the quadratic formula.",
        "ja": "因数分解できなければ解の公式を使えばいい。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "apply the quadratic formula",
        "ja": "解の公式を適用する"
      }
    ],
    "pitfalls": [
      "共通テストの問題文は判別式を D とおく（令和 5 年度 数学II ほか）。OpenStax Algebra and Trigonometry・Elementary Algebra は二次方程式の判別式を b² − 4ac のまま書き、D の文字は使わない（エントリ discriminant）。",
      "動詞と組むときは use the quadratic formula の形が用例コーパスで最も多く、apply the quadratic formula は少ない。"
    ],
    "related": [
      "discriminant",
      "completing-the-square"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二次方程式の解の公式",
        "en": "Quadratic formula"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和5年度 大学入学共通テスト 本試験 数学Ⅱ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=432&f=abm00003620.pdf&n=2023_ot_20_sugaku2.pdf",
        "note": "判別式を D とおく問題文"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quadratic formula": 126
      },
      "written": {
        "quadratic formula": 229
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadratic-function",
    "ja": {
      "term": "二次関数",
      "reading": "にじかんすう",
      "alt": []
    },
    "en": {
      "term": "quadratic function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中3",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "y が x の二次式で表される関数。学習指導要領解説では、中学校は関数 y = ax²（a ≠ 0）を扱い、高等学校（数学I）の二次関数で y = ax² + bx + c の形に広げる。",
    "definition_en": "A function whose rule is a degree-2 polynomial, such as f(x) = ax² + bx + c with a ≠ 0. Its graph is a parabola.",
    "latex": "y = ax^2 + bx + c \\quad (a \\neq 0)",
    "spoken_en": "y equals a x squared plus b x plus c, where a is not equal to zero",
    "examples": [
      {
        "en": "If the highest power of x is 2, it's a quadratic function, and the graph is going to be a parabola.",
        "ja": "x の最高次数が 2 なら二次関数で、グラフは放物線になるよ。",
        "register": "spoken"
      },
      {
        "en": "For the quadratic function y = ax², y = 12 when x = 2. Find the value of a.",
        "ja": "二次関数 y = ax² で、x = 2 のとき y = 12 である。a の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "中 3 の「y は x の 2 乗に比例する」関数 y = ax² も、英語では quadratic function に入る（y = ax² + bx + c で b = c = 0 の場合）。",
      "quadratic は「2 次の」。quad- から「4」を連想しやすいが、4 次は quartic（エントリ quartic-equation）。二次式は quadratic expression、二次方程式は quadratic equation（エントリ quadratic-equation）。",
      "y = ax² + bx + c の形と y = a(x − p)² + q の形の呼び名は、日本語と英語で対応がずれる（エントリ standard-form、vertex-form）。"
    ],
    "related": [
      "parabola",
      "quadratic-equation",
      "quadratic-expression",
      "vertex",
      "axis-of-symmetry"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二次関数",
        "en": "Quadratic function"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "中学校 第 3 学年 関数: 関数 y = ax²"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 二次関数（中学校では関数 y = ax² を取り扱い、と書く）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quadratic function": 36
      },
      "written": {
        "quadratic function": 228
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadratic-inequality",
    "ja": {
      "term": "二次不等式",
      "reading": "にじふとうしき",
      "alt": []
    },
    "en": {
      "term": "quadratic inequality",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 2"
      ]
    },
    "definition_ja": "移項して整理すると ax² + bx + c > 0（または <, ≧, ≦）の形になる不等式（a ≠ 0）。",
    "definition_en": "An inequality that can be rearranged into the form ax² + bx + c > 0 (or with <, ≥, or ≤), where a ≠ 0.",
    "latex": "ax^2 + bx + c > 0 \\quad (a \\neq 0)",
    "spoken_en": null,
    "examples": [
      {
        "en": "To solve a quadratic inequality, find where the quadratic equals zero first, then test each interval.",
        "ja": "二次不等式を解くには、まず二次式が 0 になる x を求めてから、それぞれの区間を調べる。",
        "register": "spoken"
      },
      {
        "en": "Solve the quadratic inequality x² − x − 6 < 0.",
        "ja": "二次不等式 x² − x − 6 < 0 を解け。",
        "register": "written"
      },
      {
        "en": "The solution of x² > 4 is x < −2 or x > 2, that is, (−∞, −2) ∪ (2, ∞).",
        "ja": "x² > 4 の解は x < −2, 2 < x、すなわち (−∞, −2) ∪ (2, ∞) である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "共通テスト・センター試験の問題文と正解は解を −2 < x < 3 のように不等式で書き、OpenStax Intermediate Algebra（Solve Quadratic Inequalities）は解を区間記法 (−2, 3) で書く。",
      "共通テスト・センター試験（平成 30 年度 数学II）は解を x < −2, 2 < x のように読点で並べ、この読点は「または」の意味。英語では or を書く（x < −2 or x > 2）。"
    ],
    "related": [
      "interval",
      "sign-chart-inequality",
      "polynomial-inequality",
      "linear-inequality",
      "discriminant"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "不等式",
        "en": "Inequality (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Solve Quadratic Inequalities（23 件。解は区間記法）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "二次不等式の解は −2 < x < 3 の不等式で書き、x < −2, 2 < x と読点で並べる（平成 30 年度 数学II）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quadratic inequality": 10
      },
      "written": {
        "quadratic inequality": 28
      },
      "sources": [
        "openstax-intalg",
        "yt:nancypi",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadric-surface",
    "ja": {
      "term": "二次曲面",
      "reading": "にじきょくめん",
      "alt": []
    },
    "en": {
      "term": "quadric surface",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "x, y, z の 2 次方程式で表される曲面。楕円面・放物面・双曲面など。",
    "definition_en": "A surface given by a second-degree equation in x, y, and z, such as an ellipsoid, a paraboloid, or a hyperboloid.",
    "latex": "\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2} = 1",
    "spoken_en": null,
    "examples": [
      {
        "en": "To recognize a quadric surface, look at its traces in each coordinate plane.",
        "ja": "二次曲面を見分けるには、各座標平面での切り口を見る。",
        "register": "spoken"
      },
      {
        "en": "Identify the quadric surface z = x² + y² and sketch it.",
        "ja": "二次曲面 z = x² + y² が何かを答え、概形をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "平面で切った切り口は trace と呼ぶ。"
    ],
    "related": [
      "conic-section",
      "coordinates-in-space",
      "level-curve"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "二次曲面",
        "en": "Quadric"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "quadric surface": 36
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quadrilateral",
    "ja": {
      "term": "四角形",
      "reading": "しかくけい",
      "alt": []
    },
    "en": {
      "term": "quadrilateral",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "4 本の線分（辺）で囲まれた多角形。4 つの頂点と 4 つの角をもち、内角の和は 360° である。",
    "definition_en": "A polygon with four sides. It has four vertices and four angles, and its angles add up to 360°.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Any quadrilateral can be split into two triangles by a diagonal, so its angles add up to 2 times 180, or 360.",
        "ja": "どんな四角形も、対角線で 2 つの三角形に分けられる。だから内角の和は 180 × 2 で 360° になる。",
        "register": "spoken"
      },
      {
        "en": "In quadrilateral ABCD, m∠A = 85°, m∠B = 110°, and m∠C = 70°. Find m∠D.",
        "ja": "四角形 ABCD で、∠A = 85°、∠B = 110°、∠C = 70° である。∠D の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「四角形」の字は「4 つの角」だが、英語の quadrilateral は quad-（4）と lateral（辺）からできた「4 つの辺」の形の名前。",
      "カタカナの「スクエア」につられて、四角形一般を square と言わない。英語の square は正方形、または 2 乗（エントリ square）。",
      "m∠A は「∠A の大きさ」（the measure of angle A、エントリ measure）の書き方で、共通テスト・センター試験の問題文の ∠ABC ＝ 60° の書き方にあたる。"
    ],
    "related": [
      "polygon",
      "triangle",
      "regular-polygon",
      "side",
      "vertex"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "四角形",
        "en": "Quadrilateral"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.7 Quadrilateral Classification"
      },
      {
        "type": "reference",
        "title": "大学入試センター試験 平成30年度 本試験 数学I・数学A",
        "note": "問題文の ∠ABC ＝ の書き方"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quadrilateral": 44
      },
      "written": {
        "quadrilateral": 8
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quantifier",
    "ja": {
      "term": "量化子",
      "reading": "りょうかし",
      "alt": [
        "限量子",
        "量化記号"
      ]
    },
    "en": {
      "term": "quantifier",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I",
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "述語の変数について、「すべての x について」「ある x について」のように、どれだけの値で成り立つかを指定する語や記号のこと。∀ と ∃ がある。",
    "definition_en": "A word or symbol, such as \"for all\" (∀) or \"there exists\" (∃), that tells for how many values of a variable a predicate is true.",
    "latex": "\\forall x\\ \\exists y\\ (y > x)",
    "spoken_en": "for all x, there exists y such that y is greater than x",
    "examples": [
      {
        "en": "Order matters here. If you swap the two quantifiers, the statement means something completely different.",
        "ja": "ここは順番が大事。2 つの量化子を入れ替えると、まったく別の命題になってしまう。",
        "register": "spoken"
      },
      {
        "en": "Negate the statement ∀x ∈ ℝ (x² ≥ 0) so that no negation symbol appears in front of a quantifier.",
        "ja": "命題 ∀x ∈ ℝ (x² ≥ 0) の否定を、量化子の前に否定の記号が残らない形で書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数I では「すべての」「ある」を言葉で扱い（「すべての x について p」の否定は「ある x について p でない」）、∀ ∃ の記号や「量化子」という名前は使わない。",
      "∀x ∃y と ∃y ∀x は意味が違う。量化子の順番は入れ替えられない。",
      "用例コーパスで quantifier は話し言葉には少なく（MIT OCW）、書き言葉は大半が MIT の講義ノート。Levin の本では 54 件。"
    ],
    "related": [
      "universal-quantifier",
      "existential-quantifier",
      "predicate",
      "all",
      "exist"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "quantifier": 3
      },
      "written": {
        "quantifier": 22
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quantitative-variable",
    "ja": {
      "term": "量的データ",
      "reading": "りょうてきでーた",
      "alt": [
        "量的変数"
      ]
    },
    "en": {
      "term": "quantitative data",
      "alt": [
        "quantitative variable"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数量で表され、足したり平均したりできるデータ（またはその変量）。身長、得点など。",
    "definition_en": "Data, or a variable, recorded as numbers that can be added or averaged, such as height or a test score.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Height is quantitative, so a histogram or a box plot makes sense.",
        "ja": "身長は量的データなので、ヒストグラムや箱ひげ図が使える。",
        "register": "spoken"
      },
      {
        "en": "Is the number of siblings quantitative discrete or quantitative continuous data?",
        "ja": "きょうだいの人数は、離散的な量的データか連続的な量的データか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は量的データを quantitative discrete data と quantitative continuous data に分ける。"
    ],
    "related": [
      "categorical-variable",
      "histogram",
      "discrete-random-variable",
      "continuous-random-variable"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "quantitative variable": 2
      },
      "written": {
        "quantitative data": 15
      },
      "sources": [
        "khan-ap-stats",
        "openstax-calculus",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quantity",
    "ja": {
      "term": "数量",
      "reading": "すうりょう",
      "alt": []
    },
    "en": {
      "term": "quantity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "長さ・重さ・個数・代金など、数で表すことのできる量。",
    "definition_en": "An amount that can be measured or counted and written as a number, such as a length, a weight, a number of items, or a price.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's write the relationship between these two quantities as an equation.",
        "ja": "この 2 つの数量の関係を、等式で表してみよう。",
        "register": "spoken"
      },
      {
        "en": "A car travels at 50 miles per hour for t hours and goes d miles. Write an equation relating the quantities d and t.",
        "ja": "車が時速 50 マイルで t 時間走り、d マイル進んだ。数量 d と t の関係を等式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の quantity は、かっこでくくった式の一部をまとめて指すこともある。this quantity squared は、かっこの中全体の 2 乗のこと（(x + 1)² は the quantity x plus 1, squared と読む）。",
      "日常の英語の quantity は「量」「分量」の意味で、in large quantities（大量に）のようにも使う。"
    ],
    "related": [
      "variable",
      "algebraic-expression",
      "expression",
      "parentheses",
      "square"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "a quantity !squared !cubed | two quantities !squared !cubed | these quantities | both quantities | each quantity | another quantity | unknown quantity | known quantity | same quantity | related quantities": 47
      },
      "written": {
        "a quantity !squared !cubed | two quantities !squared !cubed | these quantities | both quantities | each quantity | another quantity | unknown quantity | known quantity | same quantity | related quantities": 131
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quartic-equation",
    "ja": {
      "term": "四次方程式",
      "reading": "よじほうていしき",
      "alt": [
        "4 次方程式"
      ]
    },
    "en": {
      "term": "fourth-degree equation",
      "alt": [
        "quartic equation"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "未知数の最高次数が 4 の方程式。",
    "definition_en": "An equation whose highest power of the unknown is 4.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let t equal x squared, and this quartic turns into a quadratic in t.",
        "ja": "t = x² とおくと、この 4 次方程式は t の 2 次方程式になる。",
        "register": "spoken"
      },
      {
        "en": "Solve the fourth-degree equation x⁴ − 3x² − 4 = 0 over the complex numbers.",
        "ja": "4 次方程式 x⁴ − 3x² − 4 = 0 を複素数の範囲で解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語版 Wikipedia の記事名は Quartic equation。OpenStax Algebra and Trigonometry（2.6 Other Types of Equations）は fourth-degree equation と書く（例題の題 Solving a Fourth-degree Equation in Quadratic Form）。",
      "学習指導要領解説（数学II）は、x⁴ + x² − 2 = 0 のような複二次方程式（x² = t とおくと t の 2 次方程式になる）を取り上げる。"
    ],
    "related": [
      "polynomial-equation-of-higher-degree",
      "cubic-equation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "四次方程式",
        "en": "Quartic equation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "2.6 Other Types of Equations（fourth-degree equation）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式: x⁴＋x²−2＝0 のような複二次方程式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "quartic equation": 2
      },
      "written": {
        "fourth-degree equation": 2
      },
      "sources": [
        "mit-18.01",
        "openstax-algtrig",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは OpenStax の呼び方 fourth-degree equation（本文 2 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "quartile-deviation",
    "ja": {
      "term": "四分位偏差",
      "reading": "しぶんいへんさ",
      "alt": []
    },
    "en": {
      "term": "quartile deviation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスでは quartile deviation は話し言葉にわずかにあるだけで、semi-interquartile range は出てこない）。OpenStax Introductory Statistics には出てこず、IQR だけを使う。日本語版 Wikipedia「四分位範囲」は英語名として quartile deviation と semi-interquartile range を挙げる。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1"
      ]
    },
    "definition_ja": "四分位範囲の半分、(Q3 − Q1)/2。",
    "definition_en": "Half the interquartile range, (Q3 − Q1)/2.",
    "latex": "\\frac{Q_3 - Q_1}{2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The quartile deviation is just half of the IQR.",
        "ja": "四分位偏差は IQR の半分にすぎない。",
        "register": "spoken"
      },
      {
        "en": "Find the quartile deviation of the data.",
        "ja": "データの四分位偏差を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "interquartile-range",
      "quartile"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「四分位範囲」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%9B%E5%88%86%E4%BD%8D%E7%AF%84%E5%9B%B2",
        "note": "英語名 quartile deviation・semi-interquartile range"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quartile deviation": 3
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 3 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "quartile",
    "ja": {
      "term": "四分位数",
      "reading": "しぶんいすう",
      "alt": []
    },
    "en": {
      "term": "quartile",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データを小さい順に並べて 4 等分する位置にある 3 つの値。小さいほうから第 1 四分位数（Q1）、第 2 四分位数（中央値）、第 3 四分位数（Q3）。",
    "definition_en": "The three values that divide the ordered data into four equal parts: the first quartile Q1, the second quartile (the median), and the third quartile Q3.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To get Q1, take the median of the bottom half of the data; for Q3, take the median of the top half.",
        "ja": "Q1 は下半分の中央値、Q3 は上半分の中央値。",
        "register": "spoken"
      },
      {
        "en": "Find the first and third quartiles of the data.",
        "ja": "データの第 1 四分位数と第 3 四分位数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "第 1 四分位数は first quartile（Q1）、第 3 四分位数は third quartile（Q3）。lower quartile ／ upper quartile とも言う。",
      "Q1 は Q one、Q3 は Q three と読む。",
      "データの個数が奇数のとき、中央値を下半分・上半分に含めるかどうかで四分位数の値が変わる（教科書・ソフトウェアによって決め方が違う）。"
    ],
    "related": [
      "interquartile-range",
      "median",
      "box-plot",
      "five-number-summary"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "四分位数",
        "en": "Quartile"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quartile": 73
      },
      "written": {
        "quartile": 176
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quotient-rule",
    "ja": {
      "term": "商の微分",
      "reading": "しょうのびぶん",
      "alt": [
        "商の微分法"
      ]
    },
    "en": {
      "term": "quotient rule",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "商の導関数 {f(x)/g(x)}′ = {f′(x)g(x) − f(x)g′(x)}/{g(x)}² を与える公式。",
    "definition_en": "The rule (f/g)′ = (f′g − fg′)/g² for the derivative of a quotient.",
    "latex": "\\left(\\frac{f}{g}\\right)' = \\frac{f'g - fg'}{g^2}",
    "spoken_en": "the derivative of f over g is f prime g minus f g prime, over g squared",
    "examples": [
      {
        "en": "With the quotient rule, the minus sign means the order matters.",
        "ja": "商の微分では引き算があるので、順番が大事。",
        "register": "spoken"
      },
      {
        "en": "Use the quotient rule to differentiate y = (x + 1)/(x − 1).",
        "ja": "商の微分を用いて y = (x + 1)/(x − 1) を微分せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "product-rule",
      "chain-rule"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "商の微分法則",
        "en": "Quotient rule"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 2.9（quotient rule）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quotient rule": 157
      },
      "written": {
        "quotient rule": 111
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "quotient",
    "ja": {
      "term": "商",
      "reading": "しょう",
      "alt": []
    },
    "en": {
      "term": "quotient",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "割り算の答え。余りのある割り算では、17 ÷ 5 = 3 余り 2 の 3 を商という。",
    "definition_en": "The answer to a division problem. When there is a remainder, as in 17 divided by 5 giving 3 with a remainder of 2, the quotient is 3.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "20 divided by 4 is 5, so the quotient is 5.",
        "ja": "20 ÷ 4 = 5 だから、商は 5 だね。",
        "register": "spoken"
      },
      {
        "en": "Find the quotient and the remainder when 47 is divided by 6.",
        "ja": "47 を 6 で割ったときの商と余りを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「a を b で割った商」は the quotient when a is divided by b、または the quotient of a and b（a ÷ b の順）。",
      "余りは remainder（エントリ remainder）。整式の割り算でも、商と余りは quotient と remainder。",
      "微分の「商の微分公式」は quotient rule（エントリ quotient-rule）。quotient はそこでも割り算の結果 f(x)/g(x) の意味。"
    ],
    "related": [
      "remainder",
      "division",
      "dividend",
      "divisor-in-division",
      "quotient-rule"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "商 (数学)",
        "en": "Quotient"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "quotient !rule": 167
      },
      "written": {
        "quotient !rule": 929
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "radian-measure",
    "ja": {
      "term": "弧度法",
      "reading": "こどほう",
      "alt": []
    },
    "en": {
      "term": "radian measure",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3",
        "Geometry"
      ]
    },
    "definition_ja": "半径と同じ長さの弧に対する中心角を 1 ラジアンとして、角の大きさを表す方法。180° = π ラジアン。",
    "definition_en": "Measuring angles with 1 radian as the central angle of an arc as long as the radius; 180° = π radians.",
    "latex": "180^\\circ = \\pi\\ \\text{rad}",
    "spoken_en": "one hundred eighty degrees equals pi radians",
    "examples": [
      {
        "en": "In calculus we always use radian measure, so switch your calculator to radians.",
        "ja": "微積分ではいつも弧度法を使うので、電卓をラジアンに切り替えて。",
        "register": "spoken"
      },
      {
        "en": "Express 150° in radian measure.",
        "ja": "150° を弧度法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "in radian measure",
        "ja": "弧度法で"
      }
    ],
    "pitfalls": [
      "「弧度法で表せ」は、OpenStax Algebra and Trigonometry（7.1）では Convert … to radians と書く。",
      "用例コーパスの話し言葉では、in radians（ラジアンで）のほうが radian measure よりずっと多い（radian を参照）。"
    ],
    "related": [
      "radian",
      "degree-measure",
      "area-of-a-sector",
      "general-angle",
      "arc-length"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles（convert to radians）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.11 A New Way to Measure Angles〜7.14（radian measure）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "radian measure": 2
      },
      "written": {
        "radian measure": 39
      },
      "sources": [
        "khan-algebra",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "radian",
    "ja": {
      "term": "ラジアン",
      "reading": "らじあん",
      "alt": [
        "弧度"
      ]
    },
    "en": {
      "term": "radian",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3",
        "Geometry"
      ]
    },
    "definition_ja": "弧度法での角の単位。半径 1 の円で長さ 1 の弧に対する中心角が 1 ラジアン。",
    "definition_en": "The unit of angle in radian measure: the central angle of an arc of length 1 on a circle of radius 1.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Pi over 2 radians is 90 degrees.",
        "ja": "π/2 ラジアンは 90°。",
        "register": "spoken"
      },
      {
        "en": "Convert 3π/4 radians to degrees.",
        "ja": "3π/4 ラジアンを度数法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "センター試験（平成 30 年度 本試験 数学II 第 1 問）は、度との換算では … π ラジアンと書き、θ の範囲には単位を付けない。OpenStax Algebra and Trigonometry（7.1）も、単位の書かれていない角はラジアンを指すと書く。"
    ],
    "related": [
      "radian-measure",
      "degree-measure",
      "area-of-a-sector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ラジアン",
        "en": "Radian"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles（when an angle is described without a specific unit, it refers to radian measure）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成30年度 大学入試センター試験 本試験 数学Ⅱ（問題）",
        "url": "https://web.archive.org/web/20190601/https://www.dnc.ac.jp/albums/abm.php?f=abm00033143.pdf&n=h30+suugaku2+01suugaku2.pdf",
        "note": "第1問: 1 ラジアン、144° を弧度で表す、… π ラジアン"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.11 A New Way to Measure Angles（glossary: radian）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "radian": 299
      },
      "written": {
        "radian": 233
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "radical-equation",
    "ja": {
      "term": "無理方程式",
      "reading": "むりほうていしき",
      "alt": [
        "根号方程式"
      ]
    },
    "en": {
      "term": "radical equation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic",
      "functions"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "根号の中に未知数を含む方程式。両辺を 2 乗して解き、出てきた解をもとの方程式に代入して確かめる。",
    "definition_en": "An equation with the variable under a radical sign, such as √(x + 3) = x − 3. You solve it by squaring both sides, then check every answer in the original equation.",
    "latex": "\\sqrt{x + 3} = x - 3",
    "spoken_en": "the square root of x plus three equals x minus three",
    "examples": [
      {
        "en": "It's a radical equation, so square both sides, and then check your answers.",
        "ja": "無理方程式だから、両辺を 2 乗して、そのあと答えを確かめよう。",
        "register": "spoken"
      },
      {
        "en": "Solve the radical equation √(x + 3) = x − 3. Check for extraneous solutions.",
        "ja": "無理方程式 √(x + 3) = x − 3 を解け。もとの方程式を満たさない解がないか確かめること。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "両辺を 2 乗すると、もとの方程式を満たさない解（extraneous solution、エントリ extraneous-solution）が出ることがある。√(x + 3) = x − 3 を 2 乗して解くと x = 1, 6 が出るが、x = 1 は左辺が 2、右辺が −2 で成り立たず、解は x = 6 だけ。",
      "日本では数III で扱うが、米国の教科書では Algebra 1 の OpenStax Elementary Algebra から扱う（Intermediate Algebra、Algebra and Trigonometry にも出てくる）。",
      "radical は根号のこと（エントリ radical-sign）。"
    ],
    "related": [
      "extraneous-solution",
      "radical-sign",
      "square-root-of-a-number",
      "quadratic-equation",
      "radical-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学I (3) 二次関数（発展・拡充）: 簡単な分数方程式や無理方程式"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "radical equation": 7
      },
      "written": {
        "radical equation": 65
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "radical-expression",
    "ja": {
      "term": "根号式",
      "reading": "こんごうしき",
      "alt": []
    },
    "en": {
      "term": "radical expression",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「根号式」は学習指導要領解説に無い、本プロジェクトの訳語。日本語版 Wikipedia「有理化」は「根号を含む式」と書く。OpenStax Intermediate Algebra は radical expression と呼び（8.2 Simplify Radical Expressions、75 件）、簡単にすることを simplify radical expressions と言う（エントリ simplify-radicals）。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "根号 √ をふくむ式。√12 や 3√2 + √8 など。",
    "definition_en": "An expression that contains a radical, such as a square root. For example, √12 and 3√2 + √8 are radical expressions.",
    "latex": "3\\sqrt{2} + \\sqrt{8} = 5\\sqrt{2}",
    "spoken_en": "three root two plus root eight equals five root two",
    "examples": [
      {
        "en": "This is a radical expression, so before you add anything, simplify each radical: √8 is 2√2.",
        "ja": "これは根号をふくむ式だから、たす前にそれぞれの根号を簡単にしよう。√8 は 2√2 だ。",
        "register": "spoken"
      },
      {
        "en": "Simplify the radical expression √12 + √27 − √3.",
        "ja": "√12 + √27 − √3 を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "radical は根号（√、radical sign、エントリ radical-sign）や、根号のついた数・式のこと。radical expression は根号をふくむ式全体を指す。",
      "英語の radical は日常語では「過激な」「根本的な」の意味で使う単語。",
      "根号をふくむ式を簡単にすることは simplify radical expressions と言う（OpenStax Intermediate Algebra 8.2。エントリ simplify-radicals）。分母の根号をなくすのは rationalize the denominator（エントリ rationalizing-the-denominator）。"
    ],
    "related": [
      "radical-sign",
      "square-root-of-a-number",
      "simplify-radicals",
      "rationalizing-the-denominator",
      "expression"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "中学校 第 3 学年 数と式: 「根号式」「根号を含む式」は 0 件（見出しは本プロジェクトの訳語）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「有理化」",
        "url": "https://ja.wikipedia.org/wiki/%E6%9C%89%E7%90%86%E5%8C%96",
        "note": "冒頭の定義に「根号を含む式」"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "radical expression": 9
      },
      "written": {
        "radical expression": 122
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:nancypi"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "radical-function",
    "ja": {
      "term": "無理関数",
      "reading": "むりかんすう",
      "alt": [
        "根号関数"
      ]
    },
    "en": {
      "term": "square root function",
      "alt": [],
      "variants": [
        {
          "term": "radical function",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry の節の名前（Inverses and Radical Functions）の言い方。話し言葉にはほとんど出てこず（blackpenredpen だけ）、書き言葉は OpenStax Algebra and Trigonometry と Intermediate Algebra。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学III）の「無理関数」は √ を含む式で表される関数で、y = √(ax + b) の形のものを中心に扱う。英語の話し言葉では square root function が多く（radical function はほとんど出てこない）、立方根なども含めた呼び方は radical function（書き言葉では square root function よりやや多い）。ja.alt の「根号関数」は学習指導要領解説に無い（日本語版 Wikipedia「冪根」は n 乗根をとる関数を根号関数と呼ぶ）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "根号の中に変数を含む式で表される関数。",
    "definition_en": "A function defined by an expression with the variable under a radical, such as y = √(x + 1).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The square root function only makes sense for x ≥ 0, so watch the domain.",
        "ja": "√x は x ≥ 0 でしか定義されないので、定義域に気をつける。",
        "register": "spoken"
      },
      {
        "en": "Find the domain and range of the radical function f(x) = √(2x − 4).",
        "ja": "無理関数 f(x) = √(2x − 4) の定義域と値域を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "rational-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 極限: 分数関数と無理関数"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「冪根」",
        "url": "https://ja.wikipedia.org/wiki/%E5%86%AA%E6%A0%B9",
        "note": "n 乗根をとる関数を「根号関数」と呼ぶ"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "square root function": 18,
        "radical function": 1
      },
      "written": {
        "square root function": 17,
        "radical function": 25
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "radical-sign",
    "ja": {
      "term": "根号",
      "reading": "こんごう",
      "alt": [
        "ルート"
      ]
    },
    "en": {
      "term": "radical sign",
      "alt": [
        "radical symbol",
        "square root sign",
        "square root symbol"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "平方根を表すときに使う記号 √。√ の下に書いた数や式の平方根を表す。",
    "definition_en": "The symbol √ used to write square roots, as in √5. The number or expression under it is called the radicand.",
    "latex": "\\sqrt{a}",
    "spoken_en": "the square root of a",
    "examples": [
      {
        "en": "Everything under the radical sign gets simplified first, so do 16 plus 9 before you take the square root.",
        "ja": "根号の中身を先に計算するから、平方根をとる前に 16 + 9 をやって。",
        "register": "spoken"
      },
      {
        "en": "Use a radical sign to write the positive number whose square is 11.",
        "ja": "2 乗すると 11 になる正の数を、根号を使って表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語版 Wikipedia「根号」は √2 に「ルート 2」と読みを添える。英語では記号そのものを radical sign（radical symbol、IM は square root symbol）と呼び、√a は the square root of a と読む（IM Grade 8）。話し言葉では root two のように root だけでも読む（用例コーパス。エントリ square-root）。英語で root と言うと、方程式の解（roots of an equation）の意味にもなる。",
      "根号の中の数や式は radicand と呼ぶ。根号を含む式は radical expression（エントリ radical-expression）。",
      "根号は横線の下にある部分全体にかかる。√(16 + 9) = √25 = 5 で、√16 + √9 = 4 + 3 = 7 とは違う。",
      "3 乗根は √ の左上に小さく 3 を書いて、∛8 = 2 のように表す（エントリ nth-root）。"
    ],
    "related": [
      "square-root-of-a-number",
      "radical-expression",
      "principal-square-root",
      "nth-root",
      "simplify-radicals"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "根号",
        "en": "Radical symbol"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「根号」",
        "url": "https://ja.wikipedia.org/wiki/%E6%A0%B9%E5%8F%B7",
        "note": "√2（ルート 2）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "the square root of 73 の読み、square root symbol"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 7.15・Algebra 2 3.18: square root symbol"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "square root sign": 3,
        "radical sign": 16,
        "radical symbol": 5,
        "square root symbol": 12
      },
      "written": {
        "radical sign": 36,
        "radical symbol": 8
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "radius-of-convergence",
    "ja": {
      "term": "収束半径",
      "reading": "しゅうそくはんけい",
      "alt": []
    },
    "en": {
      "term": "radius of convergence",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "べき級数 Σcₙ(x − a)ⁿ が |x − a| < R で収束し、|x − a| > R で発散するような R。",
    "definition_en": "The number R such that a power series Σcₙ(x − a)ⁿ converges when |x − a| < R and diverges when |x − a| > R.",
    "latex": "|x - a| < R",
    "spoken_en": null,
    "examples": [
      {
        "en": "The ratio test gives you the radius of convergence, but not what happens at the endpoints.",
        "ja": "比判定法で収束半径は出るが、端点でどうなるかは分からない。",
        "register": "spoken"
      },
      {
        "en": "Find the radius of convergence of Σ xⁿ/(2ⁿ n).",
        "ja": "Σ xⁿ/(2ⁿ n) の収束半径を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "interval-of-convergence",
      "power-series",
      "ratio-test"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "収束半径",
        "en": "Radius of convergence"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.13（radius of convergence）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "radius of convergence": 76
      },
      "written": {
        "radius of convergence": 35
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "radius",
    "ja": {
      "term": "半径",
      "reading": "はんけい",
      "alt": []
    },
    "en": {
      "term": "radius",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円や球の中心と、周上（表面上）の点を結ぶ線分。また、その長さ。直径は半径の 2 倍。",
    "definition_en": "A segment from the center of a circle or sphere to a point on it, or the length of that segment. The diameter is twice the radius.",
    "latex": "d = 2r",
    "spoken_en": "d equals two r",
    "examples": [
      {
        "en": "The radius is half the diameter, so if the diameter is 10, the radius is 5.",
        "ja": "半径は直径の半分だから、直径が 10 なら半径は 5。",
        "register": "spoken"
      },
      {
        "en": "The circumference of a circle is 12π cm. Find the radius.",
        "ja": "円周の長さが 12π cm である円の半径を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形は radii（レイディアイ）。2 つの円の半径は the radii of the two circles。",
      "べき級数の収束半径 radius of convergence（エントリ radius-of-convergence）も同じ radius。"
    ],
    "related": [
      "diameter",
      "circle",
      "center",
      "circumference",
      "radius-of-convergence"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "半径",
        "en": "Radius"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "radius": 891
      },
      "written": {
        "radius": 887
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "raise-to-the-nth-power",
    "ja": {
      "term": "n 乗する",
      "reading": "えぬじょうする",
      "alt": []
    },
    "en": {
      "term": "raise … to the power",
      "alt": [
        "raise … to the nth power"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Algebra 1",
        "Precalculus"
      ]
    },
    "definition_ja": "ある数や式を n 回掛け合わせる。",
    "definition_en": "To multiply a number or expression by itself n times.",
    "latex": "z^n",
    "spoken_en": "z to the n",
    "examples": [
      {
        "en": "If you raise e to the power ln 3, you just get 3.",
        "ja": "e を ln 3 乗すれば、ちょうど 3 になる。",
        "register": "spoken"
      },
      {
        "en": "Raise z = 2(cos 20° + i sin 20°) to the sixth power.",
        "ja": "z = 2(cos 20° + i sin 20°) を 6 乗せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "raise … to the nth power",
        "ja": "n 乗する"
      }
    ],
    "pitfalls": [
      "話し言葉は半分ほどが 3Blue1Brown。書き言葉は raise … to the power のほうが raise … to the nth power（すべて OpenStax Algebra and Trigonometry）より多い。",
      "「z の n 乗」は式の読みでは z to the n（z to the nth power）。"
    ],
    "related": [
      "de-moivres-theorem",
      "nth",
      "nth-root"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "raise … to the power": 17,
        "raise … to the nth power": 1
      },
      "written": {
        "raise … to the power": 11,
        "raise … to the nth power": 4
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "random-number-table",
    "ja": {
      "term": "乱数表",
      "reading": "らんすうひょう",
      "alt": []
    },
    "en": {
      "term": "table of random numbers",
      "alt": [
        "random number table",
        "random digit table"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Intro Statistics"
      ]
    },
    "definition_ja": "0 から 9 までの数字を、どれも同じ確率で現れるようにでたらめに並べた表。",
    "definition_en": "A table of the digits 0 through 9 arranged so that every digit is equally likely and there is no pattern.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Start anywhere in the random digit table and read off two digits at a time.",
        "ja": "乱数表の好きなところから始めて、2 けたずつ読み取る。",
        "register": "spoken"
      },
      {
        "en": "Use the table of random numbers to choose 5 of the 40 members.",
        "ja": "乱数表を使って、40 人から 5 人を選べ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は、乱数表は統計の本にあるとし、電卓やソフトウェアの random number generator でも代用する（エントリ random-number）。"
    ],
    "related": [
      "random-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "乱数表",
        "en": "Random number table"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "random digit table": 1
      },
      "written": {
        "table of random numbers": 2
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは OpenStax の呼び方 table of random numbers（本文 2 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "random-number",
    "ja": {
      "term": "乱数",
      "reading": "らんすう",
      "alt": []
    },
    "en": {
      "term": "random number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "でたらめに、規則性なく決めた数。標本抽出では、どの数字も同じ確率で現れるように作った乱数（乱数表や乱数さい）を使う。",
    "definition_en": "A number chosen with no pattern, so that it cannot be predicted. In sampling, random numbers are produced so that every digit is equally likely.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Use the random number generator on your calculator to pick the students.",
        "ja": "電卓の乱数機能を使って生徒を選ぼう。",
        "register": "spoken"
      },
      {
        "en": "Use random numbers to select a sample of 10 students from the class list.",
        "ja": "乱数を使って、名簿から 10 人の標本を選べ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "乱数を出す電卓・ソフトウェアの機能は random number generator（OpenStax Introductory Statistics）。"
    ],
    "related": [
      "random-number-table",
      "random-sampling"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "乱数列",
        "en": "Random number"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "random number": 60
      },
      "written": {
        "random number": 53
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "random-sampling",
    "ja": {
      "term": "無作為抽出",
      "reading": "むさくいちゅうしゅつ",
      "alt": [
        "単純無作為抽出"
      ]
    },
    "en": {
      "term": "random sampling",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "母集団のどの個体も同じ確率で選ばれるように標本を取り出すこと。",
    "definition_en": "Selecting a sample so that every member of the population has the same chance of being chosen.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Random sampling protects you from picking only the people who are easy to reach.",
        "ja": "無作為抽出をすれば、声をかけやすい人ばかり選ぶことを防げる。",
        "register": "spoken"
      },
      {
        "en": "Explain how to take a simple random sample of 30 students from a school of 900.",
        "ja": "900 人の学校から 30 人の単純無作為標本をとる方法を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は simple random sample（31 件）・simple random sampling（18 件）と書く。略語 SRS は AP Statistics の CED（topic 1.11）にある（CED は答案で SRS とだけ書かず、単純無作為標本をとったことを説明するよう求める）。用例コーパスでは random sampling の語形変化 random sample(s) の中に入るので、別の候補にしていない。"
    ],
    "related": [
      "sampling",
      "select-at-random",
      "sample",
      "random-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "無作為抽出",
        "en": "Simple random sample"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.11（SRS）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "random sampling": 87
      },
      "written": {
        "random sampling": 171
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "random-variable",
    "ja": {
      "term": "確率変数",
      "reading": "かくりつへんすう",
      "alt": []
    },
    "en": {
      "term": "random variable",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "試行の結果によって値が決まり、それぞれの値をとる確率が定まっている変数。",
    "definition_en": "A variable whose value is determined by the outcome of a random experiment, each value having a definite probability.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let X be the random variable for the number of heads.",
        "ja": "表の出る回数を確率変数 X とする。",
        "register": "spoken"
      },
      {
        "en": "The random variable X is the number of defective items in a sample of 10. Find its probability distribution.",
        "ja": "確率変数 X を、10 個の標本に含まれる不良品の個数とする。X の確率分布を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "確率変数は大文字（X）、そのとる値は小文字（x）で書き分ける。"
    ],
    "related": [
      "probability-distribution",
      "expected-value",
      "discrete-random-variable",
      "continuous-random-variable",
      "independent-random-variables",
      "binomial-distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "確率変数",
        "en": "Random variable"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "4 章の導入（Random Variable Notation: X と x）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.8（random variable）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "random variable": 491
      },
      "written": {
        "random variable": 530
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "randomization",
    "ja": {
      "term": "無作為化",
      "reading": "むさくいか",
      "alt": []
    },
    "en": {
      "term": "randomization",
      "alt": [
        "random assignment"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "実験で、どの被験者にどの処理を割り当てるかを、くじや乱数などの偶然に任せること。処理以外の違いがグループ間でならされるので、因果関係の結論を支える。",
    "definition_en": "Using a chance process to decide which experimental units get which treatment. Random assignment is what lets an experiment support a cause-and-effect conclusion.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We flip a coin for each volunteer to pick their group, and that randomization is what lets us talk about cause and effect.",
        "ja": "ボランティア 1 人ずつコインを投げてグループを決める。この無作為化があるから、因果関係の話ができる。",
        "register": "spoken"
      },
      {
        "en": "Describe a method of randomization that divides the volunteers into a treatment group and a control group.",
        "ja": "ボランティアを処理群と対照群に分ける無作為化の方法を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "random assignment（処理を無作為に割り当てること）と random selection（標本を無作為に選ぶこと、エントリ random-sampling）は別。AP Statistics の CED は、この 2 つを区別するよう注意する。無作為割り当ては因果関係の結論を、無作為抽出は標本から母集団への一般化を支える。",
      "「無作為」は「適当に選ぶ」ことではない。くじ・コイン・乱数のような偶然の仕組みで決めること。"
    ],
    "related": [
      "experiment",
      "random-sampling",
      "completely-randomized-design",
      "control-group",
      "treatment"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.12・3.2・3.3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "random assignment": 1
      },
      "written": {
        "random assignment": 8,
        "randomization": 1
      },
      "sources": [
        "khan-ap-stats",
        "mit-notes",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは AP Statistics の CED の呼び方 randomization（topic 2.12・topic 3.2・topic 3.3・topic 3.5・topic 3.9・topic 3.10・topic 3.12・topic 3.14・topic 4.1・topic 4.2・topic 4.4・topic 4.6・topic 4.7・topic 4.9）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "range-of-data",
    "ja": {
      "term": "範囲",
      "reading": "はんい",
      "alt": []
    },
    "en": {
      "term": "range",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データの最大値から最小値を引いた値。データの散らばりの大きさを表す目安の 1 つ。",
    "definition_en": "The difference between the greatest and the least values in a data set. It gives a rough idea of the spread of the data.",
    "latex": "\\text{range} = \\text{maximum} - \\text{minimum}",
    "spoken_en": "the range equals the maximum minus the minimum",
    "examples": [
      {
        "en": "The top score was 95 and the lowest was 62, so the range is 33 points.",
        "ja": "最高点が 95 点で最低点が 62 点だから、範囲は 33 点。",
        "register": "spoken"
      },
      {
        "en": "The numbers of books read by eight students last month were 3, 7, 2, 9, 4, 6, 1, and 5. Find the range of the data.",
        "ja": "8 人の生徒が先月読んだ本の冊数は 3, 7, 2, 9, 4, 6, 1, 5 冊であった。このデータの範囲を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "関数の値域も英語は range で、同じ単語（エントリ range、domain-and-range）。データの range は 1 つの数（最大値 − 最小値）で、値域は値の集まり。",
      "四分位範囲は interquartile range（IQR、エントリ interquartile-range）。範囲は最大値と最小値だけで決まるので、外れ値が 1 つあると大きく変わる。",
      "動詞の range は「（値が）〜から〜にわたる」の意味で、The scores range from 62 to 95. のように使う。"
    ],
    "related": [
      "spread",
      "interquartile-range",
      "maximum",
      "minimum",
      "box-plot",
      "range"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "range of the data | range of a data | range of the scores | range of each data set | range of the data set": 5
      },
      "written": {
        "range of the data | range of a data | range of the scores | range of each data set | range of the data set": 10
      },
      "sources": [
        "khan-ap-stats",
        "openstax-algtrig",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "range",
    "ja": {
      "term": "値域",
      "reading": "ちいき",
      "alt": []
    },
    "en": {
      "term": "range",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "関数で、x が定義域の値をとるときに y（出力）がとる値全体。中学校学習指導要領解説の「y の変域」のこと。",
    "definition_en": "The set of all output values (y-values) that a function takes. On a graph, it is all the y-values the graph covers.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Domain is the x-values going in, and range is the y-values coming out.",
        "ja": "定義域は入っていく x の値、値域は出てくる y の値。",
        "register": "spoken"
      },
      {
        "en": "The domain of the function y = −2x + 4 is 0 ≤ x ≤ 3. Find the range.",
        "ja": "関数 y = −2x + 4 の x の変域が 0 ≦ x ≦ 3 のとき、y の変域を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "統計の「範囲」（最大値 − 最小値、エントリ range-of-data）も同じ range という単語。データの range は 1 つの数で、関数の range は y がとる値の集まり。",
      "x（入力）の側は domain（エントリ domain）、y（出力）の側が range で、取り違えやすい。中学の「y の変域」も range と言い、x の変域と合わせて言うときは domain and range（エントリ domain-and-range）。",
      "傾きが負の 1 次関数では、x の変域の小さいほうの端が、y の変域の大きいほうの端に対応する（y = −2x + 4、0 ≦ x ≦ 3 なら −2 ≦ y ≦ 4）。"
    ],
    "related": [
      "domain",
      "domain-and-range",
      "range-of-data",
      "function",
      "find-the-domain"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "値域",
        "en": "Range of a function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "変域（第 1 学年 比例・反比例ほか）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "range of the function | range of a function | range of f | range of g | range of this function": 2
      },
      "written": {
        "range of the function | range of a function | range of f | range of g | range of this function": 91
      },
      "sources": [
        "khan-algebra",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rank-nullity-theorem",
    "ja": {
      "term": "階数・退化次数の定理",
      "reading": "かいすう・たいかじすうのていり",
      "alt": [
        "次元定理"
      ]
    },
    "en": {
      "term": "dimension theorem",
      "alt": [
        "rank-nullity theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（dimension theorem ／ rank-nullity theorem は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "m × n 行列 A について、階数と零空間の次元の和が列の数 n に等しいという定理。",
    "definition_en": "For an m × n matrix A, the rank plus the dimension of the null space equals the number of columns n.",
    "latex": "\\operatorname{rank} A + \\dim \\operatorname{Nul} A = n",
    "spoken_en": null,
    "examples": [
      {
        "en": "Five columns and rank three, so by the dimension theorem the null space is two-dimensional.",
        "ja": "列が 5 つで階数が 3 なので、次元定理から零空間は 2 次元だ。",
        "register": "spoken"
      },
      {
        "en": "Use the rank-nullity theorem to find the dimension of the kernel.",
        "ja": "階数・退化次数の定理を用いて核の次元を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスには rank-nullity theorem・dimension theorem とも出てこない。Nicholson は dimension theorem（7.2 ほか、19 件）と呼ぶ。rank theorem は教科書によって別の定理（行の階数と列の階数が等しい）を指すので候補にしなかった。",
      "Wikipedia の記事名は Rank–nullity theorem。"
    ],
    "related": [
      "rank",
      "null-space",
      "dimension"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "階数・退化次数の定理",
        "en": "Rank–nullity theorem"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "7.2 Kernel and Image of a Linear Transformation"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 dimension theorem（7.2 Kernel and Image of a Linear Transformation・7.3 Isomorphisms and Composition・6.6 An Application to Differential Equations）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "rank",
    "ja": {
      "term": "階数",
      "reading": "かいすう",
      "alt": [
        "ランク"
      ]
    },
    "en": {
      "term": "rank",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列の 1 次独立な列（行）の最大の個数。行階段形にしたときのピボットの個数に等しい。",
    "definition_en": "The largest number of linearly independent columns (or rows) of a matrix; it equals the number of pivots in a row-echelon form.",
    "latex": "\\operatorname{rank} A",
    "spoken_en": null,
    "examples": [
      {
        "en": "Three pivots, so the rank is three.",
        "ja": "ピボットが 3 つなので、階数は 3。",
        "register": "spoken"
      },
      {
        "en": "Find the rank of the matrix and the dimension of its null space.",
        "ja": "行列の階数と、零空間の次元を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "dimension",
      "column-space",
      "row-space",
      "rank-nullity-theorem",
      "pivot"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.4 Rank of a Matrix"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rank of the matrix | rank of a matrix | full rank | full column rank | full row rank | rank one matrix | rank r | the rank is | rank of a": 140
      },
      "written": {
        "rank of the matrix | rank of a matrix | full rank | full column rank | full row rank | rank one matrix | rank r | the rank is | rank of a": 1
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.06",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rate-of-change",
    "ja": {
      "term": "変化の割合",
      "reading": "へんかのわりあい",
      "alt": [
        "変化率"
      ]
    },
    "en": {
      "term": "rate of change",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学の「変化の割合」は x の増加量に対する y の増加量の比。一次関数ではこれが傾き（slope）に等しく、英語では rate of change と言う。一次関数でない関数（y = ax² など）で 2 点間の変化の割合を言うときは、英語では average rate of change（平均変化率）と言い分ける。微積分の rate of change は瞬間の変化率（導関数）も指す。",
    "domains": [
      "functions",
      "calculus"
    ],
    "level": {
      "jp": [
        "中2",
        "中3",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "x の増加量に対する y の増加量の割合。一次関数では傾きに等しい。",
    "definition_en": "How much the output changes per unit change in the input; for a linear function it equals the slope.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The rate of change is 3, so y goes up 3 every time x goes up 1.",
        "ja": "変化の割合は 3 だから、x が 1 増えるごとに y は 3 増える。",
        "register": "spoken"
      },
      {
        "en": "The water level rises at a constant rate of change of 2 cm per minute.",
        "ja": "水位は毎分 2 cm の一定の割合で上がる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "変化率には単位を付けて答える（毎分 2 cm なら 2 cm/min、centimeters per minute と読む）。"
    ],
    "related": [
      "average-rate-of-change",
      "derivative",
      "slope-of-the-tangent-line"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 2.1・4.1（rates of change）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "中2 一次関数: 変化の割合（x の増加量に対する y の増加量の割合）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rate of change": 509
      },
      "written": {
        "rate of change": 349
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rate",
    "ja": {
      "term": "割合",
      "reading": "わりあい",
      "alt": []
    },
    "en": {
      "term": "ratio to the base amount",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。参照に「割合」1 語に当たる語は出てこず、場面で ratio ／ rate ／ fraction ／ percent を使い分ける（2 つの量の比べ方は ratio、全体のうちの部分は fraction、2 割・3 割引きは percent、利率・税率は rate）。OpenStax Prealgebra（6.2）は percent の問題で、もとにする量を the base と呼ぶ。見出しの ratio to the base amount は「比べる量 ÷ もとにする量」を英語で言い表した本プロジェクトの説明的な訳で、英語の名前ではない。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "小学校",
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "比べる量が、もとにする量の何倍にあたるかを表す数。割合 = 比べる量 ÷ もとにする量 で、小数・分数・百分率（%）・歩合（割・分・厘）で表す。",
    "definition_en": "The number that tells what multiple of a base amount another amount is: the compared amount divided by the base amount. It can be written as a decimal, a fraction, a percent, or in the Japanese units wari, bu and rin.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Twenty out of fifty students walk to school. That's two fifths of the class, or 40 percent.",
        "ja": "50 人中 20 人が歩いて通学している。割合はクラスの 5 分の 2、つまり 40 % だ。",
        "register": "spoken"
      },
      {
        "en": "A shirt that costs $30 is on sale for 20% off. Find the sale price.",
        "ja": "定価 30 ドルのシャツが 2 割引きで売られている。売値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "参照に「割合」1 語に当たる語は出てこない。「A の B に対する割合」は the ratio of A to B か A is what percent of B、「クラスの 5 分の 2」は two fifths of the class（fraction）、「2 割」は 20 percent、「利率」「税率」は interest rate ／ tax rate と、場面で ratio ／ rate ／ fraction ／ percent を使い分ける。IM は ratio・percent・percentage を Grade 6 の glossary の見出しにしている。",
      "rate を「割合」全般の意味で使わない。OpenStax Prealgebra（Ratios and Rate）は、同じ単位の 2 つの量の比べ方を ratio、単位の違う 2 つの量の比べ方（miles per hour、単価）を rate と分けている。",
      "proportion も「割合」と訳せるのは統計の用法（sample proportion 標本比率。AP Statistics の CED）で、OpenStax Prealgebra（Solve Proportions and their Applications）の proportion は 2 つの比が等しいという式（比例式）の意味。"
    ],
    "related": [
      "relative-frequency",
      "sample-proportion"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "割合",
        "en": "Rate (mathematics)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "5.6 Ratios and Rate・6.2 Solve General Applications of Percent（the base。mapping_note・pitfalls[1]）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 6)",
        "url": "https://im.kendallhunt.com/MS/students/1/index.html",
        "note": "Grade 6 glossary（ratio・percent・percentage）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "sample proportion（標本比率）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "総説: 小学校 第 5 学年の内容に割合（百分率）（level.jp 小学校）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "ratio-of-areas-of-similar-figures",
    "ja": {
      "term": "相似な図形の面積比",
      "reading": "そうじなずけいのめんせきひ",
      "alt": [
        "相似比と面積比"
      ]
    },
    "en": {
      "term": "ratio of the areas",
      "alt": [
        "area of similar polygons theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "CK-12 Geometry は ratio of the areas と呼ぶ（5.22 Area and Perimeter of Similar Polygons ほか、21 件）。用例コーパスには話し言葉・書き言葉ともほとんど出てこない。CK-12 Geometry 5.22 はこの事実を Area of Similar Polygons Theorem と名づけ（2 件）、相似な多角形の辺の比が m/n なら面積の比は (m/n)² になる（the ratio of the areas is (m/n)²）と文で述べる。相似比に当たる語は scale factor。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "相似な 2 つの図形の面積の比。相似比が m : n なら、面積比は m² : n² になる。",
    "definition_en": "For two similar figures, the ratio between their areas. With a scale factor of m : n, the areas come out in the ratio m² : n².",
    "latex": "\\frac{A_1}{A_2} = k^2",
    "spoken_en": "A one over A two equals k squared",
    "examples": [
      {
        "en": "If every side doubles, the ratio of the areas isn't 2 to 1, it's 4 to 1.",
        "ja": "辺がすべて 2 倍になっても、面積比は 2 : 1 ではなく 4 : 1。",
        "register": "spoken"
      },
      {
        "en": "Triangles ABC and DEF are similar with AB : DE = 2 : 3. Find the ratio of the areas of the two triangles.",
        "ja": "△ABC ∽ △DEF で AB : DE = 2 : 3 のとき、2 つの三角形の面積比を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "体積比は相似比の 3 乗（m³ : n³）。面積比の 2 乗と混同しない。",
      "相似の記号は、中学校学習指導要領解説（〔用語・記号〕）では ∽、IM Geometry・CK-12 Geometry では ∼（△ABC ∼ △DEF）。"
    ],
    "related": [
      "ratio-of-volumes-of-similar-solids",
      "scale-factor",
      "similar"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.22 Area and Perimeter of Similar Polygons・8.8 Translation Applications in Circle Similarity"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形〔用語・記号〕: ∽"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Geometry: △DHF ∼ △EHG と書く"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）〔用語・記号〕",
        "note": "中3 B 図形: ∽"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "ratio of the areas": 1
      },
      "written": {
        "ratio of the areas": 2
      },
      "sources": [
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは CK-12 の呼び方 ratio of the areas（CK-12 Geometry 5.22 Area and Perimeter of Similar Polygons・CK-12 Geometry 8.8 Translation Applications in Circle Similarity・計 21 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "ratio-of-volumes-of-similar-solids",
    "ja": {
      "term": "相似な立体の体積比",
      "reading": "そうじなりったいのたいせきひ",
      "alt": [
        "体積比"
      ]
    },
    "en": {
      "term": "volume ratio",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（IM の volume ratio は surface area to volume ratio の一部で別の概念）。IM Geometry 5.6 Scaling Solids は the volume is multiplied by k³ と文で言い、CK-12 Geometry 9.8 も名前を付けない。相似比に当たる語は scale factor（エントリ scale-factor）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "相似な 2 つの立体の体積の比。相似比が m : n なら、体積比は m³ : n³ になる。",
    "definition_en": "The ratio of the volumes of two similar solids. If their corresponding lengths are in the ratio m : n, the volume ratio is m³ : n³.",
    "latex": "V_1 : V_2 = m^3 : n^3",
    "spoken_en": "V one to V two equals m cubed to n cubed",
    "examples": [
      {
        "en": "If you double every edge of a cube, the volume ratio is 8 to 1, not 2 to 1.",
        "ja": "立方体の辺をすべて 2 倍にすると、体積比は 2 : 1 ではなく 8 : 1 になる。",
        "register": "spoken"
      },
      {
        "en": "Two similar cones have heights of 6 cm and 9 cm. Find the volume ratio of the smaller cone to the larger cone.",
        "ja": "相似な 2 つの円錐の高さは、それぞれ 6 cm、9 cm である。小さい円錐と大きい円錐の体積比を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "表面積の比は相似比の 2 乗（m² : n²、エントリ ratio-of-areas-of-similar-figures）、体積比は 3 乗（m³ : n³）。取り違えない。",
      "相似比が 1 つの数 k（scale factor、エントリ scale-factor）で与えられたときは、体積は k³ 倍になる。k = 2 なら 8 倍。",
      "相似比の 3 乗になるのは相似な立体どうしのときだけ。高さだけを 2 倍にした円柱のように、相似でない立体には使えない。"
    ],
    "related": [
      "ratio-of-areas-of-similar-figures",
      "scale-factor",
      "volume",
      "similar",
      "surface-area"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Geometry 5.6 Scaling Solids: the volume is multiplied by k³（名前は無い）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "9.8: 相似な立体の体積（名前は無い）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!to volume ratio": 1
      },
      "sources": [
        "mit-18.01"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 1 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "ratio-test",
    "ja": {
      "term": "比判定法",
      "reading": "ひはんていほう",
      "alt": [
        "ダランベールの判定法"
      ]
    },
    "en": {
      "term": "ratio test",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "L = lim |aₙ₊₁/aₙ| が 1 より小さければ級数は絶対収束し、1 より大きければ発散するという判定法。L = 1 では判定できない。",
    "definition_en": "If L = lim |aₙ₊₁/aₙ| is less than 1 the series converges absolutely, and if L is greater than 1 it diverges; L = 1 is inconclusive.",
    "latex": "L = \\lim_{n\\to\\infty} \\left| \\frac{a_{n+1}}{a_n} \\right|",
    "spoken_en": "L equals the limit as n goes to infinity of the absolute value of a sub n plus one over a sub n",
    "examples": [
      {
        "en": "Factorials in the terms? Reach for the ratio test.",
        "ja": "項に階乗があったら比判定法を使おう。",
        "register": "spoken"
      },
      {
        "en": "Use the ratio test to determine whether Σ 3ⁿ/n! converges.",
        "ja": "比判定法を用いて Σ 3ⁿ/n! が収束するかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "L = 1 のときは inconclusive（判定できない）。別の判定法を使う。"
    ],
    "related": [
      "radius-of-convergence",
      "interval-of-convergence",
      "absolute-convergence",
      "power-series"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.8（ratio test）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "ratio test": 41
      },
      "written": {
        "ratio test": 51
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "ratio",
    "ja": {
      "term": "比",
      "reading": "ひ",
      "alt": []
    },
    "en": {
      "term": "ratio",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "2 つの数量 a と b の割合の関係を、a : b の形で表したもの。",
    "definition_en": "A comparison of two quantities by division, written as a : b, a to b, or a/b.",
    "latex": "a : b",
    "spoken_en": "a to b",
    "examples": [
      {
        "en": "The ratio of boys to girls is 3 to 4, so for every 3 boys there are 4 girls.",
        "ja": "男子と女子の人数の比は 3 : 4。男子 3 人につき女子 4 人ということだ。",
        "register": "spoken"
      },
      {
        "en": "Find the number of red marbles in a bag of 40 red and blue marbles if the ratio of red marbles to blue marbles is 5 : 3.",
        "ja": "袋の中に赤玉と青玉が合わせて 40 個あり、赤玉と青玉の個数の比は 5 : 3 である。赤玉の個数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a : b は a to b と読む（3 : 4 は three to four）。a/b と分数の形で書くこともある。",
      "「a と b の比」は the ratio of a to b と言い、of の後に a : b の前の数、to の後に後ろの数を置く。",
      "等比数列の公比 common ratio（エントリ common-ratio）や比判定法 ratio test（エントリ ratio-test）にも ratio が出てくるが、別の用語。"
    ],
    "related": [
      "value-of-a-ratio",
      "proportion",
      "fraction",
      "percent",
      "common-ratio"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "比",
        "en": "Ratio"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!common !golden ratio !test": 733
      },
      "written": {
        "!common !golden ratio !test": 459
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rational-equation",
    "ja": {
      "term": "分数方程式",
      "reading": "ぶんすうほうていしき",
      "alt": [
        "有理方程式"
      ]
    },
    "en": {
      "term": "rational equation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "1/x + 1/(x − 2) = 1 のように、分母に未知数を含む方程式。",
    "definition_en": "An equation with a variable in a denominator, such as 1/x + 1/(x − 2) = 1.",
    "latex": "\\frac{1}{x} + \\frac{1}{x - 2} = 1",
    "spoken_en": "one over x plus one over the quantity x minus two equals one",
    "examples": [
      {
        "en": "With a rational equation, multiply everything by the LCD first to clear the fractions.",
        "ja": "分数方程式は、まず全体に最小公分母を掛けて分母を払おう。",
        "register": "spoken"
      },
      {
        "en": "Solve the rational equation x/(x − 3) = 3/(x − 3) + 2, and check for extraneous solutions.",
        "ja": "分数方程式 x/(x − 3) = 3/(x − 3) + 2 を解け。また、無縁解がないか確かめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "分母を払って出た解のうち、もとの分母を 0 にするものは解ではない（エントリ extraneous-solution）。もとの式に戻して確かめる。",
      "英語の rational は「分数の形の（比の）」の意味で、「合理的な」ではない（エントリ rational-function、rational-expression）。",
      "両辺に掛ける最小公分母は LCD（least common denominator。エントリ least-common-denominator）と言う。"
    ],
    "related": [
      "least-common-denominator",
      "extraneous-solution",
      "rational-expression",
      "rational-function",
      "clear-the-denominators"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学I (3) 二次関数（発展・拡充）: 簡単な分数方程式や無理方程式"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "rational equation": 7
      },
      "written": {
        "rational equation": 91
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rational-exponent",
    "ja": {
      "term": "分数の指数",
      "reading": "ぶんすうのしすう",
      "alt": []
    },
    "en": {
      "term": "rational exponent",
      "alt": [],
      "variants": [
        {
          "term": "fractional exponent",
          "register": "spoken",
          "note": "授業の言い方。話し言葉では大半が Professor Leonard で、書き言葉にはほとんど出てこない（OpenStax Algebra and Trigonometry だけ）。"
        },
        {
          "term": "rational exponent",
          "register": "both",
          "note": "話し言葉ではほとんどが Professor Leonard。書き言葉ではこちらがほとんどで、OpenStax Algebra and Trigonometry・Intermediate Algebra・Elementary Algebra に多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "a > 0 のとき a^(m/n) = ⁿ√(aᵐ) と定めた、有理数の指数。",
    "definition_en": "An exponent that is a rational number, defined by a^(m/n) = ⁿ√(aᵐ) for a > 0.",
    "latex": "a^{\\frac{m}{n}} = \\sqrt[n]{a^m}",
    "spoken_en": "a to the m over n equals the nth root of a to the m",
    "examples": [
      {
        "en": "The denominator of a rational exponent gives the root: x to the one-half is the square root of x, and x to the two-thirds is the cube root of x squared.",
        "ja": "分数の指数の分母が累乗根を表す。x の 1/2 乗は √x、x の 2/3 乗は x² の 3 乗根。",
        "register": "spoken"
      },
      {
        "en": "Evaluate 27^(2/3) and 16^(−3/4).",
        "ja": "27^(2/3) と 16^(−3/4) の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では fractional exponent（大半が Professor Leonard）と rational exponent が並ぶ。書き言葉は rational exponent（OpenStax Algebra and Trigonometry・Intermediate Algebra・Elementary Algebra ほか）。"
    ],
    "related": [
      "nth-root",
      "negative-exponent",
      "laws-of-exponents"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "rational exponent": 20,
        "fractional exponent": 24
      },
      "written": {
        "rational exponent": 111,
        "fractional exponent": 2
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rational-expression",
    "ja": {
      "term": "分数式",
      "reading": "ぶんすうしき",
      "alt": []
    },
    "en": {
      "term": "rational expression",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "OpenStax Elementary Algebra（Simplify Rational Expressions）は rational expression を多項式の商 p/q（q ≠ 0）と定義し、−13/42 のような数の分数も例に挙げる。日本語版 Wikipedia「数式」は分数式を「文字の分母を含む式」とし、多項式の商全体は有理式と呼ぶ（「多項式」。「部分分数分解」は有理式を分数式とも呼ぶ）。学習指導要領解説（数学II）は多項式の除法と分数式の四則計算を分けて扱い、「簡単な場合」を分数式の分母の次数が二次程度までのものとする。英語の rational expression は、この有理式に当たる。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "分母に文字を含む式で、整式を整式で割った形のもの。約分・通分して計算する。",
    "definition_en": "An expression that is a quotient of two polynomials with a nonzero denominator, such as (x + 1)/(x − 2); such expressions are simplified by canceling common factors and are added or subtracted over a common denominator.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To add these rational expressions, find a common denominator first.",
        "ja": "これらの分数式を足すには、まず通分する。",
        "register": "spoken"
      },
      {
        "en": "Simplify the rational expression (x² − 1)/(x² + 3x + 2).",
        "ja": "分数式 (x² − 1)/(x² + 3x + 2) を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "分数式の値が定義されない x（分母が 0 になる x）を除くことを忘れない。OpenStax Algebra and Trigonometry（Linear Equations in One Variable）は、有理方程式を解くときに分母を 0 にする値を excluded values と呼ぶ。OpenStax Elementary Algebra（Simplify Rational Expressions）は、rational expression が定義されない値を先に求めさせる。"
    ],
    "related": [
      "complex-fraction",
      "reduce",
      "find-a-common-denominator",
      "rational-function",
      "rational-equation",
      "numerator",
      "denominator"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Rational Expressions・Linear Equations in One Variable（excluded values）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Simplify Rational Expressions（p(x)/q(x)、−13/42）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数式」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%BC%8F",
        "note": "分数式: 文字の分母を含む式"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「多項式」",
        "url": "https://ja.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F",
        "note": "有理式: 二つの多項式 P, Q の商"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「部分分数分解」",
        "url": "https://ja.wikipedia.org/wiki/%E9%83%A8%E5%88%86%E5%88%86%E6%95%B0%E5%88%86%E8%A7%A3",
        "note": "有理式（あるいは分数式ともいう）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式: 多項式の除法や分数式の四則計算（簡単な場合は分数式の分母の次数が二次程度まで）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "rational expression": 93
      },
      "written": {
        "rational expression": 386
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rational-function",
    "ja": {
      "term": "分数関数",
      "reading": "ぶんすうかんすう",
      "alt": [
        "有理関数"
      ]
    },
    "en": {
      "term": "rational function",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の数III の「分数関数」は、学習指導要領解説では「簡単な分数関数」のグラフを平行移動と結びつけて扱う。米国の rational function は多項式 ÷ 多項式の関数全般を指し、日本語では有理関数に当たる。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "分母に x を含む分数式で表される関数。",
    "definition_en": "A function that is a ratio of two polynomials.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A rational function has a vertical asymptote where the denominator is zero, unless that factor cancels.",
        "ja": "分数関数は、約分で消えない限り、分母が 0 になるところに垂直な漸近線をもつ。",
        "register": "spoken"
      },
      {
        "en": "Find the horizontal asymptote of the rational function f(x) = (2x + 1)/(x − 3).",
        "ja": "分数関数 f(x) = (2x + 1)/(x − 3) の水平な漸近線を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "radical-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "有理関数",
        "en": "Rational function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 極限: 簡単な分数関数と無理関数のグラフ（平行移動）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rational function": 84
      },
      "written": {
        "rational function": 187
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rational-number",
    "ja": {
      "term": "有理数",
      "reading": "ゆうりすう",
      "alt": []
    },
    "en": {
      "term": "rational number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "整数 m と 0 でない整数 n を使って、分数 m/n の形に表せる数。小数で表すと、有限小数か循環小数になる。",
    "definition_en": "Any number expressible as a quotient m/n of two integers with n ≠ 0. Its decimal expansion either terminates or repeats.",
    "latex": "\\frac{m}{n} \\quad (m, n \\text{ integers},\\ n \\neq 0)",
    "spoken_en": "m over n, where m and n are integers and n is not zero",
    "examples": [
      {
        "en": "Negative 0.75 is a rational number, because you can write it as negative 3 over 4.",
        "ja": "−0.75 は有理数。−3/4 と分数で書けるからね。",
        "register": "spoken"
      },
      {
        "en": "Write 0.125 as a fraction to show that it is a rational number.",
        "ja": "0.125 を分数で表して、有理数であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "整数も有理数（5 = 5/1）。√ が付いていても、√16 = 4 のように整数や分数になるものは有理数。",
      "有限小数（エントリ terminating-decimal）と循環小数（エントリ repeating-decimal）は有理数。循環しない無限小数は無理数（エントリ irrational-number）。"
    ],
    "related": [
      "irrational-number",
      "real-number",
      "fraction",
      "terminating-decimal",
      "repeating-decimal"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "有理数",
        "en": "Rational number"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rational number": 51
      },
      "written": {
        "rational number": 132
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rational-root-theorem",
    "ja": {
      "term": "有理根定理",
      "reading": "ゆうりこんていり",
      "alt": []
    },
    "en": {
      "term": "rational zero theorem",
      "alt": [
        "rational root theorem"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3",
        "Precalculus"
      ]
    },
    "definition_ja": "整数係数の方程式 aₙ xⁿ + … + a₀ = 0 の有理数解は、±（a₀ の約数）/（aₙ の約数）の形に限られるという定理。",
    "definition_en": "The theorem that any rational root of a polynomial equation with integer coefficients has the form ±(factor of the constant term)/(factor of the leading coefficient).",
    "latex": "x = \\pm\\frac{p}{q}\\quad(p \\mid a_0,\\ q \\mid a_n)",
    "spoken_en": null,
    "examples": [
      {
        "en": "The rational root theorem gives you a short list of candidates: plus or minus 1, 2, 3, 6.",
        "ja": "有理根定理で候補が絞れる。±1, ±2, ±3, ±6。",
        "register": "spoken"
      },
      {
        "en": "By the rational zero theorem, what are the possible rational zeros of f(x) = 2x³ + x² − 7x − 6?",
        "ja": "有理根定理によると、f(x) = 2x³ + x² − 7x − 6 の有理数解の候補は何か。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は rational zero theorem と書く。話し言葉では rational root theorem がわずかに出てくるだけ。"
    ],
    "related": [
      "factor-theorem",
      "polynomial-equation-of-higher-degree",
      "zeros-of-a-polynomial",
      "descartes-rule-of-signs"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "有理根定理",
        "en": "Rational root theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "5.5 Zeros of Polynomial Functions（Rational Zero Theorem）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "3.6 Zeros of Polynomial Functions（Rational Zero Theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "rational root theorem": 2
      },
      "written": {
        "rational root theorem": 1,
        "rational zero theorem": 22
      },
      "sources": [
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rationalizing-the-denominator",
    "ja": {
      "term": "分母の有理化",
      "reading": "ぶんぼのゆうりか",
      "alt": [
        "有理化する"
      ]
    },
    "en": {
      "term": "rationalizing the denominator",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "分母に根号をふくむ分数の分母と分子に同じ数を掛けて、分母に根号をふくまない形に直すこと。1/√2 = √2/2 など。",
    "definition_en": "Rewriting a fraction so that no radical is left in the denominator, done by multiplying top and bottom by a suitable factor. For example, 1/√2 = √2/2.",
    "latex": "\\frac{1}{\\sqrt{2}} = \\frac{\\sqrt{2}}{2}",
    "spoken_en": "one over the square root of two equals the square root of two over two",
    "examples": [
      {
        "en": "Rationalizing the denominator just means getting the radical out of the bottom, so multiply the top and the bottom by √3.",
        "ja": "分母の有理化というのは、分母から根号をなくすこと。だから分母と分子に √3 を掛けよう。",
        "register": "spoken"
      },
      {
        "en": "Simplify 6/√3 by rationalizing the denominator.",
        "ja": "6/√3 の分母を有理化せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "問題文は動詞句の命令形 Rationalize the denominator.（分母を有理化せよ）の形で出る（OpenStax Elementary Algebra）。名詞句 rationalizing the denominator は手順を説明するときの言い方。",
      "根号の中の数を小さくする操作（simplify the radical、エントリ simplify-radicals）とは別の操作。6/√3 = 6√3/3 = 2√3 のように、有理化したあとで約分できることがある。",
      "分母が √3 + √2 のような 2 項の和のときは、√3 − √2 を分母と分子に掛けて有理化する（高等学校学習指導要領解説は数学I で「分母が二項程度までの分数の形に表された数の分母の有理化」を挙げる）。"
    ],
    "related": [
      "simplify-radicals",
      "square-root-of-a-number",
      "denominator",
      "irrational-number",
      "radical-expression"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 数と式: 分母が二項程度までの分数の形に表された数の分母の有理化"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rationalizing the denominator": 13
      },
      "written": {
        "rationalizing the denominator": 36
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "ray",
    "ja": {
      "term": "半直線",
      "reading": "はんちょくせん",
      "alt": []
    },
    "en": {
      "term": "ray",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直線の一部で、1 つの端点から一方の向きにだけ限りなくのびているもの。",
    "definition_en": "Part of a line that starts at one endpoint and goes on forever in one direction.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Ray AB starts at A, goes through B, and keeps going forever.",
        "ja": "半直線 AB は A から始まって B を通り、そのままずっとのびていく。",
        "register": "spoken"
      },
      {
        "en": "Name the two rays that form angle PQR.",
        "ja": "∠PQR をつくる 2 本の半直線を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ray AB と ray BA は別の半直線。名前の最初の文字が端点で、ray AB は A から B の向きに、ray BA は B から A の向きにのびる。",
      "端点の数で呼び分ける。端点が 2 つなら line segment（エントリ segment）、1 つなら ray、端点がなく両方向にのびるなら line（エントリ line）。",
      "角の 2 辺は 2 本の ray。∠PQR の辺は ray QP と ray QR で、どちらも頂点 Q が端点。"
    ],
    "related": [
      "line",
      "segment",
      "endpoint",
      "angle",
      "angle-bisector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.8 Identification of Angles by Vertex and Ray"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "ray": 164
      },
      "written": {
        "ray": 71
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "read-off",
    "ja": {
      "term": "読み取る",
      "reading": "よみとる",
      "alt": []
    },
    "en": {
      "term": "read off",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語の「読み取る」は、値を拾うことにも、グラフから傾向をつかむことにも使う。英語の read off は、式・グラフ・表から値をそのまま拾うときに使う。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "グラフ・表・式から、必要な値や特徴を見つけ出す。",
    "definition_en": "To get a value or a feature directly from a graph, a table, or a formula.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "From this form you can read off the center and the radius right away.",
        "ja": "この形からは、中心と半径がすぐに読み取れる。",
        "register": "spoken"
      },
      {
        "en": "Read off the number of students who scored between 60 and 70 from the histogram.",
        "ja": "ヒストグラムから、60 点以上 70 点未満の生徒の人数を読み取れ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "histogram"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "read off": 44
      },
      "written": {
        "read off": 8
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "real-axis",
    "ja": {
      "term": "実軸",
      "reading": "じつじく",
      "alt": []
    },
    "en": {
      "term": "real axis",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "複素数平面の横軸。実数を表す点が並ぶ。",
    "definition_en": "The horizontal axis of the complex plane, on which the real numbers lie.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Reflecting across the real axis gives you the conjugate.",
        "ja": "実軸について折り返すと共役複素数になる。",
        "register": "spoken"
      },
      {
        "en": "Show that the points z and z̄ are symmetric with respect to the real axis.",
        "ja": "点 z と z̄ が実軸について対称であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "imaginary-axis",
      "complex-plane",
      "complex-conjugate"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "実数直線",
        "en": "Real line"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "real axis": 20
      },
      "written": {
        "real axis": 11
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "real-number",
    "ja": {
      "term": "実数",
      "reading": "じっすう",
      "alt": []
    },
    "en": {
      "term": "real number",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "有理数と無理数をあわせた数。数直線上の点と 1 対 1 に対応する。",
    "definition_en": "A rational or an irrational number: any number that has a place on the number line. The real numbers match up one to one with the points of the number line.",
    "latex": "x \\in \\mathbb{R}",
    "spoken_en": "x is in R, or x is a real number",
    "examples": [
      {
        "en": "For any real number x, x squared can never be negative.",
        "ja": "どんな実数 x についても、x² は負にならない。",
        "register": "spoken"
      },
      {
        "en": "Find all real numbers x such that x² − 2x − 1 = 0.",
        "ja": "x² − 2x − 1 = 0 を満たす実数 x をすべて求めよ。",
        "register": "written"
      },
      {
        "en": "The domain of f(x) = x³ − 4x is all real numbers.",
        "ja": "f(x) = x³ − 4x の定義域は実数全体である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the set of real numbers",
        "ja": "実数全体の集合"
      }
    ],
    "pitfalls": [
      "方程式の「実数解」は real solution（real root）。real number は数そのもの、real solution は解について言う。",
      "実数全体の集合は ℝ（太字や黒板太字の R）で表す。"
    ],
    "related": [
      "number-line",
      "repeating-decimal",
      "complex-number",
      "imaginary-number",
      "rational-number",
      "irrational-number",
      "natural-number"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "実数",
        "en": "Real number"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "real number": 335
      },
      "written": {
        "real number": 1118
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "real-part",
    "ja": {
      "term": "実部",
      "reading": "じつぶ",
      "alt": []
    },
    "en": {
      "term": "real part",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数II",
        "数C"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "複素数 a + bi の a。",
    "definition_en": "The number a in the complex number a + bi.",
    "latex": "\\operatorname{Re}(a + bi) = a",
    "spoken_en": null,
    "examples": [
      {
        "en": "The real part is 3 and the imaginary part is negative 2.",
        "ja": "実部は 3、虚部は −2。",
        "register": "spoken"
      },
      {
        "en": "Find the real number k such that the real part of (1 + ki)(2 − i) is 0.",
        "ja": "(1 + ki)(2 − i) の実部が 0 となるような実数 k を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉は大半が MIT OCW 18.03（微分方程式の複素数解）。",
      "記号 Re z（Re(z)）は the real part of z と読む。"
    ],
    "related": [
      "imaginary-part",
      "complex-number",
      "equality-of-complex-numbers"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和7年度 大学入学共通テスト 本試験 数学Ⅱ，数学Ｂ，数学Ｃ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=771&f=abm00005945.pdf&n=2025_op_30_sugaku2BC.pdf",
        "note": "第7問: 純虚数（実部が 0 である虚数）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "real part": 244
      },
      "written": {
        "real part": 51
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "real-solution",
    "ja": {
      "term": "実数解",
      "reading": "じっすうかい",
      "alt": []
    },
    "en": {
      "term": "real solution",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "real solution",
          "register": "both",
          "note": "話し言葉・書き言葉とも多い。話し言葉は MIT OCW が半分ほど、書き言葉は OpenStax Elementary Algebra・Intermediate Algebra が中心。"
        },
        {
          "term": "real root",
          "register": "both",
          "note": "書き言葉では OpenStax Calculus に多い。話し言葉・書き言葉とも real solution の半分ほど。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "方程式の解のうち、実数であるもの。二次方程式 ax² + bx + c = 0 は判別式 b² − 4ac ≧ 0 のとき実数解をもつ。",
    "definition_en": "A solution of an equation that is a real number; the quadratic equation ax² + bx + c = 0 has real solutions exactly when b² − 4ac ≥ 0.",
    "latex": "b^2 - 4ac \\geq 0",
    "spoken_en": "b squared minus four a c is greater than or equal to zero",
    "examples": [
      {
        "en": "The discriminant is negative, so there are no real solutions.",
        "ja": "判別式が負なので、実数解はない。",
        "register": "spoken"
      },
      {
        "en": "If b² − 4ac is positive, you get two distinct real roots.",
        "ja": "b² − 4ac が正なら、異なる 2 つの実数解をもつ。",
        "register": "spoken"
      },
      {
        "en": "Find the range of k for which x² + 2x + k = 0 has two distinct real solutions.",
        "ja": "x² + 2x + k = 0 が異なる 2 つの実数解をもつような k の値の範囲を求めよ。",
        "register": "written"
      },
      {
        "en": "Show that the equation x² − x + 1 = 0 has no real roots.",
        "ja": "方程式 x² − x + 1 = 0 は実数解をもたないことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「実数解をもたない」は has no real solutions（has no real roots）。複素数の解まで考える場面で real を落として has no solutions と言うと、解がまったく無い意味になる。",
      "「異なる 2 つの実数解」は two distinct real solutions。distinct を落とすと重解の場合と区別できない。",
      "英語では方程式の解を solution、多項式の根を root と使い分け、実数解にも real solution と real root の 2 つが当たる。関数の零点として OpenStax Algebra and Trigonometry・Precalculus は real zero と書く（各 30 件前後）。"
    ],
    "related": [
      "discriminant",
      "imaginary-solution",
      "number-of-real-solutions",
      "quadratic-formula",
      "solution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "real root": 15,
        "real solution": 34
      },
      "written": {
        "real solution": 73,
        "real root": 36
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rearranging-an-equation",
    "ja": {
      "term": "等式の変形",
      "reading": "とうしきのへんけい",
      "alt": [
        "公式を変形する"
      ]
    },
    "en": {
      "term": "solve the formula for",
      "alt": [
        "rearrange the equation"
      ],
      "variants": [
        {
          "term": "solve for a variable",
          "register": "spoken",
          "note": "話し言葉ではこちらが多い（大半が Professor Leonard）。CK-12 Algebra の節の名前（Solving for a Variable）。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語では、V = Bh を B について解くことを solve the formula for B と言うように、「〜について解く」（エントリ solve-for）の形で言う（話し言葉では solve for a variable とも）。文字だけでできた等式は literal equation と呼ぶ（OpenStax Intermediate Algebra ほか）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "等式の性質を使って、等式を「ある文字 = 〜」の形に変えること。たとえば V = Bh を B について解くと B = V/h となる。",
    "definition_en": "Using the properties of equality to rewrite an equation or formula so that a chosen variable is alone on one side. For example, solving V = Bh for B gives B = V/h.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To solve for a variable in a formula, treat the other letters like numbers: V = Bh, so divide both sides by h, and B = V over h.",
        "ja": "公式をある文字について解くときは、ほかの文字を数と同じように扱う。V = Bh なら両辺を h で割って、B = V/h。",
        "register": "spoken"
      },
      {
        "en": "Solve the formula A = ½bh for h.",
        "ja": "等式 A = ½bh を h について解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「B について解く」は solve for B で、前置詞 for を落とさない。",
      "英語の問題文は Solve A = ½bh for h. のように、解く文字が文の最後に来る。日本語の「h について解け」とは語順が逆。"
    ],
    "related": [
      "solve-for",
      "properties-of-equality",
      "both-sides",
      "equation",
      "variable"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "literal equation（solving for a variable in literal equations）"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "2.4.8 Solving for a Variable"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "solve for a variable | solve for the variable": 9,
        "rearrange the equation | rearrange the formula | rearrange an equation | rearrange a formula": 1
      },
      "written": {
        "solve the formula for | solve a formula for | solve each formula for": 38,
        "solve for a variable | solve for the variable": 9,
        "rearrange the equation | rearrange the formula | rearrange an equation | rearrange a formula": 2
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "reciprocal",
    "ja": {
      "term": "逆数",
      "reading": "ぎゃくすう",
      "alt": []
    },
    "en": {
      "term": "reciprocal",
      "alt": [
        "multiplicative inverse"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある数に掛けると積が 1 になる数。0 でない数 a の逆数は 1/a で、0 には逆数がない。",
    "definition_en": "The number you multiply a given number by to get 1; the reciprocal of a nonzero number a is 1/a. Zero has no reciprocal.",
    "latex": "a \\times \\frac{1}{a} = 1 \\quad (a \\neq 0)",
    "spoken_en": "a times one over a equals one, where a is not zero",
    "examples": [
      {
        "en": "The reciprocal of negative 2/3 is negative 3/2: you flip the fraction, but the sign stays the same.",
        "ja": "−2/3 の逆数は −3/2。分数はひっくり返すけど、符号はそのまま。",
        "register": "spoken"
      },
      {
        "en": "Find the reciprocal of each number: −6, 2/5, 0.25.",
        "ja": "次の数の逆数を求めよ。−6、2/5、0.25",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "逆数（reciprocal）と、符号を変えた数（opposite、反数）を取り違えない。−2/3 の reciprocal は −3/2、opposite は 2/3（エントリ opposite）。",
      "0 には逆数がない（0 has no reciprocal）。0 に何を掛けても 1 にならないため。",
      "multiplicative inverse は reciprocal と同じ意味。inverse だけだと逆関数（inverse function）や命題の裏（エントリ inverse）など別の意味になる。"
    ],
    "related": [
      "opposite",
      "division",
      "divide",
      "fraction",
      "negative-exponent"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "逆数",
        "en": "Multiplicative inverse"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "reciprocal": 255,
        "multiplicative inverse": 35
      },
      "written": {
        "reciprocal": 328,
        "multiplicative inverse": 86
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rectangle",
    "ja": {
      "term": "長方形",
      "reading": "ちょうほうけい",
      "alt": []
    },
    "en": {
      "term": "rectangle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "4 つの角がすべて直角の四角形。平行四辺形の特別な場合で、2 本の対角線の長さが等しい。",
    "definition_en": "A quadrilateral with four right angles. It is a special kind of parallelogram, and its two diagonals are the same length.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A square is a rectangle too, since all four of its angles are still right angles.",
        "ja": "正方形も長方形なんだよ。4 つの角がやっぱり全部直角だからね。",
        "register": "spoken"
      },
      {
        "en": "In rectangle ABCD, prove that AC = BD.",
        "ja": "長方形 ABCD で、AC = BD であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "正方形は長方形の特別な場合（エントリ square-shape）。4 つの角がすべて直角なので、「正方形は長方形か」の答えは yes。",
      "長方形は平行四辺形の特別な場合（エントリ parallelogram）。対辺・対角が等しい、対角線がそれぞれの中点で交わるという平行四辺形の性質をすべてもち、さらに対角線の長さが等しい。",
      "rectangle は平面図形の名前。立体の直方体は rectangular prism（エントリ rectangular-prism）。"
    ],
    "related": [
      "parallelogram",
      "square-shape",
      "rhombus",
      "quadrilateral",
      "rectangular-prism"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "長方形",
        "en": "Rectangle"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rectangle": 796
      },
      "written": {
        "rectangle": 631
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rectangular-coordinates",
    "ja": {
      "term": "直交座標",
      "reading": "ちょっこうざひょう",
      "alt": []
    },
    "en": {
      "term": "rectangular coordinates",
      "alt": [
        "Cartesian coordinates"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "AP Calculus BC",
        "Calculus II",
        "Calculus III"
      ]
    },
    "definition_ja": "互いに直交する座標軸を使って、点を (x, y) や (x, y, z) で表す座標。極座標と区別するときの呼び方。",
    "definition_en": "Coordinates (x, y) or (x, y, z) measured along perpendicular axes; the name used to contrast them with polar coordinates.",
    "latex": "(r, \\theta) \\mapsto (r\\cos\\theta, r\\sin\\theta)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's convert back to rectangular coordinates to see what curve this is.",
        "ja": "直交座標に戻して、どんな曲線か見てみよう。",
        "register": "spoken"
      },
      {
        "en": "Convert the polar equation r = 4 cos θ to rectangular coordinates.",
        "ja": "極方程式 r = 4 cos θ を直交座標の方程式に直せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "convert to rectangular coordinates",
        "ja": "直交座標に直す"
      }
    ],
    "pitfalls": [
      "Cartesian coordinates とも言うが少ない（話し言葉・書き言葉とも rectangular coordinates のほうがずっと多い）。"
    ],
    "related": [
      "polar-coordinates",
      "polar-equation",
      "coordinates-in-space"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 9 の概要、Topic 9.8（rectangular coordinates）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "rectangular coordinates": 48,
        "Cartesian coordinates": 11
      },
      "written": {
        "rectangular coordinates": 265,
        "Cartesian coordinates": 40
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rectangular-prism",
    "ja": {
      "term": "直方体",
      "reading": "ちょくほうたい",
      "alt": []
    },
    "en": {
      "term": "rectangular prism",
      "alt": [],
      "variants": [
        {
          "term": "rectangular box",
          "register": "written",
          "note": "教科書（OpenStax Calculus）の書き方。中学・Geometry の教材（IM・CK-12）は rectangular prism と呼ぶ。"
        }
      ],
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "6 つの面がすべて長方形の角柱。向かい合う面どうしは合同で平行である。",
    "definition_en": "A prism whose six faces are all rectangles. Each pair of opposite faces is congruent and parallel.",
    "latex": "V = lwh",
    "spoken_en": "V equals l times w times h",
    "examples": [
      {
        "en": "A cereal box is a rectangular prism, so its volume is just length times width times height.",
        "ja": "シリアルの箱は直方体だから、体積は縦 × 横 × 高さで求められるよ。",
        "register": "spoken"
      },
      {
        "en": "A rectangular prism is 8 cm long, 5 cm wide, and 3 cm tall. Find its volume and its surface area.",
        "ja": "縦 8 cm、横 5 cm、高さ 3 cm の直方体の体積と表面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "立方体（cube、エントリ cube-solid）は、6 つの面がすべて正方形の特別な直方体。cube は数の 3 乗（エントリ cube）と同じ語。",
      "V = lwh の l、w、h は length、width、height の頭文字で、日本語の「縦 × 横 × 高さ」と同じ計算。",
      "中学・Geometry の教材（IM・CK-12）も rectangular prism と呼ぶ。rectangular box は教科書（OpenStax Calculus）の書き方。"
    ],
    "related": [
      "prism",
      "cube-solid",
      "volume",
      "surface-area",
      "cubic-units"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "直方体",
        "en": "Rectangular cuboid"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "rectangular prism"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "rectangular prism"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rectangular box": 1,
        "rectangular prism": 24
      },
      "written": {
        "rectangular box": 26,
        "rectangular prism": 3
      },
      "sources": [
        "khan-middle",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rectilinear-motion",
    "ja": {
      "term": "直線運動",
      "reading": "ちょくせんうんどう",
      "alt": []
    },
    "en": {
      "term": "straight-line motion",
      "alt": [
        "rectilinear motion",
        "motion along a line"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "直線（数直線）上を動く点の運動。位置・速度・加速度が微分で結びつく。",
    "definition_en": "Motion of a point along a straight line, in which position, velocity, and acceleration are related by derivatives.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For straight-line motion, the particle moves to the right when the velocity is positive.",
        "ja": "直線運動では、速度が正のとき点は右に動く。",
        "register": "spoken"
      },
      {
        "en": "A particle moves along the x-axis with position x(t) = t³ − 6t² + 9t. When is it moving to the left?",
        "ja": "x 軸上を動く点の位置が x(t) = t³ − 6t² + 9t のとき、点が左に動くのはいつか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は topic 4.2 の名前で straight-line motion と言い、本文では rectilinear motion も使う。"
    ],
    "related": [
      "velocity",
      "acceleration",
      "instantaneous-velocity"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 4.2（straight-line motion ／ rectilinear motion）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "motion along a line": 3,
        "rectilinear motion": 3
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは CED の呼び方 straight-line motion（topic 4.2）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "recurrence-relation",
    "ja": {
      "term": "漸化式",
      "reading": "ぜんかしき",
      "alt": []
    },
    "en": {
      "term": "recursive formula",
      "alt": [],
      "variants": [
        {
          "term": "recursive formula",
          "register": "written",
          "note": "高校レベル（OpenStax Algebra and Trigonometry）の言い方で、書き言葉はほとんどがこの本。話し言葉にはほとんど出てこない（大半が Professor Leonard）。"
        },
        {
          "term": "recurrence equation",
          "register": "written",
          "note": "MIT 6.042（離散数学）のテキストの言い方。話し言葉にはほとんど出てこず（MIT OCW だけ）、書き言葉はすべて MIT の講義ノート。"
        },
        {
          "term": "recurrence relation",
          "register": "written",
          "note": "大学（OpenStax Calculus・離散数学）の言い方。話し言葉には出てこず、書き言葉は大半が OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語は本によって呼び方が割れる。高校レベルの教科書（OpenStax Algebra and Trigonometry）は recursive formula、大学初年次の教科書（OpenStax Calculus Volume 2）は recurrence relation と呼ぶ。学習指導要領解説（数学B）は初項と漸化式を分けて書き、漸化式は項の間の関係式だけを指す。OpenStax Algebra and Trigonometry（13.1）の recursive formula は初項と式の 2 つを合わせたもので、Levin（4.1）は式だけを recurrence relation、初項と合わせたものを recursive definition（recursive-definition）と呼ぶ。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Discrete Math",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "数列の項を、その前の項を使って表す式。初項と合わせると数列が決まる。",
    "definition_en": "An equation that gives each term of a sequence from the terms before it; with the first term, it determines the sequence.",
    "latex": "a_{n+1} = 2a_n + 1, \\quad a_1 = 1",
    "spoken_en": "a sub n plus one equals two a sub n plus one, with a sub one equal to one",
    "examples": [
      {
        "en": "A recursive formula tells you how to get the next term from the one before.",
        "ja": "漸化式は、前の項から次の項を作る方法を教えてくれる。",
        "register": "spoken"
      },
      {
        "en": "Write a recursive formula for the sequence 2, 6, 18, 54, ….",
        "ja": "数列 2, 6, 18, 54, … を表す漸化式を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a_{n+1} = pa_n + q を α = pα + q で解く方法は characteristic-equation を参照。"
    ],
    "related": [
      "recursive-definition",
      "solve-the-recurrence",
      "set-up-a-recurrence",
      "characteristic-equation",
      "system-of-recurrences",
      "linear-recurrence-relation",
      "fibonacci-sequence",
      "sequence-of-differences",
      "find-the-nth-term"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "漸化式",
        "en": "Recurrence relation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.1 Sequences and Their Notations（recursive formula）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences（recurrence relation）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B (1) 数列（漸化式）と課題学習の例（初項 a₁ と漸化式で定義される数列）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.1 Describing Sequences（Definition 4.1.4: recurrence relation と recursive definition）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "recurrence equation": 1,
        "recursive formula": 4
      },
      "written": {
        "recurrence relation": 12,
        "recursive formula": 74,
        "recurrence equation": 14
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "recursive-algorithm",
    "ja": {
      "term": "再帰アルゴリズム",
      "reading": "さいきあるごりずむ",
      "alt": [
        "再帰"
      ]
    },
    "en": {
      "term": "recursion",
      "alt": [
        "recursive algorithm"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "問題を同じ形のより小さな問題に分け、自分自身を呼び出して解く手順（再帰）。いちばん小さな場合（基底）は直接答えを返す。",
    "definition_en": "A procedure that solves a problem by calling itself on smaller instances of the same problem. The smallest cases, called base cases, are answered directly.",
    "latex": "\\gcd(a, b) = \\gcd(b,\\ a \\bmod b)",
    "spoken_en": "the gcd of a and b equals the gcd of b and a mod b",
    "examples": [
      {
        "en": "The idea of recursion is that the function calls itself on a smaller input, so you always need a base case or it never stops.",
        "ja": "再帰の考え方は、関数がより小さな入力で自分自身を呼び出すこと。だから基底の場合が必ず要る。無いと止まらない。",
        "register": "spoken"
      },
      {
        "en": "Write a recursive algorithm that computes gcd(a, b), and use it to find gcd(252, 198).",
        "ja": "gcd(a, b) を計算する再帰アルゴリズムを書き、それを用いて gcd(252, 198) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "recursion は「再帰」という考え方全体、recursive algorithm はその考え方で書いた手順。Levin は数列の節（4.1 Describing Sequences・4.4 Exponential Sequences）でも recursion を使う（各 1 件）。",
      "日本語の「帰納的」も「再帰的」も英語では recursive になることが多い（帰納的定義 = recursive definition）。",
      "用例コーパスで recursion は話し言葉のほうが多く（ほとんどが MIT OCW）、書き言葉には少ない。recursive algorithm は話し言葉・書き言葉とも少ない（すべて MIT）。"
    ],
    "related": [
      "recursive-definition",
      "recurrence-relation",
      "structural-induction",
      "euclidean-algorithm",
      "base-case"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "recursion": 13,
        "recursive algorithm": 2
      },
      "written": {
        "recursion": 3,
        "recursive algorithm": 4
      },
      "sources": [
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "recursive-definition",
    "ja": {
      "term": "帰納的定義",
      "reading": "きのうてきていぎ",
      "alt": [
        "再帰的定義"
      ]
    },
    "en": {
      "term": "recursive definition",
      "alt": [
        "recursively defined"
      ],
      "variants": [
        {
          "term": "defined recursively",
          "register": "written",
          "note": "受け身の形（the sequence is defined recursively by …）で、書き言葉で最も多い。話し言葉にはほとんど出てこない。書き言葉は MIT の講義ノートが半分以上で、ほかは OpenStax Calculus・Algebra and Trigonometry。"
        },
        {
          "term": "recursive definition",
          "register": "written",
          "note": "名詞の形。書き言葉はすべて MIT 6.042 のテキスト。話し言葉では少なく、大半が Khan Academy。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "discrete"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Discrete Math",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "初項（初めのいくつかの項）と漸化式によって数列を定めること。",
    "definition_en": "Defining a sequence by giving its first term (or first few terms) and a rule that gives each later term from earlier ones.",
    "latex": "a_1 = 1, \\quad a_{n+1} = a_n + 2n + 1",
    "spoken_en": null,
    "examples": [
      {
        "en": "It's defined recursively, so you can't jump straight to the 100th term.",
        "ja": "帰納的に定義されているから、いきなり第 100 項にはいけない。",
        "register": "spoken"
      },
      {
        "en": "The sequence {aₙ} is defined recursively by a₁ = 2 and aₙ₊₁ = 3aₙ − 1. Find a₄.",
        "ja": "数列 {aₙ} は a₁ = 2、aₙ₊₁ = 3aₙ − 1 で帰納的に定義されている。a₄ を求めよ。",
        "register": "written"
      },
      {
        "en": "Give a recursive definition of the sequence 2, 5, 8, 11, …",
        "ja": "数列 2, 5, 8, 11, … の帰納的定義を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の recursive は、情報科学の「再帰」と同じ語。",
      "帰納的定義の「帰納的」は英語では recursive。Levin（4.1）は inductive definition という別名も挙げる。"
    ],
    "related": [
      "recurrence-relation",
      "sequence",
      "proof-by-induction",
      "set-up-a-recurrence",
      "recursive-algorithm"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "再帰的定義",
        "en": "Recursive definition"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences（defined recursively）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 1.5 Sequences are Functions〜1.9（recursive definition）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.1 Describing Sequences（Definition 4.1.4 Recursive definition、inductive definition）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "recursive definition": 7,
        "defined recursively": 3
      },
      "written": {
        "recursively defined": 6,
        "defined recursively": 20,
        "recursive definition": 15
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "reduce",
    "ja": {
      "term": "約分する",
      "reading": "やくぶんする",
      "alt": [
        "約分"
      ]
    },
    "en": {
      "term": "simplify the fraction",
      "alt": [
        "cancel the common factor",
        "reduce the fraction"
      ],
      "uk": null,
      "register": "written",
      "variants": [
        {
          "term": "simplify the fraction",
          "register": "written",
          "note": "OpenStax Prealgebra・Elementary Algebra・Intermediate Algebra の言い方。話し言葉では少ない。"
        },
        {
          "term": "remove common factors",
          "register": "written",
          "note": "同じ OpenStax の 3 冊が、約分の手順（simplify by removing common factors）として書く。話し言葉には出てこない。"
        }
      ]
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "「約分（する）」は、用例コーパスと参照では動詞句で言う（名詞にするときは CK-12 Algebra の reducing fractions）。書き言葉では simplify the fraction と、その手順を言う remove common factors が並ぶ（OpenStax Prealgebra・Elementary Algebra・Intermediate Algebra）。分数式では cancel the common factor(s) とも書く（OpenStax Calculus・Algebra and Trigonometry）。話し言葉では simplify the fraction・reduce the fraction は少なく、約分も cancel で言う（the x + 1's cancel）。結果の形は in lowest terms（既約）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "小学校",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "分数（分数式）の分子と分母を共通な因数で割って簡単にする。",
    "definition_en": "To simplify a fraction or rational expression by dividing the numerator and denominator by a common factor.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Factor the top and bottom, and then the x + 1's cancel.",
        "ja": "分子と分母を因数分解すると、x + 1 が約分できる。",
        "register": "spoken"
      },
      {
        "en": "Simplify (x² − 4)/(x² + x − 6) by canceling the common factor.",
        "ja": "(x² − 4)/(x² + x − 6) を約分して簡単にせよ。",
        "register": "written"
      },
      {
        "en": "Simplify the fraction 12 over 18: divide the top and the bottom by 6 and you get 2 over 3.",
        "ja": "12/18 を約分しよう。分子と分母を 6 で割って 2/3。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "約分しきった形は in lowest terms（既約）。",
      "cancel だけでは「打ち消し合う」（x − x が消える）意味も多い。約分の意味をはっきりさせるときは cancel the common factor。",
      "約分できる因数は掛け算の因数だけ。(x + 2)/2 の 2 を消すのは誤り（You can only cancel factors, not terms.）。",
      "OpenStax Prealgebra・Elementary Algebra は、約分の根拠 a/b = (a·c)/(b·c) を Equivalent Fractions Property と呼ぶ。"
    ],
    "related": [
      "rational-expression",
      "find-a-common-denominator",
      "numerator",
      "denominator",
      "cancel-out",
      "simplify",
      "fraction"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Multiply and Divide Fractions（removing common factors、Equivalent Fractions Property）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "Add and Subtract Rational Expressions with a Common Denominator（removing common factors）"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "1.6.1 Real Number Properties and Axioms（reducing fractions）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "simplify the fraction": 6,
        "reduce the fraction": 2
      },
      "written": {
        "cancel the common factor": 8,
        "reduce the fraction": 4,
        "simplify the fraction": 73,
        "remove common factors": 70
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "reduced-row-echelon-form",
    "ja": {
      "term": "既約行階段形",
      "reading": "きやくぎょうかいだんけい",
      "alt": [
        "簡約階段形"
      ]
    },
    "en": {
      "term": "reduced row-echelon form",
      "alt": [
        "RREF"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "行階段形のうち、各行の先頭の 0 でない成分（ピボット）が 1 で、その列のほかの成分がすべて 0 である形。行列ごとにただ 1 つに決まる。",
    "definition_en": "A row-echelon form in which every leading entry is 1 and is the only nonzero entry in its column; each matrix has exactly one.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Keep going until you get the reduced row-echelon form, with zeros above the pivots too.",
        "ja": "ピボットの上も 0 にして、既約行階段形になるまで続けよう。",
        "register": "spoken"
      },
      {
        "en": "Transform the augmented matrix into reduced row-echelon form and read off the solution.",
        "ja": "拡大係数行列を既約行階段形に変形し、解を読み取れ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて MIT OCW 18.06。教科書では Nicholson が使う。",
      "日本語は教科書により簡約階段形・既約行階段形など呼び方が分かれる。"
    ],
    "related": [
      "row-echelon-form",
      "gaussian-elimination",
      "pivot",
      "augmented-matrix"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "1.2 Gaussian Elimination"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "reduced row-echelon form": 17,
        "RREF": 1
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "reference-angle",
    "ja": {
      "term": "基準角",
      "reading": "きじゅんかく",
      "alt": []
    },
    "en": {
      "term": "reference angle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「基準角」は学習指導要領解説に無い、本プロジェクトの訳語（「基準角」「参照角」は学習指導要領解説・共通テスト／センター試験・日本語版 Wikipedia に出てこない）。OpenStax Algebra and Trigonometry（7.1 Angles・7.3 Unit Circle）は、動径と x 軸がつくる鋭角を reference angle と呼び、三角関数の値を reference angle の値と象限の符号から求める（用例コーパスでは話し言葉・書き言葉ともよく出てくる）。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "角の動径と x 軸がつくる鋭角（0° < θ' < 90°）。三角関数の値は、基準角の値に象限で決まる符号をつけたものになる。",
    "definition_en": "The acute angle between the terminal side of an angle and the x-axis; a trigonometric value equals the value at the reference angle with the sign set by the quadrant.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The reference angle for 150 degrees is 30 degrees, and sine is positive in quadrant II.",
        "ja": "150° の基準角は 30° で、第 2 象限では sin は正。",
        "register": "spoken"
      },
      {
        "en": "Find the reference angle for 5π/4, and use it to find cos(5π/4).",
        "ja": "5π/4 の基準角を求め、それを使って cos(5π/4) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉はほとんどが The Organic Chemistry Tutor、書き言葉はほとんどが OpenStax Algebra and Trigonometry。"
    ],
    "related": [
      "terminal-side",
      "quadrantal-angle",
      "trigonometric-function",
      "unit-circle",
      "coterminal-angle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles・7.3 Unit Circle（reference angle）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「基準角」「参照角」は出てこない"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "「基準角」「参照角」は 0 件"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "reference angle": 76
      },
      "written": {
        "reference angle": 84
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "reflect",
    "ja": {
      "term": "折り返す",
      "reading": "おりかえす",
      "alt": []
    },
    "en": {
      "term": "reflect",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語の「折り返す」は、直線を折り目にして紙を折る動作で対称移動を言う。英語では鏡に映す動詞 reflect を使い、reflect the figure over line ℓ（across line ℓ とも）で「直線 ℓ について対称移動する」を表す。名詞は reflection（対称移動）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "図形を、1 本の直線を折り目として折り返し、その直線について対称な位置に移す。",
    "definition_en": "To flip a figure over a line so that each point lands the same distance away on the other side of the line, giving a mirror image.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When you reflect a point over the x-axis, x stays the same and y changes sign, so (2, 5) goes to (2, −5).",
        "ja": "x 軸を折り目にして点を折り返すと、x はそのままで y の符号が変わる。だから (2, 5) は (2, −5) に移る。",
        "register": "spoken"
      },
      {
        "en": "Reflect quadrilateral ABCD across line ℓ, and draw its image A′B′C′D′.",
        "ja": "四角形 ABCD を、直線 ℓ を折り目として折り返した四角形 A′B′C′D′ をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "前置詞は over か across（reflect the figure over line ℓ ／ across line ℓ）。日本語の「直線 ℓ について」「直線 ℓ を折り目として」に当たる。",
      "名詞は reflection（エントリ reflection）。折り目になる直線は line of reflection（エントリ line-of-reflection）。",
      "reflect は日常語で「（光を）反射する」「反映する」「よく考える」の意味でも使う。"
    ],
    "related": [
      "reflection",
      "line-of-reflection",
      "translate",
      "rotate",
      "symmetric"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 1.13 ほか"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "reflect over | reflect across | reflect … over | reflect … across | reflect it over | reflect it across | reflect about": 74
      },
      "written": {
        "reflect over | reflect across | reflect … over | reflect … across | reflect it over | reflect it across | reflect about": 42
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "reflection",
    "ja": {
      "term": "対称移動",
      "reading": "たいしょういどう",
      "alt": [
        "反射",
        "折り返し"
      ]
    },
    "en": {
      "term": "reflection",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数I",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "図形を直線（または点）に関して対称な位置に移すこと。座標平面では x 軸・y 軸・直線 y = x・原点に関する対称移動をよく扱う。",
    "definition_en": "Moving a figure to its mirror image across a line (or through a point); on the coordinate plane, usually across an axis, the line y = x, or the origin.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A reflection across the y-axis just changes x to negative x.",
        "ja": "y 軸に関する対称移動は、x を −x にするだけ。",
        "register": "spoken"
      },
      {
        "en": "Find the point that is the reflection of P(2, 5) across the line y = x + 1.",
        "ja": "直線 y = x + 1 に関して点 P(2, 5) と対称な点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "reflection across the x-axis",
        "ja": "x 軸に関する対称移動"
      }
    ],
    "pitfalls": [
      "「〜に関して対称な点」は the reflection of P across the line、または the point symmetric to P about the line。",
      "前置詞は across ／ over ／ about が使われる（用例コーパスでは OpenStax が about、Khan Academy が over を使う）。",
      "話し言葉では flip とも言う。"
    ],
    "related": [
      "symmetric",
      "rotation",
      "perpendicular-bisector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "reflection": 75
      },
      "written": {
        "reflection": 87
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "reflex-angle",
    "ja": {
      "term": "優角",
      "reading": "ゆうかく",
      "alt": []
    },
    "en": {
      "term": "reflex angle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）ではほぼ扱わない（reflex angle は CED・OpenStax・IM・CK-12 とも 0 件。CK-12 Geometry は 180° より大きい角を凹多角形の説明で 2 回言うだけで、名前は付けない）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": []
    },
    "definition_ja": "大きさが 180° より大きく 360° より小さい角。2 本の半直線がつくる 2 つの角のうち、大きいほうを指す。",
    "definition_en": "An angle that measures more than 180° but less than 360°. Of the two angles formed by two rays from the same point, it is the larger one.",
    "latex": "180^\\circ < \\theta < 360^\\circ",
    "spoken_en": "theta is greater than one hundred eighty degrees and less than three hundred sixty degrees",
    "examples": [
      {
        "en": "When I say angle AOB, I mean the smaller one. If you want the one that goes the long way around, that's the reflex angle.",
        "ja": "角 AOB と言ったら小さいほうのこと。遠回りのほうが欲しいなら、それは優角だ。",
        "register": "spoken"
      },
      {
        "en": "The measure of ∠AOB is 130°. Find the measure of the reflex angle AOB.",
        "ja": "∠AOB = 130° のとき、優角 AOB の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「劣角」（180° より小さいほう）に当たる決まった英語の名前は無い。区別したいときは the angle less than 180° のように説明する。",
      "用例コーパスにも reflex angle は出てこない。英語版 Wikipedia にも独立した記事は無く、Angle の記事の一部で扱われている。"
    ],
    "related": [
      "angle",
      "straight-angle",
      "major-arc"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。そのまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "reflexive-property",
    "ja": {
      "term": "反射律",
      "reading": "はんしゃりつ",
      "alt": []
    },
    "en": {
      "term": "reflexive property",
      "alt": [
        "reflexive property of congruence"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof",
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "どんな a についても a = a が成り立つという性質。図形では、どの線分や角も自分自身と合同であることを言う。",
    "definition_en": "The property that every quantity is equal to itself: a = a. In geometry it says that any segment or angle is congruent to itself.",
    "latex": "a = a",
    "spoken_en": "a equals a",
    "examples": [
      {
        "en": "These two triangles share side BD, so BD is congruent to BD by the reflexive property.",
        "ja": "この 2 つの三角形は辺 BD を共有しているので、反射律より BD ≅ BD。",
        "register": "spoken"
      },
      {
        "en": "In the figure, △ABD and △CBD share side BD. Give the reason for the statement BD ≅ BD.",
        "ja": "図で、△ABD と △CBD は辺 BD を共有している。BD = BD と言える理由を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "センター試験の問題文（平成 27 年度 数学Ⅰ・数学Ａ）は「∠C は共通」のように書き、反射律という名前は使わない。CK-12 Geometry の 2 列証明は、理由の列に Reflexive Property of Congruence と書く（5.8 Parallelogram Classification ほか）。"
    ],
    "related": [
      "transitive-property",
      "symmetric-property",
      "congruent",
      "two-column-proof"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.12 Constructions and Bisectors・4.12 Congruent Triangles・4.25 Comparing Angles and Sides in Triangles"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成27年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20160601/https://www.dnc.ac.jp/albums/abm.php?f=abm00004808.pdf&n=2701-0402+suugaku1A.pdf",
        "note": "「∠C は共通」"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "reflexive property は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "reflexive property": 5
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 reflexive property（CK-12 Geometry 1.12 Constructions and Bisectors・CK-12 Geometry 4.12 Congruent Triangles・CK-12 Geometry 4.25 Comparing Angles and Sides in Triangles・計 4 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "region",
    "ja": {
      "term": "領域",
      "reading": "りょういき",
      "alt": []
    },
    "en": {
      "term": "region",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Precalculus"
      ]
    },
    "definition_ja": "不等式を満たす点 (x, y) 全体の集合。座標平面上に図示すると、境界線で区切られた部分になる。",
    "definition_en": "All the points (x, y) that make an inequality true, taken as a set; on the coordinate plane it is a part of the plane cut off by a boundary.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Test the origin: if it satisfies the inequality, shade the region that contains it.",
        "ja": "原点で試して、不等式を満たせば原点を含む側の領域を塗る。",
        "register": "spoken"
      },
      {
        "en": "Shade the region represented by the system x² + y² ≤ 9, y ≥ x + 1.",
        "ja": "連立不等式 x² + y² ≦ 9, y ≧ x + 1 の表す領域を図示せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "shade the region",
        "ja": "領域を図示する"
      }
    ],
    "pitfalls": [
      "用例コーパスの region は話し言葉・書き言葉ともよく出てきて、その多くは微積分の「曲線で囲まれた部分」（the region bounded by …）。学習指導要領解説（数学II 微分・積分の考え）が「直線や関数のグラフで囲まれた図形の面積」と書くところも、英語は region。",
      "不等式・連立不等式の表す領域を、IM Algebra 1（2.21〜2.26）は solution region、OpenStax Algebra and Trigonometry（11.3）は feasible region と呼ぶ。"
    ],
    "related": [
      "boundary",
      "linear-programming",
      "sketch",
      "feasible-region",
      "system-of-linear-inequalities"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.3 Systems of Nonlinear Equations and Inequalities: Two Variables（feasible region）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 1 2.21〜2.26（solution region）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 微分・積分の考え: 直線や関数のグラフで囲まれた図形の面積"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "region": 960
      },
      "written": {
        "region": 1407
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "regression-line",
    "ja": {
      "term": "回帰直線",
      "reading": "かいきちょくせん",
      "alt": [
        "最小二乗回帰直線",
        "回帰式"
      ]
    },
    "en": {
      "term": "regression line",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "line of best fit",
          "register": "written",
          "note": "話し言葉にはほとんど出てこず、書き言葉は大半が OpenStax Introductory Statistics。"
        },
        {
          "term": "regression equation",
          "register": "written",
          "note": "話し言葉にはほとんど出てこず、書き言葉で使う（OpenStax Introductory Statistics・Algebra and Trigonometry）。"
        },
        {
          "term": "least-squares line",
          "register": "written",
          "note": "書き言葉だけで使う（すべて OpenStax Introductory Statistics）。"
        },
        {
          "term": "best-fit line",
          "register": "written",
          "note": "書き言葉だけで使う（大半が OpenStax Introductory Statistics）。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 1",
        "Precalculus"
      ]
    },
    "definition_ja": "散布図の点の近くを通るように、最小二乗法で決めた直線 ŷ = a + bx。一方の変量から他方を予測するのに使う。",
    "definition_en": "The line ŷ = a + bx fitted to a scatter plot by least squares, used to predict one variable from the other.",
    "latex": "\\hat{y} = a + bx",
    "spoken_en": "y hat equals a plus b x",
    "examples": [
      {
        "en": "The regression line predicts about 72 for someone who studied 5 hours.",
        "ja": "回帰直線によると、5 時間勉強した人は約 72 点と予測される。",
        "register": "spoken"
      },
      {
        "en": "Find the least-squares regression line for these data, then say in words what its slope means.",
        "ja": "このデータの最小二乗法による回帰直線を求め、傾きが何を表すかを言葉で説明せよ。",
        "register": "written"
      },
      {
        "en": "Plug 70 into the regression equation to predict the score; the predicted value is y-hat.",
        "ja": "得点を予測するには回帰式に 70 を代入する。予測値が y ハットだ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "書き言葉では regression line・line of best fit・regression equation・least-squares line・best-fit line が並ぶ。話し言葉は regression line（すべて Khan Academy）。最小二乗法で決めたことを示すときは least-squares regression line。",
      "ŷ は y hat と読む。OpenStax Introductory Statistics（12.3）は回帰直線を ŷ = a + bx と書き（16 件）、AP Statistics の CED（topic 5.5）も傾きを b、y 切片を a とする。中学校学習指導要領解説（第 2 学年 一次関数）の y = ax + b とは a と b の役割が逆になる。OpenStax Algebra and Trigonometry（4.3）の練習問題は、電卓の出力に合わせて y = ax + b（a が傾き）と書く。",
      "回帰直線で求めた値を予測値（predicted value）と言い、ŷ と書く。predicted value は書き言葉では大半が OpenStax Introductory Statistics。回帰直線を使ってデータの範囲外を予測するのは外挿（extrapolation）。",
      "IM Algebra 1（3.4〜3.10）と IM 6–8 Math（Grade 8 の 6.6）は line of best fit だけを使う（regression line は IM で 0 件）。"
    ],
    "related": [
      "least-squares",
      "scatter-plot",
      "correlation-coefficient",
      "extrapolation",
      "residual",
      "predict",
      "interpolation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "線形回帰",
        "en": "Linear regression"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 5.5（least-squares regression line、slope b、y-intercept a）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第2学年 一次関数: y=ax+b"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "4.3 Fitting Linear Models to Data（y = ax + b）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 1 3.4〜3.10（line of best fit）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 6.6 The Slope of a Fitted Line（line of best fit）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "regression line": 67,
        "regression equation": 2,
        "line of best fit": 3
      },
      "written": {
        "regression line": 69,
        "line of best fit": 55,
        "regression equation": 24,
        "best-fit line": 21,
        "least-squares line": 23
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "regular-polygon",
    "ja": {
      "term": "正多角形",
      "reading": "せいたかくけい",
      "alt": []
    },
    "en": {
      "term": "regular polygon",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "辺の長さがすべて等しく、角の大きさもすべて等しい多角形。正三角形・正方形・正六角形など。",
    "definition_en": "A polygon whose sides are all the same length and whose angles are all the same size.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A regular polygon needs equal sides and equal angles, so a rhombus that isn't a square doesn't count.",
        "ja": "正多角形は辺も角も全部等しくないといけないから、正方形でないひし形は正多角形じゃない。",
        "register": "spoken"
      },
      {
        "en": "Each interior angle of a regular polygon measures 140°. How many sides does the polygon have?",
        "ja": "ある正多角形の 1 つの内角の大きさは 140° である。この正多角形は正何角形か。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "正三角形は equilateral triangle、正方形は square という名前がある。正五角形・正六角形は regular pentagon ／ regular hexagon のように regular を付けて言う。",
      "辺だけ、または角だけが等しくても正多角形ではない。ひし形は辺がすべて等しいが角が等しいとは限らず、長方形は角がすべて等しいが辺が等しいとは限らない。",
      "regular は日常語の「ふつうの・いつもの」と同じ語だが、regular polygon の regular は「辺も角もすべて等しい」という意味。"
    ],
    "related": [
      "polygon",
      "area-of-a-regular-polygon",
      "apothem",
      "equiangular-triangle",
      "tessellation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正多角形",
        "en": "Regular polygon"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.20 Regular and Irregular Polygons・5.21 Area of Regular and Irregular Polygons・5.23 Construct Regular Polygons"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "regular polygon": 3
      },
      "written": {
        "regular polygon": 1
      },
      "sources": [
        "mit-notes",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CK-12 の呼び方 regular polygon（CK-12 Geometry 5.23 Construct Regular Polygons・CK-12 Geometry 5.21 Area of Regular and Irregular Polygons・CK-12 Geometry 5.20 Regular and Irregular Polygons・計 54 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "regular-polyhedron",
    "ja": {
      "term": "正多面体",
      "reading": "せいためんたい",
      "alt": []
    },
    "en": {
      "term": "Platonic solid",
      "alt": [
        "regular polyhedron"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語ではプラトン（Plato）にちなんで Platonic solid と呼ぶ（IM Geometry）。直訳の regular polyhedron も同じ 5 つの立体を指し、CK-12 Geometry は「正多面体は 5 つだけで、Platonic solids と呼ぶ」と書き、Levin は regular polyhedron (also known as a Platonic solid) と書く。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "どの面も合同な正多角形で、どの頂点にも同じ数の面が集まっている、へこみのない多面体。正四面体・正六面体（立方体）・正八面体・正十二面体・正二十面体の 5 種類しかない。",
    "definition_en": "A convex polyhedron whose faces are all congruent regular polygons, with the same number of faces meeting at every vertex. There are exactly five of them, with 4, 6, 8, 12, or 20 faces.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "There are only five Platonic solids, and the cube is one of them.",
        "ja": "正多面体は 5 種類しかなくて、立方体はそのうちの 1 つだよ。",
        "register": "spoken"
      },
      {
        "en": "A regular polyhedron has 20 faces, and each face is an equilateral triangle. How many edges does it have?",
        "ja": "面の数が 20 で、どの面も正三角形である正多面体がある。この正多面体の辺の数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "5 つの英語の名前は regular tetrahedron（正四面体）、cube（正六面体・立方体）、regular octahedron（正八面体）、regular dodecahedron（正十二面体）、regular icosahedron（正二十面体）。regular を付けない tetrahedron などは、面が正多角形でないものも含む。",
      "どの面も合同な正多角形というだけでは正多面体にならない。正三角形 6 つでできる双三角錐は、3 つの面が集まる頂点と 4 つの面が集まる頂点があるので、正多面体ではない。",
      "Platonic は哲学者プラトン（Plato）の名前から来ているので、大文字で書き始める。"
    ],
    "related": [
      "polyhedron",
      "regular-polygon",
      "tetrahedron",
      "eulers-formula-for-polyhedra",
      "congruent"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正多面体",
        "en": "Platonic solid"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 5.12 Prisms and Pyramids ほか"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "9.1 Polyhedrons: There are only five regular polyhedra, called the Platonic solids（regular polyhedron 1 件）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction (4th ed.)",
        "url": "https://discrete.openmathbooks.org/dmoi4.pdf",
        "note": "2.3 Planar Graphs: regular polyhedron (also known as a Platonic solid)（7 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "Platonic solid": 2
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは IM の呼び方 Platonic solid（Algebra 1 2.2 Writing Equations to Model Relationships (Part 1)・Algebra 1 2.9 Which Variable to Solve for? (Part 2)・Geometry 5.12 Prisms and Pyramids・計 13 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "regular-tetrahedron",
    "ja": {
      "term": "正四面体",
      "reading": "せいしめんたい",
      "alt": []
    },
    "en": {
      "term": "regular tetrahedron",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "4 つの面がすべて合同な正三角形でできている立体。辺は 6 本、頂点は 4 つで、正多面体の 1 つ。",
    "definition_en": "A solid with four faces that are all congruent equilateral triangles. It has 6 edges and 4 vertices and is one of the five regular polyhedra.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A regular tetrahedron is a triangular pyramid where every face is an equilateral triangle, even the base.",
        "ja": "正四面体は、底面も含めて、すべての面が正三角形の三角錐だよ。",
        "register": "spoken"
      },
      {
        "en": "How many edges and vertices does a regular tetrahedron have?",
        "ja": "正四面体の辺の数と頂点の数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "regular を付けない tetrahedron（四面体、エントリ tetrahedron）は、4 つの面が三角形であればよく、正三角形でなくてもよい。三角錐（triangular pyramid）と同じ立体。",
      "tetrahedron は「テトラヒードロン」に近く、he の音節に強勢がある（Merriam-Webster: ˌte-trə-ˈhē-drən）。複数形は tetrahedrons または tetrahedra（Merriam-Webster）。"
    ],
    "related": [
      "tetrahedron",
      "regular-polyhedron",
      "pyramid",
      "polyhedron",
      "cube-solid"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正四面体",
        "en": "Regular tetrahedron"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 6)",
        "url": "https://im.kendallhunt.com/MS/students/1/index.html",
        "note": "Grade 6 6.19 Tables, Equations, and Graphs, Oh My!（regular tetrahedron 1 件）"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「tetrahedron」",
        "url": "https://www.merriam-webster.com/dictionary/tetrahedron",
        "note": "発音 ˌte-trə-ˈhē-drən、複数形 tetrahedrons または tetrahedra（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 regular tetrahedron（Grade 6 6.19 Tables, Equations, and Graphs, Oh My!・計 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "reject",
    "ja": {
      "term": "棄却する",
      "reading": "ききゃくする",
      "alt": []
    },
    "en": {
      "term": "reject",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I",
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "仮説検定で、データが帰無仮説のもとでは起こりにくいとして、帰無仮説を否定する。",
    "definition_en": "In a hypothesis test, to rule out the null hypothesis because the data would be unlikely if it were true.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The p-value is less than alpha, so we reject the null hypothesis.",
        "ja": "p 値が α より小さいので、帰無仮説を棄却する。",
        "register": "spoken"
      },
      {
        "en": "At the 1% significance level, do you reject the null hypothesis? Explain.",
        "ja": "有意水準 1% で帰無仮説を棄却するか。説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ふつうは reject the null hypothesis（帰無仮説を棄却する）の形で使う。",
      "棄却しないときは fail to reject と言う。accept the null hypothesis（帰無仮説を採択する）は用例コーパスにほとんど出てこない。"
    ],
    "related": [
      "fail-to-reject",
      "null-hypothesis",
      "hypothesis-testing",
      "significance-level"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "令和 7・8 年度 数学II・B・C 第5問: 帰無仮説が棄却（否定）される（高等学校学習指導要領解説に「棄却」は 0 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "reject": 209
      },
      "written": {
        "reject": 260
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rejection-region",
    "ja": {
      "term": "棄却域",
      "reading": "ききゃくいき",
      "alt": []
    },
    "en": {
      "term": "rejection region",
      "alt": [
        "critical region"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "OpenStax Introductory Statistics と AP Statistics は p 値で判断する（OpenStax Introductory Statistics は p-value 475 件・rejection region 0 件、AP Statistics の CED は p-value 100 件・rejection region 0 件）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "仮説検定で、検定統計量がそこに入ったら帰無仮説を棄却する値の範囲。",
    "definition_en": "In a hypothesis test, the set of values of the test statistic for which the null hypothesis is rejected.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Our z of 2.4 falls in the rejection region, so we reject the null.",
        "ja": "z = 2.4 は棄却域に入るので、帰無仮説を棄却する。",
        "register": "spoken"
      },
      {
        "en": "Find the rejection region for a two-tailed test at the 5% significance level.",
        "ja": "有意水準 5% の両側検定の棄却域を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは rejection region は話し言葉にわずかにあるだけ（The Organic Chemistry Tutor・Professor Leonard）。"
    ],
    "related": [
      "hypothesis-testing",
      "significance-level",
      "critical-value"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "rejection region": 6
      },
      "sources": [
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元 2 の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "related-rates",
    "ja": {
      "term": "関連変化率",
      "reading": "かんれんへんかりつ",
      "alt": [
        "関連変化率の問題"
      ]
    },
    "en": {
      "term": "related rates",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学III）には、時刻とともに変わる 2 つの量の変化率の関係を求める問題の名前が出てこない。AP の CED は related rates を独立した topic（4.4・4.5）にしている。見出しの「関連変化率」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "時刻とともに変わる量どうしの関係式を t で微分し、一方の変化率から他方の変化率を求める問題。",
    "definition_en": "Problems in which the rate of change of one quantity is found from the rate of change of another, by differentiating an equation that relates them with respect to time.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In related rates, differentiate with respect to t before you plug in any numbers.",
        "ja": "関連変化率の問題では、数値を代入する前に t で微分する。",
        "register": "spoken"
      },
      {
        "en": "A 10-ft ladder slides down a wall. If the bottom moves away from the wall at 2 ft/s, how fast is the top sliding down when the bottom is 6 ft from the wall?",
        "ja": "長さ 10 ft のはしごが壁をすべり落ちる。下端が毎秒 2 ft で壁から離れるとき、下端が壁から 6 ft の位置にある瞬間の、上端が下がる速さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "implicit-differentiation",
      "rate-of-change",
      "chain-rule"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 4.4・4.5（related rates）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III: 時刻とともに変わる 2 つの量の変化率の関係を求める問題の名前が出てこない・「関連変化率」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "related rates": 20
      },
      "written": {
        "related rates": 9
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "relation",
    "ja": {
      "term": "二項関係",
      "reading": "にこうかんけい",
      "alt": [
        "関係"
      ]
    },
    "en": {
      "term": "relation",
      "alt": [
        "binary relation"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete",
      "set-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "集合 A から集合 B への二項関係とは、直積 A × B の部分集合 R のこと。(a, b) ∈ R のとき a R b と書き、A から A への関係を A 上の関係という。",
    "definition_en": "A relation from a set A to a set B is a subset R of the Cartesian product A × B; we write a R b when (a, b) is in R. A relation from A to itself is called a relation on A.",
    "latex": "R \\subseteq A \\times B",
    "spoken_en": "R is a subset of A cross B",
    "examples": [
      {
        "en": "A relation on A is really just a set of ordered pairs, so you can list it or draw it as arrows.",
        "ja": "A 上の関係は結局は順序対の集合なので、並べて書いてもいいし、矢印でかいてもいい。",
        "register": "spoken"
      },
      {
        "en": "Let R be the relation on {1, 2, 3, 4} defined by a R b if a divides b. List all ordered pairs in R.",
        "ja": "集合 {1, 2, 3, 4} 上の関係 R を、a が b を割り切るとき a R b と定める。R に属する順序対をすべて挙げよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「解と係数の関係」「関係式」の「関係」は relation の意味ではない。数学用語の二項関係は relation で、日常語の relationship とも区別する。",
      "関数は関係の特別な場合で、各 a ∈ A に対して a R b となる b がちょうど 1 つあるもの。"
    ],
    "related": [
      "equivalence-relation",
      "partial-order",
      "cartesian-product",
      "subset",
      "directed-graph"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "binary relation | relation on | relations on | the relation r | a relation r": 5
      },
      "written": {
        "binary relation | relation on | relations on | the relation r | a relation r": 39,
        "binary relation": 8
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "relationship-between-roots-and-coefficients",
    "ja": {
      "term": "解と係数の関係",
      "reading": "かいとけいすうのかんけい",
      "alt": [
        "根と係数の関係"
      ]
    },
    "en": {
      "term": "Vieta's formulas",
      "alt": [
        "relationship between roots and coefficients"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（Vieta's formulas ／ relationship between roots and coefficients は CED・OpenStax・IM・CK-12 とも 0 件）。見出しは英語版 Wikipedia の記事名「Vieta's formulas」（ja 記事「根と係数の関係」の langlink 先。冒頭で太字で定義し、n 次の多項式の係数と根の和・積の関係として書く）。用例コーパスにも名前は出てこず、the sum of the roots ／ the product of the roots と中身を言う講義がわずかにある（すべて 3Blue1Brown）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": []
    },
    "definition_ja": "2 次方程式 ax² + bx + c = 0 の 2 つの解 α, β について、α + β = −b/a、αβ = c/a が成り立つこと。3 次以上の方程式にも同様の関係がある。",
    "definition_en": "For the roots α and β of ax² + bx + c = 0, α + β = −b/a and αβ = c/a; similar relations hold for higher-degree equations.",
    "latex": "\\alpha + \\beta = -\\frac{b}{a},\\quad \\alpha\\beta = \\frac{c}{a}",
    "spoken_en": "alpha plus beta equals negative b over a, alpha beta equals c over a",
    "examples": [
      {
        "en": "By Vieta's formulas, the sum of the roots is negative b over a, and the product is c over a.",
        "ja": "解と係数の関係より、2 つの解の和は −b/a、積は c/a。",
        "register": "spoken"
      },
      {
        "en": "Let α and β be the roots of 2x² − 3x + 5 = 0. Find the value of α² + β².",
        "ja": "2 次方程式 2x² − 3x + 5 = 0 の 2 つの解を α, β とする。α² + β² の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "sum of the roots",
        "ja": "2 つの解の和"
      },
      {
        "en": "product of the roots",
        "ja": "2 つの解の積"
      }
    ],
    "pitfalls": [
      "2 次式の因数分解 ax² + bx + c = a(x − α)(x − β)（複素数の範囲）は、OpenStax Algebra and Trigonometry（Zeros of Polynomial Functions）の Linear Factorization Theorem に当たる。",
      "日本語の「解」と「根」はここでは同じ。英語版 Wikipedia「Vieta's formulas」も roots と書く。"
    ],
    "related": [
      "quadratic-formula",
      "discriminant",
      "imaginary-solution",
      "elementary-symmetric-polynomial",
      "symmetric-expression"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "根と係数の関係",
        "en": "Vieta's formulas"
      },
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Zeros of Polynomial Functions（Linear Factorization Theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "product of the roots": 3,
        "sum of the roots": 1
      },
      "sources": [
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 Vieta's formulas（英語版「Vieta's formulas」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "relative-frequency",
    "ja": {
      "term": "相対度数",
      "reading": "そうたいどすう",
      "alt": []
    },
    "en": {
      "term": "relative frequency",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "各階級の度数を度数の合計で割った値。全体に対する割合を表す。",
    "definition_en": "The frequency of a class as a fraction of the whole: the frequency divided by how many data values there are.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The relative frequency is the frequency over the total, so they all add up to one.",
        "ja": "相対度数は度数を合計で割ったものなので、全部足すと 1 になる。",
        "register": "spoken"
      },
      {
        "en": "Find the relative frequency of each class and round to two decimal places.",
        "ja": "各階級の相対度数を求め、小数第 2 位まで答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "相対度数を確率とみなしたものが統計的確率（empirical probability。エントリ experimental-probability）。"
    ],
    "related": [
      "frequency",
      "cumulative-relative-frequency",
      "experimental-probability"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!cumulative relative frequency": 19
      },
      "written": {
        "!cumulative relative frequency": 119
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "relative-positions-of-two-circles",
    "ja": {
      "term": "2 円の位置関係",
      "reading": "にえんのいちかんけい",
      "alt": []
    },
    "en": {
      "term": "relative position of two circles",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（relative position of two circles は参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）にも用例コーパスにも出てこない）。CK-12 Geometry（6.2 Identify Circle Components）は、1 点で接する 2 円を tangent circles、中心が同じで半径が違う 2 円を concentric circles と呼び、練習問題で externally tangent を使う。英語版 Wikipedia「Tangent」（節 Tangent circles）は、中心間の距離が半径の和なら externally tangent、差なら internally tangent と書く。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "2 つの円の関係を、中心間の距離 d と半径 r₁, r₂ の大小で分けたもの。互いに外部にある・外接する・2 点で交わる・内接する・一方が他方の内部にある、の 5 通り。",
    "definition_en": "How two circles sit relative to each other, classified by comparing the distance d between their centers with the radii r₁ and r₂: separate, externally tangent, intersecting at two points, internally tangent, or one inside the other.",
    "latex": "|r_1 - r_2| < d < r_1 + r_2",
    "spoken_en": "the absolute value of r one minus r two is less than d, which is less than r one plus r two",
    "examples": [
      {
        "en": "If the distance between the centers equals r₁ plus r₂, the circles are externally tangent — they touch at exactly one point from the outside.",
        "ja": "中心間の距離が r₁ + r₂ に等しければ、2 円は外接する。外側から 1 点で接している。",
        "register": "spoken"
      },
      {
        "en": "When d equals the difference of the radii, one circle is internally tangent to the other.",
        "ja": "d が半径の差に等しいとき、一方の円が他方の円に内接する。",
        "register": "spoken"
      },
      {
        "en": "Circle C₁ has center (0, 0) and radius 2, and circle C₂ has center (5, 0) and radius r. Find the range of r for which C₁ and C₂ intersect at two points.",
        "ja": "円 C₁ は中心 (0, 0)、半径 2、円 C₂ は中心 (5, 0)、半径 r である。C₁ と C₂ が異なる 2 点で交わるような r の値の範囲を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「2 円が外接する」は externally tangent、「2 円が内接する」は internally tangent（英語版 Wikipedia「Tangent」の節 Tangent circles）。多角形と円の関係に使う circumscribed ／ inscribed とは別の言い方になる。",
      "2 点で交わる条件は |r₁ − r₂| < d < r₁ + r₂。等号が入ると接する場合になる。"
    ],
    "related": [
      "common-tangent",
      "position-of-a-line-relative-to-a-circle",
      "circle-through-the-intersections-of-two-circles",
      "equation-of-a-circle",
      "distance"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.2 Identify Circle Components（tangent circles、concentric circles、externally tangent）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Tangent",
        "note": "記事「Tangent」の節 Tangent circles（externally ／ internally tangent。数学カテゴリから 4 段）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "relative-positions-of-two-lines",
    "ja": {
      "term": "2 直線の位置関係",
      "reading": "にちょくせんのいちかんけい",
      "alt": []
    },
    "en": {
      "term": "relative position of two lines",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（relative position of two lines は参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）にも用例コーパスにも出てこない）。OpenStax Calculus Volume 3（2.5 Equations of Lines and Planes in Space）は小見出し Relationships between Lines を立て、空間の 2 直線を parallel but not equal ／ equal ／ intersecting ／ skew に分ける。CK-12 Geometry（3.2 Parallel and Skew Lines）は平行と skew lines を並べて扱う。「ねじれの位置にある」は are skew（skew lines）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数A"
      ],
      "us": [
        "Geometry",
        "Calculus III"
      ]
    },
    "definition_ja": "空間の 2 直線の関係。1 点で交わる・平行である・ねじれの位置にある（同じ平面上にない）の 3 通りがある。",
    "definition_en": "How two lines in space are related: they intersect in one point, are parallel, or are skew (not in any common plane).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These two edges of the cube never meet, but they're not parallel either — they're skew lines.",
        "ja": "立方体のこの 2 辺は交わらないが、平行でもない。ねじれの位置にある。",
        "register": "spoken"
      },
      {
        "en": "In the cube ABCD-EFGH, name every edge that is skew to edge AB.",
        "ja": "立方体 ABCD-EFGH において、辺 AB とねじれの位置にある辺をすべて答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ねじれの位置の skew lines と、統計の分布が「歪んだ」skewed は別の意味。",
      "平面上なら 2 直線は交わるか平行かのどちらか。skew は空間にしかない。"
    ],
    "related": [
      "parallel-lines",
      "perpendicular-lines",
      "intersect",
      "coplanarity-condition",
      "line-perpendicular-to-a-plane",
      "skew-lines"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.5 Equations of Lines and Planes in Space（Relationships between Lines）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "3.2 Parallel and Skew Lines"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 B 図形（空間図形）: 交わる・平行・ねじれの位置"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "relatively-prime",
    "ja": {
      "term": "互いに素",
      "reading": "たがいにそ",
      "alt": []
    },
    "en": {
      "term": "relatively prime",
      "alt": [
        "coprime"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "number-theory"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "2 つの整数の最大公約数が 1 であること。共通の素因数をもたない。",
    "definition_en": "Describes two integers whose greatest common divisor is 1, so they share no prime factor.",
    "latex": "\\gcd(a, b) = 1",
    "spoken_en": "the GCD of a and b is 1",
    "examples": [
      {
        "en": "Since 8 and 15 are relatively prime, their LCM is just 8 times 15.",
        "ja": "8 と 15 は互いに素だから、最小公倍数はそのまま 8 × 15 だ。",
        "register": "spoken"
      },
      {
        "en": "Prove that n and n + 1 are relatively prime for every positive integer n.",
        "ja": "すべての正の整数 n について、n と n + 1 は互いに素であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "relatively prime は形容詞。「a と b は互いに素」は a and b are relatively prime、「a は b と互いに素」は a is relatively prime to b。",
      "3 つ以上の整数では、全体の最大公約数が 1 であることと、どの 2 つも互いに素であることを言い分ける。6, 10, 15 は全体の最大公約数は 1 だが、どの 2 つも互いに素ではない。英語版 Wikipedia「Coprime integers」は setwise coprime と pairwise coprime、日本語版 Wikipedia「互いに素 (整数論)」は「互いに素」と「対ごとに素」と呼び分ける。",
      "日本語の「互いに素」は集合（共通部分が空）にも使い、そのときの英語は disjoint（日本語版 Wikipedia「互いに素 (整数論)」の冒頭。エントリ intersection）。"
    ],
    "related": [
      "greatest-common-divisor",
      "euclidean-algorithm",
      "linear-diophantine-equation",
      "least-common-multiple",
      "bezouts-identity",
      "chinese-remainder-theorem",
      "prime-number"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "互いに素 (整数論)",
        "en": "Coprime integers"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和4年度 大学入学共通テスト 本試験 数学Ⅰ・数学Ａ（問題）",
        "note": "整数の問題: 互いに素"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "relatively prime": 37
      },
      "written": {
        "relatively prime": 24
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "remainder-theorem",
    "ja": {
      "term": "剰余の定理",
      "reading": "じょうよのていり",
      "alt": [
        "剰余定理"
      ]
    },
    "en": {
      "term": "remainder theorem",
      "alt": [
        "polynomial remainder theorem"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "整式 P(x) を x − a で割った余りは P(a) に等しいという定理。",
    "definition_en": "The theorem that the remainder when a polynomial P(x) is divided by x − a is P(a).",
    "latex": "P(x) = (x - a)Q(x) + P(a)",
    "spoken_en": "P of x equals x minus a times Q of x, plus P of a",
    "examples": [
      {
        "en": "By the remainder theorem, just plug in 2 — you don't have to divide.",
        "ja": "剰余の定理から、2 を代入するだけ。割らなくていい。",
        "register": "spoken"
      },
      {
        "en": "Use the remainder theorem to find the remainder when x^{100} + 1 is divided by x + 1.",
        "ja": "剰余の定理を用いて、x¹⁰⁰ + 1 を x + 1 で割った余りを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry・Intermediate Algebra と IM Algebra 2（2.15 The Remainder Theorem）は、剰余の定理を 1 次式 x − k で割る場合だけで扱い、2 次式で割った余り（ax + b の形）を求める問題は出てこない。学習指導要領解説（数学II）は、A = BQ + R（R の次数は B の次数より低い）から剰余の定理を導く。",
      "AP Calculus BC で出てくる Taylor's remainder theorem（CED topic 10.12 の呼び方は Lagrange error bound）は、テイラー多項式の誤差の定理で、別の定理。"
    ],
    "related": [
      "remainder",
      "synthetic-division",
      "division-algorithm",
      "factor-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "剰余の定理",
        "en": "Polynomial remainder theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 2.15 The Remainder Theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Use the Remainder and Factor Theorems"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式: A=BQ+R から剰余の定理や因数定理を導く"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.12（Lagrange error bound）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "remainder theorem": 19
      },
      "written": {
        "remainder theorem": 25
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "remainder",
    "ja": {
      "term": "余り",
      "reading": "あまり",
      "alt": [
        "剰余"
      ]
    },
    "en": {
      "term": "remainder",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "小学校",
        "中2",
        "数A",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Discrete Math",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "割り算で割り切れずに残る部分。整式の除法では、余りの次数は割る式の次数より小さい。",
    "definition_en": "What is left over in a division; in polynomial division, the remainder has lower degree than the divisor.",
    "latex": "A = BQ + R",
    "spoken_en": "A equals B Q plus R",
    "examples": [
      {
        "en": "The remainder is 3, so x − 2 is not a factor.",
        "ja": "余りが 3 なので、x − 2 は因数ではない。",
        "register": "spoken"
      },
      {
        "en": "Find the remainder when x³ − 2x + 5 is divided by x + 1.",
        "ja": "x³ − 2x + 5 を x + 1 で割ったときの余りを求めよ。",
        "register": "written"
      },
      {
        "en": "When an integer is divided by 3, the remainder is 0, 1, or 2, so every integer is 3k, 3k + 1, or 3k + 2.",
        "ja": "整数を 3 で割った余りは 0, 1, 2 のどれかなので、すべての整数は 3k, 3k + 1, 3k + 2 のどれかで表せる。",
        "register": "written"
      },
      {
        "en": "Show that when the sum of three consecutive integers is divided by 3, the remainder is 0.",
        "ja": "連続する 3 つの整数の和を 3 で割ったときの余りは 0 であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「〜で割ったときの余り」は the remainder when … is divided by …。",
      "remainder はテイラーの剰余項（the remainder of a Taylor polynomial）にも使う。",
      "日本語版 Wikipedia「除法」は余りを剰余とも呼ぶ。合同式 a ≡ b (mod n) は a と b を n で割った余りが等しいこと（エントリ modulus）。",
      "「余りは 0」は the remainder is zero（書き言葉・話し言葉とも使う）。見出しの形 the remainder is を中に含むので、候補表現には数えない。"
    ],
    "related": [
      "remainder-theorem",
      "polynomial-division",
      "division-algorithm",
      "modulus"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「除法」",
        "url": "https://ja.wikipedia.org/wiki/%E9%99%A4%E6%B3%95",
        "note": "剰余は余りとも呼ばれる"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the remainder is": 27
      },
      "written": {
        "the remainder is": 32
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "removable-discontinuity",
    "ja": {
      "term": "除去可能な不連続点",
      "reading": "じょきょかのうなふれんぞくてん",
      "alt": [
        "除去可能な不連続"
      ]
    },
    "en": {
      "term": "removable discontinuity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "1 点での値を定め直せば連続になる不連続点。グラフでは穴になる。",
    "definition_en": "A discontinuity that could be removed by redefining the function at a single point; it shows up as a hole in the graph.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "(x² − 1)/(x − 1) has a removable discontinuity at x = 1. It's just a hole.",
        "ja": "(x² − 1)/(x − 1) は x = 1 で除去可能な不連続点をもつ。ただの穴だ。",
        "register": "spoken"
      },
      {
        "en": "Identify the removable discontinuity of f(x) = (x² − 4)/(x − 2).",
        "ja": "f(x) = (x² − 4)/(x − 2) の除去可能な不連続点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（数学III）には不連続点の種類の名前が出てこない。グラフの穴は英語で hole。"
    ],
    "related": [
      "discontinuous",
      "jump-discontinuity",
      "infinite-discontinuity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III: 不連続点の種類の名前が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "removable discontinuity": 31
      },
      "written": {
        "removable discontinuity": 60
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "repeated-trials",
    "ja": {
      "term": "反復試行",
      "reading": "はんぷくしこう",
      "alt": []
    },
    "en": {
      "term": "independent trials",
      "alt": [
        "repeated trials"
      ],
      "uk": null,
      "variants": [
        {
          "term": "Bernoulli trial",
          "register": "written",
          "note": "成功・失敗の 2 通りの結果しかない 1 回の試行の名前で、OpenStax Introductory Statistics の書き方。話し言葉にはほとんど出てこず（Khan Academy）、書き言葉はすべて OpenStax Introductory Statistics。書き言葉では independent trials（こちらもすべて OpenStax Introductory Statistics）と並ぶ。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学B）は反復試行を、ある試行を独立に何回も繰り返すものとして書く。用例コーパスでは independent trials が話し言葉（すべて Khan Academy）にも書き言葉（すべて OpenStax Introductory Statistics）にも出てくるが、repeated trials はほとんど出てこない。Bernoulli trial（ベルヌーイ試行）は繰り返す 1 回 1 回の、結果が 2 通りの試行を指すので、反復試行そのものとは少しずれる。",
    "domains": [
      "combinatorics",
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "同じ条件の試行を、各回の結果が互いに影響しないように何回も繰り返すこと。",
    "definition_en": "Repeating the same experiment several times under the same conditions, so that the result of each run does not affect the others.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We're flipping the same coin ten times, so these are independent trials, and the probability of heads is one half every time.",
        "ja": "同じ硬貨を 10 回投げるので反復試行で、表の出る確率は毎回 1/2 だ。",
        "register": "spoken"
      },
      {
        "en": "Each of the 8 free throws is a Bernoulli trial with probability of success 0.7.",
        "ja": "8 回のフリースローは、1 回ごとに成功の確率が 0.7 のベルヌーイ試行である。",
        "register": "written"
      },
      {
        "en": "A die is rolled 5 times, and the rolls are independent trials. Find the probability that a 6 appears exactly twice.",
        "ja": "さいころを 5 回投げる反復試行で、6 の目がちょうど 2 回出る確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「独立な試行」（2 つの試行が互いに影響しない）は independence-of-events の項。反復試行はそれを同じ試行で n 回続けたもの。",
      "Bernoulli trial は 1 回の試行の名前なので、「n 回のベルヌーイ試行」は n Bernoulli trials と複数形にする。"
    ],
    "related": [
      "trial",
      "binomial-probability",
      "binomial-distribution",
      "independence-of-events",
      "sampling-with-replacement"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「ベルヌーイ試行」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%AB%E3%83%8C%E3%83%BC%E3%82%A4%E8%A9%A6%E8%A1%8C",
        "note": "1 回の試行（結果が 2 通り）: variant Bernoulli trial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Bernoulli_trial",
        "note": "記事「Bernoulli trial」の冒頭（1 回の試行。数学カテゴリから 5 段）: variant Bernoulli trial"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B 統計的な推測: 試行を独立に何回も繰り返すような反復試行"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "independent trials": 17,
        "Bernoulli trial": 1
      },
      "written": {
        "independent trials": 13,
        "Bernoulli trial": 15,
        "repeated trials": 1
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "repeating-decimal",
    "ja": {
      "term": "循環小数",
      "reading": "じゅんかんしょうすう",
      "alt": []
    },
    "en": {
      "term": "repeating decimal",
      "alt": [],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic",
      "number-theory"
    ],
    "level": {
      "jp": [
        "中3",
        "数A",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "小数点以下のある位から先で、同じ数字の並びが限りなく繰り返される無限小数。分数で表せる数のうち、有限小数にならないものはこの形になる。",
    "definition_en": "A decimal in which a block of digits repeats forever from some point on. Every fraction that does not give a terminating decimal gives one of these.",
    "latex": "0.\\overline{142857} = \\frac{1}{7}",
    "spoken_en": "zero point one four two eight five seven repeating equals one seventh",
    "examples": [
      {
        "en": "Write the repeating decimal 0.363636… as a fraction in lowest terms.",
        "ja": "循環小数 0.3̇6̇ を既約分数で表せ。",
        "register": "written"
      },
      {
        "en": "A fraction in lowest terms gives a repeating decimal exactly when its denominator has a prime factor other than 2 and 5.",
        "ja": "既約分数は、分母が 2 と 5 以外の素因数をもつとき、またそのときに限り循環小数になる。",
        "register": "written"
      },
      {
        "en": "One seventh is a repeating decimal: the six digits 142857 just keep going.",
        "ja": "7 分の 1 は循環小数で、142857 の 6 桁がずっと続く。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "write the repeating decimal as a fraction",
        "ja": "循環小数を分数で表す"
      }
    ],
    "pitfalls": [
      "記号が違う。英語版 Wikipedia「Repeating decimal」の Notation の節は、米国などは繰り返す部分の上に横線（vinculum）を引き、日本・英国などは両端の数字の上に点を打つ、と書く。日本語版 Wikipedia「循環小数」も点・上線・下線の書き方を挙げる。OpenStax Intermediate Algebra と IM Grade 8 は横線（bar）を引く（0.1̅2̅3̅）。",
      "循環小数は有理数。√2 や π は循環しない無限小数で、無理数。",
      "英語版 Wikipedia「Repeating decimal」は recurring decimal とも呼ぶ。"
    ],
    "related": [
      "real-number",
      "prime-factorization",
      "infinite-geometric-series"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "循環小数",
        "en": "Repeating decimal"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Repeating_decimal",
        "note": "記事「Repeating decimal」の冒頭: repeating decimal or recurring decimal。Notation の節: 米国などは横線、日本・英国などは点"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「循環小数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%BE%AA%E7%92%B0%E5%B0%8F%E6%95%B0",
        "note": "表記法: 循環節の先頭と末尾の数に点をつける ／ 循環節全体に上線をつける"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "Grade 8 8.14 Decimal Representations of Rational Numbers: the bar over part of the expansion"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "repeating decimal": 17
      },
      "written": {
        "repeating decimal": 35
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "represent",
    "ja": {
      "term": "表す",
      "reading": "あらわす",
      "alt": []
    },
    "en": {
      "term": "represent",
      "alt": [
        "stand for"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "文字や式、グラフなどが、ある数量や数量の関係を示す。「x を〜の数とする」のように、文字が何を指すかを決めるときにも使う。",
    "definition_en": "To stand for a number, a quantity, or a relationship. For example, a variable can represent an unknown number, and a graph can represent a function.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "So what does x represent here? Right, the number of hours she worked.",
        "ja": "じゃあ、ここの x は何を表している？ そう、彼女が働いた時間だね。",
        "register": "spoken"
      },
      {
        "en": "Let a represent Maya's age now, and write an expression for her age 5 years from now.",
        "ja": "マヤの現在の年齢を a 歳として、5 年後の年齢を表す式を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「x を〜の数とする」は let x represent the number of … と言う。stand for も同じ意味で、x stands for … とも言える。",
      "「式で表す」は write an equation ／ write an expression と言う（エントリ write-an-equation）。",
      "represent には「代表する」の意味もあるが、数学の文では「（文字・式・グラフが）〜を表す」の意味で使う。"
    ],
    "related": [
      "variable",
      "unknown",
      "write-an-equation",
      "expression",
      "word-problem"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "represent": 1372,
        "stand for": 90
      },
      "written": {
        "represent": 1623,
        "stand for": 14
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "representations-of-a-function",
    "ja": {
      "term": "関数の表現",
      "reading": "かんすうのひょうげん",
      "alt": []
    },
    "en": {
      "term": "multiple representations",
      "alt": [
        "representations of a function",
        "tables, graphs, and equations"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説は、関数を表，式，グラフを相互に関連付けて考えることを挙げる。英語では、同じ関係を表・グラフ・式・言葉などいくつもの形で表すことを multiple representations と言い、関数に限らず使う。AP Calculus の CED（topic 1.9）は極限を表・グラフ・式で表すことをこの言い方で扱っており、見出しはこの CED の言い方にした。",
    "domains": [
      "algebra",
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Integrated Math 1",
        "Algebra 1"
      ]
    },
    "definition_ja": "1 つの関数（2 つの数量の関係）を、表・式・グラフなどいくつかの形で表すこと。どの形も同じ関係を表しているので、1 つの形からほかの形へ行き来できる。",
    "definition_en": "Showing the same function or relationship in several forms, such as a table, a graph, an equation, or words. Every form shows the same relationship, so you can go from any one of them to the others.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Table, graph, equation: those are multiple representations of the same function, so if your table says y is 7 when x is 2, your graph better go through (2, 7).",
        "ja": "表・グラフ・式は、同じ関数をいろいろな形で表したもの。だから表で x = 2 のとき y = 7 なら、グラフは (2, 7) を通っていないとおかしい。",
        "register": "spoken"
      },
      {
        "en": "A 20 cm candle burns down 2 cm every hour. Let y be its height in centimeters after x hours. Use multiple representations to show the relationship: make a table, write an equation, and draw a graph.",
        "ja": "長さ 20 cm のろうそくが、1 時間に 2 cm ずつ短くなる。火をつけてから x 時間後のろうそくの長さを y cm として、x と y の関係を表・式・グラフで表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「関数の表現」を function expression と直訳しない。英語の expression は「式」（エントリ expression）の意味で、表やグラフは含まない。",
      "multiple representations は表し方をまとめて言う言い方で、1 つ 1 つの表し方は a representation（represent の名詞形、エントリ represent）と言う。",
      "AP の CED は、表・グラフ・式に加えて、言葉（verbally）で表すことも表し方に数える（topic 2.2）。"
    ],
    "related": [
      "function",
      "table-of-values",
      "graph",
      "equation",
      "represent"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.9（multiple representations）・2.2（verbally）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "関数: 表，式，グラフを相互に関連付けて"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "tables, graphs, and equations": 1
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CED の呼び方 multiple representations（topic 1.9）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "residual-plot",
    "ja": {
      "term": "残差プロット",
      "reading": "ざんさぷろっと",
      "alt": []
    },
    "en": {
      "term": "residual plot",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "各データの残差（実際の値 − 予測値）を、説明変数に対してかいた散布図。模様がなければ直線のモデルが合っている。",
    "definition_en": "A scatter plot of the residuals (actual minus predicted values) against the explanatory variable; no pattern means a linear model fits.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "There's a curve in the residual plot, so a line isn't the best model here.",
        "ja": "残差プロットに曲がった模様があるので、ここでは直線は最適なモデルではない。",
        "register": "spoken"
      },
      {
        "en": "Make a residual plot and use it to assess whether a linear model is appropriate.",
        "ja": "残差プロットをかき、直線のモデルが適切かどうか判断せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "残差（residual）は y − ŷ（実際の値 − 予測値）。"
    ],
    "related": [
      "regression-line",
      "coefficient-of-determination",
      "influential-point"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "residual plot": 8
      },
      "written": {
        "residual plot": 9
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 8 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは AP Statistics の CED の呼び方 residual plot（topic 5.4）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "residual",
    "ja": {
      "term": "残差",
      "reading": "ざんさ",
      "alt": []
    },
    "en": {
      "term": "residual",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Intro Statistics"
      ]
    },
    "definition_ja": "実際に観測した値と、回帰直線で予測した値との差（実際の y − 予測した ŷ）。点が回帰直線より上にあれば正、下にあれば負になる。",
    "definition_en": "The difference between an observed value and the value predicted by the regression line: observed minus predicted. It is positive when the point lies above the line and negative when it lies below.",
    "latex": "e = y - \\hat{y}",
    "spoken_en": "e equals y minus y hat",
    "examples": [
      {
        "en": "This point is above the line, so its residual is positive.",
        "ja": "この点は直線より上にあるから、残差は正だね。",
        "register": "spoken"
      },
      {
        "en": "The regression line predicts ŷ = 52 when x = 6, but the observed value is 49. Find the residual.",
        "ja": "x = 6 のとき回帰直線による予測値は ŷ = 52 だが、実際の値は 49 である。残差を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "引く順番は「実際 − 予測」（observed − predicted）。逆にすると符号が反対になる。上の例では 49 − 52 = −3。",
      "ŷ は y hat と読む。",
      "残差を点で表した図は residual plot（エントリ residual-plot）。複素関数論の留数（residue）は別の語。"
    ],
    "related": [
      "residual-plot",
      "regression-line",
      "least-squares",
      "scatter-plot",
      "predict"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "residual !plot !plots !sum !sums": 115
      },
      "written": {
        "residual !plot !plots !sum !sums": 53
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "resistant-statistic",
    "ja": {
      "term": "抵抗性がある",
      "reading": "ていこうせいがある",
      "alt": [
        "抵抗性のある統計量"
      ]
    },
    "en": {
      "term": "resistant",
      "alt": [],
      "uk": null
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": "見出しの「抵抗性がある」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "統計量が外れ値の影響をほとんど受けないこと。中央値や四分位範囲は抵抗性があり、平均値や標準偏差はない。",
    "definition_en": "Of a statistic: hardly affected by outliers. The median and the IQR are resistant; the mean and the standard deviation are not.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The median is resistant, so one crazy value doesn't move it much.",
        "ja": "中央値は抵抗性があるので、極端な値が 1 つあってもあまり動かない。",
        "register": "spoken"
      },
      {
        "en": "Which measure of center is resistant to outliers? Explain.",
        "ja": "外れ値に対して抵抗性があるのはどちらの代表値か。説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは resistant は書き言葉にわずかにあるだけ（OpenStax Introductory Statistics）。"
    ],
    "related": [
      "outlier",
      "median",
      "interquartile-range"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "resistant": 1
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは AP Statistics の CED の呼び方 resistant（topic 1.7）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "response-variable",
    "ja": {
      "term": "応答変数",
      "reading": "おうとうへんすう",
      "alt": [
        "目的変数"
      ]
    },
    "en": {
      "term": "response variable",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "説明変数によって説明・予測される側の変量。散布図の縦軸にとる。",
    "definition_en": "The variable that is explained or predicted by the explanatory variable; it goes on the vertical axis of a scatter plot.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The response variable goes on the y-axis.",
        "ja": "応答変数は y 軸にとる。",
        "register": "spoken"
      },
      {
        "en": "In the study of fertilizer and plant height, what is the response variable?",
        "ja": "肥料と植物の高さの研究で、応答変数は何か。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数学の dependent variable（従属変数）に当たる役割。"
    ],
    "related": [
      "explanatory-variable",
      "scatter-plot",
      "regression-line"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "response variable": 6
      },
      "written": {
        "response variable": 25
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "restricted-domain",
    "ja": {
      "term": "定義域の制限",
      "reading": "ていぎいきのせいげん",
      "alt": []
    },
    "en": {
      "term": "restricted domain",
      "alt": [],
      "variants": [
        {
          "term": "restrict the domain",
          "register": "written",
          "note": "動詞の言い方。話し言葉にはほとんど出てこない。書き言葉では restricted domain と同じくらい使う（OpenStax Algebra and Trigonometry・Calculus）。"
        },
        {
          "term": "restricted domain",
          "register": "written",
          "note": "話し言葉にはほとんど出てこない。書き言葉では restrict the domain と同じくらい使う（OpenStax Algebra and Trigonometry・Calculus）。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「定義域の制限」は学習指導要領解説に無い（日本語版 Wikipedia「定義域」の節「定義域の制限と延長」にはある）。高等学校学習指導要領解説は同じ内容を「区間が制限された関数の最大値や最小値」（数学II 微分）、逆関数は「元の関数が 1 対 1 の対応であるとき」（数学III）と書く。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I",
        "数II",
        "数III",
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "関数の定義域を一部に限ること。1 対 1 でない関数に逆関数を考えるときなどに使う。",
    "definition_en": "Cutting a function's domain down to part of it, for instance so that a function which fails the horizontal line test can be given an inverse.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "x squared isn't one-to-one, so restrict the domain to x ≥ 0.",
        "ja": "x² は 1 対 1 ではないので、定義域を x ≧ 0 に制限する。",
        "register": "spoken"
      },
      {
        "en": "On the restricted domain x ≥ 2, find the inverse of f(x) = (x − 2)² + 1.",
        "ja": "定義域を x ≧ 2 に制限したとき、f(x) = (x − 2)² + 1 の逆関数を求めよ。",
        "register": "written"
      },
      {
        "en": "With the domain restricted to 0 ≤ x ≤ 3, the vertex is inside the interval, so check the vertex and both endpoints.",
        "ja": "定義域が 0 ≦ x ≦ 3 に制限されていて頂点がその中にあるので、頂点と両端の値を調べる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "逆三角関数は、三角関数の定義域を制限して逆関数をつくったもの（sin は [−π/2, π/2]）。"
    ],
    "related": [
      "inverse-function",
      "one-to-one",
      "inverse-trigonometric-function",
      "find-the-domain"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「定義域」",
        "url": "https://ja.wikipedia.org/wiki/%E5%AE%9A%E7%BE%A9%E5%9F%9F",
        "note": "節「定義域の制限と延長」"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 微分: 区間が制限された関数の最大値や最小値（数学I の二次関数を振り返る）。数学III: 逆関数は元の関数が 1 対 1 の対応であるとき"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "restrict the domain": 4,
        "restricted domain": 3
      },
      "written": {
        "restricted domain": 22,
        "restrict the domain": 24
      },
      "sources": [
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "resultant",
    "ja": {
      "term": "合力",
      "reading": "ごうりょく",
      "alt": [
        "和ベクトル"
      ]
    },
    "en": {
      "term": "resultant",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「合力」は力の和に限るが、英語の resultant は力に限らずベクトルの和一般に使う（resultant force ／ resultant vector）。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "2 つ以上の力（ベクトル）を合わせた 1 つの力（ベクトル）。ベクトルの和。",
    "definition_en": "The single force (vector) equivalent to two or more forces (vectors) acting together: their vector sum.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Add the two force vectors tip to tail, and the resultant goes from the start to the end.",
        "ja": "2 つの力のベクトルをつないで足すと、合力は始点から終点へのベクトル。",
        "register": "spoken"
      },
      {
        "en": "Two forces of 30 N and 40 N act on an object at a right angle. Find the magnitude of the resultant force.",
        "ja": "30 N と 40 N の力が直角に働くとき、合力の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector",
      "work"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "resultant": 23
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "reverse-of-differentiation",
    "ja": {
      "term": "微分の逆",
      "reading": "びぶんのぎゃく",
      "alt": [
        "逆演算"
      ]
    },
    "en": {
      "term": "antidifferentiation",
      "alt": [
        "reverse of differentiation"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "微分の逆の操作。微分すると与えられた関数になる関数（原始関数）を求めること。積分は微分の逆演算にあたる。",
    "definition_en": "The reverse of differentiation: finding a function whose derivative is the given function. Integration is the inverse operation of differentiation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Antidifferentiation is just differentiation run backwards.",
        "ja": "微分の逆は、微分を逆向きにたどるだけ。",
        "register": "spoken"
      },
      {
        "en": "Integration and differentiation are inverse operations. One undoes the other.",
        "ja": "積分と微分は互いに逆演算。一方がもう一方を打ち消す。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞 antidifferentiate もあるが、授業では take the antiderivative と言うほうがずっと多い。",
      "米国の Pre-Algebra でも、方程式を解くときの足し算と引き算、掛け算と割り算の関係を inverse operations と呼ぶ（OpenStax Prealgebra ほか）。微分の逆の意味だけを数える形がないので、inverse operation は候補表現に数えない。"
    ],
    "related": [
      "antiderivative",
      "integration",
      "fundamental-theorem-of-calculus"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.14（antidifferentiation の技法の選び方）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "inverse operations"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "antidifferentiation": 6,
        "reverse of differentiation": 1
      },
      "written": {
        "antidifferentiation": 1,
        "reverse of differentiation": 1
      },
      "sources": [
        "mit-18.01",
        "openstax-calculus",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 7 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは CED の呼び方 antidifferentiation（topic 6.14・topic 7.6）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "revolve-around-the-x-axis",
    "ja": {
      "term": "x 軸のまわりに回転させる",
      "reading": "えっくすじくのまわりにかいてんさせる",
      "alt": []
    },
    "en": {
      "term": "rotate … around the x-axis",
      "alt": [
        "rotate around the x-axis",
        "revolve around the x-axis",
        "rotate about the x-axis",
        "revolve about the x-axis"
      ],
      "variants": [
        {
          "term": "rotate … around the x-axis",
          "register": "both",
          "note": "話し言葉の主見出し（Khan Academy が中心）。書き言葉では revolve と並ぶ。書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "revolve … around the x-axis",
          "register": "written",
          "note": "教科書（OpenStax Calculus）の書き方で、書き言葉では最も多い。is revolved around the x-axis のような受け身を含む。話し言葉では rotate より少なく（MIT OCW・Professor Leonard）、書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "revolve … about the x-axis",
          "register": "written",
          "note": "about を使う書き方。話し言葉にはほとんど出てこず（patrickJMT）、書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "rotate … about the x-axis",
          "register": "both",
          "note": "話し言葉・書き言葉とも少ない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "平面の図形を、x 軸を軸として 1 回転させて立体（回転体）をつくること。",
    "definition_en": "To spin a plane region one full turn about the x-axis, producing a solid of revolution.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Take this region and rotate it around the x-axis. Each slice becomes a disk.",
        "ja": "この部分を x 軸のまわりに回転させる。切り口はどれも円板になる。",
        "register": "spoken"
      },
      {
        "en": "The region under y = √x on [0, 4] is revolved around the x-axis. Find the volume of the solid.",
        "ja": "区間 [0, 4] で y = √x の下の部分を x 軸のまわりに 1 回転させてできる立体の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞は rotate も revolve も、前置詞は around も about も使う。… の位置には it ／ this region ／ the shaded region などの目的語が入る。問題文では is revolved about the x-axis のように受け身でも書く。"
    ],
    "related": [
      "revolve-around-the-y-axis",
      "volume-of-a-solid-of-revolution",
      "disk",
      "axis-of-revolution"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rotate … around the x-axis": 24,
        "revolve … around the x-axis": 9,
        "rotate about the x-axis": 4,
        "revolve about the x-axis": 1
      },
      "written": {
        "rotate … around the x-axis": 15,
        "revolve … around the x-axis": 27,
        "rotate about the x-axis": 4,
        "revolve … about the x-axis": 9
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "revolve-around-the-y-axis",
    "ja": {
      "term": "y 軸のまわりに回転させる",
      "reading": "わいじくのまわりにかいてんさせる",
      "alt": []
    },
    "en": {
      "term": "rotate … around the y-axis",
      "alt": [
        "rotate around the y-axis",
        "revolve around the y-axis",
        "rotate about the y-axis",
        "revolve about the y-axis"
      ],
      "variants": [
        {
          "term": "rotate … around the y-axis",
          "register": "both",
          "note": "話し言葉は MIT OCW が半分以上。書き言葉は大半が OpenStax Calculus。"
        },
        {
          "term": "revolve … around the y-axis",
          "register": "both",
          "note": "書き言葉では rotate とほぼ同じ数（すべて OpenStax Calculus）。話し言葉では少ない（すべて MIT OCW）。"
        },
        {
          "term": "revolve … about the y-axis",
          "register": "written",
          "note": "話し言葉には出てこない。書き言葉でも少なく、すべて OpenStax Calculus。"
        },
        {
          "term": "rotate … about the y-axis",
          "register": "both",
          "note": "話し言葉・書き言葉ともほとんど出てこない（blackpenredpen・MIT の講義ノート）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "平面の図形を、y 軸を軸として 1 回転させて立体（回転体）をつくること。",
    "definition_en": "To spin a plane region one full turn about the y-axis, producing a solid of revolution.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you rotate it around the y-axis instead, you'll want x in terms of y.",
        "ja": "代わりに y 軸のまわりに回すなら、x を y の式で表したい。",
        "register": "spoken"
      },
      {
        "en": "The region is revolved about the y-axis. Set up an integral for the volume.",
        "ja": "その部分を y 軸のまわりに 1 回転させる。体積を表す積分の式を立てよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞は rotate も revolve も、前置詞は around も about も使う。… の位置には it ／ this region ／ the shaded region などの目的語が入る。問題文では is revolved about the y-axis のように受け身でも書く。",
      "y 軸のまわりの回転体は、x = g(y) と書き直して π∫x²dy で求めるほか、OpenStax Calculus Volume 1 は x のまま 2π∫x f(x)dx とする shell method も扱う。"
    ],
    "related": [
      "revolve-around-the-x-axis",
      "shell-method",
      "volume-of-a-solid-of-revolution",
      "axis-of-revolution"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "6.3 Volumes of Revolution: Cylindrical Shells"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "revolve around the y-axis": 4,
        "rotate … around the y-axis": 17,
        "rotate about the y-axis": 1
      },
      "written": {
        "rotate … around the y-axis": 24,
        "revolve … around the y-axis": 22,
        "revolve … about the y-axis": 6,
        "rotate about the y-axis": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rewrite-in-exponential-form",
    "ja": {
      "term": "指数の形に直す",
      "reading": "しすうのかたちになおす",
      "alt": []
    },
    "en": {
      "term": "convert … to exponential form",
      "alt": [],
      "variants": [
        {
          "term": "convert … to exponential form",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry（6.3）・Precalculus（4.3）の節の題 Converting from Logarithmic to Exponential Form と学習目標、Intermediate Algebra（10.3）の言い方。"
        },
        {
          "term": "rewrite … in exponential form",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry（6.3 の例と練習問題）・Intermediate Algebra（10.3）の言い方。IM Algebra 2（4.10〜4.18）は write … in exponential form と書く。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "log_a M = p を aᵖ = M の形に書き直す。",
    "definition_en": "To rewrite log_a M = p as aᵖ = M.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Rewrite it in exponential form: log base 2 of x equals 5 means 2 to the fifth equals x.",
        "ja": "指数の形に直そう。log₂ x = 5 は 2⁵ = x のこと。",
        "register": "spoken"
      },
      {
        "en": "Convert to exponential form: log₄ 64 = 3.",
        "ja": "log₄ 64 = 3 を指数の形に直せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "in exponential form は、同じ数のかけ算を累乗で書く（3 · 3 · 3 · 3 を 3⁴ と書く）意味（OpenStax Prealgebra 2.1、CK-12 Algebra 1.4.3）や、科学的記数法で 10 の累乗を書く形（4 × 10³ の 10³。OpenStax Prealgebra 10.5・Elementary Algebra 6.7・Intermediate Algebra 5.2）にも使う。"
    ],
    "related": [
      "rewrite-in-logarithmic-form",
      "logarithm",
      "undo-the-log",
      "logarithmic-equation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.3 Logarithmic Functions（Converting from Logarithmic to Exponential Form、rewrite … in exponential form）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "2.1 Use the Language of Algebra（write … in exponential form）・10.5 Integer Exponents and Scientific Notation（a power of 10 written in exponential form）"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "1.4.3（write … in exponential form）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "10.3 Evaluate and Graph Logarithmic Functions（convert between exponential and logarithmic form）・5.2 Properties of Exponents and Scientific Notation（a power of 10 written in exponential form）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "4.3 Logarithmic Functions（Converting from Logarithmic to Exponential Form）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "6.7 Integer Exponents and Scientific Notation（a power of 10 written in exponential form）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 4.10〜4.18（write … in exponential form）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!expression !write !factorization !answer !ten !written !n in exponential form": 7,
        "to exponential form": 2
      },
      "written": {
        "to exponential form": 28,
        "!expression !write !factorization !answer !ten !written !n in exponential form": 18
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rewrite-in-logarithmic-form",
    "ja": {
      "term": "対数の形に直す",
      "reading": "たいすうのかたちになおす",
      "alt": []
    },
    "en": {
      "term": "convert … to logarithmic form",
      "alt": [],
      "variants": [
        {
          "term": "rewrite … in logarithmic form",
          "register": "written",
          "note": "OpenStax Algebra and Trigonometry（6.3 の例と練習問題）の言い方。IM Algebra 2（4.10〜4.14）は write … in logarithmic form と書く。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "aᵖ = M を log_a M = p の形に書き直す。",
    "definition_en": "To rewrite aᵖ = M as log_a M = p.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Rewrite 10 squared equals 100 in logarithmic form: log of 100 equals 2.",
        "ja": "10² = 100 を対数の形に直すと log 100 = 2。",
        "register": "spoken"
      },
      {
        "en": "Convert 3^(−2) = 1/9 to logarithmic form.",
        "ja": "3⁻² = 1/9 を対数の形に直せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "rewrite-in-exponential-form",
      "logarithm"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.3 Logarithmic Functions（Converting from Exponential to Logarithmic Form、rewrite … in logarithmic form）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "10.3 Evaluate and Graph Logarithmic Functions（convert from exponential to logarithmic form）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "4.3 Logarithmic Functions（Converting from Exponential to Logarithmic Form）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 4.10〜4.14（in logarithmic form）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "in logarithmic form": 1
      },
      "written": {
        "to logarithmic form": 14,
        "in logarithmic form": 5
      },
      "sources": [
        "openstax-algtrig",
        "openstax-intalg",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rhombus",
    "ja": {
      "term": "ひし形",
      "reading": "ひしがた",
      "alt": []
    },
    "en": {
      "term": "rhombus",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "4 つの辺の長さがすべて等しい四角形。平行四辺形の特別な場合で、2 本の対角線は垂直に交わる。",
    "definition_en": "A quadrilateral with all four sides the same length. It is a special kind of parallelogram, and its diagonals are perpendicular.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A rhombus is like a square that got pushed over: all four sides are still equal, but the angles don't have to be 90 degrees.",
        "ja": "ひし形は、正方形を横から押して傾けたような形。4 つの辺は等しいままだけど、角は 90° でなくていい。",
        "register": "spoken"
      },
      {
        "en": "The diagonals of a rhombus are 6 cm and 8 cm long. Find the area of the rhombus.",
        "ja": "対角線の長さが 6 cm と 8 cm のひし形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形は rhombi または rhombuses。",
      "正方形もひし形の特別な場合（エントリ square-shape）。ひし形の角は直角とは限らず、4 つの角も直角なら正方形になる。",
      "ひし形の対角線は垂直に交わるが、長さが等しいとは限らない。対角線の長さも等しいのは正方形。"
    ],
    "related": [
      "parallelogram",
      "square-shape",
      "rectangle",
      "diagonal",
      "quadrilateral"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.7 Quadrilateral Classification"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rhombus": 60
      },
      "written": {
        "rhombus": 1
      },
      "sources": [
        "khan-middle",
        "openstax-calculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "riemann-sum",
    "ja": {
      "term": "区分求積法",
      "reading": "くぶんきゅうせきほう",
      "alt": [
        "リーマン和"
      ]
    },
    "en": {
      "term": "Riemann sum",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "区分求積法は、区間を n 等分して長方形の面積の和をつくり、その極限を定積分として求める手法の名前（学習指導要領解説も区分求積法の考えで定積分を導入する扱いに触れる）。AP の CED と OpenStax Calculus は手法の名前を立てず、長方形の和を Riemann sum、その極限を定積分の定義として扱う。日本の「lim (1/n)Σ f(k/n) を定積分で表せ」型の問題は、OpenStax Calculus Volume 1（5.2 The Definite Integral）では express the limits as integrals と指示される（AP の CED topic 6.3 は、リーマン和の極限を定積分として表すことを学習目標に挙げる）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "区間を細かく分けて長方形の面積の和をつくり、その極限として面積や定積分を求める方法。lim_{n→∞} (1/n)Σf(k/n) = ∫₀¹ f(x)dx を使う。",
    "definition_en": "A sum Σ f(xᵢ)Δx of areas of thin rectangles that estimates the area under a graph; its limit is the definite integral.",
    "latex": "\\lim_{n \\to \\infty} \\frac{1}{n}\\sum_{k=1}^{n} f\\!\\left(\\frac{k}{n}\\right) = \\int_0^1 f(x)\\,dx",
    "spoken_en": "the limit as n goes to infinity of one over n times the sum from k equals one to n of f of k over n equals the integral from zero to one of f of x dx",
    "respelling": "REE-mahn sum",
    "examples": [
      {
        "en": "Each rectangle is one piece of the Riemann sum. Make them thinner and the sum gets closer to the integral.",
        "ja": "長方形 1 つ 1 つがリーマン和の一部。細くするほど和は積分に近づく。",
        "register": "spoken"
      },
      {
        "en": "Recognize the limit as a Riemann sum and write it as a definite integral.",
        "ja": "この極限をリーマン和とみて、定積分で表せ。",
        "register": "written"
      },
      {
        "en": "Distance is velocity times time, so on a velocity graph each rectangle's area is a little piece of distance.",
        "ja": "距離は速度 × 時間だから、速度のグラフでは長方形 1 つ 1 つの面積が距離の小さなかけらになる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "区分求積法の直訳 piecewise quadrature は用例コーパスにも OpenStax にも出てこない。",
      "長方形の和で近似する考え方は、曲線の下の面積だけでなく、速度のグラフから移動距離を求めるときにも使う。"
    ],
    "related": [
      "summation-notation",
      "limit-of-a-riemann-sum",
      "definite-integral",
      "area-under-the-curve",
      "left-riemann-sum",
      "right-riemann-sum",
      "midpoint-riemann-sum",
      "total-distance-traveled"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "リーマン和",
        "en": "Riemann sum"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "5.2 The Definite Integral（express the limits as integrals）"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.2（left ／ right ／ midpoint Riemann sums と trapezoidal sums）、6.3"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II・III: 区分求積法の考えで定積分を導入"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "Riemann sum": 174
      },
      "written": {
        "Riemann sum": 137
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "right-angle",
    "ja": {
      "term": "直角",
      "reading": "ちょっかく",
      "alt": []
    },
    "en": {
      "term": "right angle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "大きさが 90° の角。2 本の直線が垂直に交わるとき、そこにできる角はすべて直角になる。",
    "definition_en": "An angle that measures exactly 90°. Perpendicular lines meet to form right angles.",
    "latex": "\\angle C = 90^\\circ",
    "spoken_en": "angle C equals ninety degrees",
    "examples": [
      {
        "en": "See the little square in the corner? That tells you it's a right angle, so it's exactly 90 degrees.",
        "ja": "角のところに小さな四角があるよね。これは直角の印で、ちょうど 90° ということ。",
        "register": "spoken"
      },
      {
        "en": "In the figure, ∠ABC is a right angle and m∠ABD = 27°. Find m∠DBC.",
        "ja": "図で、∠ABC は直角で、∠ABD = 27° である。∠DBC の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "right angle の right は「右」ではない。英語版 Wikipedia「Right angle」（Etymology）は、ラテン語 angulus rectus の rectus（直立した・まっすぐな）の訳だとする。直角三角形 right triangle（エントリ right-triangle）の right も同じ。",
      "図で角に小さな正方形の印がかいてあれば、その角が直角であることを表す。",
      "「2 直線が直角に交わる」は、英語では形容詞 perpendicular（エントリ perpendicular）を使って the lines are perpendicular と言える。"
    ],
    "related": [
      "acute-angle",
      "obtuse-angle",
      "right-triangle",
      "perpendicular",
      "straight-angle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "直角",
        "en": "Right angle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Right_angle",
        "note": "節 Etymology（angulus rectus、rectus は upright）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "right angle": 180
      },
      "written": {
        "right angle": 37
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "right-hand-limit",
    "ja": {
      "term": "右側極限",
      "reading": "みぎがわきょくげん",
      "alt": []
    },
    "en": {
      "term": "limit from the right",
      "alt": [],
      "variants": [
        {
          "term": "right-hand limit",
          "register": "written",
          "note": "書き言葉ではこちらが多いが、ほぼ OpenStax Precalculus。話し言葉はすべて MIT OCW。"
        },
        {
          "term": "right-sided limit",
          "register": "spoken",
          "note": "少数の言い方。話し言葉はすべて NancyPi で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x を a より大きい側から a に近づけたときの極限。",
    "definition_en": "The limit of f(x) as x approaches a from values greater than a.",
    "latex": "\\lim_{x \\to a^+} f(x)",
    "spoken_en": "the limit as x approaches a from the right of f of x",
    "examples": [
      {
        "en": "The limit from the right is 3, but from the left it's 1, so the limit doesn't exist.",
        "ja": "右側極限は 3、左側極限は 1 なので、極限は存在しない。",
        "register": "spoken"
      },
      {
        "en": "Evaluate lim_{x→2⁺} (x − 2)/|x − 2|.",
        "ja": "lim_{x→2+0} (x − 2)/|x − 2| を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（数学III）は lim_{x→+0} と書き、日本語版 Wikipedia「片側極限」は x → a⁺ などと並べて x → a + 0 の書き方を挙げる。OpenStax は x → a⁺ と書き、as x approaches a from the right と言う（講義では from the positive direction とも言う。Khan Academy）。"
    ],
    "related": [
      "left-hand-limit",
      "one-sided-limit",
      "limit"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.3（limit from the right）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 微分法: lim_{x→+0} の書き方"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「片側極限」",
        "url": "https://ja.wikipedia.org/wiki/%E7%89%87%E5%81%B4%E6%A5%B5%E9%99%90",
        "note": "lim_{x→a+0} f(x)"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "right-hand limit": 8,
        "limit from the right": 12,
        "right-sided limit": 4
      },
      "written": {
        "limit from the right": 12,
        "right-hand limit": 60
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "right-hand-side",
    "ja": {
      "term": "右辺",
      "reading": "うへん",
      "alt": []
    },
    "en": {
      "term": "right-hand side",
      "alt": [],
      "variants": [
        {
          "term": "right side",
          "register": "written",
          "note": "書き言葉では right-hand side と並んで使い、こちらが多い（OpenStax の中学・Algebra の教科書）。話し言葉では right-hand side のほうがずっと多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "等式や不等式で、等号や不等号の右側にある式。",
    "definition_en": "The expression to the right of the equal sign or inequality sign in an equation or inequality.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Now everything's on the left, so the right-hand side is just zero.",
        "ja": "これで全部左に来たから、右辺は 0 だけだね。",
        "register": "spoken"
      },
      {
        "en": "Rewrite 5x + 2 = 3x + 10 so that the right-hand side is a constant, and then solve for x.",
        "ja": "5x + 2 = 3x + 10 を右辺が定数だけになるように変形し、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "略して RHS と書くことがある（左辺は LHS）。",
      "右辺・左辺の「辺」も、三角形の辺も、英語では同じ side。どちらの意味かは文脈で区別する。"
    ],
    "related": [
      "left-hand-side",
      "both-sides",
      "equation",
      "equal-sign",
      "inequality"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "right-hand side": 1086,
        "right side": 247
      },
      "written": {
        "right-hand side": 83,
        "right side": 143
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "right-riemann-sum",
    "ja": {
      "term": "右リーマン和",
      "reading": "みぎりーまんわ",
      "alt": []
    },
    "en": {
      "term": "right Riemann sum",
      "alt": [],
      "variants": [
        {
          "term": "right Riemann sum",
          "register": "spoken",
          "note": "AP の呼び方（CED topic 6.2）。話し言葉はほぼ Khan Academy。書き言葉（OpenStax Calculus）は right-endpoint approximation が多く、こちらは少ない。"
        },
        {
          "term": "right-endpoint approximation",
          "register": "written",
          "note": "OpenStax Calculus の書き方で、話し言葉には出ない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「右リーマン和」は学習指導要領解説に無い（日本語版 Wikipedia「リーマン和」にはある）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "区間をいくつかの小区間に分け、各小区間の右端での関数の値を高さとする長方形の面積の和。小区間の幅はそろっていなくてもよい。",
    "definition_en": "A Riemann sum whose rectangles take their heights from the right endpoint of each subinterval.",
    "latex": "R_n = \\sum_{i=1}^{n} f(x_i)\\,\\Delta x_i",
    "spoken_en": "R sub n equals the sum from i equals one to n of f of x sub i delta x sub i",
    "examples": [
      {
        "en": "For an increasing function, the right Riemann sum overestimates the area.",
        "ja": "増加関数なら、右リーマン和は面積を多めに見積もる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "区分求積法の (1/n)Σ_{k=1}^{n} f(k/n) は、区間 [0, 1] の right Riemann sum にあたる。",
      "AP の CED（topic 6.2）は、小区間の幅がそろわない分け方（nonuniform partitions）でもリーマン和で近似する。表で与えられた関数では、表の x の値で区切るので幅がそろわないことがある。"
    ],
    "related": [
      "left-riemann-sum",
      "midpoint-riemann-sum",
      "riemann-sum",
      "overestimate-and-underestimate",
      "trapezoidal-rule"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.2（left ／ right ／ midpoint Riemann sums、uniform or nonuniform partitions）。Topic 6.3（the width of the ith subinterval）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「リーマン和」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E5%92%8C",
        "note": "左リーマン和・右リーマン和"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「右リーマン和」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "right Riemann sum": 40
      },
      "written": {
        "right Riemann sum": 9,
        "right-endpoint approximation": 33
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "right-triangle-similarity",
    "ja": {
      "term": "直角三角形の相似",
      "reading": "ちょっかくさんかくけいのそうじ",
      "alt": [
        "斜辺への垂線"
      ]
    },
    "en": {
      "term": "right triangle similarity theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「直角三角形の相似」は学習指導要領解説に無い、本プロジェクトの訳語。日本では直角三角形の直角の頂点から斜辺に垂線を引いてできる 3 つの三角形が相似であることを、定理の名前を付けずに使う。 用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直角三角形の直角の頂点から斜辺に垂線を下ろすと、できる 2 つの小さな三角形はどちらももとの三角形と相似になるという定理。",
    "definition_en": "If you draw the altitude from the right angle of a right triangle to the hypotenuse, the two smaller triangles are each similar to the original triangle and to each other.",
    "latex": "\\triangle ABC \\sim \\triangle ACD \\sim \\triangle CBD",
    "spoken_en": "Triangle ABC is similar to triangle ACD, which is similar to triangle CBD.",
    "examples": [
      {
        "en": "Once you drop the altitude to the hypotenuse, you get three similar right triangles, so match up the corresponding sides carefully.",
        "ja": "斜辺に垂線を下ろすと相似な直角三角形が 3 つできるので、対応する辺を注意して合わせよう。",
        "register": "spoken"
      },
      {
        "en": "In right triangle ABC, ∠C = 90° and CD is the altitude to the hypotenuse. If AD = 4 and DB = 9, find CD.",
        "ja": "∠C = 90° の直角三角形 ABC で、CD は斜辺への垂線である。AD = 4, DB = 9 のとき、CD を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "この相似から CD² = AD × DB、AC² = AD × AB、BC² = BD × AB が出る。垂線 CD は AD と DB の相乗平均（geometric mean）になる。",
      "3 つの三角形の頂点の対応は角で決める（直角どうし、共通の鋭角どうし）。△ABC ∼ △ACD ∼ △CBD のように対応順に書かないと、比を取り違える。",
      "用例コーパス・IM とも出てこない。定理の英語名は教科書による。"
    ],
    "related": [
      "geometric-mean",
      "similar-triangles",
      "foot-of-the-perpendicular"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "right-triangle",
    "ja": {
      "term": "直角三角形",
      "reading": "ちょっかくさんかくけい",
      "alt": []
    },
    "en": {
      "term": "right triangle",
      "alt": [],
      "uk": "right-angled triangle",
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "1 つの角が直角である三角形。直角の向かいの辺を斜辺という。",
    "definition_en": "A triangle with one right angle. Its longest side, across from the right angle, is the hypotenuse; the other two sides are the legs.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The side across from the right angle is the hypotenuse, and it's always the longest side of a right triangle.",
        "ja": "直角の向かいの辺が斜辺で、直角三角形ではいつもいちばん長い辺になる。",
        "register": "spoken"
      },
      {
        "en": "In right triangle ABC, m∠C = 90° and m∠A = 34°. Find m∠B.",
        "ja": "∠C = 90° の直角三角形 ABC で、∠A = 34° のとき、∠B の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "直角をはさむ 2 辺は legs、直角の向かいの辺は hypotenuse（エントリ hypotenuse）と呼ぶ。日本語の「斜辺以外の辺」に当たる語が leg。",
      "right は「右」ではなく直角の意味（エントリ right-angle）。英国では right-angled triangle と言う。",
      "中学校学習指導要領解説（第 2 学年）は、直角三角形だけに使える合同条件（斜辺と一つの鋭角、斜辺と他の 1 辺）を別に扱う（エントリ congruence-criteria-for-right-triangles）。"
    ],
    "related": [
      "hypotenuse",
      "right-angle",
      "congruence-criteria-for-right-triangles",
      "acute-triangle",
      "obtuse-triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "直角三角形",
        "en": "Right triangle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 B 図形: 直角三角形の合同条件（斜辺と一つの鋭角、斜辺と他の１辺）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Right_angle",
        "note": "記事「Right angle」: right triangle（英国英語では right-angled triangle）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "right triangle": 351
      },
      "written": {
        "right triangle": 273
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rigid-motion",
    "ja": {
      "term": "合同変換",
      "reading": "ごうどうへんかん",
      "alt": [
        "剛体変換",
        "移動"
      ]
    },
    "en": {
      "term": "rigid transformation",
      "alt": [
        "rigid motion"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の中1 は平行移動・回転移動・対称移動をまとめて「移動」と呼ぶ（中学校学習指導要領解説の「図形の移動」）。英語の rigid transformation（rigid motion）は距離を変えない変換で、映進（glide reflection）も含む。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "図形上のどの 2 点の距離も変えない変換。平行移動・回転移動・対称移動とそれらの組み合わせがあり、移した図形はもとの図形と合同になる。",
    "definition_en": "A transformation that preserves the distance between every pair of points, such as a translation, a rotation, a reflection, or a sequence of these. The image is always congruent to the original figure.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A rigid transformation doesn't change any lengths or angles, so the image is congruent to the original.",
        "ja": "合同変換では長さも角も変わらないから、移した図形はもとの図形と合同だ。",
        "register": "spoken"
      },
      {
        "en": "Describe a sequence of rigid transformations that takes △ABC onto △DEF.",
        "ja": "△ABC を △DEF に重ねる移動の組み合わせを説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "a sequence of rigid transformations",
        "ja": "移動の組み合わせ"
      }
    ],
    "pitfalls": [
      "IM Geometry は、一方の図形を rigid transformation の組み合わせで他方に重ねられることを合同の定義にしている。合同条件（SSS・SAS・ASA）もこの定義から証明する。",
      "拡大・縮小（dilation）は長さを変えるので rigid transformation に入らない。",
      "IM では rigid transformation（97 件。Grade 8 と Geometry の glossary の見出し）と rigid motion（61 件。Geometry のみ）を両方使う。用例コーパスでは話し言葉に rigid transformation が多く（すべて Khan Academy の中学の講義）、rigid motion はほとんど出てこない（3Blue1Brown）。"
    ],
    "related": [
      "translation",
      "rotation",
      "reflection",
      "congruent",
      "dilation",
      "composition-of-transformations"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "glossary の見出し rigid transformation（Grade 8・Geometry）。Geometry 1.10 Rigid Transformations ほか"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "中1 平面図形: 図形の移動（平行移動、対称移動及び回転移動）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rigid motion": 1,
        "rigid transformation": 22
      },
      "sources": [
        "khan-middle",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rise-over-run",
    "ja": {
      "term": "上昇分と水平移動分の比",
      "reading": "じょうしょうぶんとすいへいいどうぶんのひ",
      "alt": []
    },
    "en": {
      "term": "rise over run",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "none",
    "mapping_note": "見出しの「上昇分と水平移動分の比」は学習指導要領解説に無い、本プロジェクトの訳語。日本では「x が 1 増えると y がいくつ増えるか」「y の増加量 ÷ x の増加量」と言う。OpenStax Algebra and Trigonometry は傾きを rise over run と説明することがある。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直線の傾きを、縦に上がった量（rise）を横に進んだ量（run）で割ったものとして覚える言い方。グラフ上で点から点へ「上に 2、右に 3」と数えれば、傾きは 2/3。",
    "definition_en": "A way to remember slope as the vertical change (rise) divided by the horizontal change (run) between two points on a line. Going up 2 and right 3 from one point to another gives a slope of 2/3.",
    "latex": "m = \\frac{\\text{rise}}{\\text{run}}",
    "spoken_en": "m equals rise over run",
    "examples": [
      {
        "en": "Start at this point, go up 2 and over 3 to the next point, so rise over run gives us a slope of 2/3.",
        "ja": "この点から上に 2、右に 3 進むと次の点に着く。だから縦 ÷ 横で、傾きは 2/3。",
        "register": "spoken"
      },
      {
        "en": "Use rise over run to find the slope of the line through the points (1, 4) and (3, 0) on the graph.",
        "ja": "グラフ上の 2 点 (1, 4)、(3, 0) を通る直線の傾きを、縦の変化 ÷ 横の変化で求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "右下がりの直線では rise が負になる。右に 2 進んで下に 4 下がれば、rise は −4、run は 2 で、傾きは −2。",
      "over は分数の線を読む語で、rise over run は「rise ÷ run」（run 分の rise）。順番を逆にしない。",
      "run は右向きに数える。左向きに数えるときは run も負にして、rise の符号と合わせて割る。"
    ],
    "related": [
      "slope",
      "slope-formula",
      "rate-of-change",
      "slope-intercept-form-of-a-line",
      "linear-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rise over run": 36
      },
      "written": {
        "rise over run": 4
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rolles-theorem",
    "ja": {
      "term": "ロルの定理",
      "reading": "ろるのていり",
      "alt": []
    },
    "en": {
      "term": "Rolle's theorem",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "f が [a, b] で連続、(a, b) で微分可能で f(a) = f(b) なら、f′(c) = 0 となる c が a と b の間にあるという定理。",
    "definition_en": "If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then f′(c) = 0 for some c between a and b.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Rolle's theorem is just the mean value theorem when both ends are at the same height.",
        "ja": "ロルの定理は、両端の高さが同じときの平均値の定理にすぎない。",
        "register": "spoken"
      },
      {
        "en": "Use Rolle's theorem to show that x³ + x − 1 = 0 has exactly one real root.",
        "ja": "ロルの定理を用いて、x³ + x − 1 = 0 の実数解がちょうど 1 つであることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "mean-value-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ロルの定理",
        "en": "Rolle's theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Rolle's theorem": 1
      },
      "written": {
        "Rolle's theorem": 28
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rose-curve",
    "ja": {
      "term": "バラ曲線",
      "reading": "ばらきょくせん",
      "alt": [
        "正葉曲線"
      ]
    },
    "en": {
      "term": "rose curve",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "極方程式 r = a cos nθ（または a sin nθ）で表される花びら形の曲線。n が奇数なら n 枚、偶数なら 2n 枚の花びら。",
    "definition_en": "The petal-shaped curve r = a cos nθ (or a sin nθ); it has n petals when n is odd and 2n when n is even.",
    "latex": "r = a\\cos n\\theta",
    "spoken_en": "r equals a cosine n theta",
    "examples": [
      {
        "en": "With n = 3 you get three petals, but n = 2 gives four.",
        "ja": "n = 3 なら花びらは 3 枚、n = 2 なら 4 枚。",
        "register": "spoken"
      },
      {
        "en": "Sketch the rose curve r = 3 sin 2θ and state the number of petals.",
        "ja": "バラ曲線 r = 3 sin 2θ をかき、花びらの枚数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは書き言葉はすべて OpenStax Algebra and Trigonometry で、話し言葉には出てこない。"
    ],
    "related": [
      "cardioid",
      "limacon",
      "polar-equation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "バラ曲線",
        "en": "Rose (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "rose curve": 17
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rotate",
    "ja": {
      "term": "回転させる",
      "reading": "かいてんさせる",
      "alt": [
        "回転する"
      ]
    },
    "en": {
      "term": "rotate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "中1",
        "数C"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "図形や点を、ある点を中心に決まった角だけ回す。",
    "definition_en": "To turn a figure or a point through a given angle about a fixed point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you rotate the point 1 + i by 90 degrees about the origin, you land on −1 + i.",
        "ja": "点 1 + i を原点のまわりに 90° 回転させると −1 + i に移る。",
        "register": "spoken"
      },
      {
        "en": "Rotate the point z = 2 + i about the origin through π/3.",
        "ja": "点 z = 2 + i を原点のまわりに π/3 だけ回転させよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "rotate … about the origin",
        "ja": "原点のまわりに回転させる"
      },
      {
        "en": "rotate … through an angle θ",
        "ja": "角 θ だけ回転させる"
      }
    ],
    "pitfalls": [
      "「〜のまわりに」は about か around。「θ だけ」は through θ または by θ。"
    ],
    "related": [
      "rotation",
      "revolve-around-the-x-axis",
      "complex-plane"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "rotate": 419
      },
      "written": {
        "rotate": 223
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-10-09"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rotated-conics",
    "ja": {
      "term": "座標軸の回転",
      "reading": "ざひょうじくのかいてん",
      "alt": [
        "回転した二次曲線",
        "回転した円錐曲線"
      ]
    },
    "en": {
      "term": "rotation of axes",
      "alt": [
        "rotated conic"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "xy の項を含む二次曲線 Ax² + Bxy + Cy² + … = 0 を、座標軸を角 θ だけ回転して xy の項のない形にすること。",
    "definition_en": "Rotating the coordinate axes through an angle θ so that a conic Ax² + Bxy + Cy² + … = 0 loses its xy-term.",
    "latex": "\\cot 2\\theta = \\frac{A - C}{B}",
    "spoken_en": "cotangent of two theta equals A minus C over B",
    "examples": [
      {
        "en": "The xy term means the conic is tilted, so rotate the axes to get rid of it.",
        "ja": "xy の項があるのは二次曲線が傾いているから。軸を回転してそれを消す。",
        "register": "spoken"
      },
      {
        "en": "Use a rotation of axes to eliminate the xy-term in x² + xy + y² = 6, and identify the conic.",
        "ja": "座標軸の回転を用いて x² + xy + y² = 6 の xy の項を消し、二次曲線の種類を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "軸の回転の公式は日本の高校の学習指導要領には含まれない。書き言葉は大半が OpenStax Algebra and Trigonometry。"
    ],
    "related": [
      "conic-section",
      "rotation",
      "standard-form-of-a-conic"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "rotation of axes": 14,
        "rotated conic": 4
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rotation",
    "ja": {
      "term": "回転移動",
      "reading": "かいてんいどう",
      "alt": [
        "回転"
      ]
    },
    "en": {
      "term": "rotation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "complex-numbers"
    ],
    "level": {
      "jp": [
        "中1",
        "数C"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "図形の各点を、ある点を中心に同じ角だけ回して移すこと。複素数平面では、ある複素数を掛けることで表せる。",
    "definition_en": "A transformation that turns every point of a figure through the same angle about a fixed point; in the complex plane it can be done by multiplying by a complex number.",
    "latex": "z \\mapsto (\\cos\\theta + i\\sin\\theta)\\,z",
    "spoken_en": "z goes to cosine theta plus i sine theta, times z",
    "examples": [
      {
        "en": "Multiplying by i is a rotation by 90 degrees counterclockwise.",
        "ja": "i を掛けるのは、反時計回りに 90° の回転移動。",
        "register": "spoken"
      },
      {
        "en": "Describe the rotation that maps the triangle ABC onto the triangle A′B′C′.",
        "ja": "三角形 ABC を三角形 A′B′C′ に移す回転移動を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "a rotation of 90° about the origin",
        "ja": "原点のまわりの 90° の回転"
      }
    ],
    "pitfalls": [
      "向きは counterclockwise（反時計回り）／ clockwise（時計回り）で言う。角の正の向きは counterclockwise。",
      "回転の中心は center of rotation、回転角は angle of rotation。"
    ],
    "related": [
      "rotate",
      "complex-plane",
      "polar-form"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "回転",
        "en": "Rotation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "rotation": 206
      },
      "written": {
        "rotation": 139
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "rotational-symmetry",
    "ja": {
      "term": "回転対称",
      "reading": "かいてんたいしょう",
      "alt": []
    },
    "en": {
      "term": "rotational symmetry",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "小学校",
        "中1"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "図形をある点のまわりに 0° より大きく 360° より小さい角度だけ回転させると、もとの図形にぴったり重なる性質。180° の回転で重なる点対称はその特別な場合である。",
    "definition_en": "The property of a figure that maps onto itself when it is rotated about a point by an angle greater than 0° and less than 360°. Point symmetry is the special case of a 180° rotation.",
    "latex": "\\frac{360^\\circ}{n}",
    "spoken_en": "360 degrees divided by n",
    "examples": [
      {
        "en": "A regular hexagon has rotational symmetry: turn it 60 degrees about its center and it lands right on top of itself.",
        "ja": "正六角形は回転対称だ。中心のまわりに 60° 回すと、もとの図形にぴったり重なる。",
        "register": "spoken"
      },
      {
        "en": "Does a regular pentagon have rotational symmetry? If so, find the smallest angle of rotation that maps it onto itself.",
        "ja": "正五角形は回転対称か。回転対称ならば、もとの図形に重なる最小の回転角を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "小6 の点対称（180° の回転で重なる）は英語で point symmetry。rotational symmetry はそれより広く、120° や 72° で重なる図形も含む。",
      "0° や 360° の回転ならどの図形も重なるので、それだけでは rotational symmetry があるとは言わない。正 n 角形は 360°/n の回転で重なる。",
      "用例コーパスでは話し言葉・書き言葉とも少ない（話し言葉はすべて 3Blue1Brown、書き言葉は OpenStax Calculus）。CK-12 Geometry では 8.4 Rotation Symmetry に 14 件。"
    ],
    "related": [
      "rotation",
      "symmetric",
      "tessellation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "回転対称",
        "en": "Rotational symmetry"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.4 Rotation Symmetry・8.2 Lines of Symmetry"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "rotational symmetry": 4
      },
      "written": {
        "rotational symmetry": 1
      },
      "sources": [
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CK-12 の呼び方 rotational symmetry（CK-12 Geometry 8.4 Rotation Symmetry・CK-12 Geometry 8.2 Lines of Symmetry・計 15 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "round",
    "ja": {
      "term": "四捨五入する",
      "reading": "ししゃごにゅうする",
      "alt": []
    },
    "en": {
      "term": "round",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "英語の round は「丸める」一般を指し、round up（切り上げ）や round down（切り捨て）も含む。四捨五入に当たるのは round to the nearest …（最も近い値に丸める）の形で、round to the nearest tenth は小数第 1 位までの概数にすること。日本語は捨てる位（小数第 2 位を四捨五入）を言い、英語は残す位（the nearest tenth）を言うので、指す位がずれる。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある位までの概数にするとき、その 1 つ下の位の数字が 0〜4 なら切り捨て、5〜9 なら切り上げる。",
    "definition_en": "To replace a number with a nearby, simpler number at a given place value, such as the nearest ten or the nearest tenth. If the next digit is 5 or more, round up; otherwise, round down.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Round 3.47 to the nearest tenth: the 7 means you round up, so it's 3.5.",
        "ja": "3.47 の小数第 2 位を四捨五入しよう。7 だから切り上げて 3.5。",
        "register": "spoken"
      },
      {
        "en": "A circle has a radius of 4 cm. Find its circumference, and round your answer to the nearest hundredth.",
        "ja": "半径 4 cm の円の周の長さを求め、小数第 3 位を四捨五入して答えよ。",
        "register": "written"
      },
      {
        "en": "You can't rent 3.2 buses, so round up to 4.",
        "ja": "バスを 3.2 台は借りられないから、切り上げて 4 台。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「小数第 2 位を四捨五入する」は round to the nearest tenth（小数第 1 位まで残す）。日本語は捨てる位、英語は残す位を言うので、位が 1 つずれる。",
      "round は「丸める」一般で、round up（切り上げ）、round down（切り捨て）にも使う。四捨五入に当たるのは round to the nearest … の形。",
      "残す位の言い方は、nearest tenth が小数第 1 位、nearest hundredth が小数第 2 位、nearest whole number が整数（一の位）、nearest ten が十の位。"
    ],
    "related": [
      "approximate-value",
      "decimal",
      "digit",
      "floor-function",
      "ceiling-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "round to | round up | round down | round … to the nearest": 172
      },
      "written": {
        "round to | round up | round down | round … to the nearest": 616
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "row-echelon-form",
    "ja": {
      "term": "行階段形",
      "reading": "ぎょうかいだんけい",
      "alt": [
        "階段行列"
      ]
    },
    "en": {
      "term": "row-echelon form",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "各行の最初の 0 でない成分（先頭の 1）が、下の行ほど右にある形の行列。0 だけの行は下にまとめる。",
    "definition_en": "The form of a matrix in which each row's first nonzero entry (the leading 1) lies to the right of the one above, with any zero rows at the bottom.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Once it's in row-echelon form, solve from the bottom row up.",
        "ja": "行階段形になったら、一番下の行から解いていく。",
        "register": "spoken"
      },
      {
        "en": "Write the augmented matrix in row-echelon form and solve the system.",
        "ja": "拡大係数行列を行階段形にして、連立方程式を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "字幕は row echelon form、OpenStax は row-echelon form と書く（用例コーパスでは同じ言い方としてまとめた）。",
      "話し言葉はほとんどが MIT 18.06 で、多くは reduced row echelon form（簡約階段形）の一部。"
    ],
    "related": [
      "gaussian-elimination",
      "row-operations",
      "augmented-matrix"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "行階段形",
        "en": "Row echelon form"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "row-echelon form": 21
      },
      "written": {
        "row-echelon form": 38
      },
      "sources": [
        "mit-18.06",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "row-operations",
    "ja": {
      "term": "行基本変形",
      "reading": "ぎょうきほんへんけい",
      "alt": []
    },
    "en": {
      "term": "row operations",
      "alt": [
        "elementary row operations"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列の 2 つの行を入れかえる、ある行を 0 でない数倍する、ある行に他の行の数倍を加える、の 3 つの変形。",
    "definition_en": "The three operations on a matrix: swapping two rows, multiplying a row by a nonzero number, and adding a multiple of one row to another.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Use row operations to get zeros below the first entry.",
        "ja": "行基本変形で、最初の成分の下を 0 にする。",
        "register": "spoken"
      },
      {
        "en": "Use row operations to write the augmented matrix in row-echelon form.",
        "ja": "行基本変形を用いて、拡大係数行列を行階段形にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉はすべて MIT 18.06。書き言葉は大半が OpenStax Algebra and Trigonometry で、残りは OpenStax Intermediate Algebra。"
    ],
    "related": [
      "augmented-matrix",
      "gaussian-elimination",
      "row-echelon-form"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "row operations": 11
      },
      "written": {
        "row operations": 61,
        "elementary row operations": 3
      },
      "sources": [
        "mit-18.06",
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "row-space",
    "ja": {
      "term": "行空間",
      "reading": "ぎょうくうかん",
      "alt": []
    },
    "en": {
      "term": "row space",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列の行ベクトルが張る空間。行基本変形で変わらない。",
    "definition_en": "The space spanned by the rows of a matrix; row operations do not change it.",
    "latex": "\\operatorname{Row} A",
    "spoken_en": null,
    "examples": [
      {
        "en": "Row operations change the column space but not the row space.",
        "ja": "行基本変形で列空間は変わるが、行空間は変わらない。",
        "register": "spoken"
      },
      {
        "en": "Find a basis for the row space of the matrix.",
        "ja": "行列の行空間の基底を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて MIT 18.06。"
    ],
    "related": [
      "column-space",
      "rank",
      "null-space"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "行空間",
        "en": "Row and column spaces"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.4 Rank of a Matrix"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "row space": 160
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "row",
    "ja": {
      "term": "行",
      "reading": "ぎょう",
      "alt": []
    },
    "en": {
      "term": "row",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列の横の並び。上から第 1 行、第 2 行と数える。",
    "definition_en": "A horizontal line of entries in a matrix, counted from the top as the first row, the second row, and so on.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Multiply the first row by the first column to get that entry.",
        "ja": "第 1 行と第 1 列を掛けて、その成分を出す。",
        "register": "spoken"
      },
      {
        "en": "Interchange the first and second rows of the matrix.",
        "ja": "行列の第 1 行と第 2 行を入れかえよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は「行」と「列」を取り違えやすいが、英語は row（横）と column（縦）。"
    ],
    "related": [
      "column",
      "matrix",
      "row-operations"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "first row | second row | row of the matrix | row of a matrix": 157
      },
      "written": {
        "first row | second row | row of the matrix | row of a matrix": 64
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.02",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "ruler-postulate",
    "ja": {
      "term": "数直線上の距離",
      "reading": "すうちょくせんじょうのきょり",
      "alt": []
    },
    "en": {
      "term": "ruler postulate",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「数直線上の距離」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書は 2 点間の距離を数直線上の座標の差の絶対値として扱い、公準として名前を付けない。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "直線上の点と実数（座標）を 1 対 1 に対応させることができ、2 点間の距離はその座標の差の絶対値になる、という公準。",
    "definition_en": "The postulate that the points on a line can be paired one-to-one with the real numbers, so that the distance between two points is the absolute value of the difference of their coordinates.",
    "latex": "AB = |b - a|",
    "spoken_en": "AB equals the absolute value of b minus a",
    "examples": [
      {
        "en": "By the ruler postulate, if A is at negative 3 and B is at 5, then AB is the absolute value of 5 minus negative 3, which is 8.",
        "ja": "数直線上の距離の考え方で、A が −3、B が 5 にあるなら、AB は |5 − (−3)| で 8 になる。",
        "register": "spoken"
      },
      {
        "en": "Points A and B on a number line have coordinates −4 and 7. Use the ruler postulate to find AB.",
        "ja": "数直線上の 2 点 A, B の座標はそれぞれ −4, 7 である。AB を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語名の ruler は目盛りの付いた定規のこと。直線に定規を当てて目盛りを読むことを公準にしたもの。",
      "線分の長さの和 AB + BC = AC は別の公準（segment addition postulate）。"
    ],
    "related": [
      "segment-addition-postulate",
      "postulate",
      "number-line",
      "distance"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.3 Definition of Line segments"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 ruler postulate（CK-12 Geometry 1.3 Definition of Line segments・計 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "rules-of-inference",
    "ja": {
      "term": "推論規則",
      "reading": "すいろんきそく",
      "alt": []
    },
    "en": {
      "term": "rules of inference",
      "alt": [
        "rule of inference",
        "inference rule"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（rules of inference ／ rule of inference ／ inference rule は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "logic",
      "proof"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "真であるとわかっている前提から、どのような結論を正しく導けるかを定めた推論の型のこと。肯定式や三段論法などがある。",
    "definition_en": "The valid patterns of reasoning that let you draw a conclusion from premises already known to be true. Examples include the law of detachment and the law of syllogism.",
    "latex": "\\frac{p \\to q \\qquad p}{\\therefore\\ q}",
    "spoken_en": "p implies q, p, therefore q",
    "examples": [
      {
        "en": "Every line of the proof has to follow from earlier lines by one of the rules of inference.",
        "ja": "証明のどの行も、前の行から推論規則のどれかで導かれていなければならない。",
        "register": "spoken"
      },
      {
        "en": "State which of the rules of inference justifies each step of the argument below.",
        "ja": "次の論証の各段階を正当化している推論規則を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形の rules of inference は規則の集まり全体を指す。1 つの規則は a rule of inference（inference rule）と言う。",
      "英語の inference は統計の「推測」（statistical inference）にも使うので、論理の話か統計の話かを文脈で見分ける。",
      "用例コーパスでは書き言葉に inference rule が少しある（すべて MIT の講義ノート）だけで、rules of inference は出てこない。話し言葉にはどちらも出てこない。"
    ],
    "related": [
      "law-of-detachment",
      "law-of-syllogism",
      "propositional-logic",
      "proof"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "推論規則",
        "en": "Rule of inference"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "inference rule": 9
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは Wikipedia の記事名 rules of inference（英語版「Rule of inference」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "saddle-point",
    "ja": {
      "term": "鞍点",
      "reading": "あんてん",
      "alt": []
    },
    "en": {
      "term": "saddle point",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "偏導関数がともに 0 になる点（停留点）のうち、極大でも極小でもない点。",
    "definition_en": "A critical point of a function of several variables that is neither a local maximum nor a local minimum.",
    "latex": "f(x, y) = x^2 - y^2 \\text{ at } (0, 0)",
    "spoken_en": null,
    "examples": [
      {
        "en": "It goes up in one direction and down in the other — that's a saddle point.",
        "ja": "ある向きには上がり、別の向きには下がる。それが鞍点。",
        "register": "spoken"
      },
      {
        "en": "Find and classify the critical points of f(x, y) = x³ − 3x + y², including any saddle points.",
        "ja": "f(x, y) = x³ − 3x + y² の停留点を求め、鞍点を含めて分類せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "判定には 2 変数の second derivative test（D = f_xx f_yy − f_xy²）を使う。D < 0 なら鞍点。"
    ],
    "related": [
      "local-extremum",
      "partial-derivative",
      "lagrange-multipliers"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "鞍点",
        "en": "Saddle point"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "saddle point": 21
      },
      "written": {
        "saddle point": 25
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "same-side-exterior-angles",
    "ja": {
      "term": "同側外角",
      "reading": "どうそくがいかく",
      "alt": []
    },
    "en": {
      "term": "same-side exterior angles",
      "alt": [
        "consecutive exterior angles"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「同側外角」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書には名前がない。平行線なら和が 180°。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 直線に 1 本の直線が交わってできる角のうち、2 直線の外側にあり、交わる直線に対して同じ側にある 2 つの角。2 直線が平行なら和は 180° になる。",
    "definition_en": "A pair of angles formed when a transversal crosses two lines, lying outside the two lines and on the same side of the transversal. If the two lines are parallel, the angles are supplementary.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These are same-side exterior angles, so if the lines are parallel, they're not equal — they add up to 180.",
        "ja": "これは同側外角だから、2 直線が平行なら等しくなるんじゃなくて、足して 180° になる。",
        "register": "spoken"
      },
      {
        "en": "Lines a and b are parallel and are cut by a transversal. Two same-side exterior angles measure (2x + 14)° and (3x − 9)°. Find x.",
        "ja": "直線 a, b は平行で、1 本の直線がこれらと交わっている。同側外角の 2 つの角の大きさが (2x + 14)° と (3x − 9)° のとき、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "錯角や同位角と違い、平行線の同側外角は等しくならず、補角（和が 180°）になる。",
      "内側の組は same-side interior angles（consecutive interior angles）と呼び、平行線なら同じく和が 180°。same-side と consecutive のどちらを使うかは教科書による。",
      "same-side exterior angles・consecutive exterior angles は、用例コーパスにも参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）にも出てこない。"
    ],
    "related": [
      "transversal",
      "alternate-exterior-angles",
      "supplementary-angle",
      "parallel-lines"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。consecutive exterior angles を候補に入れて数え直しても決まらず、そのまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "same-side-interior-angles",
    "ja": {
      "term": "同側内角",
      "reading": "どうそくないかく",
      "alt": []
    },
    "en": {
      "term": "same-side interior angles",
      "alt": [
        "consecutive interior angles"
      ]
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "米国の Geometry は、2 直線の内側で交わる直線の同じ側にある角の組に same-side interior angles という名前を付け、2 直線が平行なら和が 180°（supplementary）になることを定理として扱う（CK-12 Geometry 3.7 Same Side Interior Angles）。同じ組を consecutive interior angles とも呼ぶ（英語版 Wikipedia「Transversal (geometry)」の節の名前）。中学校学習指導要領解説（第 2 学年 B 図形）は平行線の性質として同位角・錯角を挙げ、「同側内角」は出てこない（日本語版 Wikipedia にも無い）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 本の直線に 1 本の直線が交わってできる角のうち、2 直線の内側にあり、交わる直線に対して同じ側にある 2 つの角。2 直線が平行なら、和は 180° になる。",
    "definition_en": "A pair of angles formed when a transversal crosses two lines, lying between the two lines and on the same side of the transversal. If the two lines are parallel, same-side interior angles are supplementary.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The lines are parallel, so these same-side interior angles add up to 180: this one's 110, so that one's 70.",
        "ja": "2 直線は平行だから、この同側内角は足して 180°。こっちが 110° なら、あっちは 70°。",
        "register": "spoken"
      },
      {
        "en": "Lines p and q are parallel and are cut by a transversal. A pair of same-side interior angles measure (3x + 20)° and (2x + 10)°. Find x.",
        "ja": "直線 p と q は平行で、1 本の直線がこれらと交わっている。同側内角の組の大きさが (3x + 20)°、(2x + 10)° のとき、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "錯角や同位角と違い、平行線の同側内角は等しくならず、和が 180°（supplementary、エントリ supplementary-angle）になる。",
      "same-side interior angles と consecutive interior angles は同じ角の組を指す。",
      "2 直線の外側で同じ側にある組は same-side exterior angles（エントリ same-side-exterior-angles）。"
    ],
    "related": [
      "transversal",
      "supplementary-angle",
      "alternate-interior-angles",
      "same-side-exterior-angles",
      "parallel-lines"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "3.7 Same Side Interior Angles・2.15 Proofs- Angle Pairs and Segments"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Transversal_(geometry)#Consecutive_interior_angles",
        "note": "記事「Transversal (geometry)」の節 Consecutive interior angles"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 B 図形: 平行線の性質（同位角・錯角。「同側内角」は出てこない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "consecutive interior angles": 8,
        "same-side interior angles": 1
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 9 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 same-side interior angles（CK-12 Geometry 3.7 Same Side Interior Angles・CK-12 Geometry 2.15 Proofs- Angle Pairs and Segments・計 23 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sample-mean",
    "ja": {
      "term": "標本平均",
      "reading": "ひょうほんへいきん",
      "alt": []
    },
    "en": {
      "term": "sample mean",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3",
        "数B"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "標本のデータの平均値。x̄ と書く。",
    "definition_en": "The mean of the data in a sample, written x̄.",
    "latex": "\\bar{x} = \\frac{1}{n}\\sum_{i=1}^{n} x_i",
    "spoken_en": "x bar equals one over n times the sum of x sub i",
    "examples": [
      {
        "en": "The sample mean changes from sample to sample, but the population mean doesn't.",
        "ja": "標本平均は標本ごとに変わるが、母平均は変わらない。",
        "register": "spoken"
      },
      {
        "en": "The sample mean of 36 measurements is 12.4, and the population standard deviation is 3. Find a 95% confidence interval for the population mean.",
        "ja": "36 個の測定値の標本平均は 12.4、母標準偏差は 3 である。母平均の 95% 信頼区間を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "x̄ は x bar と読む（用例コーパスの話し言葉にも出てくる）。"
    ],
    "related": [
      "population-mean",
      "arithmetic-mean",
      "sample"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "標本平均・標本共分散",
        "en": "Sample mean and covariance"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sample mean": 209
      },
      "written": {
        "sample mean": 272
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sample-proportion",
    "ja": {
      "term": "標本比率",
      "reading": "ひょうほんひりつ",
      "alt": []
    },
    "en": {
      "term": "sample proportion",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "標本の中で、ある性質をもつものの割合。p̂ と書く。",
    "definition_en": "The fraction of the sample that has a given characteristic, written p̂.",
    "latex": "\\hat{p} = \\frac{x}{n}",
    "spoken_en": "p hat equals x over n",
    "examples": [
      {
        "en": "Out of 200 people, 58 said yes, so the sample proportion is 0.29.",
        "ja": "200 人中 58 人がはいと答えたので、標本比率は 0.29。",
        "register": "spoken"
      },
      {
        "en": "Find the sample proportion and use it to test the claim that p = 0.25.",
        "ja": "標本比率を求め、p = 0.25 という主張を検定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "p̂ は p hat と読む。"
    ],
    "related": [
      "population-proportion",
      "sample"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "sample proportion": 154
      },
      "written": {
        "sample proportion": 19
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sample-size",
    "ja": {
      "term": "標本の大きさ",
      "reading": "ひょうほんのおおきさ",
      "alt": [
        "標本サイズ"
      ]
    },
    "en": {
      "term": "sample size",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "標本に含まれるデータの個数。ふつう n で表す。",
    "definition_en": "The number of individuals or data values in a sample, usually written n.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you quadruple the sample size, you cut the standard error in half.",
        "ja": "標本の大きさを 4 倍にすると、標準誤差は半分になる。",
        "register": "spoken"
      },
      {
        "en": "What sample size is needed so that the margin of error is at most 3 percentage points with 95% confidence?",
        "ja": "信頼度 95% で誤差の範囲を 3 ポイント以下にするには、標本の大きさはいくら必要か。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "n は sample size の記号。n が大きいほど標本平均のばらつき（標準誤差）は小さくなる。"
    ],
    "related": [
      "sample",
      "standard-error",
      "error-bound"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sample size": 210
      },
      "written": {
        "sample size": 203
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sample-space",
    "ja": {
      "term": "全事象",
      "reading": "ぜんじしょう",
      "alt": [
        "標本空間"
      ]
    },
    "en": {
      "term": "sample space",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Intro Statistics",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "ある試行で起こりうる結果全体の集合。",
    "definition_en": "The collection of every outcome that an experiment can produce.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For two coin flips, the sample space is heads-heads, heads-tails, tails-heads, and tails-tails.",
        "ja": "硬貨を 2 回投げるときの全事象は、表表・表裏・裏表・裏裏。",
        "register": "spoken"
      },
      {
        "en": "List the sample space for rolling two dice.",
        "ja": "さいころを 2 つ振るときの全事象を書き出せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高等学校学習指導要領解説（数学A）は全事象を U で表す（P(U) = 1）。"
    ],
    "related": [
      "expected-value",
      "tree-diagram"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "標本空間",
        "en": "Sample space"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 場合の数と確率: 全事象 U の確率 P(U) = 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sample space": 95
      },
      "written": {
        "sample space": 167
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sample-survey",
    "ja": {
      "term": "標本調査",
      "reading": "ひょうほんちょうさ",
      "alt": []
    },
    "en": {
      "term": "sample survey",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（sample survey は用例コーパスに出てこない）。take a sample（標本をとる。話し言葉ではほとんどが Khan Academy）のように動詞で言う。IM Grade 7 と OpenStax Introductory Statistics も take a sample ／ select a sample と動詞で言い、AP Statistics の CED は sampling method（topic 1.11）と言う。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "集団の一部を取り出して調べ、その結果から集団全体の傾向を推測する調査。",
    "definition_en": "A survey of part of a group, used to infer the characteristics of the whole group.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We can't ask every student, so we'll survey a sample and use it to estimate.",
        "ja": "全員には聞けないので、一部を調査して推定する。",
        "register": "spoken"
      },
      {
        "en": "Explain why a sample survey is used instead of a census in this situation.",
        "ja": "この場合に全数調査ではなく標本調査を行う理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "母集団から標本を取り出すことは sampling（エントリ sampling）、全員を調べる調査は census（エントリ census。AP Statistics の CED topic 1.10）。"
    ],
    "related": [
      "census",
      "sample",
      "population",
      "sampling"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "標本調査",
        "en": "Sampling (statistics)"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Grade 7 8.14・8.16: take a sample ／ select a sample"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.11（sampling method）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sample",
    "ja": {
      "term": "標本",
      "reading": "ひょうほん",
      "alt": [
        "サンプル"
      ]
    },
    "en": {
      "term": "sample",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Pre-Algebra",
        "Algebra 2",
        "AP Statistics"
      ]
    },
    "definition_ja": "母集団から調査のために取り出した一部。",
    "definition_en": "The part of a population that is actually selected and studied.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A bigger sample usually gives you a better estimate.",
        "ja": "ふつうは標本が大きいほど、よい推定になる。",
        "register": "spoken"
      },
      {
        "en": "A random sample of 200 voters was taken. Identify the population and the sample.",
        "ja": "有権者 200 人の無作為標本をとった。母集団と標本を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "take a sample",
        "ja": "標本をとる"
      },
      {
        "en": "random sample",
        "ja": "無作為標本"
      }
    ],
    "pitfalls": [
      "sample space（標本空間・全事象）の sample は別の概念。"
    ],
    "related": [
      "population",
      "sample-space",
      "sampling",
      "random-sampling"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "標本 (統計学)",
        "en": "Sample (statistics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sample": 1927
      },
      "written": {
        "sample": 2105
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sampling-distribution-of-a-proportion",
    "ja": {
      "term": "標本比率の分布",
      "reading": "ひょうほんひりつのぶんぷ",
      "alt": []
    },
    "en": {
      "term": "sampling distribution of the sample proportion",
      "alt": [
        "sampling distribution of a proportion"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "同じ大きさ n の標本を繰り返し取ったときの標本比率 p̂ の分布。平均は p、標準偏差は √(p(1 − p)/n)。",
    "definition_en": "The distribution of the sample proportion p̂ over all samples of size n; its mean is p and its standard deviation is √(p(1 − p)/n).",
    "latex": "\\sigma_{\\hat{p}} = \\sqrt{\\frac{p(1-p)}{n}}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The sampling distribution of the sample proportion is centered at the true p.",
        "ja": "標本比率の分布の中心は真の p。",
        "register": "spoken"
      },
      {
        "en": "Describe the sampling distribution of p̂ for samples of size 100 when p = 0.3.",
        "ja": "p = 0.3 のとき、大きさ 100 の標本について p̂ の分布を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて Khan Academy。OpenStax Introductory Statistics には出てこない。"
    ],
    "related": [
      "sample-proportion",
      "sampling-distribution",
      "normal-approximation-to-the-binomial"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "sampling distribution of the sample proportion": 21
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sampling-distribution-of-the-mean",
    "ja": {
      "term": "標本平均の分布",
      "reading": "ひょうほんへいきんのぶんぷ",
      "alt": []
    },
    "en": {
      "term": "sampling distribution of the sample mean",
      "alt": [
        "sampling distribution of the mean"
      ],
      "uk": null,
      "register": "spoken",
      "variants": [
        {
          "term": "distribution of the sample mean",
          "register": "written",
          "note": "OpenStax Introductory Statistics（7.1 The Central Limit Theorem for Sample Means）の書き方。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "同じ大きさ n の標本を繰り返し取ったときの標本平均 x̄ の分布。平均は母平均 μ、標準偏差は σ/√n。",
    "definition_en": "The distribution of the sample mean x̄ over all samples of size n; its mean is the population mean μ and its standard deviation is σ/√n.",
    "latex": "\\sigma_{\\bar{x}} = \\frac{\\sigma}{\\sqrt{n}}",
    "spoken_en": "sigma sub x bar equals sigma over the square root of n",
    "examples": [
      {
        "en": "The sampling distribution of the sample mean gets narrower as n gets bigger.",
        "ja": "標本平均の分布は、n が大きくなるほど幅が狭くなる。",
        "register": "spoken"
      },
      {
        "en": "Describe the shape, center, and spread of the sampling distribution of the mean for samples of size 36.",
        "ja": "大きさ 36 の標本について、標本平均の分布の形・中心・散らばりを述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて Khan Academy。書き言葉は of the sample mean と of the mean が同じくらい。"
    ],
    "related": [
      "sample-mean",
      "central-limit-theorem",
      "standard-error",
      "sampling-distribution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "7.1 The Central Limit Theorem for Sample Means（distribution of the sample mean）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "sampling distribution of the sample mean": 38,
        "distribution of the sample mean": 1
      },
      "written": {
        "sampling distribution of the mean": 7,
        "distribution of the sample mean": 11,
        "sampling distribution of the sample mean": 7
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sampling-distribution",
    "ja": {
      "term": "標本分布",
      "reading": "ひょうほんぶんぷ",
      "alt": []
    },
    "en": {
      "term": "sampling distribution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "同じ大きさの標本を繰り返し取ったときの、標本平均や標本比率などの統計量の分布。",
    "definition_en": "The distribution of a statistic, such as the sample mean or the sample proportion, over all samples of the same size.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The sampling distribution shows how the statistic would vary if we took sample after sample.",
        "ja": "標本分布は、標本を何度も取ったときに統計量がどうばらつくかを表す。",
        "register": "spoken"
      },
      {
        "en": "Describe the sampling distribution of the sample mean for samples of size 25.",
        "ja": "大きさ 25 の標本について、標本平均の標本分布を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "標本のデータの分布（distribution of the sample）とは別。標本分布は統計量の分布。",
      "用例コーパスの話し言葉はすべて Khan Academy。"
    ],
    "related": [
      "sampling-distribution-of-the-mean",
      "sampling-distribution-of-a-proportion",
      "standard-error",
      "central-limit-theorem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "sampling distribution": 147
      },
      "written": {
        "sampling distribution": 31
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sampling-error",
    "ja": {
      "term": "標本誤差",
      "reading": "ひょうほんごさ",
      "alt": [
        "標本抽出の誤り",
        "標本の変動"
      ]
    },
    "en": {
      "term": "sampling error",
      "alt": [
        "sampling variability"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "標本を無作為に選ぶことで生じる、統計量と母集団の値とのずれ。標本ごとに統計量がばらつくこと。",
    "definition_en": "The difference between a statistic and the population value that comes from taking a random sample; statistics vary from sample to sample.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Even a perfectly random sample won't hit the true mean exactly; that's sampling error.",
        "ja": "完全に無作為な標本でも真の平均にぴったりは当たらない。それが標本誤差だ。",
        "register": "spoken"
      },
      {
        "en": "Explain how increasing the sample size affects the sampling error.",
        "ja": "標本の大きさを大きくすると標本誤差がどうなるか説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の error は「誤り」ではなく「ずれ」。回答の偏りなどの非標本誤差（nonsampling error）とは別。",
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics。AP の sampling variability は en.alt（書き言葉では sampling error より少ない）。"
    ],
    "related": [
      "sampling-distribution",
      "sample-size",
      "standard-error",
      "is-biased"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "sampling error": 16,
        "sampling variability": 5
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sampling-with-replacement",
    "ja": {
      "term": "復元抽出",
      "reading": "ふくげんちゅうしゅつ",
      "alt": []
    },
    "en": {
      "term": "with replacement",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "取り出したものを毎回元に戻してから次を取り出す抽出。どの回も同じ条件で取り出すので、各回の結果は独立になる。",
    "definition_en": "Drawing items one at a time and putting each one back before the next draw, so every draw is made from the same full set and the draws are independent.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We're drawing with replacement, so the chance of red is 3/5 on every single draw.",
        "ja": "復元抽出なので、赤が出る確率はどの回も 3/5 のまま。",
        "register": "spoken"
      },
      {
        "en": "A ball is drawn at random from the bag and then replaced, 4 times in all (sampling with replacement). Find the probability that exactly 2 of the draws are red.",
        "ja": "袋から玉を 1 個取り出して色を調べてから元に戻すことを 4 回繰り返す。赤玉がちょうど 2 回出る確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「復元抽出」に当たる 1 語の名詞は用例コーパスと参照に出てこず、動詞に副詞句を付けて draw … with replacement ／ sampling with replacement と言う。",
      "復元抽出は各回が独立なので反復試行（二項分布）になる。元に戻さない非復元抽出（without replacement）とは確率の計算が変わる。"
    ],
    "related": [
      "sampling-without-replacement",
      "repeated-trials",
      "independence-of-events",
      "binomial-probability",
      "sampling"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "with replacement": 26
      },
      "written": {
        "with replacement": 38
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sampling-without-replacement",
    "ja": {
      "term": "非復元抽出",
      "reading": "ひふくげんちゅうしゅつ",
      "alt": []
    },
    "en": {
      "term": "without replacement",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "取り出したものを元に戻さずに、続けて次を取り出す抽出。残りの個数が回ごとに変わるので、各回の結果は独立でない。",
    "definition_en": "Drawing items one at a time without putting any back, so the set shrinks after each draw and the draws are not independent.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's without replacement, so after the first red one comes out there are only 2 reds left out of 4.",
        "ja": "非復元抽出だから、最初に赤が出たあとは残り 4 個のうち赤は 2 個しかない。",
        "register": "spoken"
      },
      {
        "en": "Two cards are drawn without replacement from a standard 52-card deck. Find the probability that both are aces.",
        "ja": "52 枚のトランプから、元に戻さずに 2 枚続けて引く。2 枚ともエースである確率を求めよ。",
        "register": "written"
      },
      {
        "en": "Ten people draw lots without replacement from a box with 3 winning tickets. Each person has a 3/10 chance of winning, whether they draw first or last.",
        "ja": "当たり 3 本を含む 10 本のくじを 10 人が順に引く（引いたくじは戻さない）。何番目に引いても当たる確率は 3/10 である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "くじ引きは drawing lots。引いたくじを戻さないので非復元抽出だが、何番目に引いても当たる確率は同じになる（順番に有利・不利はない）。",
      "非復元抽出では 2 回目以降の確率が前の結果で変わるので、条件付き確率を掛けて計算する（3/10 × 2/9 のように）。"
    ],
    "related": [
      "sampling-with-replacement",
      "conditional-probability",
      "sampling",
      "probability"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "without replacement": 19
      },
      "written": {
        "without replacement": 42
      },
      "sources": [
        "khan-ap-stats",
        "mit-notes",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sampling",
    "ja": {
      "term": "標本抽出",
      "reading": "ひょうほんちゅうしゅつ",
      "alt": [
        "標本抽出法"
      ]
    },
    "en": {
      "term": "sampling",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 3",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "母集団から標本を取り出すこと。また、その方法。",
    "definition_en": "Selecting a sample from a population, or the method used to do it.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The sampling here was biased because only people who answered the phone were counted.",
        "ja": "電話に出た人しか数えなかったので、この標本抽出は偏っていた。",
        "register": "spoken"
      },
      {
        "en": "Identify the sampling method used in each study: simple random, stratified, cluster, or systematic.",
        "ja": "それぞれの調査で使われた標本抽出の方法（単純無作為・層化・集落・系統）を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "random-sampling",
      "sample",
      "sample-survey"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.11（sampling method）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "sampling method | sampling technique": 26
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sas-congruence",
    "ja": {
      "term": "2 組の辺とその間の角がそれぞれ等しい",
      "reading": "にくみのへんとそのあいだのかくがそれぞれひとしい",
      "alt": [
        "二辺夾角相等",
        "SAS"
      ]
    },
    "en": {
      "term": "side-angle-side",
      "alt": [
        "SAS",
        "side-angle-side triangle congruence theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "proof",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの三角形で、2 組の辺とその間の角がそれぞれ等しければ合同であるという合同条件。三角比では、2 辺とその間の角が分かっている場合を指し、余弦定理で残りの辺を求める。",
    "definition_en": "A congruence criterion: two triangles are congruent when two sides and the angle between them in one triangle match two sides and the angle between them in the other. In trigonometry, SAS also names the case where two sides and the included angle are known, which the law of cosines solves.",
    "latex": "AB = DE,\\ \\angle B = \\angle E,\\ BC = EF \\implies \\triangle ABC \\cong \\triangle DEF",
    "spoken_en": "if AB equals DE, angle B equals angle E, and BC equals EF, then triangle ABC is congruent to triangle DEF",
    "examples": [
      {
        "en": "We've got two pairs of congruent sides, and the angle between them matches too, so these triangles are congruent by side-angle-side, SAS.",
        "ja": "2 組の辺が等しく、その間の角も等しいから、この 2 つの三角形は合同だ。",
        "register": "spoken"
      },
      {
        "en": "Make sure the angle is the one between the two sides. That's what the A in the middle of SAS tells you.",
        "ja": "角が 2 辺の間にあるかを確かめよう。SAS の真ん中の A はそういう意味だ。",
        "register": "spoken"
      },
      {
        "en": "In the figure, AB = AD and ∠BAC = ∠DAC. Prove that △ABC ≅ △ADC.",
        "ja": "図で AB = AD, ∠BAC = ∠DAC のとき、△ABC ≡ △ADC であることを証明せよ。",
        "register": "written"
      },
      {
        "en": "Since AB = AD, ∠BAC = ∠DAC, and AC = AC, △ABC ≅ △ADC by the Side-Angle-Side Triangle Congruence Theorem.",
        "ja": "AB = AD、∠BAC = ∠DAC、AC は共通なので、2 組の辺とその間の角がそれぞれ等しく、△ABC ≡ △ADC。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "by SAS",
        "ja": "2 組の辺とその間の角がそれぞれ等しいから"
      }
    ],
    "pitfalls": [
      "英語の並び side-angle-side は、角が 2 辺の間にあることを表す。2 辺とその間にない角（SSA）は合同条件にならない（正弦定理の ambiguous case）。",
      "三角比の教科書（OpenStax Algebra and Trigonometry など）では、SAS は「2 辺とその間の角が分かっている三角形」を解く場合の名前としても使う（余弦定理を使う）。同じ文字だが、合同の証明とは別の使い方。",
      "CK-12 Geometry は by SAS などの略号の形が 13 件、side-angle-side が 3 件。",
      "IM Geometry はこの合同条件を Side-Angle-Side Triangle Congruence Theorem と呼ぶ（Geometry 2.6 ほか、17 件）。CK-12 Geometry は SAS Postulate（4.14 SAS）。"
    ],
    "related": [
      "sss-congruence",
      "asa-congruence",
      "aas-congruence",
      "congruent",
      "law-of-cosines",
      "ambiguous-case"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "2.6 Side-Angle-Side Triangle Congruence ほか（27 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.14 SAS ほか（by SAS 13 件・side-angle-side 3 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "SAS（2 辺とその間の角が分かっている三角形。余弦定理）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "by SAS | SAS congruence | (SAS) congruence | SAS postulate | SAS criterion | SAS theorem | SAS triangle congruence": 2
      },
      "written": {
        "side-angle-side": 1
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 side-angle-side（Geometry 2.6 Side-Angle-Side Triangle Congruence・Geometry 3.10 Other Conditions for Triangle Similarity・Geometry 2.7 Angle-Side-Angle Triangle Congruence・計 27 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "satisfy",
    "ja": {
      "term": "満たす",
      "reading": "みたす",
      "alt": [
        "方程式を満たす"
      ]
    },
    "en": {
      "term": "satisfy",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1",
        "Geometry"
      ]
    },
    "definition_ja": "ある値を方程式や不等式の文字に代入したとき、その式が成り立つこと。値が条件に当てはまることにも使う。",
    "definition_en": "A value satisfies an equation or inequality if substituting it makes the statement true. It is also used for a number that meets a given condition.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Both sides come out to 5, so x = 2 does satisfy the equation.",
        "ja": "両辺とも 5 になるから、x = 2 はこの方程式を満たしているね。",
        "register": "spoken"
      },
      {
        "en": "Determine whether (x, y) = (3, 1) satisfies both equations 2x + y = 7 and x − y = 2.",
        "ja": "(x, y) = (3, 1) が 2 つの方程式 2x + y = 7、x − y = 2 をともに満たすかどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "satisfy のふつうの意味は「満足させる」。数学では主語が値、目的語が方程式・不等式・条件になる（x = 2 satisfies the equation.、satisfy the condition）。",
      "「値が方程式を満たす」（satisfy）と「方程式がその値で成り立つ」（hold、エントリ hold）は主語が逆になる。",
      "連立方程式の解は、両方の式を同時に満たす（satisfies both equations）値の組。片方の式だけを満たす組は解ではない。"
    ],
    "related": [
      "hold",
      "solution",
      "check",
      "system-of-linear-equations",
      "condition"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "satisfy": 631
      },
      "written": {
        "satisfy": 396
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "scalar-multiplication",
    "ja": {
      "term": "ベクトルの実数倍",
      "reading": "べくとるのじっすうばい",
      "alt": []
    },
    "en": {
      "term": "scalar multiplication",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "scalar multiple",
          "register": "spoken",
          "note": "実数倍したベクトルのほうを指す言い方。話し言葉では大半が Professor Leonard。書き言葉にも出てくるが、scalar multiplication のほうが多い。"
        }
      ],
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトル a と実数 k から、大きさが |k| 倍で、k > 0 なら同じ向き、k < 0 なら反対向きのベクトル ka を作ること。",
    "definition_en": "Multiplying a vector a by a real number k to get ka, whose magnitude is |k| times as large and whose direction is the same as a if k > 0 and opposite if k < 0.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If one vector is a scalar multiple of the other, they're parallel.",
        "ja": "一方が他方の実数倍なら、2 つは平行。",
        "register": "spoken"
      },
      {
        "en": "Find 3a − 2b for a = ⟨1, −2⟩ and b = ⟨4, 0⟩.",
        "ja": "a = ⟨1, −2⟩, b = ⟨4, 0⟩ について 3a − 2b を求めよ。",
        "register": "written"
      },
      {
        "en": "Scalar multiplication just stretches or shrinks the vector: 3a points the same way as a but is three times as long.",
        "ja": "ベクトルの実数倍は、ベクトルを伸ばしたり縮めたりするだけ。3a は a と同じ向きで、長さが 3 倍。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "scalar",
      "parallel-vectors",
      "vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "scalar multiple": 17,
        "scalar multiplication": 7
      },
      "written": {
        "scalar multiplication": 43,
        "scalar multiple": 14
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "scalar-triple-product",
    "ja": {
      "term": "スカラー三重積",
      "reading": "すからーさんじゅうせき",
      "alt": []
    },
    "en": {
      "term": "triple scalar product",
      "alt": [
        "scalar triple product"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "3 つのベクトルについての a · (b × c)。絶対値は 3 つのベクトルがつくる平行六面体の体積になる。",
    "definition_en": "For three vectors, the number a · (b × c); its absolute value is the volume of the parallelepiped they span.",
    "latex": "\\mathbf{a} \\cdot (\\mathbf{b} \\times \\mathbf{c})",
    "spoken_en": "a dot b cross c",
    "examples": [
      {
        "en": "If the triple scalar product is zero, the three vectors lie in one plane.",
        "ja": "スカラー三重積が 0 なら、3 つのベクトルは同じ平面上にある。",
        "register": "spoken"
      },
      {
        "en": "Use the triple scalar product to find the volume of the parallelepiped determined by u, v, and w.",
        "ja": "スカラー三重積を用いて、u、v、w がつくる平行六面体の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 3 は triple scalar product と書く（30 件）。scalar triple product の語順はほとんど出てこない。"
    ],
    "related": [
      "coordinates-in-space",
      "equation-of-a-plane"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "三重積 (ベクトル解析)",
        "en": "Triple product"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "triple scalar product": 30,
        "scalar triple product": 1
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "scalar",
    "ja": {
      "term": "スカラー",
      "reading": "すからー",
      "alt": []
    },
    "en": {
      "term": "scalar",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトルに対して、向きをもたず大きさだけで表される量。ふつうの実数。",
    "definition_en": "A quantity described by a single number, with no direction, as opposed to a vector; an ordinary real number.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The dot product gives you a scalar, not a vector.",
        "ja": "内積はベクトルではなくスカラーになる。",
        "register": "spoken"
      },
      {
        "en": "Determine whether each quantity is a scalar or a vector: speed, velocity, mass, force.",
        "ja": "速さ・速度・質量・力のそれぞれがスカラーかベクトルか答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector",
      "scalar-multiplication",
      "dot-product"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "スカラー (数学)",
        "en": "Scalar (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "scalar": 123
      },
      "written": {
        "scalar": 252
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "scale-drawing",
    "ja": {
      "term": "縮図",
      "reading": "しゅくず",
      "alt": []
    },
    "en": {
      "term": "scale drawing",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "IM Grade 7 1.7 Scale Drawings は、図の長さを同じ scale factor で enlarged or reduced した図をまとめて scale drawing と呼ぶ。日本語の「縮図」は縮めた図だけで、拡大した図は拡大図（学習指導要領解説は縮図・拡大図を小学校第 6 学年の内容とする）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "小学校",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "実物の形を変えずに、すべての長さを同じ割合で縮めてかいた図。縮めた割合を縮尺という。",
    "definition_en": "A drawing of a real object or place in which every length is multiplied by the same number, so the shape stays the same. The scale tells how a length on the drawing compares with the actual length.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "On this scale drawing, 1 inch stands for 10 feet, so a wall that's 3 inches long on paper is really 30 feet.",
        "ja": "この縮図では 1 インチが 10 フィートを表すから、紙の上で 3 インチの壁は実際には 30 フィートだね。",
        "register": "spoken"
      },
      {
        "en": "A scale drawing of a park uses a scale of 1 cm to 50 m, and the park is 7 cm long on the drawing. Find the actual length of the park.",
        "ja": "1 cm が 50 m を表す縮尺でかいた公園の縮図で、公園の長さは 7 cm である。公園の実際の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "IM Grade 7 1.7 の scale drawing は、縮めた図にも拡大した図にも使う。日本語の「縮図」と「拡大図」の両方に当たる。",
      "縮尺（エントリ scale）は 1 : 5000 のように単位のない比でも、1 cm で 50 m のように単位付きでも表せる。単位付きのときは単位をそろえてから比に直す（1 cm : 50 m = 1 cm : 5000 cm = 1 : 5000）。",
      "scale は「縮尺」のほかに、グラフの軸の「目盛り」や「はかり」の意味でも使う。"
    ],
    "related": [
      "scale",
      "scale-factor",
      "similar",
      "dilation",
      "ratio"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Grade 7 1.7 Scale Drawings・1.9・1.10 と glossary（enlarged or reduced by the same scale factor）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "縮図や拡大図は小学校第 6 学年の内容"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "scale drawing": 11
      },
      "sources": [
        "khan-middle"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "scale-factor",
    "ja": {
      "term": "相似比",
      "reading": "そうじひ",
      "alt": [
        "拡大率"
      ]
    },
    "en": {
      "term": "scale factor",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（中学校）は対応する線分の長さの比を相似比とし、日本語版 Wikipedia「図形の相似」は F を r 倍して G と合同になるとき F と G の相似比を 1 : r と定義する（2 つの図形を挙げた順に比で書き、1 つの数にはしない）。共通テスト（令和 7 年度 数学I・A）の問題文も「その相似比は □ : □」。英語の scale factor は元の図形から像への 1 つの数（3/2）で、向きがある（IM Grade 7 の問題文は the scale factor from Polygon A to Polygon B と書く）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "相似な 2 つの図形の、対応する長さの比。英語の scale factor は、もとの図形の長さを何倍すると像の長さになるかを 1 つの数で表す。",
    "definition_en": "The number by which every length of a figure is multiplied to produce the matching length of its image under a dilation or in a scaled copy. Above 1 it enlarges the figure; between 0 and 1 it shrinks it.",
    "latex": "k = \\frac{A'B'}{AB}",
    "spoken_en": "k equals A prime B prime over A B",
    "examples": [
      {
        "en": "The new side is 12 and the original side is 8, so the scale factor is 12 over 8, which is three halves.",
        "ja": "像の辺が 12、もとの辺が 8 だから、相似比は 8 : 12 = 2 : 3 で、3/2 倍に拡大している。",
        "register": "spoken"
      },
      {
        "en": "Rectangle A is 4 cm by 6 cm, and rectangle B is 10 cm by 15 cm. What is the scale factor from rectangle A to rectangle B?",
        "ja": "長方形 A は 4 cm × 6 cm、長方形 B は 10 cm × 15 cm である。A に対する B の拡大率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "a scale factor of 2",
        "ja": "2 倍（相似比 1 : 2）"
      },
      {
        "en": "the scale factor from A to B",
        "ja": "A に対する B の拡大率"
      }
    ],
    "pitfalls": [
      "向きに注意。A から B への scale factor が 3/2 なら、B から A へは 2/3。日本語の相似比 2 : 3 は 2 つの図形を挙げた順に比で書き、1 つの数にはしない（日本語版 Wikipedia「図形の相似」）。",
      "面積は scale factor の 2 乗倍、体積は 3 乗倍になる（相似比 m : n なら面積比 m² : n²、体積比 m³ : n³）。",
      "用例コーパスでは、話し言葉はほとんどが Khan Academy の中学の講義で、書き言葉は少ない（MIT の講義ノートと OpenStax）。IM では 522 件で、Grade 7・Grade 8・Geometry の glossary の見出し。"
    ],
    "related": [
      "dilation",
      "ratio-of-areas-of-similar-figures",
      "similar-triangles",
      "similarity-transformation"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "glossary の見出し scale factor（Grade 7・Grade 8・Geometry）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形: 相似比と面積比・体積比"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「図形の相似」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%B3%E5%BD%A2%E3%81%AE%E7%9B%B8%E4%BC%BC",
        "note": "相似比 1 : r（欧米は ∼、日本は ∽ の記号）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和7年度 大学入学共通テスト 本試験 数学Ⅰ，数学Ａ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=771&f=abm00005956.pdf&n=2025_op_26_sugaku1A.pdf",
        "note": "問題文「その相似比は □ : □」"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Unit 1 Scale Drawings・glossary: the scale factor from Polygon A to Polygon B"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "scale factor": 87
      },
      "written": {
        "scale factor": 4
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "scale",
    "ja": {
      "term": "縮尺",
      "reading": "しゅくしゃく",
      "alt": []
    },
    "en": {
      "term": "scale",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "IM Grade 7（1.7 Scale Drawings）は、長さを同じ割合で拡大した scale drawing にも縮小した scale drawing にも scale を使う。日本語の「縮尺」は縮めて表した割合で、日本語版 Wikipedia「縮尺」は実物より大きい図面・模型の比を倍尺、実物大を現尺と呼んで縮尺と分ける。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "小学校",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "地図や縮図で、実際の長さをどれだけの割合に縮めて表したかを示すもの。1 : 25000 のような比で書く。",
    "definition_en": "On a map or a scale drawing, the relationship between lengths in the drawing and the actual lengths they stand for, such as 1 : 25,000 or 1 inch = 10 miles.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The scale of the map is 1 inch to 10 miles, so 3 inches on the map is 30 miles in real life.",
        "ja": "この地図の縮尺は 1 インチが 10 マイルだから、地図上の 3 インチは実際には 30 マイルだ。",
        "register": "spoken"
      },
      {
        "en": "A map has a scale of 1 : 25,000. Two towns are 8 cm apart on the map. Find the actual distance between them in kilometers.",
        "ja": "縮尺 1 : 25000 の地図上で、2 つの町の間の距離は 8 cm である。実際の距離は何 km か求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "拡大・縮小の比（何倍にするか）は scale factor（エントリ scale-factor）で、別の用語。縮尺の scale は、図の上の長さと実際の長さの対応を言う。",
      "IM Grade 7（1.11・1.12）は、単位のない縮尺（1 to 633,600 のような scale without units）と、1 inch to 10 miles のような単位のある縮尺の両方を扱う。実際の長さを求めるときは、単位をそろえてから計算する。",
      "scale は、グラフの軸の目盛り（the scale on the y-axis）や、はかりの意味にもなる。「縮尺」の意味かどうかは文脈で決まる。"
    ],
    "related": [
      "scale-drawing",
      "scale-factor",
      "ratio",
      "similar",
      "dilation"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "Unit 1: 1.7 Scale Drawings（enlarged or reduced。mapping_note）・1.8 Scale Drawings and Maps・1.11 Scales without Units・1.12 Units in Scale Drawings（scale of the map）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「縮尺」",
        "url": "https://ja.wikipedia.org/wiki/%E7%B8%AE%E5%B0%BA",
        "note": "節「縮尺以外の尺度用語」: 倍尺・現尺（mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "scale of the map | scale of the drawing | map scale | scale on the map | scale of a map | scale of a drawing": 1
      },
      "sources": [
        "khan-middle"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 scale of the map | scale of the drawing | map scale | scale on the map | scale of a map | scale of a drawing（Grade 7 1.12 Units in Scale Drawings・Grade 7 1.7 Scale Drawings・Grade 7 1.8 Scale Drawings and Maps・計 10 件）。register は主張しない。",
        "raised": "2026-10-09"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "scalene-triangle",
    "ja": {
      "term": "不等辺三角形",
      "reading": "ふとうへんさんかくけい",
      "alt": []
    },
    "en": {
      "term": "scalene triangle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "3 辺の長さがすべて異なる三角形。",
    "definition_en": "A triangle whose three sides all have different lengths.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The sides are 4, 6, and 7. No two are the same, so it's a scalene triangle.",
        "ja": "辺は 4, 6, 7 で、同じ長さの辺がないから不等辺三角形。",
        "register": "spoken"
      },
      {
        "en": "A triangle has side lengths 5 cm, 7 cm, and 9 cm. Classify it as scalene, isosceles, or equilateral, and as acute, right, or obtuse.",
        "ja": "3 辺の長さが 5 cm, 7 cm, 9 cm の三角形がある。不等辺三角形・二等辺三角形・正三角形のどれか、また鋭角三角形・直角三角形・鈍角三角形のどれかを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本では「3 辺の長さがすべて異なる三角形」をふつう名前で呼ばない。英語では辺による分け方（scalene・isosceles・equilateral）と角による分け方（acute・right・obtuse）を組み合わせて、an obtuse scalene triangle のように言える。",
      "scalene は「スケイリーン」のように読む（Merriam-Webster: ˈskā-ˌlēn。第 2 音節に強勢を置く skā-ˈlēn も挙げる）。"
    ],
    "related": [
      "triangle",
      "isosceles-triangle",
      "equiangular-triangle"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "7 年 1.3 Making Scaled Copies（1 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.3 Classify Triangles by Side Measurement・4.1 Classify Triangles"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「scalene」",
        "url": "https://www.merriam-webster.com/dictionary/scalene",
        "note": "発音 ˈskā-ˌlēn、skā-ˈlēn（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 scalene triangle（CK-12 Geometry 4.3 Classify Triangles by Side Measurement・CK-12 Geometry 4.1 Classify Triangles・計 11 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "scatter-plot",
    "ja": {
      "term": "散布図",
      "reading": "さんぷず",
      "alt": []
    },
    "en": {
      "term": "scatter plot",
      "alt": [
        "scatter diagram"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "AP Statistics",
        "Integrated Math 1",
        "Intro Statistics"
      ]
    },
    "definition_ja": "2 つの変量の値の組を座標とする点を、座標平面上にかき込んだ図。",
    "definition_en": "A graph that plots each pair of values of two variables as a point in the coordinate plane.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Make a scatter plot first, and see if there's any pattern.",
        "ja": "まず散布図をかいて、何か傾向があるか見てみよう。",
        "register": "spoken"
      },
      {
        "en": "Draw a scatter plot of the data and describe the relationship between the two variables.",
        "ja": "データの散布図をかき、2 つの変量の関係を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP Statistics の CED は 1 語の scatterplot だけを使う（16 件。topic 5.1 に 12 件、Unit 5 の概要と topic 5.4 にも）。OpenStax Algebra and Trigonometry は scatter diagram とも書く（22 件。Precalculus 21 件、Introductory Statistics は 1 件）。"
    ],
    "related": [
      "correlation",
      "correlation-coefficient",
      "trend"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "散布図",
        "en": "Scatter plot"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Unit 5 の概要・Topic 5.1・5.4（scatter plot。原文の綴りは 1 語の scatterplot で 16 件: 概要 3 件、5.1 に 12 件、5.4 に 1 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "scatter plot": 32
      },
      "written": {
        "scatter plot": 138,
        "scatter diagram": 22
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "openstax-algtrig",
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "scientific-notation",
    "ja": {
      "term": "科学的記数法",
      "reading": "かがくてききすうほう",
      "alt": [
        "指数表記"
      ]
    },
    "en": {
      "term": "scientific notation",
      "alt": [],
      "uk": "standard form",
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の中 3 では、近似値を（整数部分が 1 桁の数）×（10 の累乗）の形で書くことを学ぶが、この書き方に名前を付けない。理科では「指数表記」と言うこともある。英語では scientific notation という名前で、OpenStax Algebra and Trigonometry は 1 ≦ |a| < 10、n は整数という条件で定義する。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1"
      ]
    },
    "definition_ja": "数を a × 10ⁿ（1 ≦ |a| < 10、n は整数）の形で書く書き方。とても大きい数や小さい数を短く書ける。",
    "definition_en": "A way of writing a number as a × 10^n, where 1 ≤ |a| < 10 and n is an integer. It is a short way to write very large or very small numbers.",
    "latex": "a \\times 10^{n}, \\quad 1 \\le |a| < 10",
    "spoken_en": "a times ten to the n, where the absolute value of a is greater than or equal to one and less than ten",
    "examples": [
      {
        "en": "In scientific notation, 3,200,000 is 3.2 times 10 to the sixth.",
        "ja": "3,200,000 を科学的記数法で書くと、3.2 × 10⁶ だ。",
        "register": "spoken"
      },
      {
        "en": "Write 0.00045 in scientific notation.",
        "ja": "0.00045 を科学的記数法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英国では standard form と言うが、米国の standard form は別の意味（直線の式 Ax + By = C など、エントリ standard-form-of-a-line）。",
      "a は 1 ≦ |a| < 10 でなければならない。32 × 10⁵ は値は正しくても scientific notation ではなく、3.2 × 10⁶ と書く。",
      "1 より小さい数は指数が負になる（0.00045 = 4.5 × 10⁻⁴）。10⁻⁴ は ten to the negative fourth と読む。"
    ],
    "related": [
      "significant-figures",
      "exponent",
      "negative-exponent",
      "approximate-value",
      "decimal"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Scientific_notation",
        "note": "記事「Scientific notation」の冒頭と節 Normalized notation: 英国では standard form と呼ぶ"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "scientific notation": 105
      },
      "written": {
        "scientific notation": 172
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "secant-line",
    "ja": {
      "term": "割線",
      "reading": "かっせん",
      "alt": []
    },
    "en": {
      "term": "secant line",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「割線」は、円と 2 点で交わる直線（方べきの定理で使う。日本語版 Wikipedia「割線」）。英語の secant line は、円に限らず曲線上の 2 点を通る直線にも使い、微積分では平均変化率を表す直線（the slope of the secant line）として多く出てくる。学習指導要領解説（数学II・III）には「割線」が出てこない。",
    "domains": [
      "geometry",
      "calculus"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "AP Calculus AB",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "曲線上の 2 点を通る直線のこと。円では円と 2 点で交わる直線を指し、微積分ではその傾きが 2 点間の平均変化率になる。",
    "definition_en": "A line through two points of a curve. For a circle, a line that crosses the circle at two points; in calculus, the slope of this line gives the average rate of change between those two x-values.",
    "latex": "m_{\\text{sec}} = \\frac{f(b) - f(a)}{b - a}",
    "spoken_en": "the slope of the secant line equals f of b minus f of a, over b minus a",
    "examples": [
      {
        "en": "As h goes to zero, the secant line gets closer and closer to the tangent line.",
        "ja": "h を 0 に近づけると、割線は接線にどんどん近づいていく。",
        "register": "spoken"
      },
      {
        "en": "Find the slope of the secant line to the graph of f(x) = x² between x = 1 and x = 3.",
        "ja": "関数 f(x) = x² のグラフ上の、x 座標が 1 と 3 である 2 点を通る直線の傾きを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「2 点を通る直線の傾き」「平均変化率」は、英語では secant line の傾きとして言う。用例コーパスでは secant line などの形が話し言葉・書き言葉ともよく出てくる。話し言葉は大半が Khan Academy、書き言葉は大半が OpenStax Calculus。",
      "secant は三角関数 sec θ（正割）と同じ語。直線を指すときは secant line と line を付けると紛れない。",
      "弦（chord）は円周上の 2 点を結ぶ線分で、割線はそれを両側に延ばした直線。"
    ],
    "related": [
      "tangent-line",
      "average-rate-of-change",
      "power-of-a-point",
      "secant-tangent-theorem",
      "chord",
      "difference-quotient"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "割線",
        "en": "Secant line"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II・III: 「割線」が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "secant line | a secant of | secant of the circle | secants of the circle": 134
      },
      "written": {
        "secant line | a secant of | secant of the circle | secants of the circle": 78
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "secant-tangent-theorem",
    "ja": {
      "term": "割線と接線の定理",
      "reading": "かっせんとせっせんのていり",
      "alt": []
    },
    "en": {
      "term": "tangent secant segment theorem",
      "alt": [
        "tangent secant theorem",
        "tangent-secant theorem",
        "secant-tangent theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「割線と接線の定理」は学習指導要領解説に無い、本プロジェクトの訳語。日本では方べきの定理の、円の外の点 P から引いた割線と接線の場合（PT² = PA·PB）として扱う。CK-12 Geometry は tangent secant segment theorem と呼ぶ（6.20 Tangent Secant Theorem、6 件。節の名前は Tangent Secant Theorem）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円の外の点 P から引いた接線が円に点 T で接し、P を通る割線が円と 2 点 A, B で交わるとき、PT² = PA·PB が成り立つという定理。方べきの定理のうち、一方の直線が接線になる場合にあたる。",
    "definition_en": "If a tangent from an outside point P touches a circle at T and a secant from P meets the circle at A and B, then PT² = PA · PB. The square of the tangent segment equals the outside part of the secant times the whole secant.",
    "latex": "PT^2 = PA \\cdot PB",
    "spoken_en": "P T squared equals P A times P B",
    "examples": [
      {
        "en": "With the tangent secant segment theorem, the tangent piece gets squared, and for the secant it's the outside part times the whole thing.",
        "ja": "割線と接線の定理では、接線の部分は 2 乗する。割線のほうは外側の部分 × 全体だ。",
        "register": "spoken"
      },
      {
        "en": "From a point P outside a circle, a tangent touches the circle at T, and a secant meets the circle at A and B, with A between P and B. If PA = 4 and AB = 5, use the tangent secant segment theorem to find PT.",
        "ja": "円の外の点 P から引いた接線が円と点 T で接し、P を通る割線が円と 2 点 A, B で交わる（A は P と B の間にある）。PA = 4, AB = 5 のとき、PT の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "割線の側は「外側の部分 PA × 割線全体 PB」。AB（円の内側の部分）を掛けないこと。AB が与えられたら、先に PB = PA + AB を求める。",
      "secant は「割線」の意味だが、三角関数の secant（sec θ）と同じ語。図形の問題か三角関数の問題かで区別する。",
      "呼び方は揺れる。CK-12 Geometry は定理を tangent secant segment theorem と呼び（6 件）、節の名前は 6.20 Tangent Secant Theorem（本文では 1 件）。"
    ],
    "related": [
      "power-of-a-point",
      "tangent-line",
      "intersecting-chords-theorem",
      "secant-line"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.20 Tangent Secant Theorem"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 tangent secant segment theorem（CK-12 Geometry 6.20 Tangent Secant Theorem・計 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "secant",
    "ja": {
      "term": "セカント",
      "reading": "せかんと",
      "alt": []
    },
    "en": {
      "term": "secant",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の高校では sec を使わず 1/cos θ と書く（1 + tan²θ = 1/cos²θ）。米国の教科書（OpenStax Algebra and Trigonometry）は sec を cos の逆数として定義し、微積分でも tan x の導関数を sec²x と書くように sec で表す。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "cos θ の逆数で定まる三角関数で、sec θ = 1/cos θ。直角三角形では、斜辺を角 θ の隣の辺で割った値になる。",
    "definition_en": "The trig function equal to one over cosine, so sec θ = 1/cos θ. In a right triangle, it is the hypotenuse divided by the side adjacent to the angle.",
    "latex": "\\sec\\theta = \\frac{1}{\\cos\\theta}",
    "spoken_en": "secant theta equals one over cosine theta",
    "examples": [
      {
        "en": "Secant is just one over cosine, so if cosine is one half, the secant is 2.",
        "ja": "セカントは cos の逆数にすぎないから、cos が 1/2 なら sec は 2 だ。",
        "register": "spoken"
      },
      {
        "en": "Given that θ is in Quadrant I and tan θ = 3/4, use the identity 1 + tan²θ = sec²θ to find the secant of θ.",
        "ja": "θ は第 1 象限の角で tan θ = 3/4 である。恒等式 1 + tan²θ = sec²θ を用いて、sec θ の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "同じ英語の secant は、円や曲線と 2 点で交わる直線（割線）の意味にもなる（エントリ secant-line）。三角関数の sec か直線かは文脈で区別する。",
      "sec は cos の逆数で、sin の逆数ではない。sin の逆数は cosecant（エントリ cosecant）。",
      "sec θ は secant theta と読む。日本の 1 + tan²θ = 1/cos²θ は英語の式では 1 + tan²θ = sec²θ と書き、tan x の導関数も sec²x と書く。"
    ],
    "related": [
      "cosine",
      "cosecant",
      "cotangent",
      "pythagorean-identity",
      "secant-line"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "secant !line !lines !method !slope !slopes": 684
      },
      "written": {
        "secant !line !lines !method !slope !slopes": 69
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "second-derivative-test",
    "ja": {
      "term": "第 2 次導関数判定法",
      "reading": "だいにじどうかんすうはんていほう",
      "alt": [
        "第 2 次導関数テスト"
      ]
    },
    "en": {
      "term": "second derivative test",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説には、f′(a) = 0 かつ f″(a) の符号で極大・極小を判定する方法の名前が出てこない。AP の CED（topic 5.7）と OpenStax Calculus は second derivative test と呼ぶ。見出しの「第 2 次導関数判定法」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "f′(c) = 0 のとき、f″(c) < 0 なら極大、f″(c) > 0 なら極小と判定する方法。",
    "definition_en": "A test used at a critical point c with f′(c) = 0: f″(c) < 0 means a local maximum at c, and f″(c) > 0 means a local minimum.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Second derivative test: f double prime is negative there, so it's a max.",
        "ja": "第 2 次導関数判定法で、そこで f″ が負なので極大。",
        "register": "spoken"
      },
      {
        "en": "Use the second derivative test to classify the critical points of f(x) = x³ − 3x.",
        "ja": "第 2 次導関数判定法を用いて、f(x) = x³ − 3x の臨界点が極大か極小かを判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "f″(c) = 0 のときは判定できない（first derivative test に戻る）。"
    ],
    "related": [
      "first-derivative-test",
      "second-derivative",
      "critical-point",
      "concavity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 5.7（second derivative test）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "f′(a) = 0 かつ f″(a) の符号で極大・極小を判定する方法の名前が出てこない・「第 2 次導関数判定法」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "second derivative test": 39
      },
      "written": {
        "second derivative test": 36
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "second-derivative",
    "ja": {
      "term": "第 2 次導関数",
      "reading": "だいにじどうかんすう",
      "alt": []
    },
    "en": {
      "term": "second derivative",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "導関数 f′(x) をさらに微分した関数 f″(x)。",
    "definition_en": "The derivative of the derivative, f″(x).",
    "latex": "f''(x) = \\frac{d^2y}{dx^2}",
    "spoken_en": "f double prime of x equals d squared y d x squared",
    "examples": [
      {
        "en": "The second derivative tells you about concavity.",
        "ja": "第 2 次導関数から凹凸が分かる。",
        "register": "spoken"
      },
      {
        "en": "Find the second derivative of y = x³ − 4x.",
        "ja": "y = x³ − 4x の第 2 次導関数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "d²y/dx² は d squared y d x squared と読む。"
    ],
    "related": [
      "higher-order-derivative",
      "concavity",
      "inflection-point",
      "differentiate-twice"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 3.6（second derivative）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "second derivative": 691
      },
      "written": {
        "second derivative": 95
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sector",
    "ja": {
      "term": "おうぎ形",
      "reading": "おうぎがた",
      "alt": [
        "扇形"
      ]
    },
    "en": {
      "term": "sector",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円の 2 本の半径と、その間の弧で囲まれた図形。",
    "definition_en": "The region inside a circle between two radii and the arc that joins their endpoints. It is shaped like a slice of pizza.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A sector is like a slice of pizza: two radii and the arc between them.",
        "ja": "おうぎ形はピザの 1 切れみたいな形で、2 本の半径とその間の弧でできている。",
        "register": "spoken"
      },
      {
        "en": "Find the area of a sector with a radius of 10 cm and a central angle of 72°. Leave your answer in terms of π.",
        "ja": "半径 10 cm、中心角 72° のおうぎ形の面積を求めよ。答えは π を使って表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "sector は日常語では「部門・分野」（the public sector）の意味でも使う。",
      "弦と弧で囲まれた部分（弓形）は sector ではなく circular segment と呼ぶ。",
      "おうぎ形の面積は area of a sector（エントリ area-of-a-sector）、弧の長さは arc length（エントリ arc-length）。"
    ],
    "related": [
      "area-of-a-sector",
      "arc",
      "central-angle",
      "radius",
      "arc-length"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sector": 17
      },
      "written": {
        "sector": 56
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "segment-addition-postulate",
    "ja": {
      "term": "線分の加法公理",
      "reading": "せんぶんのかほうこうり",
      "alt": []
    },
    "en": {
      "term": "segment addition postulate",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「線分の加法公理」は学習指導要領解説に無い、本プロジェクトの訳語。日本の教科書は AB + BC = AC に名前をつけない。英語名は、点 B が線分 AC 上にあれば AB + BC = AC になるという性質を公理として立てたときの名前。CK-12 Geometry は segment addition postulate と呼ぶ（1.3 Definition of Line segments、4 件）。用例コーパスには出てこない。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "点 B が線分 AC 上にあるとき AB + BC = AC が成り立つ、という公理。逆に AB + BC = AC なら B は線分 AC 上にある。",
    "definition_en": "The postulate that if point B lies on segment AC, then AB + BC = AC, and conversely.",
    "latex": "AB + BC = AC",
    "spoken_en": null,
    "examples": [
      {
        "en": "B is between A and C, so by the segment addition postulate, AB plus BC has to equal AC.",
        "ja": "B は A と C の間にあるから、線分の加法公理より AB + BC = AC になる。",
        "register": "spoken"
      },
      {
        "en": "Point B lies on AC with AB = 2x + 1, BC = x + 4, and AC = 23. Use the segment addition postulate to find x.",
        "ja": "点 B は線分 AC 上にあり、AB = 2x + 1、BC = x + 4、AC = 23 である。x を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "この公理が使えるのは B が線分 AC 上にあるときだけ。B が線分 AC の外にあれば AB + BC > AC になる（三角不等式）。"
    ],
    "related": [
      "segment",
      "axiom",
      "angle-addition-postulate",
      "undefined-terms",
      "midpoint"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Segment_addition_postulate",
        "note": "記事名 Segment addition postulate（en.term のリダイレクト先）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.3 Definition of Line segments"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 segment addition postulate（CK-12 Geometry 1.3 Definition of Line segments・計 4 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "segment",
    "ja": {
      "term": "線分",
      "reading": "せんぶん",
      "alt": []
    },
    "en": {
      "term": "line segment",
      "alt": [],
      "variants": [
        {
          "term": "segment",
          "register": "spoken",
          "note": "直前に line が付かない segment として数えた（line segment の segment は数えない）。話し言葉では MIT OCW・The Organic Chemistry Tutor・Khan Academy が多い。書き言葉では OpenStax Calculus と MIT の講義ノートが中心。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスでは、書き言葉は line segment が多く、話し言葉は line を省いた segment のほうが多い（line segment も使う）。英語の line は両方向に限りなく伸びる直線で、線分の意味にはならない。英語の記法では線分 AB を AB の上に横線を引いて表し、その長さは横線なしの AB と書き分ける（共通テストの問題文では線分も長さも AB と書く）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直線上の 2 点と、その間にあるすべての点からなる図形。2 点を端点という。",
    "definition_en": "The part of a line made up of two points, called its endpoints, and all the points between them.",
    "latex": "\\overline{AB}",
    "spoken_en": "segment A B",
    "examples": [
      {
        "en": "Draw a segment from A to B, and call its midpoint M.",
        "ja": "A から B へ線分を引いて、その中点を M としよう。",
        "register": "spoken"
      },
      {
        "en": "Find the length of line segment AB, where A(1, 2) and B(4, 6).",
        "ja": "A(1, 2), B(4, 6) のとき、線分 AB の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "直線 AB は line AB、半直線 AB は ray AB で、線分とは別の語。線分のつもりで line とだけ言うと、両方向に伸びた直線になる。",
      "「線分の長さ」は the length of segment AB と言うか、単に AB と書く。",
      "話し言葉の segment は、ソース別では The Organic Chemistry Tutor が最も多い。"
    ],
    "related": [
      "midpoint",
      "perpendicular-bisector",
      "distance",
      "chord",
      "triangle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "線分",
        "en": "Line segment"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和5年度 大学入学共通テスト 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20240408/https://www.dnc.ac.jp/albums/abm.php?d=432&f=abm00003627.pdf&n=2023_ot_19_sugaku1A.pdf",
        "note": "線分 AB と長さ AB = … の書き方"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!line segment": 189,
        "line segment": 80
      },
      "written": {
        "line segment": 215,
        "!line segment": 49
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "select-at-random",
    "ja": {
      "term": "無作為に抽出する",
      "reading": "むさくいにちゅうしゅつする",
      "alt": []
    },
    "en": {
      "term": "randomly select",
      "alt": [
        "select at random"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Algebra 2",
        "AP Statistics"
      ]
    },
    "definition_ja": "母集団のどの個体も同じ確率で選ばれるように、標本を取り出す。",
    "definition_en": "To choose members of a population so that each has the same chance of being chosen.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "She randomly selects 20 students and asks how long they studied.",
        "ja": "彼女は 20 人の生徒を無作為に選び、勉強時間を聞く。",
        "register": "spoken"
      },
      {
        "en": "Fifty students are randomly selected from a school where 40% of the students walk to school. Find the probability that more than 30 of them walk to school.",
        "ja": "生徒の 40% が徒歩通学の学校から、50 人の生徒を無作為に選ぶ。そのうち 30 人より多くが徒歩通学である確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（中学校・高等学校）は「無作為に抽出する」と書く。AP Statistics の CED（topic 1.10・1.11）と OpenStax Introductory Statistics は randomly select(ed) と書き、IM は selected at random のほうを多く使う（Geometry の確率の単元ほか 60 件。randomly select は 36 件）。用例コーパスでは話し言葉・書き言葉とも randomly select のほうが select at random より多い。",
      "OpenStax Introductory Statistics の問題文は受け身・分詞の randomly selected を多く使う（142 件。能動の形は 20 件）。"
    ],
    "related": [
      "random-sampling",
      "random-number"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 標本調査: 無作為に抽出する"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B 統計的な推測: 無作為に抽出する"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.10（randomly select）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Geometry Unit 8（確率）ほか: selected at random"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "randomly select": 31,
        "select at random": 1
      },
      "written": {
        "randomly select": 164,
        "select at random": 19
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "semicircle",
    "ja": {
      "term": "半円",
      "reading": "はんえん",
      "alt": []
    },
    "en": {
      "term": "semicircle",
      "alt": [
        "half circle"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円を直径で 2 つに分けたときの一方の図形（中心角 180° の扇形）。",
    "definition_en": "Half of a circle, cut off by a diameter. It can mean just the arc or the region bounded by the arc and the diameter.",
    "latex": "A = \\frac{1}{2}\\pi r^2",
    "spoken_en": "A equals one half pi r squared",
    "examples": [
      {
        "en": "A semicircle is just half a circle, so its area is half of pi r squared.",
        "ja": "半円は円の半分だから、面積は πr² の半分だ。",
        "register": "spoken"
      },
      {
        "en": "Find the area of a semicircle with a diameter of 12 cm. Leave your answer in terms of π.",
        "ja": "直径 12 cm の半円の面積を求めよ。答えは π を用いて表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "半円の周の長さ（perimeter）は、弧の長さ πr に直径 2r を足した πr + 2r。弧の長さ πr だけと取り違えない。",
      "semi- は「半分」の意味。OpenStax Calculus は弧としての半円を semicircular arc とも書き、semicircular は「半円の」という形容詞。",
      "ちょうど 180° の弧である半円の弧は、優弧（major arc）にも劣弧（minor arc）にも入れない（エントリ major-arc、minor-arc）。",
      "CK-12 Geometry 6.9 Arcs in Circles は semicircle を 180° の弧と定義し、6.7 では面積を求める図形の意味でも使う。日本語版 Wikipedia「円 (数学)」は、直径で分けた 2 つの弧を半円周と呼ぶ。"
    ],
    "related": [
      "circle",
      "diameter",
      "arc",
      "angle-inscribed-in-a-semicircle",
      "arc-length"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "半円",
        "en": "Semicircle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.9 Arcs in Circles（semicircle は 180° の弧）・6.7 Areas of Combined Figures Involving Semicircles"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「円 (数学)」",
        "url": "https://ja.wikipedia.org/wiki/%E5%86%86_(%E6%95%B0%E5%AD%A6)",
        "note": "直径で分けた 2 つの弧を半円周と呼ぶ"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "half circle"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "semicircle": 28,
        "half circle": 8
      },
      "written": {
        "semicircle": 43,
        "half circle": 2
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "separable-differential-equation",
    "ja": {
      "term": "変数分離形",
      "reading": "へんすうぶんりけい",
      "alt": []
    },
    "en": {
      "term": "separable differential equation",
      "alt": [],
      "variants": [
        {
          "term": "separable differential equation",
          "register": "spoken",
          "note": "話し言葉はほとんどが Khan Academy。書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "separable equation",
          "register": "spoken",
          "note": "略した言い方。ソースは分かれる。話し言葉では少ないが、書き言葉では separable differential equation と同じくらい出てくる（OpenStax Calculus と MIT の講義ノート）。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II"
      ]
    },
    "definition_ja": "dy/dx = f(x)g(y) の形に書ける微分方程式。y の式と x の式を両辺に分けて、それぞれ積分して解く。",
    "definition_en": "A differential equation that can be written as dy/dx = f(x)g(y), so the y's and x's can be put on opposite sides and each side integrated.",
    "latex": "\\frac{dy}{dx} = f(x)\\,g(y)",
    "spoken_en": "d y d x equals f of x times g of y",
    "examples": [
      {
        "en": "This one's separable: get all the y's on one side with dy, and all the x's on the other side with dx.",
        "ja": "これは変数分離形。y は dy と一緒に片側へ、x は dx と一緒にもう片側へ集める。",
        "register": "spoken"
      },
      {
        "en": "Since the equation is separable, we write dy/y = 2x dx and integrate both sides.",
        "ja": "変数分離形なので dy/y = 2x dx と変形し、両辺を積分する。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "解き方の名前は separation of variables（変数分離）。方程式の種類は形容詞 separable（分離できる）で言う。",
      "両辺を積分して出てくる ln|y| は、学習指導要領解説の書き方では log|y| になる（解説には ln が出てこず、数学III の課題学習の例は自然対数を lim x log x と底を書かずに書く）。"
    ],
    "related": [
      "differential-equation",
      "logistic-differential-equation",
      "integrate",
      "separation-of-variables",
      "exponential-model"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "変数分離",
        "en": "Separation of variables"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "ln が出てこない（数学III 課題学習〈微分法〉の例は自然対数を lim x log x と底を書かずに書く）／理数数学II (7) 積分法: dy/dx = ky（k は定数）程度の簡単な微分方程式の意味と解法"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "separable equation": 3,
        "separable differential equation": 29
      },
      "written": {
        "separable differential equation": 9,
        "separable equation": 7
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "separation-of-variables",
    "ja": {
      "term": "変数分離",
      "reading": "へんすうぶんり",
      "alt": []
    },
    "en": {
      "term": "separation of variables",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II"
      ]
    },
    "definition_ja": "微分方程式を g(y) dy = f(x) dx の形に分けて、両辺を積分して解く方法。",
    "definition_en": "Solving a differential equation by rewriting it as g(y) dy = f(x) dx and integrating both sides.",
    "latex": "\\frac{dy}{dx} = xy \\;\\Rightarrow\\; \\int \\frac{dy}{y} = \\int x\\,dx",
    "spoken_en": null,
    "examples": [
      {
        "en": "Get all the y's on one side with dy, and all the x's on the other with dx.",
        "ja": "y を全部 dy の側に、x を全部 dx の側に集める。",
        "register": "spoken"
      },
      {
        "en": "Use separation of variables to solve dy/dx = 2xy with y(0) = 3.",
        "ja": "変数分離を用いて、y(0) = 3 のもとで dy/dx = 2xy を解け。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "use separation of variables",
        "ja": "変数分離を使う"
      }
    ],
    "pitfalls": [
      "この方法で解ける方程式は separable differential equation（変数分離形）。",
      "両辺を y で割るときは、y = 0 が解から落ちていないか確かめる。"
    ],
    "related": [
      "separable-differential-equation",
      "differential-equation",
      "particular-solution",
      "initial-condition"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "変数分離",
        "en": "Separation of variables"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 7.6・7.7（separation of variables）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学II (7) 積分法: dy/dx = ky（k は定数）程度の簡単な微分方程式の意味と解法"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "separation of variables": 18
      },
      "written": {
        "separation of variables": 26
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sequence-of-differences",
    "ja": {
      "term": "階差数列",
      "reading": "かいさすうれつ",
      "alt": []
    },
    "en": {
      "term": "sequence of differences",
      "alt": [
        "first differences"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "Levin の Discrete Mathematics は sequence of differences と呼ぶ（4.3・6.1・4.1 ほか、36 件。first differences は 8 件）。米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（sequence of differences ／ first differences は CED・OpenStax・IM・CK-12 とも 0 件）。IM Algebra 2（1.3）や OpenStax Algebra and Trigonometry（13.2）は、等差数列かどうかを見るときに隣り合う項の差（the difference between consecutive terms）を調べ、IM Algebra 1（5.2 Patterns of Growth・6.2）も表に前の値との差（difference from previous year）を並べて線形かどうかを見るが、どれも差を並べた数列には名前を付けず、差の数列の和でもとの数列に戻すこともしない。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "数列 {aₙ} の隣り合う項の差 bₙ = aₙ₊₁ − aₙ を並べた数列。",
    "definition_en": "The sequence bₙ = aₙ₊₁ − aₙ formed by the differences between consecutive terms of {aₙ}.",
    "latex": "b_n = a_{n+1} - a_n",
    "spoken_en": null,
    "examples": [
      {
        "en": "Look at the differences between consecutive terms: 3, 5, 7, 9 — those go up by 2.",
        "ja": "隣り合う項の差を見ると 3, 5, 7, 9 で、2 ずつ増えている。",
        "register": "spoken"
      },
      {
        "en": "The sequence of differences of {aₙ} is the arithmetic sequence 3, 5, 7, …, and a₁ = 2. Find aₙ.",
        "ja": "{aₙ} の階差数列は等差数列 3, 5, 7, … で、a₁ = 2 である。aₙ を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "aₙ = a₁ + Σₖ₌₁ⁿ⁻¹ bₖ（n ≥ 2）で戻す。Levin（4.1）は、もとの数列を階差数列の部分和（partial sums）として説明する。n = 1 の確認を忘れない。"
    ],
    "related": [
      "sequence",
      "arithmetic-sequence",
      "recurrence-relation",
      "difference",
      "consecutive-terms"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "wikidata",
        "qid": "Q10855257"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.3 Polynomial Sequences・6.1 Generating Functions・4.1 Describing Sequences（もとの数列は階差数列の部分和）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 1.3 Different Types of Sequences（the differences of consecutive terms）・Algebra 1 5.2 Patterns of Growth・6.2 How Does it Change?（difference from previous year）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.2 Arithmetic Sequences（the difference between consecutive terms）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "first differences": 1
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Levin の呼び方 sequence of differences（4.3 Polynomial Sequences・6.1 Generating Functions・4.1 Describing Sequences）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sequence",
    "ja": {
      "term": "数列",
      "reading": "すうれつ",
      "alt": []
    },
    "en": {
      "term": "sequence",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus BC",
        "Calculus II",
        "Integrated Math 1",
        "Discrete Math"
      ]
    },
    "definition_ja": "数を 1 列に並べたもの。n 番目の数を第 n 項といい、aₙ のように書く。",
    "definition_en": "A list of numbers in a definite order; the nth number is the nth term, written aₙ.",
    "latex": "a_1, a_2, a_3, \\ldots, a_n, \\ldots",
    "spoken_en": "a sub one, a sub two, a sub three, dot dot dot",
    "examples": [
      {
        "en": "A sequence is just a list — the order matters.",
        "ja": "数列はただの数の並び。順番が大事。",
        "register": "spoken"
      },
      {
        "en": "Write the first five terms of the sequence aₙ = 2n − 1.",
        "ja": "数列 aₙ = 2n − 1 の初めの 5 項を書け。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "the terms of a sequence",
        "ja": "数列の項"
      }
    ],
    "pitfalls": [
      "sequence（数列）と series（級数、項の和）を取り違えない。"
    ],
    "related": [
      "first-term",
      "nth-term",
      "arithmetic-sequence",
      "geometric-sequence",
      "infinite-series",
      "limit-of-a-sequence",
      "find-the-nth-term",
      "recursive-definition",
      "general-term",
      "write-out-the-first-few-terms"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "数列",
        "en": "Number sequence"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.1 Describing Sequences（sequence）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "sequence": 378
      },
      "written": {
        "sequence": 1695
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "set-builder-notation",
    "ja": {
      "term": "集合の内包的記法",
      "reading": "しゅうごうのないほうてききほう",
      "alt": [
        "内包的記法"
      ]
    },
    "en": {
      "term": "set-builder notation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の数A（集合）でも {x | x は 12 の約数} のように書くが、書き方に名前を付けず「条件を満たすものの集まりとして表す」と言う。「内包的記法」は大学の用語。OpenStax Algebra and Trigonometry は、不等式の解を不等式・set-builder notation・区間記法（エントリ interval-notation）の 3 つの書き方で並べて示す。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1"
      ]
    },
    "definition_ja": "集合を、要素が満たす条件で表す書き方。{x | x > 2} は「x > 2 を満たす x 全体の集合」を表す。",
    "definition_en": "A way to describe a set by stating the condition its elements satisfy, as in {x | x > 2}, the set of all x such that x is greater than 2.",
    "latex": "\\{\\, x \\mid x > 2 \\,\\}",
    "spoken_en": "the set of all x such that x is greater than two",
    "examples": [
      {
        "en": "In set-builder notation, the bar means \"such that,\" so we read this as all x such that x is greater than 2.",
        "ja": "内包的記法では、縦棒は「〜を満たす」という意味だから、これは「x > 2 を満たす x 全体」と読む。",
        "register": "spoken"
      },
      {
        "en": "Write the solution set of 3x − 1 ≥ 11 in set-builder notation.",
        "ja": "不等式 3x − 1 ≧ 11 の解の集合を、集合の内包的記法で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "縦棒 | は such that と読む。{x : x > 2} のようにコロンで書く本もある。",
      "要素を書き並べる書き方（{1, 2, 3, 4} など、日本の「要素を書き並べる方法」）は roster notation と言い、set-builder notation とは別。",
      "不等式の解は区間記法（エントリ interval-notation）でも表せる。{x | x ≥ 4} は [4, ∞) と同じ。"
    ],
    "related": [
      "set",
      "interval-notation",
      "solution-set",
      "element",
      "inequality"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "set-builder notation": 12
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "set-up-a-recurrence",
    "ja": {
      "term": "漸化式を立てる",
      "reading": "ぜんかしきをたてる",
      "alt": []
    },
    "en": {
      "term": "write a recursive formula",
      "alt": [
        "set up a recurrence",
        "find a recurrence"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "文章題の状況から漸化式を作ることを、大学の講義（MIT 6.042）は set up a recurrence と言う。高校の教科書（OpenStax Algebra and Trigonometry）の write a recursive formula は、並んだ項から漸化式を書く問いで使う。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "問題の状況や数列の規則から、項の間の関係を漸化式で表す。",
    "definition_en": "To express, as a recurrence, how each term of a sequence depends on earlier ones in a given situation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let aₙ be the number of ways for n steps, and set up a recurrence by looking at the last step.",
        "ja": "n 段の上り方を aₙ として、最後の一歩で場合分けして漸化式を立てよう。",
        "register": "spoken"
      },
      {
        "en": "Write a recursive formula for the arithmetic sequence −2, 3, 8, 13, ….",
        "ja": "等差数列 −2, 3, 8, 13, … を漸化式で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "recurrence-relation",
      "solve-the-recurrence",
      "recursive-definition"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.2 Arithmetic Sequences・13.3 Geometric Sequences（write a recursive formula）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B (1) 数列 ア(ウ)（事象の変化を漸化式で表す）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.4 Exponential Sequences（find a recurrence relation）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "set up a recurrence": 3
      },
      "written": {
        "write a recursive formula": 15,
        "set up a recurrence": 1,
        "find a recurrence": 4
      },
      "sources": [
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "set",
    "ja": {
      "term": "集合",
      "reading": "しゅうごう",
      "alt": []
    },
    "en": {
      "term": "set",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "範囲がはっきり定まったものの集まり。集まりに入っている一つ一つのものを要素という。",
    "definition_en": "A collection of objects in which it is clear exactly which objects belong; each object in it is called an element.",
    "latex": "A = \\{ x \\mid x > 0 \\}",
    "spoken_en": "A is the set of all x such that x is greater than zero",
    "examples": [
      {
        "en": "So the domain is the set of all real numbers except 2.",
        "ja": "だから定義域は 2 以外のすべての実数の集合だ。",
        "register": "spoken"
      },
      {
        "en": "Let A be the set of positive divisors of 12. List the elements of A.",
        "ja": "12 の正の約数の集合を A とする。A の要素を書き並べよ。",
        "register": "written"
      },
      {
        "en": "Write the set of all real numbers x with x² < 4 in set-builder notation.",
        "ja": "x² < 4 を満たす実数 x 全体の集合を、条件を使った書き方で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "集合の演算は、共通部分 A ∩ B が intersection、和集合 A ∪ B が union、補集合が complement。補集合の記号は Ā のほか、OpenStax Algebra and Trigonometry は E′（余事象）、Levin は Ā と書き、英語版 Wikipedia「Complement (set theory)」は Aᶜ も挙げる。",
      "{ x | x > 0 } の書き方は set-builder notation と呼ぶ（OpenStax Algebra and Trigonometry）。Levin は縦棒の代わりにコロンで { x ∈ ℕ : x < 10 } と書く。"
    ],
    "related": [
      "element",
      "solution-set",
      "real-number",
      "all",
      "subspace"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "集合",
        "en": "Set (mathematics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "0.2.2・5.1 Sets: set builder notation { x ∈ ℕ : x < 10 }、Ā"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Complement_(set_theory)",
        "note": "冒頭: Aᶜ の記号"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the set of | a set of | set of all": 183
      },
      "written": {
        "the set of | a set of | set of all": 579
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "shell-method",
    "ja": {
      "term": "シェル法",
      "reading": "しぇるほう",
      "alt": [
        "バウムクーヘン積分"
      ]
    },
    "en": {
      "term": "shell method",
      "alt": [],
      "variants": [
        {
          "term": "method of cylindrical shells",
          "register": "written",
          "note": "OpenStax Calculus の書き方（節 Volumes of Revolution: Cylindrical Shells）で、書き言葉の主見出し。話し言葉には出てこず、書き言葉はすべて OpenStax Calculus。書き言葉の shell method（すべて OpenStax Calculus）はこれより少ない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語ではバウムクーヘン積分と呼ばれ（日本語版 Wikipedia の記事名）、学習指導要領解説には出てこない。OpenStax Calculus Volume 1 は disk・washer と並ぶ節（Volumes of Revolution: Cylindrical Shells）で method of cylindrical shells として扱う。見出しの「シェル法」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "y 軸などのまわりの回転体を薄い円筒の皮の集まりと考え、V = 2π∫ₐᵇ x f(x) dx で体積を求める方法。",
    "definition_en": "A way to find a solid of revolution's volume by summing thin cylindrical shells: V = 2π ∫ₐᵇ x f(x) dx when the region turns about the y-axis.",
    "latex": "V = 2\\pi \\int_a^b x\\,f(x)\\,dx",
    "spoken_en": "V equals two pi times the integral from a to b of x times f of x dx",
    "examples": [
      {
        "en": "Rotating around the y-axis with everything in terms of x? That's a job for the shell method.",
        "ja": "y 軸のまわりに回すのに式が x で書かれている？ それならシェル法の出番。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP Calculus の CED（2020 年版）は回転体の体積を disc method と washer method（topic 8.9–8.12）で扱い、shell method の topic はない。",
      "「バウムクーヘン」（Baumkuchen）は用例コーパスに出てこない。英語では shells と言う。"
    ],
    "related": [
      "cylindrical-shell",
      "disk-method",
      "washer-method",
      "revolve-around-the-y-axis"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "バウムクーヘン積分",
        "en": "Shell integration"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "topic なし（回転体の体積は 8.9–8.12 の disc ／ washer method のみ）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「シェル法」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "shell method": 29
      },
      "written": {
        "shell method": 9,
        "method of cylindrical shells": 16
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "shift-the-index",
    "ja": {
      "term": "添字をずらす",
      "reading": "そえじをずらす",
      "alt": []
    },
    "en": {
      "term": "reindex",
      "alt": [
        "shift the index"
      ],
      "uk": null
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "notation"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "Σ の添字を k → k + 1 のように置き換えて、和の範囲と式を書き直す。",
    "definition_en": "To rewrite a sum by replacing its index, such as k with k + 1, and adjusting the limits and the terms to match.",
    "latex": "\\sum_{k=2}^{n+1} a_{k-1} = \\sum_{k=1}^{n} a_k",
    "spoken_en": null,
    "examples": [
      {
        "en": "Shift the index so that both sums start at k equals 1.",
        "ja": "両方の和が k = 1 から始まるように添字をずらそう。",
        "register": "spoken"
      },
      {
        "en": "Reindex the series Σₙ₌₂^∞ 1/(n − 1)² so that it starts at n = 1.",
        "ja": "級数 Σₙ₌₂^∞ 1/(n − 1)² を n = 1 から始まるように添字をずらして書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "subscript",
      "summation-notation",
      "partial-sum"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.2 Infinite Series・5.6 Ratio and Root Tests（reindex）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "7.5 Series Solutions of Differential Equations（re-index）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "reindex | re-index": 5,
        "shift the index": 1
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 reindex | re-index（本文 5 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "shortest-distance",
    "ja": {
      "term": "最短距離",
      "reading": "さいたんきょり",
      "alt": []
    },
    "en": {
      "term": "shortest distance",
      "alt": [
        "minimum distance"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 点を結ぶ道のり（または点と図形の間の距離）のうち、最も短いものの長さ。立体の表面を通るときは、適切な展開図の上で 2 点を結ぶ線分の長さになる。",
    "definition_en": "The length of the shortest path between two points, or between a point and a figure. Along the surface of a solid, it is the length of the straight segment joining the two points on a suitable net.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To get the shortest distance along the surface of the box, unfold it into a net and draw a straight line.",
        "ja": "箱の表面を通る最短距離は、展開図に開いてまっすぐな線を引けば分かる。",
        "register": "spoken"
      },
      {
        "en": "An ant crawls on the surface of a cube with 4 cm edges from vertex A to the opposite vertex G. Find the minimum distance it must crawl.",
        "ja": "1 辺 4 cm の立方体の表面を、アリが頂点 A から向かい合う頂点 G まで進む。進む道のりの最短距離を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "点と直線の最短距離は、点から直線に下ろした垂線の長さ（エントリ distance-from-a-point-to-a-line）。CK-12 Geometry は点と直線・平行線の間の距離を shortest distance として説明する。",
      "立体の表面にかけたひもの最短の長さは、展開図の上の線分として三平方の定理で求める（エントリ net）。",
      "格子の道順の数を数える「最短経路」とは別の話（エントリ shortest-path）。こちらは道のりの長さ。"
    ],
    "related": [
      "distance",
      "net",
      "distance-from-a-point-to-a-line",
      "pythagorean-theorem",
      "space-diagonal"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.21 Angle Bisectors in Triangles・4.38 Distance Between Parallel Lines（点と直線・平行線の間の距離を shortest distance と説明）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "shortest distance": 2
      },
      "written": {
        "shortest distance": 4,
        "minimum distance": 7
      },
      "sources": [
        "mit-18.02",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満）。見出しは CK-12 の呼び方 shortest distance（CK-12 Geometry 4.21 Angle Bisectors in Triangles・CK-12 Geometry 4.38 Distance Between Parallel Lines・CK-12 Geometry 4.34 Solving Equations Using the Pythagorean Theorem・計 6 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "shortest-path",
    "ja": {
      "term": "最短経路",
      "reading": "さいたんけいろ",
      "alt": []
    },
    "en": {
      "term": "lattice path",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "Levin の Discrete Mathematics は lattice path と呼ぶ（3.1・3.6・3.2 ほか、39 件）。米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（lattice path は CED・OpenStax・IM・CK-12 にも用例コーパスにも出てこない）。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "碁盤の目状の道を、遠回りせずに右と上（など決まった向き）だけに進んで A から B へ行く道順。右 m 回・上 n 回なら ₘ₊ₙCₘ 通り。",
    "definition_en": "A route along a grid from one point to another that moves only right or up (in the fixed directions); with m moves right and n moves up there are C(m + n, m) of them.",
    "latex": "\\binom{m+n}{m}",
    "spoken_en": "m plus n choose m",
    "examples": [
      {
        "en": "Every lattice path from A to B is just a string of 4 R's and 3 U's, so pick where the U's go.",
        "ja": "A から B への最短経路はどれも R 4 個と U 3 個の並びなので、U の位置を選べばいい。",
        "register": "spoken"
      },
      {
        "en": "Find the number of lattice paths from (0, 0) to (5, 3) that pass through (2, 1).",
        "ja": "(0, 0) から (5, 3) へ行く最短経路のうち、(2, 1) を通るものの個数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の shortest path は、用例コーパスでは MIT OCW 6.042 のグラフの最短路（グラフで 2 頂点を結ぶ最も短い道、エントリ path-in-a-graph）の意味で出てくる。学習指導要領解説（数学C）の「最短経路の探索」も離散グラフの話。碁盤の目の道順は、Levin（3.1・3.2）のように how many lattice paths from (0, 0) to (m, n) … と問う。",
      "「通らない」経路は、全体から通る経路を引く（余事象）。通れない辺があるときは、その辺を通る経路を数えて引く。"
    ],
    "related": [
      "combination",
      "permutation-of-a-multiset",
      "path-in-a-graph",
      "counting"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.1 Pascal’s Arithmetical Triangle・3.6 Combinatorial Proofs・3.2 Combining Outcomes"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学C 数学的な表現の工夫: 離散グラフの最短経路の探索"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「場合の数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%A0%B4%E5%90%88%E3%81%AE%E6%95%B0",
        "note": "よく出る問題: 最短経路の問題"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Levin の呼び方 lattice path（3.1 Pascal’s Arithmetical Triangle・3.6 Combinatorial Proofs・3.2 Combining Outcomes）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "side-angle-inequality",
    "ja": {
      "term": "三角形の辺と角の大小",
      "reading": "さんかくけいのへんとかくのだいしょう",
      "alt": []
    },
    "en": {
      "term": "side-angle inequality",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。CK-12 Geometry 4.25 Comparing Angles and Sides in Triangles はこの定理に名前を付けず（this idea is actually a theorem）、中身を the angle opposite the longer side will be larger ／ the largest angle is opposite the longest side と言う。同じレッスンの SAS Inequality Theorem・SSS Inequality Theorem は 2 つの三角形を比べる別の定理（ヒンジ定理）。見出しの英語 side-angle inequality と日本語の「三角形の辺と角の大小」は本プロジェクトの言い方で、学習指導要領解説・日本語版 Wikipedia に無い。",
    "domains": [
      "geometry",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数A",
        "数I"
      ],
      "us": []
    },
    "definition_ja": "1 つの三角形で、大きい辺に向かい合う角は大きく、大きい角に向かい合う辺は大きいという関係。a > b ⇔ A > B で、特に最大の辺の対角が最大の角になる。",
    "definition_en": "In any triangle, the bigger of two angles faces the longer of the two sides opposite them, and the longer side faces the bigger angle; so the biggest angle faces the longest side.",
    "latex": "a > b \\iff A > B",
    "spoken_en": "a is greater than b if and only if A is greater than B",
    "examples": [
      {
        "en": "The largest angle is always opposite the longest side, so we only need the law of cosines once, for the angle across from 7.",
        "ja": "最大の角は必ず最大の辺の対角なので、余弦定理は長さ 7 の辺の対角に 1 回使えばよい。",
        "register": "spoken"
      },
      {
        "en": "In triangle ABC, a = 7, b = 5, and c = 3. Find the measure of the largest angle.",
        "ja": "△ABC において a = 7, b = 5, c = 3 のとき、最大の角の大きさを求めよ。",
        "register": "written"
      },
      {
        "en": "Since BC > AC, the side-angle inequality gives ∠A > ∠B.",
        "ja": "BC > AC であるから、三角形の辺と角の大小関係より ∠A > ∠B。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "等号の場合（a = b ⇔ A = B）は二等辺三角形の性質。",
      "余弦定理で最大の辺の対角を求めて鋭角・直角・鈍角を判定するときに使う。最大の辺の対角を余弦定理で求め、鋭角・直角・鈍角を判定する。",
      "CK-12 Geometry 4.25 の SAS Inequality Theorem・SSS Inequality Theorem は 2 つの三角形の辺と角を比べる別の定理で、1 つの三角形の中の辺と角の大小とは違う。"
    ],
    "related": [
      "triangle",
      "triangle-inequality",
      "law-of-cosines",
      "isosceles-triangle"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.25 Comparing Angles and Sides in Triangles・4.31 Converse of the Pythagorean Theorem"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 「辺と角の大小」は 0 件（辺と角との間に成り立つ基本的な関係は正弦・余弦定理の文）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "side",
    "ja": {
      "term": "辺",
      "reading": "へん",
      "alt": []
    },
    "en": {
      "term": "side",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "多角形のまわりをつくっている線分の 1 つ 1 つ。等式の左辺・右辺のように、等号の片側にある式を指すこともある。",
    "definition_en": "One of the segments that form a polygon. It also means either part of an equation or inequality, as in the left-hand side.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each side of this square is 4 inches, so the perimeter is 16 inches.",
        "ja": "この正方形の 1 辺は 4 インチだから、周の長さは 16 インチ。",
        "register": "spoken"
      },
      {
        "en": "Find the side length of a square whose area is 49 cm².",
        "ja": "面積が 49 cm² の正方形の 1 辺の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "立体の辺（稜）は side ではなく edge（エントリ edge）。立方体の辺は 12 本で、英語では 12 edges と言う。",
      "等式の左辺・右辺の「辺」も side で、left-hand side ／ right-hand side（エントリ left-hand-side）。両辺は both sides（エントリ both-sides）。"
    ],
    "related": [
      "edge",
      "polygon",
      "triangle",
      "perimeter",
      "left-hand-side",
      "vertex"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "辺",
        "en": "Edge (geometry)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "side of the triangle | side of a triangle | side of the square | side of a square | side length | length of the side | length of each side | three sides | four sides | all sides | side of the polygon": 196
      },
      "written": {
        "side of the triangle | side of a triangle | side of the square | side of a square | side length | length of the side | length of each side | three sides | four sides | all sides | side of the polygon": 273
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sign-chart-inequality",
    "ja": {
      "term": "符号図",
      "reading": "ふごうず",
      "alt": [
        "符号表"
      ]
    },
    "en": {
      "term": "sign chart",
      "alt": [],
      "variants": [
        {
          "term": "sign analysis",
          "register": "spoken",
          "note": "Professor Leonard の言い方。話し言葉はすべて Professor Leonard で、書き言葉にはほとんど出てこない（OpenStax Calculus）。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「符号図」は学習指導要領解説に無い、本プロジェクトの訳語。数直線に零点を並べ、区間ごとの式の符号（+ ／ −）を書き込んだ図。日本の増減表（sign-chart）と英語の名前は同じだが、こちらは不等式を解くための図で、関数の増減は表さない。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "不等式を解くために、数直線上に式の零点を並べて、各区間での式の符号を書き込んだ図。",
    "definition_en": "A number line marked with the zeros of an expression and the sign of the expression on each interval, used to solve inequalities.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Make a sign chart: the factors change sign at −2, 1, and 4.",
        "ja": "符号図をかこう。因数の符号は −2, 1, 4 で変わる。",
        "register": "spoken"
      },
      {
        "en": "Use a sign chart to solve (x − 3)/(x + 1) ≥ 0.",
        "ja": "符号図を用いて (x − 3)/(x + 1) ≧ 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスには sign chart は出てこず、sign analysis が話し言葉に出てくる（すべて Professor Leonard）。CED（Unit 9 の概要）は sign chart を答えを見つける道具として挙げる。",
      "分数の不等式では、分母が 0 になる点も区切りに入れ、その点は答えに含めない。"
    ],
    "related": [
      "sign-chart",
      "test-point",
      "polynomial-inequality"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 9 の概要（sign chart）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sign analysis": 11
      },
      "written": {
        "sign analysis": 1
      },
      "sources": [
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sign-chart",
    "ja": {
      "term": "増減表",
      "reading": "ぞうげんひょう",
      "alt": []
    },
    "en": {
      "term": "sign chart",
      "alt": [
        "sign diagram"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "日本の増減表（x の行・f′(x) の行・f(x) の行を並べ、矢印で増減を書く表）に当たる定型の表は、AP の CED にも OpenStax Calculus にも出てこない。同じ判断は first derivative test（CED topic 5.4）で行い、f′ の符号を区間ごとに書き出した sign chart を使うことがある。CED は Unit 9 の概要で、sign chart は（グラフの分析と同じく）答えを見つける道具として役に立つが、理由（justification）の点を得るには定義や定理に結びつけて書く必要がある、としている。答案では表だけで済ませず、f′ の符号の変化と定理の名前（by the first derivative test）で理由を書く。数IIIの凹凸まで入れた表も同じで、凹凸は f″ の符号で判断する（concavity、CED topic 5.6）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "導関数の符号を区間ごとに並べ、もとの関数の増減と極値を読み取る表。",
    "definition_en": "A number line, or table, marked with the sign of the derivative on each interval, used to find where a function rises and falls.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Make a sign chart for f prime and see where it changes from plus to minus.",
        "ja": "f′ の増減表を書いて、正から負に変わるところを見て。",
        "register": "spoken"
      },
      {
        "en": "Since f′ changes from positive to negative at x = 1, f has a local maximum there by the first derivative test.",
        "ja": "増減表より、x = 1 の前後で f′ が正から負に変わるので、f は x = 1 で極大となる。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "make a sign chart",
        "ja": "増減表を書く"
      },
      {
        "en": "f prime changes sign at x = 1",
        "ja": "x=1 の前後で f′ の符号が変わる"
      }
    ],
    "pitfalls": [
      "increase-decrease table は直訳で、用例コーパスにも CED・OpenStax にも出てこない。",
      "極大・極小は local maximum ／ local minimum（CED は relative extrema とも書く）。区間全体の最大・最小は absolute maximum ／ minimum（CED・OpenStax Calculus の言い方。global maximum も使われるが少ない）。",
      "凹凸は concave up ／ concave down（CED topic 5.6、OpenStax Calculus）。「上に凸」を convex upward と直訳した形は用例コーパスに出てこない。Khan Academy は concave upward ともよく言う。"
    ],
    "related": [
      "make-a-sign-chart",
      "first-derivative-test",
      "increasing-and-decreasing",
      "concavity",
      "derivative-at-a-point",
      "sign-chart-inequality"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Unit 9 の概要（sign chart）、Topic 5.4（first derivative test）・5.6（concavity）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CED の呼び方 sign chart（unit9）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sign",
    "ja": {
      "term": "符号",
      "reading": "ふごう",
      "alt": []
    },
    "en": {
      "term": "sign",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "数が正であるか負であるかを表す + と − のこと。−7 の符号は −、+7 の符号は + である。",
    "definition_en": "Whether a number is positive or negative, shown by a + or − in front of it. For example, the sign of −7 is negative.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When two numbers have the same sign, add them and keep that sign.",
        "ja": "2 つの数が同符号なら、足してその符号をそのまま付ければいい。",
        "register": "spoken"
      },
      {
        "en": "Without multiplying, determine the sign of the product (−2)(−5)(−1).",
        "ja": "計算しないで、積 (−2)(−5)(−1) の符号を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「同符号」は same sign、「異符号」は opposite signs または different signs。",
      "minus sign（− の記号）や radical sign（根号 √）の sign は記号そのものの名前で、数が正か負かを表す「符号」とは使い方が違う。",
      "0 は正の数でも負の数でもないので、0 の符号は + でも − でもない。"
    ],
    "related": [
      "positive-number",
      "negative-number",
      "changes-sign",
      "absolute-value",
      "sign-chart"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!radical !equal !equals !inequality !summation !integral !plus !minus sign": 1277
      },
      "written": {
        "!radical !equal !equals !inequality !summation !integral !plus !minus sign": 692
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "significance-level",
    "ja": {
      "term": "有意水準",
      "reading": "ゆういすいじゅん",
      "alt": []
    },
    "en": {
      "term": "significance level",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "level of significance",
          "register": "written",
          "note": "話し言葉にはほとんど出てこない。書き言葉はすべて OpenStax Introductory Statistics。書き言葉では significance level も同じくらい使われ、2 つが並ぶ。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I",
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "仮説検定で、帰無仮説を棄却する基準にする確率 α。5% や 1% がよく使われる。",
    "definition_en": "The probability α used as the cutoff for rejecting the null hypothesis, often 5% or 1%.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If the significance level is 0.05, we reject when the p-value is below 0.05.",
        "ja": "有意水準が 0.05 なら、p 値が 0.05 未満のとき棄却する。",
        "register": "spoken"
      },
      {
        "en": "Test the claim at the 5% significance level.",
        "ja": "有意水準 5% でその主張を検定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "α は alpha と読む。有意水準は第一種の過誤（Type I error）の確率でもある。"
    ],
    "related": [
      "hypothesis-testing",
      "reject"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "有意",
        "en": "Statistical significance"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B 統計的な推測〔用語・記号〕: 有意水準（数学I の節には無い）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "significance level": 85,
        "level of significance": 2
      },
      "written": {
        "level of significance": 116,
        "significance level": 108
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "significant-figures",
    "ja": {
      "term": "有効数字",
      "reading": "ゆうこうすうじ",
      "alt": []
    },
    "en": {
      "term": "significant figures",
      "alt": [],
      "variants": [
        {
          "term": "significant digits",
          "register": "both",
          "note": "書き言葉ではこちら（OpenStax Calculus ほか）。話し言葉でも significant figures に次いで使う。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "測定値などの近似値のうち、信頼できる桁の数字。2.30 × 10³ のように（整数部分が 1 桁の数）×（10 の累乗）の形で書くと、どの桁までが有効数字かがはっきりする。",
    "definition_en": "The digits of a measured or approximate value that can be trusted. For example, 2.30 × 10³ has three significant figures: 2, 3, and 0.",
    "latex": "2.30 \\times 10^{3}",
    "spoken_en": "two point three zero times ten to the third",
    "examples": [
      {
        "en": "The zero at the end of 2.30 counts, so that's three significant figures.",
        "ja": "2.30 の最後の 0 も数えるから、有効数字は 3 桁だね。",
        "register": "spoken"
      },
      {
        "en": "The length of a road was measured to the nearest 10 m as 4370 m. Write this measurement in scientific notation with three significant figures.",
        "ja": "ある道のりを、10 m 未満を四捨五入して測ったら 4370 m だった。この測定値を、有効数字を 3 桁として、（整数部分が 1 桁の数）×（10 の累乗）の形で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では、2.30 × 10³ のように（整数部分が 1 桁の数）×（10 の累乗）で書く形を scientific notation と呼ぶ（エントリ scientific-notation）。有効数字（significant figures）とは別の名前。",
      "OpenStax Calculus（Volume 1〜3）は significant digits と書く（10 件。significant figures は 2 件）。",
      "2.30 の最後の 0 は有効数字に数える。2.3 × 10³ と書くと有効数字は 2 桁で、2.30 × 10³ とは測定の精度が違う。",
      "見出しは話し言葉の首位 significant figures（すべて Khan Academy）。中学校学習指導要領解説（誤差や近似値）は、測定値を 2300 m ではなく 2.30 × 10³ m のように（整数部分が 1 桁の数）×（10 の累乗）の形で表して、どの数字までが有効数字かを明らかにする、と書く。"
    ],
    "related": [
      "measurement",
      "error",
      "approximate-value",
      "scientific-notation",
      "digit"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "有効数字",
        "en": "Significant figures"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "中学校 第 3 学年 誤差や近似値（［内容の取扱い］（1））: 2300 m を 2.30×10³ m のように表して有効数字を明らかにする"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "significant digits": 35,
        "significant figures": 57
      },
      "written": {
        "significant digits": 13,
        "significant figures": 2
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "similar-matrices",
    "ja": {
      "term": "相似な行列",
      "reading": "そうじなぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "similar matrices",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "正則行列 P を使って B = P⁻¹AP と書ける 2 つの正方行列。同じ固有値・行列式・トレースをもつ。",
    "definition_en": "Two square matrices with B = P⁻¹AP for some invertible P; they have the same eigenvalues, determinant and trace.",
    "latex": "B = P^{-1}AP",
    "spoken_en": null,
    "examples": [
      {
        "en": "Similar matrices are the same transformation written in different bases.",
        "ja": "相似な行列は、同じ変換を別の基底で書いたものだ。",
        "register": "spoken"
      },
      {
        "en": "Show that similar matrices have the same characteristic polynomial.",
        "ja": "相似な行列は同じ特性多項式をもつことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "図形の相似（similar figures）と同じ語。用例コーパスの話し言葉はすべて MIT 18.06。"
    ],
    "related": [
      "diagonalization",
      "change-of-basis",
      "characteristic-polynomial"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.5 Similarity and Diagonalization"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "similar matrices": 17
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "similar-triangles",
    "ja": {
      "term": "相似な三角形",
      "reading": "そうじなさんかくけい",
      "alt": []
    },
    "en": {
      "term": "similar triangles",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry",
      "geometry"
    ],
    "level": {
      "jp": [
        "中3",
        "数I"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "対応する角がそれぞれ等しく、対応する辺の比がすべて等しい 2 つの三角形。一方を拡大・縮小して移すともう一方に重なる。",
    "definition_en": "Two triangles whose corresponding angles are equal and whose corresponding sides are proportional, so one is a scaled copy of the other.",
    "latex": "\\triangle ABC \\sim \\triangle DEF",
    "spoken_en": "triangle ABC is similar to triangle DEF",
    "examples": [
      {
        "en": "These two are similar triangles, so the corresponding sides are proportional: x over 4 equals 6 over 3.",
        "ja": "この 2 つは相似な三角形なので、対応する辺の比は等しい。x/4 = 6/3 だ。",
        "register": "spoken"
      },
      {
        "en": "A 2 m pole casts a 3 m shadow while the tree's shadow is 15 m long. Use similar triangles to find the height of the tree.",
        "ja": "2 m の棒の影が 3 m のとき、木の影は 15 m であった。相似な三角形を使って、木の高さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "相似の記号は、中学校学習指導要領解説（〔用語・記号〕）では ∽、IM Geometry・CK-12 Geometry では ∼（△ABC ∼ △DEF）。どちらも対応する頂点の順に書く。",
      "「対応する辺の比が等しい」は corresponding sides are proportional と言う（比例する辺）。"
    ],
    "related": [
      "triangle",
      "congruent",
      "ratio-of-areas-of-similar-figures",
      "conditions-that-determine-a-triangle",
      "trigonometric-ratio"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形〔用語・記号〕: ∽"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Geometry: △DHF ∼ △EHG と書く"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「図形の相似」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%B3%E5%BD%A2%E3%81%AE%E7%9B%B8%E4%BC%BC",
        "note": "欧米は ∼、日本は ∽ の記号"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "7.3 Similar Polygons and Scale Factors・7.7 AA Similarity（記号 ∼: △ABC ∼ △YZX）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）〔用語・記号〕",
        "note": "中3 B 図形: ∽"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "similar triangles": 28
      },
      "written": {
        "similar triangles": 27
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "similar",
    "ja": {
      "term": "相似",
      "reading": "そうじ",
      "alt": [
        "相似な図形"
      ]
    },
    "en": {
      "term": "similar",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "一方の図形を拡大または縮小すると、他方の図形にぴったり重なること。対応する角はそれぞれ等しく、対応する辺の比はすべて等しい。",
    "definition_en": "Having the same shape, so that one figure can be enlarged or shrunk and then moved to fit exactly on the other. Corresponding angles are equal, and corresponding sides are proportional.",
    "latex": "\\triangle ABC \\sim \\triangle DEF",
    "spoken_en": "triangle ABC is similar to triangle DEF",
    "examples": [
      {
        "en": "The rectangles are similar because each side of the big one is 3 times the matching side of the small one.",
        "ja": "大きい長方形の各辺は、小さい長方形の対応する辺の 3 倍だから、この 2 つの長方形は相似だ。",
        "register": "spoken"
      },
      {
        "en": "Triangle ABC is similar to triangle DEF, with AB = 6, BC = 8, and DE = 9. Find EF.",
        "ja": "△ABC ∽ △DEF で、AB = 6、BC = 8、DE = 9 のとき、EF の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日常語の similar は「似ている」という広い意味。数学の similar は「拡大・縮小すると重なる」という決まった意味で、形がなんとなく似ているだけでは similar とは言わない。",
      "英語の similar は形容詞で、two figures are similar ／ triangle ABC is similar to triangle DEF と言う。相似の記号は、中学校学習指導要領解説（〔用語・記号〕）では ∽、IM Geometry・CK-12 Geometry では ∼（△ABC ∼ △DEF）。どちらも triangle ABC is similar to triangle DEF と読む。名詞の「相似」は similarity。",
      "相似な三角形はエントリ similar-triangles、相似比はエントリ scale-factor。合同（congruent）は、相似のうち大きさも同じ場合。"
    ],
    "related": [
      "similar-triangles",
      "scale-factor",
      "congruent",
      "dilation",
      "similarity-criteria"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形〔用語・記号〕: ∽"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Geometry: △DHF ∼ △EHG と書く"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「図形の相似」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%B3%E5%BD%A2%E3%81%AE%E7%9B%B8%E4%BC%BC",
        "note": "欧米は ∼、日本は ∽ の記号"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "7.3 Similar Polygons and Scale Factors・7.7 AA Similarity（記号 ∼: △ABC ∼ △YZX）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）〔用語・記号〕",
        "note": "中3 B 図形: ∽"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "triangles are similar | figures are similar | polygons are similar | rectangles are similar | shapes are similar | similar figures | similar polygons | similar solids | similar rectangles | is similar to triangle": 12
      },
      "written": {
        "triangles are similar | figures are similar | polygons are similar | rectangles are similar | shapes are similar | similar figures | similar polygons | similar solids | similar rectangles | is similar to triangle": 22
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.03",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "similarity-criteria",
    "ja": {
      "term": "相似条件",
      "reading": "そうじじょうけん",
      "alt": []
    },
    "en": {
      "term": "similarity theorems",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説は三角形の相似条件を 3 つ挙げ（解説の言い方は「3組の辺の比がすべて等しい」「2組の辺の比とその間の角がそれぞれ等しい」「2組の角がそれぞれ等しい」）、「相似条件」とまとめて呼ぶ。英語では個々の条件に名前がある: CK-12 Geometry は SSS Similarity Theorem（7.8）、SAS Similarity Theorem（7.9）、AA Similarity Postulate（7.7）、IM Geometry は Side-Side-Side ／ Side-Angle-Side ／ Angle-Angle Triangle Similarity Theorem（エントリ aa-similarity）。まとめて呼ぶときは、CK-12 が問題文で write the similarity theorem と書き（4 件）、IM のレッスン名は Conditions for Triangle Similarity（3.9・3.10）。見出しの similarity theorems はこの総称。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 つの三角形が相似であるための条件で、3 つある。3 組の辺の比がすべて等しい、2 組の辺の比が等しくその間の角も等しい、2 組の角がそれぞれ等しい。",
    "definition_en": "The sets of matching parts that are enough to prove two triangles similar. They are named SSS similarity (all three pairs of sides proportional), SAS similarity, and AA similarity.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Which of the similarity theorems works here? We've got two pairs of equal angles, so it's AA.",
        "ja": "ここで使える相似条件はどれかな。2 組の角がそれぞれ等しいから、「2 組の角がそれぞれ等しい」だね。",
        "register": "spoken"
      },
      {
        "en": "In △ABC and △DEF, AB = 4, AC = 6, DE = 6, DF = 9, and m∠A = m∠D. Which of the similarity theorems shows that △ABC ~ △DEF? Explain.",
        "ja": "△ABC と △DEF で、AB = 4、AC = 6、DE = 6、DF = 9、∠A = ∠D である。△ABC ∽ △DEF を示すには、どの相似条件を用いればよいか。理由も説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "SSS・SAS は合同条件と同じ略号（エントリ sss-congruence、sas-congruence）。相似の SSS・SAS は辺の「比」が等しいこと、合同の SSS・SAS は辺の「長さ」が等しいことなので、similarity か congruence かを付けて区別する。",
      "角の条件は 2 組で足りる。2 組の角が等しければ、内角の和が 180° なので 3 組目も等しい（エントリ aa-similarity）。",
      "英語では条件を theorem（定理）とも postulate（公準）とも呼ぶ（CK-12 は AA を postulate、SSS・SAS を theorem）。まとめた総称は similarity theorems（CK-12 の問題文）か conditions for triangle similarity（IM のレッスン名）。"
    ],
    "related": [
      "aa-similarity",
      "similar-triangles",
      "congruence-criteria",
      "similar",
      "scale-factor"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 3.9 Conditions for Triangle Similarity・3.10 Other Conditions for Triangle Similarity"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形: 三角形の相似条件（3 つ）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "7.7 AA Similarity・7.8 SSS Similarity・7.9 SAS Similarity"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 similarity theorems（Geometry 3.10 Other Conditions for Triangle Similarity・Geometry 3.9 Conditions for Triangle Similarity・Geometry 3.16 Bank Shot・計 23 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "similarity-transformation",
    "ja": {
      "term": "相似変換",
      "reading": "そうじへんかん",
      "alt": []
    },
    "en": {
      "term": "similarity transformation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "拡大・縮小と合同変換（平行移動・回転・対称移動）を組み合わせた変換。図形をそれと相似な図形に移す。",
    "definition_en": "A transformation made of a dilation combined with rigid transformations (translations, rotations, and reflections). It maps a figure onto a similar figure.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To show these two triangles are similar, find a similarity transformation: dilate by a factor of 2 from point A, then rotate 90 degrees.",
        "ja": "この 2 つの三角形が相似であることを示すには相似変換を見つければいい。点 A を中心に 2 倍に拡大してから 90° 回転させる。",
        "register": "spoken"
      },
      {
        "en": "Describe a similarity transformation that maps △ABC onto △DEF.",
        "ja": "△ABC を △DEF に移す相似変換を 1 つ述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "線形代数で similarity transformation と言うと、行列 A を P⁻¹AP に変えること（相似な行列）を指し、図形の相似変換とは別物。"
    ],
    "related": [
      "dilation",
      "rigid-motion",
      "similar-triangles",
      "similar-matrices"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "2.6 Similarity（2 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.8 Translation Applications in Circle Similarity・7.9 SAS Similarity"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 similarity transformation（CK-12 Geometry 8.8 Translation Applications in Circle Similarity・CK-12 Geometry 7.9 SAS Similarity・計 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "simple-interest",
    "ja": {
      "term": "単利",
      "reading": "たんり",
      "alt": [
        "利息"
      ]
    },
    "en": {
      "term": "simple interest",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "元金にだけ利息がつく利息のつき方。利息は 元金 × 年利率 × 年数 で求める。",
    "definition_en": "Interest earned only on the original amount of money, called the principal. It is found with I = Prt, where P is the principal, r is the annual interest rate, and t is the time in years.",
    "latex": "I = Prt",
    "spoken_en": "I equals P times r times t",
    "examples": [
      {
        "en": "With simple interest, you only earn interest on the principal, so you get the same amount every year.",
        "ja": "単利だと利息がつくのは元金だけだから、毎年同じ額の利息がもらえる。",
        "register": "spoken"
      },
      {
        "en": "You deposit $500 in an account that earns 4% simple interest per year. How much interest will you earn in 3 years?",
        "ja": "年利 4 % の単利で利息がつく口座に 500 ドルを預ける。3 年間で受け取る利息を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の interest は「利息・利子」、rate は利率、元金は principal（「原理」の principle とは綴りが違う）。",
      "r は小数に直して代入する（4 % なら 0.04）。t は年数なので、月数で与えられたら年に直す。",
      "利息にも利息がつくのは複利（エントリ compound-interest）。"
    ],
    "related": [
      "compound-interest",
      "percent",
      "percent-change",
      "substitute",
      "exponential-growth"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "simple interest": 78
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "simplify-radicals",
    "ja": {
      "term": "根号を簡単にする",
      "reading": "こんごうをかんたんにする",
      "alt": []
    },
    "en": {
      "term": "simplify the radical",
      "alt": [],
      "uk": null,
      "register": "written",
      "variants": [
        {
          "term": "simplify the radical",
          "register": "written",
          "note": "書き言葉の首位（OpenStax Elementary Algebra・Intermediate Algebra が中心）。"
        },
        {
          "term": "simplify square roots",
          "register": "written",
          "note": "OpenStax Elementary Algebra の節の名前 Simplify Square Roots。書き言葉では 2 番目。"
        },
        {
          "term": "simplify radicals",
          "register": "both",
          "note": "話し言葉の首位（NancyPi と Khan Academy の中学の講義）。書き言葉では 3 番目。"
        }
      ]
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic",
      "algebra"
    ],
    "level": {
      "jp": [
        "中3",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "根号の中の数から平方因数を外に出し、根号の中をできるだけ小さい自然数にする。根号を含む式では、そのうえで同じ根号の項をまとめる。",
    "definition_en": "To pull square factors out from under a radical so that the number left inside is as small as possible, then combine like radical terms in an expression.",
    "latex": "\\sqrt{12} = \\sqrt{4 \\cdot 3} = 2\\sqrt{3}",
    "spoken_en": "root twelve is the square root of four times three, which is two root three",
    "examples": [
      {
        "en": "To simplify the radical, look for the biggest perfect square that goes into 72. That's 36, so √72 is 6√2.",
        "ja": "根号を簡単にするには、72 を割り切る最大の平方数を探す。36 だから √72 = 6√2 になる。",
        "register": "spoken"
      },
      {
        "en": "Simplify the radical expression √50 − √18 + √8.",
        "ja": "√50 − √18 + √8 を簡単にせよ。",
        "register": "written"
      },
      {
        "en": "Write √48 in simplest radical form.",
        "ja": "√48 を簡単にせよ（根号の中をできるだけ小さい自然数にせよ）。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "write … in simplest radical form",
        "ja": "根号の中をできるだけ小さい自然数にした形で表す"
      }
    ],
    "pitfalls": [
      "simplest radical form は「簡単にした結果の形」を指す名詞句。問題文の Write … in simplest radical form が「簡単にせよ」に当たる。",
      "分母の根号を消す「有理化」は別の操作で、英語では rationalize the denominator と言う。",
      "根拠になる性質 √(ab) = √a·√b と √(a/b) = √a/√b を、OpenStax Elementary Algebra は Product Property of Square Roots ／ Quotient Property of Square Roots と呼ぶ（Intermediate Algebra は Product Property of Roots）。",
      "センター試験の注意事項は「根号の中に現れる自然数が最小となる形で答えなさい」と書く（平成 30 年度）。",
      "英語の simplify the radical は、文字を含む根号（OpenStax Elementary Algebra 9.2）や立方根・4 乗根（OpenStax Intermediate Algebra 8.2）にも使う。"
    ],
    "related": [
      "nth-root",
      "nested-radical",
      "radical-function",
      "rational-exponent"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "9.2 Simplify Square Roots（Product Property ／ Quotient Property of Square Roots。pitfalls[2]・[4]）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "8.2 Simplify Radical Expressions（Product Property of Roots。pitfalls[2]・[4]）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成30年度 大学入試センター試験 本試験 数学（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?f=abm00033137.pdf&n=h30+suugaku1+00hyousi.pdf",
        "note": "注意事項: 根号の中に現れる自然数が最小となる形で答えなさい"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "simplify the radical": 1,
        "simplify square roots": 1,
        "simplify radicals": 8
      },
      "written": {
        "simplify radicals": 29,
        "simplify the radical": 105,
        "simplify square roots": 37,
        "write … in simplest radical form": 2
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:nancypi"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "simplify",
    "ja": {
      "term": "簡単にする",
      "reading": "かんたんにする",
      "alt": [
        "整理する",
        "単純化する"
      ]
    },
    "en": {
      "term": "simplify",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "同類項をまとめる、かっこを外す、約分するなどして、式を同じ値のまま短く扱いやすい形に直す。",
    "definition_en": "To rewrite an expression in a shorter form with the same value: clear the parentheses, combine like terms, cancel common factors, and so on.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Can we simplify 2(x + 3) + 4x any more? Sure, distribute the 2 and you get 6x + 6.",
        "ja": "2(x + 3) + 4x はもっと簡単にできる？ できるね。2 を分配すると 6x + 6 だ。",
        "register": "spoken"
      },
      {
        "en": "Simplify 3(2a − 5) − 4(a − 2).",
        "ja": "3(2a − 5) − 4(a − 2) を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "方程式の解を求めるのは solve（エントリ solve）。simplify は式を同じ値の短い形に直すことで、答えも式のまま残る。",
      "「最も簡単な形で答えよ」は in simplest form。分数を約分しきった形は in lowest terms とも言う（エントリ reduce）。",
      "ja.alt の「整理する」でも、「降べきの順に整理する」のように項を並べ替える意味のものは write in descending order（エントリ write-in-descending-order）。"
    ],
    "related": [
      "combine-like-terms",
      "distributive-property",
      "reduce",
      "simplify-radicals",
      "solve"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "simplify": 1611
      },
      "written": {
        "simplify": 4886
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "simpsons-rule",
    "ja": {
      "term": "シンプソンの公式",
      "reading": "しんぷそんのこうしき",
      "alt": []
    },
    "en": {
      "term": "Simpson's rule",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "区間を偶数個に分け、隣り合う 2 つの小区間ごとに放物線で近似して定積分の近似値を求める公式。",
    "definition_en": "A method that approximates a definite integral by fitting parabolas over pairs of subintervals; n must be even.",
    "latex": "S_n = \\frac{\\Delta x}{3}\\bigl(f(x_0) + 4f(x_1) + 2f(x_2) + \\cdots + 4f(x_{n-1}) + f(x_n)\\bigr)",
    "spoken_en": "S sub n equals delta x over three times the quantity f of x sub zero plus four f of x sub one plus two f of x sub two, and so on, plus four f of x sub n minus one plus f of x sub n",
    "examples": [
      {
        "en": "Simpson's rule needs an even number of subintervals, so n equals five won't work.",
        "ja": "シンプソンの公式は小区間の数が偶数でないといけないから、n = 5 は使えない。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "係数は 1, 4, 2, 4, …, 2, 4, 1 の並び（one, four, two, four, …）。"
    ],
    "related": [
      "numerical-integration",
      "trapezoidal-rule",
      "midpoint-riemann-sum"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "シンプソンの公式",
        "en": "Simpson's rule"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学II (7) 積分法（発展・拡充）: 面積を台形公式やシンプソンの公式で求める"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "Simpson's rule": 34
      },
      "written": {
        "Simpson's rule": 41
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "simulation",
    "ja": {
      "term": "シミュレーション",
      "reading": "しみゅれーしょん",
      "alt": []
    },
    "en": {
      "term": "simulation",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "Integrated Math 3",
        "AP Statistics"
      ]
    },
    "definition_ja": "実際に試行する代わりに、乱数などを使って同じ仕組みの試行を模擬的に何度も行い、結果の分布を調べること。",
    "definition_en": "Imitating a random process many times, for example with random numbers, instead of carrying it out, to see the distribution of the results.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's run a simulation: flip a virtual coin a thousand times and see what happens.",
        "ja": "シミュレーションをしてみよう。仮想の硬貨を 1000 回投げて何が起こるか見る。",
        "register": "spoken"
      },
      {
        "en": "Design a simulation using random digits to estimate the probability that a family of four children has at least two girls.",
        "ja": "乱数を使うシミュレーションを設計し、4 人きょうだいに女子が少なくとも 2 人いる確率を推定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉は少ない（ほとんどが OpenStax Introductory Statistics）。",
      "AP Statistics の CED（topic 2.3 Estimating Probabilities Using Simulation）と IM（Grade 7 8.6、Algebra 2 7.11・7.14）は、単元の題や問題文でも simulation と書く。"
    ],
    "related": [
      "random-number",
      "experimental-probability",
      "mathematical-model"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "シミュレーション",
        "en": "Simulation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.3 Estimating Probabilities Using Simulation（2.3.A.4 simulation）・2.8・2.10・2.12"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 7 8.6 Estimating Probabilities Using Simulation・8.10 Designing Simulations"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 7.11・7.14（simulation）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "simulation": 56
      },
      "written": {
        "simulation": 5
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sine",
    "ja": {
      "term": "正弦",
      "reading": "せいげん",
      "alt": [
        "サイン"
      ]
    },
    "en": {
      "term": "sine",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "角 θ に対して、直角三角形では（対辺）/（斜辺）、座標では単位円上の点の y 座標で定める三角比・三角関数 sin θ。",
    "definition_en": "The trigonometric function sin θ: in a right triangle, the length of the side opposite θ divided by the hypotenuse; on the unit circle, the height (y-coordinate) of the point reached at angle θ.",
    "latex": "\\sin\\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}}",
    "spoken_en": "sine theta equals opposite over hypotenuse",
    "examples": [
      {
        "en": "So the sine of theta is opposite over hypotenuse, which is 3 over 5.",
        "ja": "だから sin θ は対辺 / 斜辺で、3/5 になる。",
        "register": "spoken"
      },
      {
        "en": "If cos θ = 4/5 and 0° < θ < 90°, find sin θ.",
        "ja": "0° < θ < 90° で cos θ = 4/5 のとき、sin θ の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "sin θ は \"sine theta\" と読む（sin は sine の略記）。sin²θ は (sin θ)² のことで \"sine squared theta\"。",
      "sin⁻¹ x は 1/sin x ではなく逆関数 arcsin x（arcsine）。",
      "正弦定理は英語で law of sines。"
    ],
    "related": [
      "cosine",
      "tangent",
      "trigonometric-ratio",
      "arcsine",
      "pythagorean-identity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "sine": 3033
      },
      "written": {
        "sine": 560
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "singular-matrix",
    "ja": {
      "term": "特異行列",
      "reading": "とくいぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "singular matrix",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "matrix is singular",
          "register": "spoken",
          "note": "述語の形（the matrix is singular）。話し言葉はすべて MIT OCW で、書き言葉には出てこない。"
        }
      ],
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "逆行列をもたない正方行列。行列式が 0 であることと同じ。",
    "definition_en": "A square matrix with no inverse; equivalently, its determinant is zero.",
    "latex": "\\det A = 0",
    "spoken_en": null,
    "examples": [
      {
        "en": "The determinant is zero, so this is a singular matrix, and it has no inverse.",
        "ja": "行列式が 0 なので、これは特異行列で、逆行列がない。",
        "register": "spoken"
      },
      {
        "en": "Find all values of k for which the matrix is singular.",
        "ja": "行列が特異行列になる k の値をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて MIT 18.06。Nicholson の singular は特異値分解（singular value）の文脈が多い。"
    ],
    "related": [
      "invertible-matrix",
      "determinant"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "2.4 Matrix Inverses・3.2 Determinants and Matrix Inverses"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "singular matrix": 19,
        "matrix is singular": 14
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "singular-value-decomposition",
    "ja": {
      "term": "特異値分解",
      "reading": "とくいちぶんかい",
      "alt": []
    },
    "en": {
      "term": "singular value decomposition",
      "alt": [
        "SVD"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "任意の行列を A = U Σ Vᵀ（U, V は直交行列、Σ は特異値を並べた対角行列）の形に分けること。",
    "definition_en": "Writing any matrix as A = U Σ Vᵀ, where U and V are orthogonal and Σ is diagonal with the singular values.",
    "latex": "A = U\\Sigma V^{T}",
    "spoken_en": null,
    "examples": [
      {
        "en": "The SVD works for any matrix, even a rectangular one.",
        "ja": "特異値分解は、長方形の行列でもどんな行列にも使える。",
        "register": "spoken"
      },
      {
        "en": "Find a singular value decomposition of the matrix.",
        "ja": "行列の特異値分解を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "orthonormal-basis",
      "diagonalization",
      "transpose"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "特異値分解",
        "en": "Singular value decomposition"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "8.6 The Singular Value Decomposition"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "singular value decomposition": 15,
        "SVD": 19
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sinusoid",
    "ja": {
      "term": "正弦曲線",
      "reading": "せいげんきょくせん",
      "alt": [
        "正弦波"
      ]
    },
    "en": {
      "term": "sinusoidal function",
      "alt": [
        "sinusoid"
      ],
      "variants": [
        {
          "term": "sine wave",
          "register": "spoken",
          "note": "話し言葉ではこちらが多いが、ほとんどが 3Blue1Brown。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の数II では y = sin θ のグラフを「正弦曲線」と呼ぶ。英語の sinusoidal function は cos のグラフや平行移動・拡大したものもまとめて指す。「正弦波」は主に物理の言い方。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "y = A sin(Bx − C) + D（または cos の形）で表される関数と、そのグラフ。振幅、周期、位相のずれ、振動の中心線で形が決まる。",
    "definition_en": "A function of the form y = A sin(Bx − C) + D or y = A cos(Bx − C) + D, or its wave-shaped graph. Its shape is set by the amplitude, period, phase shift, and midline.",
    "latex": "y = A\\sin(Bx - C) + D",
    "spoken_en": "y equals A sine of B x minus C, plus D",
    "examples": [
      {
        "en": "A sinusoidal function is basically a sine wave that's been stretched and shifted around.",
        "ja": "正弦曲線というのは、要するに正弦波を伸ばしたりずらしたりしたもの。",
        "register": "spoken"
      },
      {
        "en": "Write an equation of the sinusoidal function with amplitude 2, period π, and midline y = −1.",
        "ja": "振幅 2、周期 π、振動の中心線 y = −1 の正弦曲線を表す式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "sinusoid は名詞、sinusoidal は形容詞。sinusoidal function ／ sinusoidal graph のように名詞の前に置く。",
      "周期は 2π/|B|、位相のずれ（エントリ phase-shift）は C ではなく C/B。先に B でくくって y = A sin(B(x − C/B)) + D の形にすると読み取りやすい。",
      "振幅は |A|、振動の中心線は y = D（エントリ midline）。"
    ],
    "related": [
      "sine",
      "amplitude",
      "period",
      "phase-shift",
      "midline"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正弦波",
        "en": "Sine wave"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "sine wave": 39,
        "sinusoidal function": 12,
        "sinusoid": 3
      },
      "written": {
        "sinusoidal function": 67,
        "sine wave": 5,
        "sinusoid": 12
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sketch-of-a-solid",
    "ja": {
      "term": "見取図",
      "reading": "みとりず",
      "alt": []
    },
    "en": {
      "term": "perspective drawing",
      "alt": [
        "three-dimensional drawing"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。見取図は a drawing of the solid、a 3D sketch、perspective drawing などと言い、見えない辺は hidden edges と呼んで dashed lines（点線）でかく。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "立体の全体の形がひと目で分かるように、斜めから見たようにかいた図。見えない辺は点線でかく。",
    "definition_en": "A drawing of a solid as seen from an angle, so that its whole shape can be seen at once. Edges that would be hidden from view are drawn with dashed lines.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In this perspective drawing, the dashed lines are the edges hidden in the back.",
        "ja": "この見取図で点線になっているのは、奥に隠れて見えない辺だよ。",
        "register": "spoken"
      },
      {
        "en": "Make a perspective drawing of a triangular prism. Show the hidden edges with dashed lines.",
        "ja": "三角柱の見取図をかけ。ただし、見えない辺は点線で示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "投影図（エントリ orthographic-projection）とは別の図。見取図は 1 枚の図で立体を斜めから表し、投影図は正面から見た図と真上から見た図を組にする。",
      "perspective は美術では遠近法（遠くのものを小さくかく方法）のことも指す。"
    ],
    "related": [
      "orthographic-projection",
      "net",
      "solid",
      "edge",
      "prism"
    ],
    "sources": [
      {
        "type": "mext-translation"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "three-dimensional drawing": 1,
        "perspective drawing": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 2 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sketch-the-graph",
    "ja": {
      "term": "グラフの概形をかく",
      "reading": "ぐらふのがいけいをかく",
      "alt": [
        "概形をかく",
        "曲線をかく"
      ]
    },
    "en": {
      "term": "sketch the graph",
      "alt": [],
      "variants": [
        {
          "term": "sketch a graph",
          "register": "written",
          "note": "a を使う書き方。話し言葉には出てこない。書き言葉は OpenStax Algebra and Trigonometry と OpenStax Calculus が中心。"
        },
        {
          "term": "sketch the curve",
          "register": "both",
          "note": "曲線の概形をかくときの言い方。話し言葉はすべて MIT OCW、書き言葉はほとんどが OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "functions"
    ],
    "level": {
      "jp": [
        "数II",
        "数III",
        "数C"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "増減・凹凸・漸近線などを調べて、グラフのおおまかな形をかく。",
    "definition_en": "To draw the general shape of a graph, showing features such as where it rises and falls, its concavity, and its asymptotes.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Before you sketch the curve, find the intercepts and the critical points.",
        "ja": "概形をかく前に、切片と臨界点を求めておく。",
        "register": "spoken"
      },
      {
        "en": "Sketch the graph of f(x) = x/(x² + 1), showing all asymptotes and extrema.",
        "ja": "漸近線と極値を示して、f(x) = x/(x² + 1) のグラフの概形をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「概形」に当たる 1 語はなく、sketch（ざっとかく）が「概形をかく」の意味を担う。"
    ],
    "related": [
      "increasing-and-decreasing",
      "local-extremum"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sketch the curve": 6,
        "sketch the graph": 4
      },
      "written": {
        "sketch the graph": 162,
        "sketch a graph": 83,
        "sketch the curve": 14
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sketch",
    "ja": {
      "term": "図示する",
      "reading": "ずしする",
      "alt": [
        "図示せよ"
      ]
    },
    "en": {
      "term": "sketch",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語の問題文の「図示せよ」「図示すると」（共通テストの問題文には「図示すると」がある）は、領域なら Sketch ／ Graph ／ Shade the region、曲線なら Sketch the graph と言う。英語の sketch はグラフや図をおおまかにかくことで、正確な作図は construct と言う（CK-12 Geometry 1.12 は construction を rough sketch にすぎない drawing と区別する）。draw は sketch と同じく手がきにも使う（IM の Draw a rough sketch）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus I",
        "Algebra 1"
      ]
    },
    "definition_ja": "条件を満たす点の集合（領域・軌跡）やグラフを、座標平面上にかいて示す。",
    "definition_en": "To show a set of points (a region or locus) or a graph by drawing it on the coordinate plane.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Just sketch it — you don't need exact points, but label the intercepts.",
        "ja": "おおまかにかけばいい。正確な点はいらないが、切片は書いておこう。",
        "register": "spoken"
      },
      {
        "en": "Sketch the region defined by x² + y² ≤ 4 and y ≥ x.",
        "ja": "x² + y² ≦ 4, y ≧ x の表す領域を図示せよ。",
        "register": "written"
      },
      {
        "en": "Sketch the graph of y = −x² + 2x + 3 and label the vertex and the intercepts.",
        "ja": "y = −x² + 2x + 3 のグラフの概形をかき、頂点と軸との交点を示せ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "sketch the region",
        "ja": "領域を図示する"
      }
    ],
    "pitfalls": [
      "用例コーパスの sketch の書き言葉（OpenStax Calculus・OpenStax Algebra and Trigonometry が中心）には、sketch the graph の形がよく出てくる。sketch the region は少ない。"
    ],
    "related": [
      "region",
      "boundary",
      "sketch-the-graph"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和8年度 大学入学共通テスト 本試験 数学Ⅱ，数学Ｂ，数学Ｃ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=2144&f=abm00017550.pdf&n=2026_ol_18_sugaku2BC.pdf",
        "note": "不等式の表す領域を図示すると"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.12 Constructions and Bisectors（construction と drawing の区別）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 2)",
        "url": "https://im.kendallhunt.com/HS/students/3/index.html",
        "note": "Algebra 2 2.10 Multiplicity ほか（Draw a rough sketch）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sketch": 192
      },
      "written": {
        "sketch": 531
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "skew-lines",
    "ja": {
      "term": "ねじれの位置",
      "reading": "ねじれのいち",
      "alt": []
    },
    "en": {
      "term": "skew lines",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「ねじれの位置」は 2 直線の位置関係の名前だが、英語では 2 直線そのものを skew lines と呼ぶ。「直線 AB と CD はねじれの位置にある」は lines AB and CD are skew と、skew を形容詞にして言う。CK-12 Geometry は平行線と並べて 3.2 Parallel and Skew Lines で扱う。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "空間で、平行でなく交わりもしない 2 直線の位置関係。ねじれの位置にある 2 直線は、同じ平面上にない。",
    "definition_en": "Two lines in space that are not parallel and never intersect. Skew lines cannot lie in the same plane.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These two edges never meet, but they're not parallel either, so they're skew lines.",
        "ja": "この 2 本の辺は交わらないけど、平行でもない。だから、ねじれの位置にある。",
        "register": "spoken"
      },
      {
        "en": "Name two pairs of skew lines in the rectangular prism shown.",
        "ja": "図の直方体で、ねじれの位置にある 2 直線の組を 2 つ答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "平面上の 2 直線は、交わるか平行（parallel lines、エントリ parallel-lines）のどちらかで、skew にはならない。skew lines は空間の中でだけ起こる。",
      "立体の辺どうしがねじれの位置かどうかは、辺を延ばした直線で考える。辺そのものが交わっていなくても、延ばして交われば skew ではない。",
      "統計で分布が偏っている skewed（エントリ be-skewed）と同じ語。"
    ],
    "related": [
      "relative-positions-of-two-lines",
      "parallel-lines",
      "coplanar",
      "intersect",
      "plane"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ねじれの位置",
        "en": "Skew lines"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "3.2 Parallel and Skew Lines"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "skew lines": 3
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは CK-12 の呼び方 skew lines（CK-12 Geometry 3.2 Parallel and Skew Lines・計 12 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "skewness",
    "ja": {
      "term": "歪度",
      "reading": "わいど",
      "alt": [
        "分布の歪み",
        "ゆがみ"
      ]
    },
    "en": {
      "term": "skewness",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "分布が左右対称からどれだけ、どちら向きにずれているかの程度。右に裾が長いと正、左に長いと負。",
    "definition_en": "The degree to which, and the direction in which, a distribution departs from symmetry; positive for a long right tail and negative for a long left tail.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Skewness tells you which way the tail goes.",
        "ja": "歪度は、裾がどちら向きに伸びているかを表す。",
        "register": "spoken"
      },
      {
        "en": "Describe the skewness of the distribution and how it affects the mean and the median.",
        "ja": "分布の歪みを述べ、それが平均値と中央値にどう影響するか説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP Statistics の授業では、数値の歪度より、分布の形を skewed right ／ skewed left と言葉で述べることが多い（用例コーパスでも skewed は多く、skewness はほとんど出てこない）。"
    ],
    "related": [
      "be-skewed",
      "distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "歪度",
        "en": "Skewness"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "skewness": 1
      },
      "written": {
        "skewness": 3
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは AP Statistics の CED の呼び方 skewness（topic 1.9・topic 4.2・topic 4.4・topic 4.7・topic 4.9）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "slant-asymptote",
    "ja": {
      "term": "斜め漸近線",
      "reading": "ななめぜんきんせん",
      "alt": []
    },
    "en": {
      "term": "slant asymptote",
      "alt": [],
      "variants": [
        {
          "term": "oblique asymptote",
          "register": "written",
          "note": "教科書（OpenStax）では slant asymptote と並んで使う。話し言葉では少ない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "分数関数で、分子の次数が分母の次数よりちょうど 1 大きいときにできる、斜めの直線の漸近線。分子を分母で割った商の一次式が、その直線の式になる。",
    "definition_en": "A slanted line that the graph of a rational function approaches when the degree of the numerator is exactly one more than the degree of the denominator. Its equation is the quotient you get by dividing the numerator by the denominator.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The top's degree is one more than the bottom's, so divide, and the quotient gives you the slant asymptote.",
        "ja": "分子の次数が分母より 1 大きいから、割り算をしよう。その商が斜め漸近線になる。",
        "register": "spoken"
      },
      {
        "en": "Find the slant asymptote of f(x) = (x² + 3x + 1)/(x + 1).",
        "ja": "f(x) = (x² + 3x + 1)/(x + 1) の斜め漸近線を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "oblique asymptote とも言う（oblique は「斜めの」）。OpenStax Algebra and Trigonometry は slant asymptote として扱う。",
      "割り算の余りは捨て、商だけが漸近線の式になる。",
      "分子の次数が分母の次数以下なら水平漸近線（エントリ horizontal-asymptote）になり、斜め漸近線はできない。"
    ],
    "related": [
      "asymptote",
      "horizontal-asymptote",
      "vertical-asymptote",
      "rational-function",
      "polynomial-division"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "slant asymptote": 13,
        "oblique asymptote": 3
      },
      "written": {
        "oblique asymptote": 13,
        "slant asymptote": 16
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "slant-height",
    "ja": {
      "term": "母線",
      "reading": "ぼせん",
      "alt": [
        "斜高"
      ]
    },
    "en": {
      "term": "slant height",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「母線」は円錐（や円柱）の側面をつくる線分そのもの。英語の slant height はその長さ、つまり円錐の頂点から底面の円周までの側面に沿った距離を指し、正角錐では側面の三角形の高さ（日本語の「斜高」）にも使う。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "円錐の頂点と底面の円周上の点を結ぶ線分。英語の slant height はその長さを指す。",
    "definition_en": "The distance from the apex of a cone to the edge of its base, measured along the lateral surface; for a regular pyramid, the height of one of its triangular faces.",
    "latex": "\\ell = \\sqrt{r^2 + h^2}",
    "spoken_en": "the slant height is the square root of r squared plus h squared",
    "examples": [
      {
        "en": "The slant height is the hypotenuse of this right triangle, so it's the square root of r squared plus h squared.",
        "ja": "母線はこの直角三角形の斜辺だから、長さは √(r² + h²) になる。",
        "register": "spoken"
      },
      {
        "en": "A cone has a base radius of 3 cm and a slant height of 5 cm. Find its lateral area.",
        "ja": "底面の半径が 3 cm、母線の長さが 5 cm の円錐の側面積を求めよ。",
        "register": "written"
      },
      {
        "en": "A square pyramid has a base edge of 6 and a slant height of 5. Find its surface area.",
        "ja": "底面の 1 辺が 6、斜高が 5 の正四角錐の表面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "slant height of the cone",
        "ja": "円錐の母線（の長さ）"
      }
    ],
    "pitfalls": [
      "日本語の「母線」は線分、英語の slant height は長さ。「母線の長さ」も英語では slant height だけで言える。",
      "height（高さ）は頂点から底面に下ろした垂線の長さで、slant height とは別。円錐では高さ・底面の半径・母線が直角三角形をつくる。",
      "用例コーパスでは話し言葉がすべて The Organic Chemistry Tutor、書き言葉は少なく、すべて OpenStax Calculus。"
    ],
    "related": [
      "area",
      "vertex",
      "surface-area-of-revolution",
      "triangle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "slant height": 13
      },
      "written": {
        "slant height": 3
      },
      "sources": [
        "openstax-calculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "slope-field",
    "ja": {
      "term": "方向場",
      "reading": "ほうこうば",
      "alt": [
        "傾きの場"
      ]
    },
    "en": {
      "term": "slope field",
      "alt": [],
      "variants": [
        {
          "term": "slope field",
          "register": "spoken",
          "note": "AP（Khan Academy）の言い方で、AP の CED もこの語を使う（topic 7.3）。話し言葉は大半が Khan Academy。MIT OCW・Professor Leonard も使う。書き言葉では direction field より少ない。"
        },
        {
          "term": "direction field",
          "register": "both",
          "note": "大学（MIT OCW・OpenStax Calculus）の言い方。書き言葉ではこちらが多い。話し言葉はすべて MIT OCW、書き言葉は OpenStax Calculus と MIT の講義ノート。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II"
      ]
    },
    "definition_ja": "微分方程式 dy/dx = f(x, y) の傾きを、平面上の各点に短い線分で描いた図。",
    "definition_en": "A picture of dy/dx = f(x, y) made by drawing a short segment with the given slope at many points of the plane.",
    "latex": "\\frac{dy}{dx} = x - y",
    "spoken_en": "d y d x equals x minus y",
    "examples": [
      {
        "en": "Just follow the little segments in the slope field, starting from the initial point.",
        "ja": "初期値の点から、方向場の小さな線分に沿ってたどればいい。",
        "register": "spoken"
      },
      {
        "en": "Sketch the slope field for dy/dx = x + y at the nine points with x, y ∈ {−1, 0, 1}.",
        "ja": "x, y ∈ {−1, 0, 1} の 9 点で、dy/dx = x + y の方向場をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説には出てこない。"
    ],
    "related": [
      "differential-equation",
      "initial-condition",
      "particular-solution",
      "eulers-method"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 7.3（slope fields）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「方向場」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "direction field": 45,
        "slope field": 73
      },
      "written": {
        "slope field": 10,
        "direction field": 70
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "slope-formula",
    "ja": {
      "term": "傾きの公式",
      "reading": "かたむきのこうしき",
      "alt": []
    },
    "en": {
      "term": "slope formula",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「傾きの公式」は学習指導要領解説に無い（日本語版 Wikipedia「中点法」は 2 点を結ぶ直線の傾きの式を「傾きの公式」と呼ぶ）。日本では「変化の割合 = y の増加量 ÷ x の増加量」（エントリ rate-of-change）として同じ計算をするが、公式の名前はない。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 点 (x₁, y₁)、(x₂, y₂) を通る直線の傾きを、y 座標の差を x 座標の差で割って求める式 m = (y₂ − y₁)/(x₂ − x₁)。x₁ = x₂ のときは分母が 0 になり、使えない。",
    "definition_en": "The formula m = (y₂ − y₁)/(x₂ − x₁) for the slope of the line through two points (x₁, y₁) and (x₂, y₂). It does not apply when x₁ = x₂, because the denominator would be zero.",
    "latex": "m = \\frac{y_2 - y_1}{x_2 - x_1}",
    "spoken_en": "m equals y two minus y one over x two minus x one",
    "examples": [
      {
        "en": "Plug the points (2, 3) and (4, 7) into the slope formula: 7 minus 3 over 4 minus 2 is 4 over 2, so the slope is 2.",
        "ja": "点 (2, 3) と (4, 7) を傾きの公式に入れる。7 − 3 を 4 − 2 で割って 4/2、だから傾きは 2。",
        "register": "spoken"
      },
      {
        "en": "Use the slope formula to find the slope of the line passing through (−1, 5) and (3, −3).",
        "ja": "傾きの公式を用いて、2 点 (−1, 5)、(3, −3) を通る直線の傾きを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "分子と分母で同じ点を先にする。(y₂ − y₁)/(x₁ − x₂) のように順番をそろえないと、符号が逆になる。",
      "x₁ = x₂ のとき（垂直な直線）は分母が 0 になり、傾きは定義されない（undefined）。水平な直線は分子が 0 で、傾きは 0。",
      "y₂ の小さい 2 は下付き文字（subscript）で、y sub two とも読む。y の 2 乗（y squared）と混同しない。"
    ],
    "related": [
      "slope",
      "rate-of-change",
      "rise-over-run",
      "point-slope-form",
      "midpoint-formula"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「中点法」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%AD%E7%82%B9%E6%B3%95",
        "note": "y(t) と y(t + h) を結ぶ傾きの式を「傾きの公式」と呼ぶ"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "slope formula": 15
      },
      "written": {
        "slope formula": 59
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "slope-intercept-form-of-a-line",
    "ja": {
      "term": "傾き切片形",
      "reading": "かたむきせっぺんけい",
      "alt": []
    },
    "en": {
      "term": "slope-intercept form",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「傾き切片形」は学習指導要領解説に無い、本プロジェクトの訳語。日本では y = ax + b を一次関数の式として扱い、形に名前を付けない。米国の教科書（OpenStax Elementary Algebra）は直線の式の形に名前を付けて区別する（point-slope form、standard form）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直線の式を y = mx + b の形に書いたもの。m が傾き、b が y 切片を表す。",
    "definition_en": "The form y = mx + b of the equation of a line, where m is the slope and b is the y-intercept.",
    "latex": "y = mx + b",
    "spoken_en": "y equals m x plus b",
    "examples": [
      {
        "en": "Once it's in slope-intercept form, you can just read off the slope and the y-intercept.",
        "ja": "傾き切片形にしてしまえば、傾きと y 切片をそのまま読み取れる。",
        "register": "spoken"
      },
      {
        "en": "Write the equation 4x + 2y = 6 in slope-intercept form.",
        "ja": "式 4x + 2y = 6 を y = mx + b の形（傾き切片形）に変形せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "米国の教科書（OpenStax）は傾きを m で書く。日本の一次関数 y = ax + b の a が m にあたる。",
      "b は y 切片（y-intercept）で、グラフは点 (0, b) を通る。",
      "直線の式の別の形は point-slope form（エントリ point-slope-form）と standard form（エントリ standard-form-of-a-line）。"
    ],
    "related": [
      "slope",
      "y-intercept",
      "point-slope-form",
      "standard-form-of-a-line",
      "linear-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "slope-intercept form": 52
      },
      "written": {
        "slope-intercept form": 183
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "slope-of-the-tangent-line",
    "ja": {
      "term": "接線の傾き",
      "reading": "せっせんのかたむき",
      "alt": []
    },
    "en": {
      "term": "slope of the tangent line",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線上の点で引いた接線の傾き。その点での微分係数に等しい。",
    "definition_en": "How steep the tangent line is at a given point of a curve; it equals the derivative there.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The slope of the tangent line at x = 1 is f prime of 1.",
        "ja": "x = 1 での接線の傾きは f′(1)。",
        "register": "spoken"
      },
      {
        "en": "Find the slope of the tangent line to y = x³ at the point (2, 8).",
        "ja": "点 (2, 8) における y = x³ の接線の傾きを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "口頭では line を省いて slope of the tangent とも言う。"
    ],
    "related": [
      "derivative-at-a-point",
      "find-the-equation-of-the-tangent-line",
      "normal-line",
      "point-of-tangency",
      "average-rate-of-change"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "slope of the tangent line": 280
      },
      "written": {
        "slope of the tangent line": 71
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "slope",
    "ja": {
      "term": "傾き",
      "reading": "かたむき",
      "alt": []
    },
    "en": {
      "term": "slope",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Geometry"
      ]
    },
    "definition_ja": "直線の傾きぐあいを表す数。直線上の 2 点について、y の増加量を x の増加量でわった値。",
    "definition_en": "A number that measures how steep a line is: take any two points on the line and divide the vertical change by the horizontal change, often said as \"rise over run.\"",
    "latex": "m = \\frac{y_2 - y_1}{x_2 - x_1}",
    "spoken_en": "m equals y two minus y one over x two minus x one",
    "examples": [
      {
        "en": "Slope is rise over run, so up 4 and over 2 gives you a slope of 2.",
        "ja": "傾きは「上がった分 ÷ 横に進んだ分」。4 上がって右に 2 進むから、傾きは 2。",
        "register": "spoken"
      },
      {
        "en": "Find the slope of the line passing through (1, −2) and (4, 7).",
        "ja": "2 点 (1, −2)、(4, 7) を通る直線の傾きを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "傾きは m で表す（y = mx + b）。中学校学習指導要領解説の y = ax + b の a に当たる。",
      "rise over run は「上がった分（rise）÷ 横に進んだ分（run）」。下がるときは rise が負になり、傾きも負になる。",
      "一次関数では、傾きと変化の割合（エントリ rate-of-change）は同じ値。"
    ],
    "related": [
      "linear-function",
      "rate-of-change",
      "change",
      "y-intercept",
      "slope-of-the-tangent-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "傾き (数学)",
        "en": "Slope"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 関数: 一次関数 y = ax + b"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "slope": 3084
      },
      "written": {
        "slope": 1537
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "small-change",
    "ja": {
      "term": "微小変化",
      "reading": "びしょうへんか",
      "alt": []
    },
    "en": {
      "term": "small change",
      "alt": [],
      "variants": [
        {
          "term": "tiny change",
          "register": "spoken",
          "note": "3Blue1Brown の言い方。話し言葉はすべて 3Blue1Brown で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "x がごくわずか変わるときの変化量 Δx と、それに伴う y の変化量 Δy。",
    "definition_en": "A very small change Δx in x, and the resulting change Δy in y.",
    "latex": "\\Delta y \\approx f'(x)\\,\\Delta x",
    "spoken_en": "delta y is approximately f prime of x times delta x",
    "examples": [
      {
        "en": "A small change in x gives a change in y of about f prime times delta x.",
        "ja": "x の微小変化に対して、y はおよそ f′ × Δx だけ変わる。",
        "register": "spoken"
      },
      {
        "en": "If x changes from 2 to 2.01, use the derivative to estimate the corresponding change in y = x³.",
        "ja": "x が 2 から 2.01 に変わるとき、y = x³ の変化量を導関数を用いて見積もれ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Δx は delta x と読む。"
    ],
    "related": [
      "linear-approximation",
      "differential"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "small change": 89,
        "tiny change": 43
      },
      "written": {
        "small change": 7
      },
      "sources": [
        "khan-ap-calc",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sohcahtoa",
    "ja": {
      "term": "SOHCAHTOA",
      "reading": "そーかとあ",
      "alt": []
    },
    "en": {
      "term": "SOHCAHTOA",
      "alt": [
        "soh cah toa"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「SOHCAHTOA」は英語の語呂合わせをそのまま使う（学習指導要領解説に無い。日本語版 Wikipedia「三角関数の暗記方法」は英語圏の覚え方として SOH-CAH-TOA を挙げる）。日本では sin・cos・tan の定義を筆記体の s・c・t の形で覚える方法があり、この語呂合わせは使わない。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直角三角形の三角比の定義を覚えるための英語の語呂合わせ。sine = opposite / hypotenuse、cosine = adjacent / hypotenuse、tangent = opposite / adjacent の頭文字を並べたもの。",
    "definition_en": "A mnemonic for the right-triangle definitions of the trigonometric ratios: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.",
    "latex": "\\sin\\theta = \\frac{\\text{opp}}{\\text{hyp}},\\quad \\cos\\theta = \\frac{\\text{adj}}{\\text{hyp}},\\quad \\tan\\theta = \\frac{\\text{opp}}{\\text{adj}}",
    "spoken_en": "sine theta equals opposite over hypotenuse, cosine theta equals adjacent over hypotenuse, tangent theta equals opposite over adjacent",
    "examples": [
      {
        "en": "Remember SOHCAHTOA. We know the opposite side and we want the hypotenuse, so that's sine.",
        "ja": "SOHCAHTOA を思い出そう。対辺がわかっていて斜辺を求めたいから、使うのは sin だ。",
        "register": "spoken"
      },
      {
        "en": "Label the opposite side, the adjacent side, and the hypotenuse relative to angle A. Then use SOHCAHTOA to find tan A.",
        "ja": "∠A から見た対辺・隣辺・斜辺を図に書き込み、tan A を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉では SOHCAHTOA（The Organic Chemistry Tutor が中心）と soh cah toa（Khan Academy が中心）が同じくらい出てくるが、同じ語呂合わせを書き起こしが 2 通りに書いたもの。書き言葉にはほとんど出てこない（OpenStax Algebra and Trigonometry）。",
      "SOH・CAH・TOA の 3 つがそれぞれ sin・cos・tan の定義に当たる。O は opposite（対辺）、A は adjacent（隣辺）、H は hypotenuse（斜辺）。",
      "直角三角形で定義できるのは鋭角の三角比だけ。90° 以上の角の sin・cos・tan は単位円（unit circle）で定める。"
    ],
    "related": [
      "sine",
      "cosine",
      "tangent",
      "trigonometric-ratio"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角関数の暗記方法」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0%E3%81%AE%E6%9A%97%E8%A8%98%E6%96%B9%E6%B3%95",
        "note": "英語圏の覚え方 SOH-CAH-TOA"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "soh cah toa": 13,
        "SOHCAHTOA": 19
      },
      "written": {
        "SOHCAHTOA": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solid-of-revolution",
    "ja": {
      "term": "回転体",
      "reading": "かいてんたい",
      "alt": []
    },
    "en": {
      "term": "solid of revolution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数III"
      ],
      "us": [
        "Geometry",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "平面図形を、その平面上の 1 本の直線を軸として 1 回転させてできる立体。円柱・円錐・球は回転体である。",
    "definition_en": "A solid formed by revolving a plane figure one full turn around a line in the same plane, called the axis of revolution. Cylinders, cones, and spheres are all solids of revolution.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Revolve this right triangle around one of its legs, and the solid of revolution you get is a cone.",
        "ja": "この直角三角形を、直角をはさむ辺の 1 つを軸にして 1 回転させると、できる回転体は円錐になる。",
        "register": "spoken"
      },
      {
        "en": "A right triangle with legs of 3 cm and 4 cm is revolved around the 4 cm leg. Find the volume of the solid of revolution.",
        "ja": "直角をはさむ 2 辺の長さが 3 cm、4 cm の直角三角形を、長さ 4 cm の辺を軸として 1 回転させてできる回転体の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「1 回転させる」は revolve … around …（rotate … around … とも言う）。回転の軸は axis of revolution（エントリ axis-of-revolution）。",
      "revolution は「革命」と同じ綴りの語で、ここでは動詞 revolve の名詞形として「回転」の意味。",
      "回転体の体積を積分で求める方法はエントリ volume-of-a-solid-of-revolution。"
    ],
    "related": [
      "axis-of-revolution",
      "volume-of-a-solid-of-revolution",
      "revolve-around-the-x-axis",
      "cylinder",
      "cone"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "回転体",
        "en": "Solid of revolution"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "solid of revolution": 27
      },
      "written": {
        "solid of revolution": 98
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solid",
    "ja": {
      "term": "立体",
      "reading": "りったい",
      "alt": [
        "空間図形"
      ]
    },
    "en": {
      "term": "solid",
      "alt": [
        "solid figure",
        "three-dimensional figure"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "空間の中で縦・横・高さの広がりをもち、体積のある図形。角柱・円柱・球など。",
    "definition_en": "A three-dimensional figure that takes up space and has volume, such as a prism, a cylinder, or a sphere.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Is a circle a solid? No, it's flat. A sphere is a solid, because it takes up space.",
        "ja": "円は立体かな？ いや、平らだよね。球は空間の中で場所をとるから立体。",
        "register": "spoken"
      },
      {
        "en": "A solid consists of a cylinder with radius 3 cm and height 5 cm topped by a cone with the same radius and a height of 4 cm. Find the volume of the solid in terms of π.",
        "ja": "半径 3 cm、高さ 5 cm の円柱の上に、同じ底面で高さ 4 cm の円錐を重ねた立体がある。この立体の体積を求めよ。ただし、円周率は π とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "solid は形容詞で「中身の詰まった」の意味もあり、solid line（実線）、solid dot（黒丸）の solid は立体とは別の意味。",
      "立体は 3D shape（three-dimensional shape）とも言う。平面図形は plane figure（エントリ plane）。",
      "ja.alt の「空間図形」は中 1 の単元名で、立体のほかに空間の直線・平面の位置関係も扱う。"
    ],
    "related": [
      "plane",
      "polyhedron",
      "volume",
      "surface-area",
      "cross-section"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "立体",
        "en": "Solid figure"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "a solid !line !dot !circle !curve !lines !dots | the solid !line !dot !circle !curve !lines !dots | solid figure": 120,
        "three-dimensional figure": 4,
        "solid figure": 1
      },
      "written": {
        "a solid !line !dot !circle !curve !lines !dots | the solid !line !dot !circle !curve !lines !dots | solid figure": 373,
        "three-dimensional figure": 2,
        "solid figure": 4
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solution-pair",
    "ja": {
      "term": "解の組",
      "reading": "かいのくみ",
      "alt": []
    },
    "en": {
      "term": "ordered pair",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「解の組」は方程式を満たすことまで含む言い方だが、英語の ordered pair は x と y の値を順に並べた組そのものを指し、座標平面上の点の座標にも使う。解であることは The solution is the ordered pair (2, 3). のように文で言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つの文字をふくむ方程式（や連立方程式）を成り立たせる、x の値と y の値の組。x = 2, y = 3 や (x, y) = (2, 3) のように書く。",
    "definition_en": "An ordered pair (x, y) of values that makes an equation in two variables, or every equation in a system, true. The x-value is written first and the y-value second.",
    "latex": "(x, y)",
    "spoken_en": "the ordered pair x comma y",
    "examples": [
      {
        "en": "x = 2 and y = 3 work in both equations, so the solution is the ordered pair (2, 3).",
        "ja": "x = 2, y = 3 は両方の式を満たすから、解の組は (2, 3) だ。",
        "register": "spoken"
      },
      {
        "en": "Which ordered pair is a solution of 3x − y = 5: (1, 2), (2, 1), or (3, 3)?",
        "ja": "(1, 2)、(2, 1)、(3, 3) のうち、方程式 3x − y = 5 の解の組になっているものを選べ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "ordered pair は座標平面上の点の座標にも使う（エントリ coordinates）。解の組 (2, 3) は、グラフでは点 (2, 3) に当たる。",
      "ordered（順序のある）は、(2, 3) と (3, 2) が別の組であることを表す。x の値を先、y の値を後に書く。",
      "センター試験の問題文は不定方程式の解を「x = ア，y = イウ」と分けて書き、組で答えるときは「(A, B) = (49 × ク，23 × ケコ)」と書く（平成31年度 本試験 数学Ⅰ・数学Ａ 第4問）。英語では The solution is the ordered pair (2, 3). のように、ordered pair 1 つで解を言う書き方もある。"
    ],
    "related": [
      "solution",
      "linear-equation-in-two-variables",
      "coordinates",
      "point-of-intersection",
      "satisfy"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成31年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20200601/https://www.dnc.ac.jp/albums/abm.php?f=abm00036453.pdf&n=k+19+suugaku1A.pdf",
        "note": "第4問: 不定方程式の解 x = ア，y = イウ、組 (A, B) = (49 × ク，23 × ケコ)"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "ordered pair": 46
      },
      "written": {
        "ordered pair": 404
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solution-set",
    "ja": {
      "term": "解集合",
      "reading": "かいしゅうごう",
      "alt": [
        "x の値の範囲"
      ]
    },
    "en": {
      "term": "solution set",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "方程式や不等式を満たす値をすべて集めたもの。不等式では 2 < x < 3 のような範囲になる。",
    "definition_en": "The set of all values that make an equation or inequality true; for an inequality it is usually an interval such as 2 < x < 3.",
    "latex": "\\{x \\mid x > 2\\} = (2, \\infty)",
    "spoken_en": "the set of all x such that x is greater than two, which is the open interval from two to infinity",
    "examples": [
      {
        "en": "So the solution set is every x bigger than 2, and in interval notation that's open paren 2, infinity.",
        "ja": "解集合は 2 より大きいすべての x。区間の記法では (2, ∞) だ。",
        "register": "spoken"
      },
      {
        "en": "Find the solution set of x² − 5x + 6 < 0 and write it in interval notation.",
        "ja": "x² − 5x + 6 < 0 の解集合を求め、区間の記法で表せ。",
        "register": "written"
      },
      {
        "en": "Over the real numbers, the solution set of x² + 1 = 0 is the empty set.",
        "ja": "実数の範囲では、x² + 1 = 0 の解集合は空集合である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "solution set of the inequality",
        "ja": "不等式の解集合"
      }
    ],
    "pitfalls": [
      "区間の記法 (2, 3)、[1, ∞) は、高等学校学習指導要領解説では数学III の積分に閉区間 [a, b] が出てくる程度で、共通テストの問題文は 2 < x < 3、x ≧ 1 と不等式で書く（慣習差 interval-notation-vs-inequalities）。(2, 3) は点の座標と同じ書き方なので読み違えに注意。読みは the open interval from 2 to 3。",
      "問題文の「〜を満たす x の範囲を求めよ」は Find the values of x that satisfy … か Find the solution set of …。集合の記法では {x | x > 2}（the set of all x such that x > 2）。",
      "解が無いときは no solution（解集合は the empty set ∅）、すべての実数が解なら all real numbers。"
    ],
    "related": [
      "solution",
      "set",
      "linear-inequality",
      "number-line",
      "real-number"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 積分: 閉区間 [a, b]（1 件。数学I の不等式の解は 2 < x < 3 の形）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "解の範囲は不等式で書く（区間の記法は無い）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「方程式」",
        "url": "https://ja.wikipedia.org/wiki/%E6%96%B9%E7%A8%8B%E5%BC%8F",
        "note": "解全体の集合を解集合という"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "solution set": 64
      },
      "written": {
        "solution set": 34
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solution",
    "ja": {
      "term": "解",
      "reading": "かい",
      "alt": [
        "方程式の解"
      ]
    },
    "en": {
      "term": "solution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "方程式を成り立たせる未知数の値。解全体の集合を解集合という。",
    "definition_en": "A value of the unknown that makes an equation true.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "So x = 3 is a solution — check it: 2 times 3 minus 1 is 5.",
        "ja": "だから x = 3 は解。確かめると 2 × 3 − 1 = 5。",
        "register": "spoken"
      },
      {
        "en": "Find all solutions to the equation x² − 5x + 6 = 0.",
        "ja": "方程式 x² − 5x + 6 = 0 の解をすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "has no real solution",
        "ja": "実数解をもたない"
      }
    ],
    "pitfalls": [
      "solution は「問題の解答」の意味でも使う。solution of the equation とも言う。",
      "多項式の方程式の解は root とも言う（the roots of the quadratic）。関数の値が 0 になる x は zero。",
      "「解をもつ」は has a solution、「実数解をもたない」は has no real solutions。"
    ],
    "related": [
      "solve",
      "imaginary-solution",
      "number-of-real-solutions"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "solution to the equation": 18
      },
      "written": {
        "solution to the equation": 142
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solve-for-y-prime",
    "ja": {
      "term": "y′ について解く",
      "reading": "わいだっしゅについてとく",
      "alt": []
    },
    "en": {
      "term": "solve for dy/dx",
      "alt": [
        "solve for y prime"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "陰関数の微分をしたあとの式を、dy/dx（y′）について解く。",
    "definition_en": "To isolate dy/dx (y′) after differentiating an equation implicitly.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Now get all the dy/dx terms on one side and solve for dy/dx.",
        "ja": "dy/dx を含む項を片側に集めて、dy/dx について解く。",
        "register": "spoken"
      },
      {
        "en": "Differentiate implicitly and solve for dy/dx: x² + xy + y² = 7.",
        "ja": "x² + xy + y² = 7 を陰関数として微分し、dy/dx について解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉では solve for dy/dx が多く、solve for y prime はほとんど出てこない。"
    ],
    "related": [
      "implicit-differentiation",
      "differentiate-both-sides"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "solve for dy/dx": 18,
        "solve for y prime": 1
      },
      "written": {
        "solve for dy/dx": 7
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solve-for",
    "ja": {
      "term": "〜について解く",
      "reading": "についてとく",
      "alt": [
        "θ を求める"
      ]
    },
    "en": {
      "term": "solve for",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "中2",
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "式を、指定した文字 = （その文字を含まない式）の形に変形する。方程式を満たすその文字の値を求めることにも言う。",
    "definition_en": "To rearrange an equation into the form (chosen variable) = (expression without it), or to find the values of that variable that satisfy the equation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Solve for y: subtract 2x and divide by 3.",
        "ja": "y について解こう。2x を引いて 3 で割る。",
        "register": "spoken"
      },
      {
        "en": "Solve for θ: 2 cos θ + 1 = 0, 0 ≤ θ < 2π.",
        "ja": "0 ≦ θ < 2π のとき、2 cos θ + 1 = 0 を満たす θ を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "読みの「〜」は漢字の読みに含めない（「x について解く」の x の部分）。"
    ],
    "related": [
      "solve",
      "solution"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "solve for": 954
      },
      "written": {
        "solve for": 849
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solve-the-recurrence",
    "ja": {
      "term": "漸化式を解く",
      "reading": "ぜんかしきをとく",
      "alt": []
    },
    "en": {
      "term": "solve the recurrence",
      "alt": [
        "solve a recurrence"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "漸化式と初めの項（初期条件）から、数列の一般項を n の式で求める。",
    "definition_en": "To get a closed formula for aₙ in terms of n from a recurrence and its initial term(s).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To solve the recurrence, subtract the fixed point from both sides first.",
        "ja": "漸化式を解くには、まず両辺から不動点（特性方程式の解）を引く。",
        "register": "spoken"
      },
      {
        "en": "Solve the recurrence aₙ₊₁ = 3aₙ + 2 with a₁ = 1.",
        "ja": "漸化式 aₙ₊₁ = 3aₙ + 2、a₁ = 1 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Levin（4.4）は solve the recurrence relation と書き、求める式を closed formula と呼ぶ。"
    ],
    "related": [
      "recurrence-relation",
      "characteristic-equation",
      "find-the-nth-term",
      "set-up-a-recurrence",
      "linear-recurrence-relation",
      "system-of-recurrences"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.4 Exponential Sequences（solve the recurrence relation、closed formula）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "solve the recurrence": 1,
        "solve a recurrence": 2
      },
      "written": {
        "solve the recurrence": 11,
        "solve a recurrence": 1
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solve-the-right-triangle",
    "ja": {
      "term": "三角比で辺を求める",
      "reading": "さんかくひでへんをもとめる",
      "alt": [
        "直角三角形を解く"
      ]
    },
    "en": {
      "term": "solving right triangles",
      "alt": [
        "solve the right triangle",
        "solve a right triangle"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "見出しの「三角比で辺を求める」は学習指導要領解説に無い、本プロジェクトの訳語。英語の solve the triangle は、足りない辺と角をすべて求めることを言う。OpenStax Algebra and Trigonometry（9.5）と Precalculus（7.5）は Solving Trigonometric Equations の節で Solving Right Triangle Problems の見出しのもとに扱う（Solve right triangle problems と合わせて本文 4 件）。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直角三角形で、わかっている辺や角から三角比（と三平方の定理）を使って、残りの辺と角を求めること。",
    "definition_en": "Using trigonometric ratios (and the Pythagorean theorem) to find the unknown sides and angles of a right triangle from the ones that are given.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Solving right triangles means finding both missing sides and the other acute angle, not just x.",
        "ja": "直角三角形を解くというのは、x だけでなく、残りの 2 辺ともう 1 つの鋭角をすべて求めることだ。",
        "register": "spoken"
      },
      {
        "en": "We've got one angle and the hypotenuse, so use sine and cosine to get the two legs.",
        "ja": "1 つの角と斜辺がわかっているから、sin と cos を使って残りの 2 辺を求めよう。",
        "register": "spoken"
      },
      {
        "en": "Solve the right triangle ABC with C = 90°, A = 32°, and c = 15. Round lengths to the nearest tenth.",
        "ja": "C = 90°, A = 32°, c = 15 の直角三角形 ABC を解け。長さは小数第 2 位を四捨五入せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "1 辺だけを聞く問題は Find x. と書く。Solve the right triangle. と言われたら、足りない辺と角をすべて答える。",
      "用例コーパスでは solving right triangles は話し言葉（NancyPi）にも書き言葉（OpenStax Algebra and Trigonometry・Precalculus）にも少ない。",
      "直角でない三角形を解くときは正弦定理（law of sines）・余弦定理（law of cosines）を使う。"
    ],
    "related": [
      "solving-triangles",
      "trigonometric-ratio",
      "sohcahtoa"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "9.5 Solving Trigonometric Equations（Solving Right Triangle Problems）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "7.5 Solving Trigonometric Equations（Solving Right Triangle Problems）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "solving right triangles": 1,
        "solve a right triangle": 1
      },
      "written": {
        "solving right triangles": 4,
        "solve the right triangle": 1
      },
      "sources": [
        "openstax-algtrig",
        "openstax-precalculus",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 solving right triangles（本文 4 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "solve",
    "ja": {
      "term": "解く",
      "reading": "とく",
      "alt": [
        "方程式を解く"
      ]
    },
    "en": {
      "term": "solve",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数I",
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "方程式・不等式を満たす未知数の値（解）をすべて求める。",
    "definition_en": "To find every value of the unknown that satisfies an equation or inequality.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's solve this for x — subtract 3 from both sides first.",
        "ja": "これを x について解こう。まず両辺から 3 を引く。",
        "register": "spoken"
      },
      {
        "en": "Solve 2 sin θ − 1 = 0 for 0 ≤ θ < 2π.",
        "ja": "0 ≦ θ < 2π のとき、2 sin θ − 1 = 0 を解け。",
        "register": "written"
      },
      {
        "en": "Solve the inequality 3x − 5 < x + 7.",
        "ja": "不等式 3x − 5 < x + 7 を解け。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "solve for",
        "ja": "〜について解く"
      },
      {
        "en": "solve the equation",
        "ja": "方程式を解く"
      }
    ],
    "pitfalls": [
      "「x について解く」は solve for x。「θ を求める」も solve for θ と言える。",
      "問題文の「解け」は Solve …、「求めよ」は Find …。"
    ],
    "related": [
      "solution",
      "compare"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "solve": 2924
      },
      "written": {
        "solve": 6314
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solving-by-factoring",
    "ja": {
      "term": "因数分解による解法",
      "reading": "いんすうぶんかいによるかいほう",
      "alt": [
        "因数分解して解く"
      ]
    },
    "en": {
      "term": "solving by factoring",
      "alt": [
        "solve … by factoring"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "二次方程式を、右辺を 0 にしてから左辺を因数分解し、「AB = 0 ならば A = 0 または B = 0」を使って解く方法。x² − 5x + 6 = 0 なら (x − 2)(x − 3) = 0 から x = 2, 3。",
    "definition_en": "A way to solve a quadratic equation by getting 0 on one side, factoring the other side, and setting each factor equal to 0. It works because of the zero product property.",
    "latex": "x^2 - 5x + 6 = 0 \\Rightarrow (x - 2)(x - 3) = 0 \\Rightarrow x = 2 \\text{ or } x = 3",
    "spoken_en": "x squared minus five x plus six equals zero, so the quantity x minus two times the quantity x minus three equals zero, so x equals two or x equals three",
    "examples": [
      {
        "en": "When you're solving by factoring, get zero on one side first, then factor and set each factor equal to zero.",
        "ja": "因数分解で解くときは、まず片側を 0 にする。それから因数分解して、それぞれの因数を 0 とおくんだ。",
        "register": "spoken"
      },
      {
        "en": "Solve the equation by factoring: x² − 7x + 10 = 0.",
        "ja": "因数分解を利用して、方程式 x² − 7x + 10 = 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "右辺を 0 にしないまま因数分解しても解は出ない。x² − 2x = 8 を x(x − 2) = 8 として x = 8 とするのは誤りで、x² − 2x − 8 = 0 に直してから (x − 4)(x + 2) = 0、x = 4, −2 とする。",
      "問題文の Factor. と Solve by factoring. は答えるものが違う。Factor x² − 5x + 6. の答えは式 (x − 2)(x − 3)、Solve x² − 5x + 6 = 0 by factoring. の答えは解 x = 2, 3（エントリ factoring）。",
      "両辺を x で割ると解を失う。x² = 3x は x で割らずに x² − 3x = 0、x(x − 3) = 0 として x = 0, 3 を出す。"
    ],
    "related": [
      "quadratic-equation",
      "zero-product-property",
      "factoring",
      "factored-form",
      "quadratic-formula"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "solving by factoring": 10
      },
      "written": {
        "solving by factoring": 61
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solving-by-graphing",
    "ja": {
      "term": "グラフによる解法",
      "reading": "ぐらふによるかいほう",
      "alt": []
    },
    "en": {
      "term": "solving by graphing",
      "alt": [
        "solve … by graphing"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "連立方程式の 2 つの式のグラフをかき、交点の座標を解として読み取る解き方。",
    "definition_en": "A method of solving a system of equations by graphing each equation and reading the coordinates of the point where the graphs intersect.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "When we're solving by graphing, the solution is just the point where the two lines cross.",
        "ja": "グラフで解くときは、2 本の直線が交わる点がそのまま解になる。",
        "register": "spoken"
      },
      {
        "en": "Solve each system by graphing: y = x + 1 and y = −2x + 7.",
        "ja": "次の連立方程式をグラフを用いて解け。y = x + 1、y = −2x + 7",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "交点の座標が整数でないと読み取りにくい。読み取った値はもとの式に代入して確かめるか、代入法（エントリ substitution-method）・加減法（エントリ elimination）で解き直す。",
      "2 直線が平行なら交点がなく解はない（エントリ no-solution）。2 直線が重なれば解は無数にある（エントリ infinitely-many-solutions）。",
      "OpenStax Elementary Algebra は solve a system of linear equations by graphing の形で、代入法・加減法と並べて扱う。"
    ],
    "related": [
      "system-of-linear-equations",
      "point-of-intersection",
      "substitution-method",
      "elimination",
      "independent-system"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "solving by graphing": 61
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solving-by-taking-square-roots",
    "ja": {
      "term": "平方根の考えを使う解法",
      "reading": "へいほうこんのかんがえをつかうかいほう",
      "alt": [
        "平方根をとって解く"
      ]
    },
    "en": {
      "term": "square root property",
      "alt": [],
      "variants": [
        {
          "term": "take the square root of both sides",
          "register": "spoken",
          "note": "用例コーパスの話し言葉では、方法の名前より手順をこの形で言うほうが多い（The Organic Chemistry Tutor・Khan Academy が中心）。"
        },
        {
          "term": "taking square roots",
          "register": "spoken",
          "note": "話し言葉の言い方（用例コーパスでは話し言葉が中心）。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "米国の教科書 OpenStax Elementary Algebra は、この解き方を square root property（k ≥ 0 のとき、x² = k ならば x = ±√k）という性質の名前で呼び、節の名前も Solve Quadratic Equations Using the Square Root Property とする。用例コーパスの話し言葉では、方法の名前 square root property より、take the square root of both sides（両辺の平方根をとる）と手順で言う形のほうが多い（書き言葉は逆で、square root property は OpenStax に多い）。",
    "domains": [
      "algebra",
      "functions"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "x² = k や (x + a)² = k の形の二次方程式を、両辺の平方根をとって解く方法。k > 0 なら、x² = k の解は x = √k と x = −√k の 2 つ。",
    "definition_en": "Solving an equation of the form x² = k or (x + a)² = k by taking the square root of each side, which rests on the square root property: for k ≥ 0, x² = k gives x = √k or x = −√k.",
    "latex": "x^2 = k \\Rightarrow x = \\pm\\sqrt{k}",
    "spoken_en": "x squared equals k, so x equals plus or minus the square root of k",
    "examples": [
      {
        "en": "There's no x term here, so just solve by taking square roots, and don't forget the plus or minus.",
        "ja": "ここは x の項がないから、平方根の考えを使って解けばいい。± を忘れないでね。",
        "register": "spoken"
      },
      {
        "en": "Solve (x − 3)² = 25 with the square root property, and check both solutions.",
        "ja": "平方根の考えを使って方程式 (x − 3)² = 25 を解き、2 つの解を確かめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "± を忘れない。x² = 9 の解は x = 3 と x = −3 の 2 つ。記号 √9 は正の方の 3 だけを表す（エントリ principal-square-root）。",
      "(x − 3)² = 25 は展開せずに、そのまま x − 3 = ±5 とする。x = 3 ± 5 から x = 8, −2。",
      "k < 0 のとき x² = k は実数の解をもたない（x² = −4 を満たす実数はない）。"
    ],
    "related": [
      "quadratic-equation",
      "square-root-of-a-number",
      "principal-square-root",
      "completing-the-square",
      "solving-by-factoring"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "take the square root of both sides": 73,
        "taking square roots": 10,
        "square root property": 4
      },
      "written": {
        "taking square roots": 3,
        "take the square root of both sides": 7,
        "square root property": 171
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "solving-triangles",
    "ja": {
      "term": "三角形を解く",
      "reading": "さんかくけいをとく",
      "alt": []
    },
    "en": {
      "term": "solve the triangle",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "三角形の辺と角のうちわかっているものから、残りの辺と角をすべて求める。正弦定理・余弦定理を使う。",
    "definition_en": "To find all the missing sides and angles of a triangle from the known ones, using the law of sines and the law of cosines.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We know two sides and the included angle, so use the law of cosines first.",
        "ja": "2 辺とその間の角がわかっているので、まず余弦定理を使う。",
        "register": "spoken"
      },
      {
        "en": "Solve the triangle ABC with a = 7, b = 5, and C = 60°.",
        "ja": "a = 7, b = 5, C = 60° の三角形 ABC を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "米国の教科書は正弦定理を law of sines、余弦定理を law of cosines と呼ぶ（OpenStax Algebra and Trigonometry）。"
    ],
    "related": [
      "ambiguous-case",
      "area-formula-with-sine"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "solve the triangle": 1
      },
      "written": {
        "solve the triangle": 13
      },
      "sources": [
        "openstax-algtrig",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "space-curve",
    "ja": {
      "term": "空間曲線",
      "reading": "くうかんきょくせん",
      "alt": []
    },
    "en": {
      "term": "space curve",
      "alt": [
        "curve in space"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "空間内の曲線。ベクトル値関数 r(t) の終点が描く軌跡として表す。",
    "definition_en": "A curve in three-dimensional space, described as the path traced by the tip of a vector-valued function r(t).",
    "latex": "\\mathbf{r}(t) = \\langle \\cos t, \\sin t, t \\rangle",
    "spoken_en": null,
    "examples": [
      {
        "en": "This space curve is a helix: it goes around and up at the same time.",
        "ja": "この空間曲線はらせん。回りながら同時に上がっていく。",
        "register": "spoken"
      },
      {
        "en": "Sketch the space curve traced by r(t) = ⟨t, t², t³⟩ for 0 ≤ t ≤ 1.",
        "ja": "0 ≤ t ≤ 1 で r(t) = ⟨t, t², t³⟩ が描く空間曲線をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉の space curve はすべて Professor Leonard。"
    ],
    "related": [
      "vector-valued-function",
      "curvature",
      "unit-tangent-vector",
      "arc-length-of-a-curve"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "space curve": 10,
        "curve in space": 1
      },
      "written": {
        "space curve": 17,
        "curve in space": 6
      },
      "sources": [
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "space-diagonal",
    "ja": {
      "term": "立体の対角線",
      "reading": "りったいのたいかくせん",
      "alt": [
        "体対角線"
      ]
    },
    "en": {
      "term": "space diagonal",
      "alt": [
        "diagonal of a rectangular prism",
        "diagonal of a box"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（space diagonal は CED・OpenStax・IM・CK-12 とも 0 件。diagonal of the rectangular prism ／ diagonal of the box は OpenStax Calculus Volume 3（2.2）に各 1 件で、CK-12 Geometry 4.33・4.35 と IM Grade 8 はどの頂点からどの頂点への diagonal かを言って説明する）。見出しは英語版 Wikipedia の記事名「Space diagonal」（数学カテゴリから 4 段、冒頭で太字）。見出しの「立体の対角線」は学習指導要領解説に無い（日本語版 Wikipedia「対角線」は、内部を通る対角線を体対角線、面の上のものを面対角線と呼ぶ）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": []
    },
    "definition_ja": "直方体や立方体で、同じ面の上にない 2 つの頂点を結ぶ線分。縦 a、横 b、高さ c の直方体では、長さは √(a² + b² + c²)。",
    "definition_en": "A segment joining two vertices of a rectangular prism that do not lie on the same face. For a prism with dimensions a, b, and c, its length is √(a² + b² + c²).",
    "latex": "\\sqrt{a^2 + b^2 + c^2}",
    "spoken_en": "the square root of a squared plus b squared plus c squared",
    "examples": [
      {
        "en": "The space diagonal goes through the inside, so to get the diagonal of a box, use the Pythagorean theorem twice: across the bottom, then up to the top corner.",
        "ja": "立体の対角線は中を通る。だから箱の対角線は三平方の定理を 2 回使って求める。まず底面を横切り、次に上の頂点まで上がる。",
        "register": "spoken"
      },
      {
        "en": "Find the length of the diagonal of a rectangular prism that measures 2 cm by 3 cm by 6 cm.",
        "ja": "縦 2 cm、横 3 cm、高さ 6 cm の直方体の対角線の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "diagonal だけでは多角形の対角線や面の対角線と区別がつかない（エントリ diagonal）。英語版 Wikipedia「Space diagonal」は面の対角線を face diagonal と呼んで区別する。",
      "長さは三平方の定理を 2 回使う。底面の対角線が √(a² + b²)、それと高さ c から √(a² + b² + c²)。"
    ],
    "related": [
      "rectangular-prism",
      "diagonal",
      "cube-solid",
      "pythagorean-theorem",
      "face"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Space_diagonal",
        "note": "記事「Space diagonal」（数学カテゴリから 4 段）の冒頭: space diagonal を太字で定義"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「対角線」",
        "url": "https://ja.wikipedia.org/wiki/%E5%AF%BE%E8%A7%92%E7%B7%9A",
        "note": "体対角線・面対角線"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.33・4.35: 直方体の頂点を結ぶ diagonal"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "「立体の対角線」は 0 件"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "直方体の頂点を結ぶ diagonal（どの頂点からどの頂点への diagonal かを言って説明する）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "diagonal of a rectangular prism | diagonal of the rectangular prism": 1
      },
      "written": {
        "diagonal of a rectangular prism | diagonal of the rectangular prism": 1,
        "diagonal of a box | diagonal of the box": 1
      },
      "sources": [
        "openstax-calculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは Wikipedia の記事名 space diagonal（英語版「Space diagonal」、en.term のリダイレクト先）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "span",
    "ja": {
      "term": "張る",
      "reading": "はる",
      "alt": [
        "生成する",
        "スパン"
      ]
    },
    "en": {
      "term": "span",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "いくつかのベクトルの 1 次結合全体が、ある空間になること。その空間を、それらのベクトルが張る空間という。",
    "definition_en": "Of a set of vectors: to have all their linear combinations make up a given space; that space is the span of the vectors.",
    "latex": "\\operatorname{span}\\{\\vec{v}_1, \\ldots, \\vec{v}_k\\}",
    "spoken_en": null,
    "examples": [
      {
        "en": "These two vectors aren't parallel, so they span the whole plane.",
        "ja": "この 2 つのベクトルは平行でないので、平面全体を張る。",
        "register": "spoken"
      },
      {
        "en": "Is the vector b in the span of v₁ and v₂?",
        "ja": "ベクトル b は v₁ と v₂ が張る空間に含まれるか。",
        "register": "spoken"
      },
      {
        "en": "Show that the columns of A span ℝ³.",
        "ja": "A の列ベクトルが ℝ³ を張ることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語は名詞（the span）と動詞（span）が同じ形。日本語は名詞に「生成する部分空間」「張る空間」と言う。"
    ],
    "related": [
      "linear-combination",
      "subspace",
      "basis",
      "linearly-independent"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.1 Subspaces and Spanning"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the span of | spans the … space | span the … space | in the span | span of the columns | span of two vectors": 29
      },
      "written": {
        "the span of | spans the … space | span the … space | in the span | span of the columns | span of two vectors": 1
      },
      "sources": [
        "mit-18.06",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "spanning-tree",
    "ja": {
      "term": "全域木",
      "reading": "ぜんいきぎ",
      "alt": []
    },
    "en": {
      "term": "spanning tree",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "連結なグラフのすべての頂点を含み、もとのグラフの辺だけでできている木のこと。頂点が n 個なら辺は n − 1 本になる。",
    "definition_en": "A tree that contains every vertex of a connected graph and uses only edges of that graph. If the graph has n vertices, a spanning tree has n − 1 edges.",
    "latex": "|E(T)| = |V(G)| - 1",
    "spoken_en": "the number of edges of T equals the number of vertices of G minus one",
    "examples": [
      {
        "en": "Keep deleting edges that lie on a cycle until no cycles are left, and what you end up with is a spanning tree.",
        "ja": "閉路に含まれる辺を、閉路がなくなるまで取り除いていこう。最後に残ったものが全域木だ。",
        "register": "spoken"
      },
      {
        "en": "Find two different spanning trees of the graph below.",
        "ja": "下のグラフの全域木を 2 つ求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "辺に重みがあるとき、重みの合計が最小の全域木を minimum spanning tree（最小全域木）という。",
      "全域木をもつのは連結なグラフだけ。1 つのグラフに全域木はふつう何通りもある。",
      "用例コーパスでは話し言葉はすべて MIT OCW、書き言葉はすべて MIT の講義ノート。Levin の本では 2.2 Trees に 13 件。"
    ],
    "related": [
      "connected",
      "vertex-graph",
      "tree",
      "graph-network"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "全域木",
        "en": "Spanning tree"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "spanning tree": 70
      },
      "written": {
        "spanning tree": 23
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "special-products",
    "ja": {
      "term": "乗法公式",
      "reading": "じょうほうこうしき",
      "alt": []
    },
    "en": {
      "term": "special products",
      "alt": [
        "special product formulas"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「乗法公式」は展開の公式をまとめた呼び名。米国の教科書 OpenStax Elementary Algebra は 6.4 Special Products で Binomial Squares Pattern（(a + b)²、(a − b)²、エントリ square-of-a-binomial）と Product of Conjugates Pattern（(a + b)(a − b)、エントリ difference-of-squares）を扱い、(x + a)(x + b) は 6.3 でふつうの二項式の積として扱う（日本の乗法公式はこれも含む）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3",
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "(x + a)(x + b)、(a + b)²、(a − b)²、(a + b)(a − b) のような、よく出てくる積を展開するための公式。左辺と右辺を入れ替えて使うと、因数分解の公式になる。",
    "definition_en": "Shortcut patterns for multiplying common kinds of binomials: (x + a)(x + b), (a + b)², (a − b)², and (a + b)(a − b). Used in reverse, they become patterns for factoring.",
    "latex": "(x + a)(x + b) = x^2 + (a + b)x + ab",
    "spoken_en": "the quantity x plus a times the quantity x plus b equals x squared plus the quantity a plus b times x, plus a b",
    "examples": [
      {
        "en": "These special products show up all the time, so learn them and you won't have to multiply everything out.",
        "ja": "この乗法公式はしょっちゅう出てくるから、覚えておけば毎回全部展開しなくて済む。",
        "register": "spoken"
      },
      {
        "en": "Use special products to multiply (3x + 2)(3x − 2).",
        "ja": "乗法公式を利用して、(3x + 2)(3x − 2) を展開せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "(x + a)(x + b) の x の係数は a と b の和、定数項は a と b の積。符号をつけたまま計算する（(x + 5)(x − 3) = x² + 2x − 15）。",
      "(a + b)² を a² + b² としない。まん中の 2ab を落とさない（エントリ square-of-a-binomial）。",
      "三角関数の積和の公式（product-to-sum formulas、エントリ product-to-sum-formulas）とは別のもの。"
    ],
    "related": [
      "square-of-a-binomial",
      "difference-of-squares",
      "expand",
      "factoring",
      "perfect-square-trinomial"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "special products": 37,
        "special product formulas": 5
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "special-right-triangles",
    "ja": {
      "term": "特別な直角三角形",
      "reading": "とくべつなちょっかくさんかくけい",
      "alt": []
    },
    "en": {
      "term": "special right triangles",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「特別な直角三角形」は学習指導要領解説に無い、本プロジェクトの訳語。CK-12 Geometry 4.41 Special Right Triangles and Ratios は special right triangles を 30-60-90・45-45-90・Pythagorean triple の 3 種類とし（4.43 は「2 つの special right triangles」と書く）、英語版 Wikipedia「Special right triangle」も角による型（angle-based）と辺による型（side-based、ピタゴラス数）に分ける。この見出しは角の型の 2 種類を指す。日本語版 Wikipedia「三角定規」は、この 2 種類を直角二等辺三角形と半正三角形の 2 枚 1 組の三角定規として書く。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "3 つの角が 45°, 45°, 90° の直角三角形と、30°, 60°, 90° の直角三角形。辺の比がそれぞれ 1 : 1 : √2、1 : √3 : 2 と決まっている。",
    "definition_en": "The 45-45-90 and 30-60-90 right triangles, whose side lengths always come in fixed ratios. Their sides are in the ratios 1 : 1 : √2 and 1 : √3 : 2.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "With special right triangles you don't need the Pythagorean theorem every time; you just use the side ratios.",
        "ja": "特別な直角三角形なら、毎回三平方の定理を使わなくていい。辺の比を使えばいいんだ。",
        "register": "spoken"
      },
      {
        "en": "A right triangle has two 45° angles and a leg of length 6. Use special right triangles to find the exact length of the hypotenuse.",
        "ja": "2 つの角が 45° で、直角をはさむ辺の長さが 6 の直角三角形がある。特別な直角三角形を用いて、斜辺の長さを正確な値で求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "CK-12 Geometry は複数形 special right triangles で 2 種類をまとめて扱う（4.41 Special Right Triangles and Ratios）。1 つだけを指すときは角の大きさで 45-45-90 triangle、30-60-90 triangle と呼ぶ（エントリ 45-45-90-triangle、30-60-90-triangle）。",
      "1 : √3 : 2 は短い辺からの順。CK-12 Geometry 4.41・4.43 は x, x√3, 2x と書く。√3 がつくのは長い方の leg で、斜辺は 2x。",
      "CK-12 Geometry は Special Right Triangles and Ratios の節で扱う。"
    ],
    "related": [
      "45-45-90-triangle",
      "30-60-90-triangle",
      "right-triangle",
      "pythagorean-theorem",
      "trigonometric-ratio"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.41 Special Right Triangles and Ratios（3 種類）・4.43 30-60-90 Right Triangles（2 つの special right triangles）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Special_right_triangle",
        "note": "記事「Special right triangle」（数学カテゴリから 4 段）: angle-based と side-based の節"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "「特別な直角三角形」は 0 件（見出しは本プロジェクトの訳語）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角定規」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E5%AE%9A%E8%A6%8F",
        "note": "直角二等辺三角形と半正三角形の 2 枚 1 組"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "special right triangles": 17
      },
      "sources": [
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "speed",
    "ja": {
      "term": "速さ",
      "reading": "はやさ",
      "alt": []
    },
    "en": {
      "term": "speed",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "単位時間あたりに進む道のり。速さ = 道のり ÷ 時間 で求める。",
    "definition_en": "The distance something travels per unit of time. It is found by dividing the distance by the time.",
    "latex": "\\text{speed} = \\frac{\\text{distance}}{\\text{time}}",
    "spoken_en": "speed equals distance divided by time",
    "examples": [
      {
        "en": "If you bike 30 kilometers in 2 hours, your speed is 15 kilometers per hour.",
        "ja": "2 時間で 30 km 自転車で走ったら、速さは時速 15 km だね。",
        "register": "spoken"
      },
      {
        "en": "A train travels 240 kilometers in 3 hours at a constant speed. Find its speed in kilometers per hour.",
        "ja": "列車が一定の速さで 3 時間に 240 km 進む。この列車の速さは時速何 km か求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「時速」「秒速」は per hour ／ per second で言う。時速 60 km は 60 kilometers per hour（60 km/h）、秒速 5 m は 5 meters per second（5 m/s）。",
      "道のり = 速さ × 時間 は、英語では distance = rate × time と書き（OpenStax Elementary Algebra は D = rt）、公式の中では速さを rate と呼ぶ。",
      "速さ（speed）は向きを含まず、負にならない。向きを含む「速度」は velocity（エントリ velocity）。"
    ],
    "related": [
      "distance",
      "velocity",
      "unit",
      "rate-of-change",
      "word-problem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "速さ",
        "en": "Speed"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "D = rt（distance = rate × time）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "speed": 338
      },
      "written": {
        "speed": 722
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sphere",
    "ja": {
      "term": "球",
      "reading": "きゅう",
      "alt": []
    },
    "en": {
      "term": "sphere",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説は球を柱体・錐体と並ぶ立体として扱い、日本語版 Wikipedia「球」は数学の球を「球体（ball）または球面（sphere）」と書く。IM Geometry（5.2）・CK-12 Geometry（9.5）・OpenStax Calculus Volume 3（2.2）は sphere を中心から等しい距離にある点の集まり（表面）と定義するが、the volume of a sphere ／ the surface area of a sphere とも言い、CK-12 Geometry 9.2 は sphere を a round, three-dimensional solid と書く。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "空間で、1 つの点（中心）から等しい距離（半径）にある点全体がつくる面（球面）で囲まれた、まるい立体。英語の sphere はこの表面（球面）を指す。",
    "definition_en": "The round three-dimensional shape made of all the points in space that are the same distance, the radius, from a center point, like the surface of a ball.",
    "latex": "V = \\frac{4}{3}\\pi r^3,\\quad S = 4\\pi r^2",
    "spoken_en": "V equals four thirds pi r cubed, and S equals four pi r squared",
    "examples": [
      {
        "en": "Every point on the sphere is the same distance from the center, and that distance is the radius.",
        "ja": "球面上のどの点も中心から同じ距離にあって、その距離が半径。",
        "register": "spoken"
      },
      {
        "en": "Find the volume and the surface area of a sphere with a radius of 6 cm. Leave your answers in terms of π.",
        "ja": "半径 6 cm の球の体積と表面積を求めよ。ただし、円周率は π とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "IM Geometry・CK-12 Geometry・OpenStax Calculus Volume 3 の定義の sphere は表面（日本語の球面）。中身を含む立体（日本語の球）も sphere と呼ぶ（the volume of a sphere）。",
      "平面で中心から等しい距離にある点の集まりは circle（円。エントリ circle）。空間なら sphere。",
      "球を半分にした立体は hemisphere（半球）。"
    ],
    "related": [
      "circle",
      "radius",
      "volume",
      "surface-area",
      "equation-of-a-sphere"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.2 Vectors in Three Dimensions: A sphere is the set of all points in space equidistant from a fixed point"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "5.2 Slicing Solids: sphere（the set of points … the same distance from some center）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "9.5 Volume: sphere（set of all points in space equidistant from a center point）、Volume of a Sphere／9.2: a round, three-dimensional solid"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「球」",
        "url": "https://ja.wikipedia.org/wiki/%E7%90%83",
        "note": "球体（ball）または球面（sphere）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第1学年 B 図形: 柱体，錐体及び球の表面積と体積"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "sphere": 191
      },
      "written": {
        "sphere": 285
      },
      "sources": [
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "spherical-coordinates",
    "ja": {
      "term": "球座標",
      "reading": "きゅうざひょう",
      "alt": [
        "球面座標"
      ]
    },
    "en": {
      "term": "spherical coordinates",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "原点からの距離 ρ と 2 つの角 θ、φ で空間の点を (ρ, θ, φ) と表す座標。",
    "definition_en": "Coordinates (ρ, θ, φ) that locate a point in space by its distance ρ from the origin and two angles θ and φ.",
    "latex": "x = \\rho\\sin\\varphi\\cos\\theta, \\; y = \\rho\\sin\\varphi\\sin\\theta, \\; z = \\rho\\cos\\varphi",
    "spoken_en": "x equals rho sine phi cosine theta, y equals rho sine phi sine theta, z equals rho cosine phi",
    "examples": [
      {
        "en": "For a ball centered at the origin, spherical coordinates are the natural choice.",
        "ja": "原点中心の球なら、球座標が自然。",
        "register": "spoken"
      },
      {
        "en": "Use spherical coordinates to find the volume of the solid inside the sphere ρ = 2 and above the cone φ = π/4.",
        "ja": "球座標を用いて、球 ρ = 2 の内部かつ円錐 φ = π/4 の上側にある立体の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "体積要素は dV = ρ² sin φ dρ dφ dθ。",
      "θ と φ の役割が数学と物理で逆になる教科書がある。どちらの約束か確かめる。"
    ],
    "related": [
      "cylindrical-coordinates",
      "triple-integral",
      "coordinates-in-space"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "球面座標系",
        "en": "Spherical coordinate system"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "spherical coordinates": 26
      },
      "written": {
        "spherical coordinates": 142
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "split-into-cases",
    "ja": {
      "term": "場合分けする",
      "reading": "ばあいわけする",
      "alt": []
    },
    "en": {
      "term": "consider … cases",
      "alt": [
        "break … into cases"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「場合分けする」は 1 語の動詞だが、英語では consider two cases ／ break it into cases のように cases を目的語にした句で言う。二次関数の最大・最小で「軸の位置で場合分けする」と言うときは、英語で説明するなら頂点（軸）が区間の左・内側・右のどこにあるかで cases を分ける、という言い方になる。",
    "domains": [
      "algebra",
      "functions",
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "条件によって結論や計算が変わるとき、起こりうる場合を漏れなく重複なく分けて、それぞれについて考える。",
    "definition_en": "To handle a problem by dividing the possibilities into separate cases that together cover everything, and working through each case on its own.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's consider two cases: x ≥ 0 and x < 0.",
        "ja": "x ≧ 0 と x < 0 の 2 つの場合に分けて考えよう。",
        "register": "spoken"
      },
      {
        "en": "To find the minimum of f(x) = x² − 2ax on 0 ≤ x ≤ 2, consider three cases: a < 0, 0 ≤ a ≤ 2, and a > 2.",
        "ja": "0 ≦ x ≦ 2 における f(x) = x² − 2ax の最小値を求めるには、a < 0、0 ≦ a ≦ 2、a > 2 の 3 つの場合に分けて考える。",
        "register": "written"
      },
      {
        "en": "To count the even three-digit numbers with distinct digits, consider two cases: the last digit is 0, or it is not.",
        "ja": "異なる 3 つの数字を並べてできる 3 桁の偶数の個数は、一の位が 0 の場合と 0 でない場合に分けて数える。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "答案では Case 1: x ≥ 0 ／ Case 2: x < 0 のように見出しを立てて書くと、どの場合を扱っているかがはっきりする。",
      "「場合の数」は number of cases ではない（分野名は counting、個数は the number of ways）。「場合分け」の case と混同しない。"
    ],
    "related": [
      "case-analysis",
      "absolute-value-equation",
      "absolute-value-inequality",
      "axis-of-symmetry",
      "maximum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "consider … cases": 15
      },
      "written": {
        "consider … cases": 27,
        "break … into cases": 2
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "spread",
    "ja": {
      "term": "散らばり",
      "reading": "ちらばり",
      "alt": [
        "データの散らばり",
        "ばらつき",
        "ばらつく",
        "散らばりの測度"
      ]
    },
    "en": {
      "term": "spread",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "spread",
          "register": "both",
          "note": "the spread の形で数えた。話し言葉は Khan Academy が中心、書き言葉は OpenStax Introductory Statistics と MIT の講義ノートが中心。"
        },
        {
          "term": "variability",
          "register": "both",
          "note": "話し言葉はすべて Khan Academy（中学の講義と AP Statistics）。書き言葉はほとんどが OpenStax Introductory Statistics。"
        }
      ],
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「散らばり」「ばらつき」は英語では spread と variability の 2 つが並ぶ（下の variants）。動詞「ばらつく」は be spread out と言う。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1",
        "中2",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Intro Statistics",
        "AP Statistics",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データの値が代表値のまわりにどれだけ広がっているかの様子。範囲・四分位範囲・分散・標準偏差などで表す。",
    "definition_en": "How widely the data values are spread around a central value, measured by the range, the interquartile range, the variance, or the standard deviation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Both classes have the same mean, but this one has a lot more spread.",
        "ja": "どちらのクラスも平均は同じだが、こちらのほうがずっと散らばりが大きい。",
        "register": "spoken"
      },
      {
        "en": "Compare the centers and spreads of the two distributions.",
        "ja": "2 つの分布の中心と散らばりを比べよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "measure of spread",
        "ja": "散らばりの指標"
      }
    ],
    "pitfalls": [
      "vary（変わる）は「ばらつく」の意味より、answers may vary や vary t（t を動かす）のような意味で使うことが多い（用例コーパスで見た文脈はすべて別の意味）。"
    ],
    "related": [
      "interquartile-range",
      "standard-deviation",
      "variance",
      "measure-of-center"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the spread": 24,
        "variability": 46
      },
      "written": {
        "the spread": 45,
        "variability": 25
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "square-matrix",
    "ja": {
      "term": "正方行列",
      "reading": "せいほうぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "square matrix",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "行の数と列の数が等しい行列。n × n 行列を n 次の正方行列という。",
    "definition_en": "A matrix with the same number of rows as columns; an n × n matrix is a square matrix of order n.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Only a square matrix can have a determinant.",
        "ja": "行列式をもつのは正方行列だけ。",
        "register": "spoken"
      },
      {
        "en": "Give an example of a 2 × 2 square matrix that has no inverse.",
        "ja": "逆行列をもたない 2 × 2 の正方行列の例を挙げよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "matrix",
      "identity-matrix",
      "determinant"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正方行列",
        "en": "Square matrix"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "square matrix": 57
      },
      "written": {
        "square matrix": 29
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "square-of-a-binomial",
    "ja": {
      "term": "平方の公式",
      "reading": "へいほうのこうしき",
      "alt": [
        "和の平方",
        "差の平方"
      ]
    },
    "en": {
      "term": "binomial squares",
      "alt": [
        "square of a binomial"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 つの項の和や差を 2 乗したものを展開する公式。(a + b)² = a² + 2ab + b²、(a − b)² = a² − 2ab + b² である。",
    "definition_en": "The patterns for squaring a binomial: (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b². The term in the middle comes from doubling the product of the two terms.",
    "latex": "(a \\pm b)^2 = a^2 \\pm 2ab + b^2",
    "spoken_en": "the quantity a plus or minus b, squared, equals a squared plus or minus two a b, plus b squared",
    "examples": [
      {
        "en": "This is the square of a binomial, so don't forget the middle term: (x + 3)² is x² + 6x + 9, not x² + 9.",
        "ja": "これは平方の公式の形だから、まん中の項を忘れないこと。(x + 3)² は x² + 6x + 9 で、x² + 9 じゃない。",
        "register": "spoken"
      },
      {
        "en": "Use the binomial squares pattern to expand (2x − 5)².",
        "ja": "平方の公式を利用して、(2x − 5)² を展開せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "(a + b)² を a² + b² としない。まん中の 2ab を落とさない。",
      "(a − b)² のまん中の項は −2ab だが、最後の b² はプラスのまま。(x − 4)² = x² − 8x + 16。",
      "OpenStax Elementary Algebra はこの公式を Binomial Squares Pattern と呼ぶ。展開した a² + 2ab + b² の形は perfect square trinomial（完全平方式、エントリ perfect-square-trinomial）。"
    ],
    "related": [
      "special-products",
      "perfect-square-trinomial",
      "difference-of-squares",
      "expand",
      "completing-the-square"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "square of a binomial": 1
      },
      "written": {
        "square of a binomial": 16,
        "binomial squares": 100
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "square-pyramid",
    "ja": {
      "term": "四角錐",
      "reading": "しかくすい",
      "alt": []
    },
    "en": {
      "term": "rectangular pyramid",
      "alt": [
        "square pyramid"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「四角錐」は底面が四角形なら何でもよいが、英語は底面の形で名前を付け、底面が長方形なら rectangular pyramid、正方形なら square pyramid と呼び分ける。「正四角錐」（底面が正方形で、頂点が底面の中心の真上にあるもの）は square pyramid に当たる。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "底面が四角形の角錐。側面は 4 つの三角形で、面は 5 つ、辺は 8 本、頂点は 5 つある。",
    "definition_en": "A pyramid whose base is a rectangle, with four triangular faces that meet at the apex. When the base is a square, it is called a square pyramid.",
    "latex": "V = \\frac{1}{3}lwh",
    "spoken_en": "V equals one third l times w times h",
    "examples": [
      {
        "en": "The base is a rectangle, not a square, so this is a rectangular pyramid, not a square pyramid.",
        "ja": "底面が正方形じゃなくて長方形だから、これは正四角錐じゃなくて、底面が長方形の四角錐だね。",
        "register": "spoken"
      },
      {
        "en": "A rectangular pyramid has a base that measures 6 cm by 4 cm and a height of 5 cm. Find its volume.",
        "ja": "底面が縦 6 cm、横 4 cm の長方形で、高さが 5 cm の四角錐の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高さ（height）は、側面の三角形が集まる頂点から底面に下ろした垂線の長さ。側面の三角形の高さ（slant height、エントリ slant-height）とは別。",
      "四角錐の頂点は 5 つ（vertices）で、そのうち側面の三角形が集まる 1 つを apex と呼んで、底面の 4 つの頂点と区別する。"
    ],
    "related": [
      "pyramid",
      "rectangular-prism",
      "slant-height",
      "tetrahedron",
      "volume"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "四角錐",
        "en": "Square pyramid"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "square pyramid": 2,
        "rectangular pyramid": 10
      },
      "written": {
        "square pyramid": 2
      },
      "sources": [
        "khan-middle",
        "openstax-calculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "square-root-of-a-negative-number",
    "ja": {
      "term": "負の数の平方根",
      "reading": "ふのかずのへいほうこん",
      "alt": []
    },
    "en": {
      "term": "square root of a negative number",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "a > 0 のとき、−a の平方根は ±√a i。√−a = √a i と定める。",
    "definition_en": "For a > 0, the square roots of −a are ±√a i; √−a is defined as √a i.",
    "latex": "\\sqrt{-a} = \\sqrt{a}\\, i \\quad (a > 0)",
    "spoken_en": "the square root of negative a equals the square root of a times i",
    "examples": [
      {
        "en": "To take the square root of a negative number, pull out the i first: the square root of negative 4 is 2i.",
        "ja": "負の数の平方根をとるときは、まず i を出す。√−4 は 2i。",
        "register": "spoken"
      },
      {
        "en": "Compute √−2 × √−8, and explain why it is not equal to √16.",
        "ja": "√−2 × √−8 を計算し、√16 と等しくない理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "√a √b = √(ab) は a, b がともに負のときには成り立たない（OpenStax Intermediate Algebra も、負の数 2 つの積には根号の積の性質を使えないと書く）。先に i を出してから計算する（Write it in terms of i first.）。"
    ],
    "related": [
      "imaginary-unit",
      "imaginary-number",
      "complex-number"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Use the Complex Number System（square roots of negative numbers）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "square root of a negative number": 8
      },
      "written": {
        "square root of a negative number": 26
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "square-root-of-a-number",
    "ja": {
      "term": "平方根",
      "reading": "へいほうこん",
      "alt": []
    },
    "en": {
      "term": "square root",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「a の平方根」は 2 乗して a になる数の両方を指す（中学校学習指導要領解説（第 3 学年）は、正の数 a の平方根を正と負の 2 つの数とし、√a はその正の方を表すとする）。英語の the square root of a は正の方の √a だけを指し（IM Grade 8 の用語集は square root を 2 乗して a になる正の数と定義する）、両方を言うときは square roots、片方は a square root of a と言う（OpenStax Elementary Algebra 1.8・9.1 は n² = m の n を a square root of m と呼び、正の数には正と負の 2 つの square roots があると書く。IM Algebra 1 7.15 は 6 と −6 を both square roots of 36 と書く）。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 乗すると a になる数を、a の平方根という。正の数 a の平方根は正と負の 2 つあり、まとめて ±√a と表す。",
    "definition_en": "Either of the numbers that give a when squared. For a > 0 there are exactly two of them, √a and −√a.",
    "latex": "x^2 = a \\Rightarrow x = \\pm\\sqrt{a} \\quad (a > 0)",
    "spoken_en": "if x squared equals a, then x equals plus or minus the square root of a, where a is greater than zero",
    "examples": [
      {
        "en": "The square root of 49 is 7, since 7 times 7 is 49.",
        "ja": "√49 は 7。7 × 7 = 49 だからね。",
        "register": "spoken"
      },
      {
        "en": "Find both square roots of 64.",
        "ja": "64 の平方根を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「9 の平方根」は 3 と −3 の 2 つ。記号 √9 は正のほう（principal square root、エントリ principal-square-root）だけを表すので √9 = 3 で、√9 = ±3 ではない。",
      "√a は the square root of a と読む。記号 √ の名前は radical sign（エントリ radical-sign）。",
      "2 乗して負になる実数はないので、負の数の平方根は実数の範囲にはない（エントリ square-root-of-a-negative-number）。",
      "英語の the square root of 49 は √49 = 7 だけを指す。±7 の両方は both square roots of 49 と言う（OpenStax Elementary Algebra は根号が正の平方根を表すと書く）。"
    ],
    "related": [
      "principal-square-root",
      "radical-sign",
      "square",
      "nth-root",
      "solving-by-taking-square-roots"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "平方根",
        "en": "Square root"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "5.7 Simplify and Use Square Roots（level.us Pre-Algebra）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.8 The Real Numbers・9.1 Simplify and Use Square Roots（a square root of m・two square roots。mapping_note・pitfalls[3]）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 A 数と式: 正の数 a の平方根は正と負の 2 つで、√a は正の方を表す（mapping_note）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "Grade 8 glossary（square root: 正の数として定義。mapping_note）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 1)",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 7.15 Quadratic Equations with Irrational Solutions（both square roots of 36。mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "square root": 3472
      },
      "written": {
        "square root": 878
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "square-shape",
    "ja": {
      "term": "正方形",
      "reading": "せいほうけい",
      "alt": []
    },
    "en": {
      "term": "square",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "4 つの辺の長さがすべて等しく、4 つの角がすべて直角である四角形。長方形でもあり、ひし形でもある。",
    "definition_en": "A quadrilateral whose four sides are equal and whose four angles are right angles. Every square is both a rectangle and a rhombus.",
    "latex": "A = s^2",
    "spoken_en": "A equals s squared",
    "examples": [
      {
        "en": "A square has four right angles and four equal sides, so it's a rectangle and a rhombus at the same time.",
        "ja": "正方形は 4 つの角が直角で 4 つの辺が等しいから、長方形でもあり、ひし形でもある。",
        "register": "spoken"
      },
      {
        "en": "The perimeter of a square is 36 cm. Find the area of the square.",
        "ja": "周の長さが 36 cm の正方形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の square は、数の 2 乗・平方（エントリ square）と同じ語。1 辺が s の正方形の面積は s² で、s squared と読む。",
      "「四角形」は quadrilateral（エントリ quadrilateral）で、square ではない。square は 4 つの辺と 4 つの角がすべてそろった正方形だけを指す。",
      "面積の単位 cm² は square centimeters と読む（エントリ square-units）。"
    ],
    "related": [
      "rectangle",
      "rhombus",
      "quadrilateral",
      "parallelogram",
      "square"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "正方形",
        "en": "Square"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "area of a square | area of the square | perimeter of a square | perimeter of the square | a square has | the square has | squares and rectangles | rectangles and squares | draw a square | shape of a square | rectangle is a square": 54
      },
      "written": {
        "area of a square | area of the square | perimeter of a square | perimeter of the square | a square has | the square has | squares and rectangles | rectangles and squares | draw a square | shape of a square | rectangle is a square": 20
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "square-units",
    "ja": {
      "term": "平方単位",
      "reading": "へいほうたんい",
      "alt": []
    },
    "en": {
      "term": "square units",
      "alt": [],
      "variants": [
        {
          "term": "square units",
          "register": "both"
        },
        {
          "term": "units squared",
          "register": "both"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「平方単位」は学習指導要領解説に無い、本プロジェクトの訳語。日本では cm²・m² のように決まった単位で答える。英語の問題では単位を決めずに、1 辺 1 の正方形いくつ分かを square units ／ units squared で答えさせることがある。",
    "domains": [
      "geometry",
      "arithmetic"
    ],
    "level": {
      "jp": [
        "小学校"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "1 辺の長さが 1 単位の正方形を 1 つと数えて面積を表すときの単位。長さの単位が決まっていない問題で、面積を「いくつ分」で答えるのに使う。",
    "definition_en": "The unit used to measure area by counting squares whose sides are 1 unit long. It is used to give an area when no specific unit of length, such as inches, is stated.",
    "latex": "A = 6 \\cdot 4 = 24 \\text{ units}^2",
    "spoken_en": "A equals 6 times 4, which is 24 square units",
    "examples": [
      {
        "en": "Each little box on the grid is one square unit, so just count them: this shape covers 12 square units.",
        "ja": "方眼の小さなマス 1 つが 1 平方単位だから、数えればいい。この図形の面積は 12 平方単位だ。",
        "register": "spoken"
      },
      {
        "en": "The base is 6 and the height is 4, so the area of the rectangle is 24 units squared.",
        "ja": "底辺が 6、高さが 4 だから、長方形の面積は 24 平方単位だ。",
        "register": "spoken"
      },
      {
        "en": "Find the area of the triangle with vertices (0, 0), (6, 0), and (0, 4). Give your answer in square units.",
        "ja": "3 点 (0, 0), (6, 0), (0, 4) を頂点とする三角形の面積を求めよ（単位は平方単位）。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "24 square units と 24 units squared は同じ意味。用例コーパスでは話し言葉・書き言葉とも square units のほうが多いが、units squared も使われる。",
      "日本の問題では面積を cm² や m² で答え、「平方単位」とは言わない。記号では units² と書く。",
      "IM は square units を 253 件（Grade 6 1.8 Area of Triangles 16 ほか）、units squared を 28 件使う。CK-12 Geometry は square units を 13 件使う。"
    ],
    "related": [
      "cubic-units",
      "area"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "units squared": 54,
        "square units": 98
      },
      "written": {
        "square units": 28,
        "units squared": 10
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "square",
    "ja": {
      "term": "平方",
      "reading": "へいほう",
      "alt": [
        "2 乗",
        "二乗",
        "2 乗する"
      ]
    },
    "en": {
      "term": "square",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "同じ数を 2 個掛け合わせること、またはその結果。a の平方（a の 2 乗）を a² と書く。",
    "definition_en": "The result of multiplying a number by itself, written with an exponent of 2, as in a². As a verb, to square a number means to multiply it by itself.",
    "latex": "a^2 = a \\times a",
    "spoken_en": "a squared equals a times a",
    "examples": [
      {
        "en": "When you square negative 6, you get positive 36, because a negative times a negative is positive.",
        "ja": "−6 を 2 乗すると 36。負 × 負は正だからね。",
        "register": "spoken"
      },
      {
        "en": "Find the square of 0.3.",
        "ja": "0.3 の 2 乗を求めよ。",
        "register": "written"
      },
      {
        "en": "Square both sides of √x = 9 to find x.",
        "ja": "√x = 9 の両辺を 2 乗して、x を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "square は「正方形」（エントリ square-shape）と同じ語で（1 辺 x の正方形の面積が x²）、平方根も square root と言う。数の 2 乗か、図形の正方形か、根号かは前後の語で見分ける。",
      "x² は x squared と読む。動詞の square は「2 乗する」で、square both sides は「両辺を 2 乗する」。",
      "(−6)² は 36、−6² は −(6²) で −36。口で negative six squared と言うとどちらか分からないので、かっこの有無を確かめる。"
    ],
    "related": [
      "power",
      "exponent",
      "cube",
      "base-of-a-power",
      "square-units",
      "square-shape"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the square of | squares of | square the | square each | square both": 483
      },
      "written": {
        "the square of | squares of | square the | square each | square both": 318
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "squeeze-theorem",
    "ja": {
      "term": "はさみうちの原理",
      "reading": "はさみうちのげんり",
      "alt": [
        "はさみうちの定理"
      ]
    },
    "en": {
      "term": "squeeze theorem",
      "alt": [],
      "uk": null,
      "register": "both",
      "variants": [
        {
          "term": "sandwich theorem",
          "register": "spoken",
          "note": "Khan Academy の言い方。話し言葉はすべて Khan Academy で、書き言葉には出てこない。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "analysis"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "g(x) ≦ f(x) ≦ h(x) で g と h が同じ極限をもつとき、f もその極限をもつという定理。",
    "definition_en": "If g(x) ≤ f(x) ≤ h(x) and g and h have the same limit, then f has that limit too.",
    "latex": "\\lim_{x \\to 0} x^2 \\sin\\frac{1}{x} = 0",
    "spoken_en": "the limit as x goes to zero of x squared sine of one over x is zero",
    "examples": [
      {
        "en": "This one is a squeeze theorem problem — bound it between negative x squared and x squared.",
        "ja": "これははさみうちの問題。−x² と x² ではさんで。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "by the squeeze theorem",
        "ja": "はさみうちの原理より"
      },
      {
        "en": "squeeze it between",
        "ja": "〜ではさむ"
      }
    ],
    "pitfalls": [
      "AP の CED（topic 1.8）と OpenStax Calculus は squeeze theorem と書き、sandwich theorem は出てこない。sandwich theorem は Khan Academy の講義で聞く。"
    ],
    "related": [
      "squeeze"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "はさみうちの原理",
        "en": "Squeeze theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.8（squeeze theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "squeeze theorem": 15,
        "sandwich theorem": 7
      },
      "written": {
        "squeeze theorem": 13
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "squeeze",
    "ja": {
      "term": "はさむ",
      "reading": "はさむ",
      "alt": []
    },
    "en": {
      "term": "squeeze",
      "alt": [],
      "variants": [
        {
          "term": "sandwich",
          "register": "spoken",
          "note": "Khan Academy の言い方。sandwich … between の形で数えて、話し言葉はすべて Khan Academy で、書き言葉には出てこない。"
        }
      ],
      "uk": null
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "英語に決まった動詞はない。ふつうは定理の名前 squeeze theorem で言う（用例コーパスでも squeeze theorem は話し言葉・書き言葉とも出てくるが、はさむ意味の squeeze … between ／ sandwich … between は話し言葉にわずかしかない）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "求めたい式を、同じ極限をもつ 2 つの式の間にはさんで、極限を求める（はさみうちの原理を使う）。",
    "definition_en": "To trap an expression between two others with the same limit, so that its limit is forced (the squeeze theorem).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We can squeeze x² sin(1/x) between −x² and x².",
        "ja": "x² sin(1/x) は −x² と x² ではさめる。",
        "register": "spoken"
      },
      {
        "en": "Squeeze the sequence between 0 and 1/n to show that it converges to 0.",
        "ja": "数列を 0 と 1/n ではさんで、0 に収束することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの squeeze はほとんどが名詞 squeeze theorem（はさみうちの原理）の squeeze か、「詰め込む」の意味。はさむ意味の squeeze … between も sandwich … between（Khan Academy）も、話し言葉にわずかに出てくるだけ。"
    ],
    "related": [
      "squeeze-theorem",
      "comparison-for-divergence",
      "limit"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.8（squeeze theorem）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sandwich … between": 4,
        "squeeze … between": 1
      },
      "sources": [
        "khan-ap-calc",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-24。docs/DECISIONS.md「Phase 2 代数 2・Precalculus の単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-24"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sss-congruence",
    "ja": {
      "term": "3 組の辺がそれぞれ等しい",
      "reading": "さんくみのへんがそれぞれひとしい",
      "alt": [
        "三辺相等",
        "SSS"
      ]
    },
    "en": {
      "term": "side-side-side",
      "alt": [
        "SSS",
        "side-side-side triangle congruence theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "proof"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "三角形の合同条件の 1 つ。2 つの三角形で 3 組の辺がそれぞれ等しければ、その 2 つの三角形は合同である。",
    "definition_en": "A triangle congruence criterion: if the three sides of one triangle are congruent to the three sides of another, the two triangles are congruent.",
    "latex": "AB = DE,\\ BC = EF,\\ CA = FD \\Rightarrow \\triangle ABC \\cong \\triangle DEF",
    "spoken_en": "if AB equals DE, BC equals EF, and CA equals FD, then triangle ABC is congruent to triangle DEF",
    "examples": [
      {
        "en": "All three pairs of sides match up, so the triangles are congruent by side-side-side, SSS.",
        "ja": "3 組の辺がそれぞれ等しいから、この 2 つの三角形は合同。",
        "register": "spoken"
      },
      {
        "en": "In quadrilateral ABCD, AB = CB and AD = CD. Prove that △ABD ≅ △CBD.",
        "ja": "四角形 ABCD で AB = CB、AD = CD である。△ABD ≡ △CBD を証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "CK-12 Geometry の証明では、理由を by SSS のように略号で書く。日本語では学習指導要領解説の言い方どおり「3 組の辺がそれぞれ等しい」と条件を文で言う。",
      "相似条件の SSS similarity（3 組の辺の比がすべて等しい）とは別。略号だけでは区別できないので、congruence か similarity かを添える。",
      "CK-12 Geometry は by SSS などの略号の形が 11 件、side-side-side が 3 件。",
      "IM Geometry はこの合同条件を Side-Side-Side Triangle Congruence Theorem と呼び（Geometry 2.9 ほか、10 件）、CK-12 Geometry は SSS Postulate（4.13 SSS）と呼ぶ。"
    ],
    "related": [
      "sas-congruence",
      "asa-congruence",
      "aas-congruence",
      "hl-congruence",
      "congruent",
      "cpctc",
      "conditions-that-determine-a-triangle"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "2.9 Side-Side-Side Triangle Congruence ほか（16 件）"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 2 学年 B 図形: 対応する３組の辺がそれぞれ等しい"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.13 SSS ほか（by SSS 11 件・side-side-side 3 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "by SSS | SSS congruence | (SSS) congruence | SSS postulate | SSS criterion | SSS theorem | SSS triangle congruence": 1
      },
      "written": {
        "side-side-side": 1
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 side-side-side（Geometry 2.9 Side-Side-Side Triangle Congruence・Geometry 2.10 Practicing Proofs・Geometry 3.10 Other Conditions for Triangle Similarity・計 16 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "stacked-bar-chart",
    "ja": {
      "term": "帯グラフ",
      "reading": "おびぐらふ",
      "alt": []
    },
    "en": {
      "term": "segmented bar graph",
      "alt": [
        "stacked bar graph",
        "stacked bar chart"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Pre-Algebra",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "長方形の帯を全体とし、各項目の割合を帯の区切りの長さで表したグラフ。",
    "definition_en": "A single bar representing the whole, divided into segments whose lengths show the share of each category.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each bar goes up to 100 percent, and the segments show the proportions.",
        "ja": "どの帯も 100% まであって、区切りが割合を表す。",
        "register": "spoken"
      },
      {
        "en": "Make a segmented bar graph comparing the responses of the two groups.",
        "ja": "2 つのグループの回答を比べる帯グラフをかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは stacked bar graph が書き言葉にわずかに出てくるだけで、segmented bar graph は出てこない。IM は Grade 8 の glossary の見出しが segmented bar graph（Grade 8・Algebra 1 のレッスンに 25 件、stacked bar graph は 3 件）で、見出しは IM の言い方にした。"
    ],
    "related": [
      "pie-chart",
      "bar-chart"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "Grade 8 glossary・Grade 8 6.10 Using Data Displays to Find Associations・Grade 8 6.9 Looking for Associations"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "stacked bar graph": 1
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは IM の呼び方 segmented bar graph（Grade 8 glossary・Grade 8 6.10 Using Data Displays to Find Associations・Grade 8 6.9 Looking for Associations・計 25 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-deviation",
    "ja": {
      "term": "標準偏差",
      "reading": "ひょうじゅんへんさ",
      "alt": []
    },
    "en": {
      "term": "standard deviation",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "AP Statistics",
        "Integrated Math 1",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "分散の 0 以上の平方根。データの散らばりを、もとのデータと同じ単位で表す。",
    "definition_en": "The nonnegative square root of the variance. It describes how spread out the data are, in the original units of the data.",
    "latex": "s = \\sqrt{\\frac{1}{n}\\sum_{i=1}^{n}(x_i - \\bar{x})^2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "A small standard deviation means the data are clustered close to the mean.",
        "ja": "標準偏差が小さいとは、データが平均の近くに集まっているということ。",
        "register": "spoken"
      },
      {
        "en": "Find the mean and standard deviation of the data. Round to one decimal place.",
        "ja": "データの平均値と標準偏差を求め、小数第 1 位まで答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics と AP Statistics の CED（topic 1.7）の sample standard deviation（s）は n − 1 で割った分散の平方根。日本語版 Wikipedia「標準偏差」は、n で割った標本分散の平方根を標本標準偏差と呼び、不偏分散の平方根と分けて書く（同じ「標本」でも式が違う。エントリ variance）。",
      "「標準偏差が大きい」は OpenStax Introductory Statistics では a larger standard deviation ／ the standard deviation is small のように言い、「散らばりが大きい」は more spread out（IM・OpenStax）。"
    ],
    "related": [
      "variance",
      "deviation",
      "spread"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "標準偏差",
        "en": "Standard deviation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.7（sample standard deviation）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 1.12 Standard Deviation・1.13 More Standard Deviation（more spread out）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "standard deviation": 764
      },
      "written": {
        "standard deviation": 1016
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-error",
    "ja": {
      "term": "標準誤差",
      "reading": "ひょうじゅんごさ",
      "alt": []
    },
    "en": {
      "term": "standard error",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "標本から計算した統計量（標本平均など）の標準偏差。標本平均では σ/√n、σ がわからないときは s/√n で見積もる。",
    "definition_en": "The standard deviation of a statistic computed from a sample, such as the sample mean: σ/√n, estimated by s/√n when σ is unknown.",
    "latex": "\\frac{s}{\\sqrt{n}}",
    "spoken_en": "s over the square root of n",
    "examples": [
      {
        "en": "The standard error tells you how much the sample mean bounces around from sample to sample.",
        "ja": "標準誤差は、標本平均が標本ごとにどれだけぶれるかを表す。",
        "register": "spoken"
      },
      {
        "en": "Compute the standard error of the mean for a sample of size 49 with s = 14.",
        "ja": "大きさ 49、s = 14 の標本について、平均の標準誤差を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "standard deviation（データの散らばり）と standard error（統計量のぶれ）は別。",
      "標本平均の標準誤差は standard error of the mean（用例コーパスにも出てくる）。"
    ],
    "related": [
      "sampling-distribution-of-the-mean",
      "standard-deviation",
      "confidence-interval"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "標準誤差",
        "en": "Standard error"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "standard error": 30
      },
      "written": {
        "standard error": 30
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-form-of-a-conic",
    "ja": {
      "term": "二次曲線の標準形",
      "reading": "にじきょくせんのひょうじゅんけい",
      "alt": []
    },
    "en": {
      "term": "standard form",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "中心や頂点を原点（平行移動したものは (h, k)）に置いたときの、二次曲線の方程式の形。",
    "definition_en": "The form of a conic's equation with its center or vertex at the origin, or at (h, k) after a shift.",
    "latex": "\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1",
    "spoken_en": "x minus h squared over a squared plus y minus k squared over b squared equals one",
    "examples": [
      {
        "en": "Complete the square in x and in y to get it into standard form.",
        "ja": "x と y それぞれで平方完成して、標準形に直そう。",
        "register": "spoken"
      },
      {
        "en": "Write the equation x² + 4y² − 6x + 8y = 3 in standard form and identify the conic.",
        "ja": "方程式 x² + 4y² − 6x + 8y = 3 を標準形に直し、どの二次曲線か答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "standard form は直線（Ax + By = C）や 2 次関数にも使う語。書き言葉では OpenStax Intermediate Algebra と OpenStax Algebra and Trigonometry が中心で、話し言葉にはほとんど出てこない。",
      "平行移動した二次曲線は、中心 (h, k) を入れた標準形で表す。"
    ],
    "related": [
      "conic-section",
      "ellipse",
      "completing-the-square"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "standard form of the equation of": 2
      },
      "written": {
        "standard form of the equation of": 48
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-form-of-a-line",
    "ja": {
      "term": "直線の標準形",
      "reading": "ちょくせんのひょうじゅんけい",
      "alt": []
    },
    "en": {
      "term": "standard form of a linear equation",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「直線の標準形」は学習指導要領解説に無い（日本語版 Wikipedia「一次関数」は「平面における直線の標準形」の記事に触れる）。日本の数II では直線の方程式の一般形を ax + by + c = 0（右辺が 0）と書くので、英語の standard form（定数が右辺）と形が違う。英語の standard form は二次関数の ax² + bx + c や英国の科学的記数法など別の意味でも使うので、「直線の」「一次方程式の」を付けて区別する。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直線の式を Ax + By = C の形（x と y の項を左辺、定数を右辺）に書いたもの。ふつう A、B、C を整数にし、A ≧ 0 にそろえる。",
    "definition_en": "The form Ax + By = C of the equation of a line, with the x- and y-terms on the left and the constant on the right. Usually A, B, and C are integers and A is not negative.",
    "latex": "Ax + By = C",
    "spoken_en": "A x plus B y equals C",
    "examples": [
      {
        "en": "Let's put y = (2/3)x + 4 in standard form of a linear equation: multiply everything by 3, move the x-term over, and you get 2x − 3y = −12.",
        "ja": "y = (2/3)x + 4 を直線の標準形にしよう。全体に 3 をかけて x の項を移すと、2x − 3y = −12 になる。",
        "register": "spoken"
      },
      {
        "en": "Write the equation y = −3x + 5 in standard form of a linear equation, Ax + By = C.",
        "ja": "式 y = −3x + 5 を、直線の標準形 Ax + By = C の形に書き直せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数II の一般形 ax + by + c = 0 から書き直すと、定数の符号が変わる。2x − 3y + 12 = 0 は standard form では 2x − 3y = −12。",
      "standard form だけでは別の意味にもなる。二次関数の ax² + bx + c の形（日本の「一般形」、エントリ standard-form）や、英国での科学的記数法（エントリ scientific-notation）にも使うので、「直線の」「一次方程式の」を付けて区別する。",
      "OpenStax Elementary Algebra は A、B、C を整数にし A ≧ 0 とすることが多い。分数の係数は払い、x の係数が負なら全体に −1 をかける。"
    ],
    "related": [
      "slope-intercept-form-of-a-line",
      "point-slope-form",
      "linear-equation-in-two-variables",
      "equation-of-a-line",
      "x-intercept"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Scientific_notation",
        "note": "記事「Scientific notation」の冒頭と節 Normalized notation: 英国では standard form と呼ぶ"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "standard form of linear equation | standard form of a linear equation | linear equation in standard form | line in standard form | standard form of a line | standard form of the line | standard form of the equation of a line | equation of a line in standard form": 4
      },
      "written": {
        "standard form of linear equation | standard form of a linear equation | linear equation in standard form | line in standard form | standard form of a line | standard form of the line | standard form of the equation of a line | equation of a line in standard form": 8
      },
      "sources": [
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは OpenStax の呼び方 standard form of linear equation | standard form of a linear equation | linear equation in standard form | line in standard form | standard form of a line | standard form of the line | standard form of the equation of a line | equation of a line in standard form（本文 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-form",
    "ja": {
      "term": "一般形",
      "reading": "いっぱんけい",
      "alt": []
    },
    "en": {
      "term": "general form",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しは OpenStax の呼び方（general form of a quadratic などの形で本文 7 件。OpenStax Algebra and Trigonometry 5・OpenStax Elementary Algebra 1・OpenStax Intermediate Algebra 1）。用例コーパスの話し言葉には出てこない。standard form は両方の形に使われる: OpenStax Algebra and Trigonometry は二次関数では ax² + bx + c を general form、a(x − h)² + k を standard form（vertex form）と呼ぶが、同じ本の二次方程式の節と IM Algebra 1（用語集 standard form (of a quadratic expression)）は ax² + bx + c を standard form と呼ぶ。用例コーパスの The Organic Chemistry Tutor なども ax² + bx + c を standard form と呼ぶ。どちらを指すかは y = ax² + bx + c と式を添えて示すのが確実。見出しの「一般形」は学習指導要領解説に無い（日本語版 Wikipedia「二次関数」にはある）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "二次関数を y = ax² + bx + c（a ≠ 0）と展開した形。",
    "definition_en": "A quadratic function written in expanded form, y = ax² + bx + c with a ≠ 0.",
    "latex": "y = ax^2 + bx + c \\quad (a \\neq 0)",
    "spoken_en": null,
    "examples": [
      {
        "en": "This quadratic is in general form, so a is 2, b is −4, and c is −1.",
        "ja": "この二次関数は一般形なので、a = 2, b = −4, c = −1 だ。",
        "register": "spoken"
      },
      {
        "en": "Write the quadratic function f(x) = 2(x − 1)² − 3 in general form.",
        "ja": "二次関数 f(x) = 2(x − 1)² − 3 を一般形で表せ。",
        "register": "written"
      },
      {
        "en": "For a quadratic in general form, the axis of symmetry is x = −b/(2a).",
        "ja": "一般形の二次関数では、軸は x = −b/(2a) である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の「一般形」を standard form と訳すと、日本の「標準形」a(x − h)² + k の意味に取られることがある（OpenStax Algebra and Trigonometry の standard form はこちら）。日本語の 2 つの名前と英語の standard form の使い方は入れ替わりうる。"
    ],
    "related": [
      "vertex-form",
      "factored-form",
      "completing-the-square",
      "quadratic-formula",
      "axis-of-symmetry"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「二次関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BA%8C%E6%AC%A1%E9%96%A2%E6%95%B0",
        "note": "一般形 f(x) = ax² + bx + c（standard form）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "general form of a quadratic（1 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "general form of a quadratic（1 件）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 1)",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "用語集: standard form (of a quadratic expression) は ax² + bx + c"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 二次関数: 「一般形」は 0 件（見出しの日本語は解説に出てこない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "general form of a quadratic | quadratic function in general form | general form of the quadratic": 7
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 general form of a quadratic | quadratic function in general form | general form of the quadratic（本文 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-matrix",
    "ja": {
      "term": "標準行列",
      "reading": "ひょうじゅんぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "standard matrix",
      "alt": [
        "matrix of the transformation"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「標準行列」は学習指導要領解説に無い、本プロジェクトの訳語（日本語は「線形写像の表現行列」と言うことが多い）。 米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（standard matrix ／ matrix of the transformation は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "ℝⁿ から ℝᵐ への線形写像 T を T(x) = Ax と書くときの行列 A。標準基底の像を列に並べたもの。",
    "definition_en": "The matrix A with T(x) = Ax for a linear map T from ℝⁿ to ℝᵐ; its columns are the images of the standard basis vectors.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To get the standard matrix, just see where e1 and e2 go and write those as columns.",
        "ja": "標準行列を作るには、e1 と e2 の行き先を列に書けばいい。",
        "register": "spoken"
      },
      {
        "en": "Find the standard matrix of the linear transformation that reflects across the line y = x.",
        "ja": "直線 y = x に関する対称移動の標準行列を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスには出てこない。3Blue1Brown は同じことを「i hat と j hat の行き先を記録した行列」と説明する。"
    ],
    "related": [
      "linear-transformation",
      "matrix",
      "standard-unit-vectors"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "9.2 Operators and Similarity"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 standard matrix（9.2 Operators and Similarity・8.8 An Application to Linear Codes over Finite Fields）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-normal-distribution",
    "ja": {
      "term": "標準正規分布",
      "reading": "ひょうじゅんせいきぶんぷ",
      "alt": []
    },
    "en": {
      "term": "standard normal distribution",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3",
        "Algebra 2"
      ]
    },
    "definition_ja": "平均 0、標準偏差 1 の正規分布 N(0, 1)。",
    "definition_en": "The normal distribution N(0, 1), centered at 0 with a standard deviation of 1.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Once you standardize, you're working with the standard normal, mean zero and standard deviation one.",
        "ja": "標準化すれば、平均 0、標準偏差 1 の標準正規分布で考えられる。",
        "register": "spoken"
      },
      {
        "en": "Use the standard normal distribution to find P(−1 < Z < 2).",
        "ja": "標準正規分布を用いて P(−1 < Z < 2) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "標準正規分布に従う確率変数は Z と書く。"
    ],
    "related": [
      "normal-distribution",
      "z-score",
      "standard-normal-table",
      "standardize",
      "empirical-rule"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "6.1 The Standard Normal Distribution（Z ~ N(0, 1)）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Normal_distribution",
        "note": "節 Standard normal distribution"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.11（standard normal distribution）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B (2) 統計的な推測（Z は標準正規分布 N(0, 1) に従う）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "standard normal distribution": 5
      },
      "written": {
        "standard normal distribution": 35
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.01",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-normal-table",
    "ja": {
      "term": "正規分布表",
      "reading": "せいきぶんぷひょう",
      "alt": []
    },
    "en": {
      "term": "standard normal table",
      "alt": [
        "standard normal probability table"
      ],
      "variants": [
        {
          "term": "z-table",
          "register": "spoken",
          "note": "話し言葉ではこちらだけが出てくるが、すべて Khan Academy の AP Statistics。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "センター試験（数学II・数学B）の問題に付く正規分布表は P(0 ≤ Z ≤ z₀) を載せる。OpenStax Introductory Statistics の表は z の左側の面積（area to the left of the z-score）を載せ、Khan Academy も使う表の種類に注意するよう言う。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3",
        "Algebra 2"
      ]
    },
    "definition_ja": "標準正規分布について、z の値とそれに対応する確率（面積）をまとめた表。",
    "definition_en": "A table that gives, for values of z, the corresponding probabilities (areas) under the standard normal distribution.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Look up 1.96 in the z-table and you'll get about 0.975.",
        "ja": "z-table で 1.96 を引くと、約 0.975 になる。",
        "register": "spoken"
      },
      {
        "en": "Use the standard normal table to find the area to the left of z = −1.25.",
        "ja": "正規分布表を使って、z = −1.25 より左の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉の z-table はすべて Khan Academy。書き言葉には normal table・standard normal table がわずかに出てくるだけ。OpenStax Introductory Statistics は確率を電卓の normalcdf で求める。AP Statistics の CED（topic 2.11）は standard normal table と書く。"
    ],
    "related": [
      "standard-normal-distribution",
      "z-score",
      "normal-distribution",
      "empirical-rule"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "6.2 Using the Normal Distribution・8.1・8.3（standard normal probability table、normalcdf）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成28年度 大学入試センター試験 本試験 数学Ⅱ・数学Ｂ（問題）",
        "url": "https://web.archive.org/web/20170601/https://www.dnc.ac.jp/albums/abm.php?f=abm00007097.pdf&n=2801-0502+suugaku2B.pdf",
        "note": "数学II・数学B の正規分布表（標準正規分布の P(0 ≤ Z ≤ z₀)）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.11（standard normal table）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 7.15（Using Tables for Normal Distribution Areas）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "z-table": 51
      },
      "written": {
        "standard normal table": 1,
        "standard normal probability table": 3
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-score",
    "ja": {
      "term": "偏差値",
      "reading": "へんさち",
      "alt": []
    },
    "en": {
      "term": "T-score",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "米国の学校では使わない（OpenStax Introductory Statistics・AP Statistics の CED に、平均 50・標準偏差 10 への換算は出てこない）。説明するなら a standardized score with mean 50 and standard deviation 10（standardized score は AP Statistics の CED topic 1.9 と OpenStax Introductory Statistics の言い方）。日本語版 Wikipedia「偏差値」は英語名を T-score とし、英語版 Wikipedia「Standard score」の節 T-score も、教育測定で平均 50・標準偏差 10 に直した標準得点を T-score と呼ぶ。近い考え方は z-score と percentile（パーセンタイル）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "得点を、平均 50、標準偏差 10 になるように換算した値。50 + 10z。",
    "definition_en": "A score rescaled to have mean 50 and standard deviation 10, that is, 50 + 10z.",
    "latex": "50 + 10z",
    "spoken_en": null,
    "examples": [
      {
        "en": "Your T-score is basically your z-score times 10, plus 50.",
        "ja": "偏差値は、だいたい z スコアを 10 倍して 50 を足したもの。",
        "register": "spoken"
      },
      {
        "en": "A student's z-score is 1.5. Find the corresponding T-score, 50 + 10z.",
        "ja": "ある生徒の z スコアは 1.5 である。対応する偏差値 50 + 10z を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の standard score は z スコアのことで、偏差値の訳には使えない。",
      "t 統計量（t-statistic）とは別物。統計の授業の t-score は t 分布の値（OpenStax Introductory Statistics）で、用例コーパスの t-score もすべてその意味。"
    ],
    "related": [
      "z-score",
      "standardize"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「偏差値」",
        "url": "https://ja.wikipedia.org/wiki/%E5%81%8F%E5%B7%AE%E5%80%A4",
        "note": "平均 50、標準偏差 10 に変換した値。英語名 T-score"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Standard_score#T-score",
        "note": "記事「Standard score」の節 T-score（太字で定義。教育測定で平均 50・標準偏差 10 に直した標準得点。en.term T-score はこの節へのリダイレクト。数学カテゴリから 5 段で、記事名で見出しを決める規則の 4 段の外）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.9（standardized score）。平均 50・標準偏差 10 の換算は無い"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "standardized score（1 件）。t-score は t 分布の値"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-26。docs/DECISIONS.md「Phase 5 監査（セッション 2）」2: en.term を T-score、mapping near）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "standard-unit-vectors",
    "ja": {
      "term": "基本ベクトル",
      "reading": "きほんべくとる",
      "alt": [
        "単位ベクトル i, j"
      ]
    },
    "en": {
      "term": "standard unit vectors",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "座標軸の正の向きの単位ベクトル。平面では e₁ = (1, 0), e₂ = (0, 1)、空間では e₃ = (0, 0, 1) も加える。",
    "definition_en": "The unit vectors in the positive directions of the coordinate axes: i = ⟨1, 0⟩ and j = ⟨0, 1⟩ in the plane, plus k = ⟨0, 0, 1⟩ in space.",
    "latex": "\\vec{a} = a_1\\vec{e_1} + a_2\\vec{e_2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Any vector in the plane is some multiple of i plus some multiple of j.",
        "ja": "平面のベクトルは、i の何倍かと j の何倍かの和で表せる。",
        "register": "spoken"
      },
      {
        "en": "Write the vector from (2, −1) to (5, 3) in terms of the standard unit vectors i and j.",
        "ja": "点 (2, −1) から (5, 3) へのベクトルを基本ベクトル i, j を用いて表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の教科書は e₁, e₂（空間では e₃）、米国の教科書は i, j, k と書く（OpenStax Calculus Volume 3）。"
    ],
    "related": [
      "vector",
      "component-form"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "standard unit vectors": 35
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "standardize",
    "ja": {
      "term": "標準化する",
      "reading": "ひょうじゅんかする",
      "alt": [
        "標準化"
      ]
    },
    "en": {
      "term": "standardize",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語は名詞「標準化」と動詞「標準化する」の両方を使うが、英語の授業ではおもに動詞 standardize で言う（名詞 standardization は用例コーパスの話し言葉では少なく、すべて Professor Leonard）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "確率変数やデータの値 x を、平均を引いて標準偏差で割り、z = (x − μ)/σ に直す。",
    "definition_en": "To convert a value x of a random variable or of data to z = (x − μ)/σ by subtracting the mean and dividing by the standard deviation.",
    "latex": "z = \\frac{x - \\mu}{\\sigma}",
    "spoken_en": "z equals x minus mu over sigma",
    "examples": [
      {
        "en": "Standardize first, then look the z-score up in the table.",
        "ja": "まず標準化して、z スコアを表で引く。",
        "register": "spoken"
      },
      {
        "en": "Standardize X = 68 if X ~ N(60, 4).",
        "ja": "X ~ N(60, 4) のとき X = 68 を標準化せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "standardized test（統一テスト）の standardized は別の意味。用例コーパスの standardize の中にも混ざる。"
    ],
    "related": [
      "z-score",
      "standard-normal-distribution",
      "normal-distribution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "standardize": 2
      },
      "written": {
        "standardize": 12
      },
      "sources": [
        "khan-middle",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "statistic",
    "ja": {
      "term": "統計量",
      "reading": "とうけいりょう",
      "alt": [
        "標本統計量"
      ]
    },
    "en": {
      "term": "statistic",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "標本のデータから計算した値。標本平均・標本比率など。母集団の値（母数）を推定するのに使う。",
    "definition_en": "A number computed from sample data, such as a sample mean or sample proportion; it is used to estimate a population parameter.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A statistic comes from the sample; a parameter describes the population.",
        "ja": "統計量は標本から、母数は母集団を表す。",
        "register": "spoken"
      },
      {
        "en": "Identify the parameter and the statistic in the study.",
        "ja": "この調査の母数と統計量を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "statistic（統計量）の複数形 statistics は、分野の statistics（統計学）と同じつづり。",
      "英語は parameter（母数）と statistic を対にして覚える（p と s の頭文字どうし）。",
      "母集団の値（母数）は英語で parameter。媒介変数の parameter（parameter エントリ）と同じ語。"
    ],
    "related": [
      "sample",
      "population",
      "test-statistic",
      "statistics"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "統計量",
        "en": "Statistic"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "a statistic | sample statistic | statistics and parameters | parameter and a statistic | summary statistics": 56
      },
      "written": {
        "a statistic | sample statistic | statistics and parameters | parameter and a statistic | summary statistics": 43
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "statistical-graph",
    "ja": {
      "term": "統計グラフ",
      "reading": "とうけいぐらふ",
      "alt": []
    },
    "en": {
      "term": "statistical graph",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Pre-Algebra",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "データの特徴がわかるように表したグラフの総称。棒グラフ・折れ線グラフ・円グラフ・帯グラフ・ヒストグラムなど。",
    "definition_en": "Any graph that displays data to show its features, such as a bar graph, a line graph, a pie chart, or a histogram.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Pick the kind of graph that fits the data: a bar graph for categories, a line graph for change over time.",
        "ja": "データに合ったグラフを選ぼう。カテゴリには棒グラフ、時間による変化には折れ線グラフ。",
        "register": "spoken"
      },
      {
        "en": "Choose an appropriate statistical graph to display the data and explain your choice.",
        "ja": "データを表すのに適した統計グラフを選び、選んだ理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語では総称より、個々のグラフの名前（bar graph、line graph、pie chart など）で呼ぶことが多い。用例コーパスの statistical graph は書き言葉にわずかに出てくるだけ。"
    ],
    "related": [
      "bar-chart",
      "line-graph",
      "pie-chart",
      "histogram",
      "stacked-bar-chart"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "statistical graph": 1
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは OpenStax の呼び方 statistical graph（本文 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "statistical-hypothesis",
    "ja": {
      "term": "仮説",
      "reading": "かせつ",
      "alt": []
    },
    "en": {
      "term": "hypothesis",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "母集団について立てる主張で、標本のデータを使って検定する対象になるもの。仮説検定では帰無仮説と対立仮説の 2 つを立てる。",
    "definition_en": "A claim about a population, usually about a parameter, that is tested using sample data. A hypothesis test sets up two of them: the null hypothesis and the alternative hypothesis.",
    "latex": "H_0: p = 0.5 \\quad H_a: p > 0.5",
    "spoken_en": "H sub zero: p equals 0.5; H sub a: p is greater than 0.5",
    "examples": [
      {
        "en": "Before you look at the data, write down both hypotheses in terms of the parameter, not the sample statistic.",
        "ja": "データを見る前に、2 つの仮説を標本の統計量ではなく母数を使って書こう。",
        "register": "spoken"
      },
      {
        "en": "A coin is tossed 100 times and lands heads up 61 times. State the hypotheses for testing whether the coin is more likely to land heads up.",
        "ja": "硬貨を 100 回投げたところ、表が 61 回出た。この硬貨は表が出やすいといえるかを検定するための仮説を立てよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "証明の「仮定」（p ならば q の p）も英語では hypothesis で、別の見出し hypothesis。統計の検定の話か、証明の話かで見分ける。用例コーパスの hypothesis 単独は大半がこの統計の意味（MIT OCW・講義ノートには帰納法や定理の仮定の意味もある）。",
      "複数形は hypotheses。仮説は H₀ ／ Hₐ のように母数（μ、p など）で書き、標本平均などの値では書かない。",
      "用例コーパスでは話し言葉・書き言葉ともよく出てくる。話し言葉は Khan Academy が中心、書き言葉は大半が OpenStax Introductory Statistics。AP Statistics の CED では hypothesis が 195 件、hypotheses が 40 件。"
    ],
    "related": [
      "null-hypothesis",
      "alternative-hypothesis",
      "hypothesis-testing",
      "hypothesis"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Unit 3・4（hypothesis 195 件・hypotheses 40 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "hypothesis": 541
      },
      "written": {
        "hypothesis": 754
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "statistical-power",
    "ja": {
      "term": "検出力",
      "reading": "けんしゅつりょく",
      "alt": []
    },
    "en": {
      "term": "power of the test",
      "alt": [
        "power of a test",
        "statistical power"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "対立仮説が正しいときに、帰無仮説を正しく棄却する確率。1 から第二種の過誤の確率を引いた値。",
    "definition_en": "The probability that a test correctly rejects the null hypothesis when the alternative is true; 1 minus the probability of a Type II error.",
    "latex": "\\text{power} = 1 - \\beta",
    "spoken_en": null,
    "examples": [
      {
        "en": "A bigger sample gives the test more power to detect a real difference.",
        "ja": "標本を大きくすると、本当の差を見つける検出力が上がる。",
        "register": "spoken"
      },
      {
        "en": "Explain how increasing the significance level affects the power of the test.",
        "ja": "有意水準を大きくすると検出力がどう変わるか説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は power of the test を単に power とも書く（a high power）。累乗の power と同じ語。"
    ],
    "related": [
      "type-ii-error",
      "type-i-error",
      "significance-level",
      "hypothesis-testing"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "power of the test": 2
      },
      "written": {
        "power of the test": 5,
        "power of a test": 2
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 power of the test（本文 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "statistical-variable",
    "ja": {
      "term": "変量",
      "reading": "へんりょう",
      "alt": []
    },
    "en": {
      "term": "variable",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "高等学校学習指導要領解説（数学I データの分析）はデータの項目（身長・点数など）を変量と呼び（共通テストも「英語の得点を変量 x」のように書く）、式の文字の「変数」とは言葉を分ける。英語はどちらも variable と言い、AP Statistics の CED は quantitative variable（52 件）・categorical variable（50 件）と種類を付けて言う。用例コーパスの variable の件数は、式の変数の意味も合わせたもの。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "調査の対象ごとに値が決まる特性（身長・テストの点数など）のこと。データはこの値を対象の数だけ集めたものである。",
    "definition_en": "A characteristic that can take different values for different individuals in a data set, such as height or a test score. A data set records the values of one or more variables for each individual.",
    "examples": [
      {
        "en": "Before you pick a graph, decide whether each variable is categorical or quantitative.",
        "ja": "グラフを選ぶ前に、それぞれの変量が質的か量的かを決めよう。",
        "register": "spoken"
      },
      {
        "en": "A survey records the age, height, and favorite sport of 50 students. Which variables are quantitative?",
        "ja": "50 人の生徒について、年齢・身長・好きなスポーツを調べた。これらの変量のうち、量的なものはどれか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の variable は、式の文字の「変数」と統計の「変量」の両方を指す。どちらの意味かは文脈と、quantitative ／ categorical のような前に付く語で読み分ける。",
      "確率変数（random variable）とは別の語。変量はデータの項目で、確率変数は試行の結果に数を対応させるもの。"
    ],
    "related": [
      "quantitative-variable",
      "categorical-variable",
      "random-variable",
      "data"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I データの分析: 変量（14 件）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "数学I データの分析: 変量 x（31 件）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Unit 1 ほか（variable 354 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "variable": 2388
      },
      "written": {
        "variable": 3499
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "statistically-significant",
    "ja": {
      "term": "有意",
      "reading": "ゆうい",
      "alt": [
        "有意である",
        "統計的に有意"
      ]
    },
    "en": {
      "term": "statistically significant",
      "alt": [],
      "uk": null
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学",
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "観測した結果が偶然だけでは起こりにくい（p 値が有意水準より小さい）こと。",
    "definition_en": "Of a result: unlikely to have happened by chance alone, so that the p-value is below the significance level.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The difference is statistically significant, so it's probably not just chance.",
        "ja": "差は統計的に有意なので、たぶん偶然ではない。",
        "register": "spoken"
      },
      {
        "en": "Is the result statistically significant at the 1% level? Explain.",
        "ja": "結果は有意水準 1% で有意か。説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「有意」は重要という意味ではない。統計的に有意でも、差が小さく実用上は意味がないこともある。"
    ],
    "related": [
      "significance-level",
      "p-value",
      "hypothesis-testing"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "有意",
        "en": "Statistical significance"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "statistically significant": 3
      },
      "written": {
        "statistically significant": 4
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは AP Statistics の CED の呼び方 statistically significant（topic 3.7）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "statistics",
    "ja": {
      "term": "統計学",
      "reading": "とうけいがく",
      "alt": []
    },
    "en": {
      "term": "statistics",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "データを集め、整理し、分析して、不確かさを考えながら結論を引き出す方法を研究する分野。",
    "definition_en": "The science of collecting, organizing and analyzing data and drawing conclusions from it while accounting for uncertainty.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In statistics we never prove anything for sure; we measure how unlikely something is.",
        "ja": "統計学では何かを確実に証明することはない。どれくらい起こりにくいかを測るのだ。",
        "register": "spoken"
      },
      {
        "en": "Explain the difference between descriptive statistics and inferential statistics.",
        "ja": "記述統計と推測統計の違いを説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の statistics は分野（単数扱い）と statistic（統計量）の複数形の両方。",
      "AP Statistics・Intro Statistics のように科目名にも使う。"
    ],
    "related": [
      "statistic",
      "descriptive-statistics",
      "inferential-statistics",
      "data"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "統計学",
        "en": "Statistics"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "in statistics | field of statistics | study of statistics | statistics is the | statistics class | statistics course | statistics students": 45
      },
      "written": {
        "in statistics | field of statistics | study of statistics | statistics is the | statistics class | statistics course | statistics students": 81
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "stem-and-leaf-plot",
    "ja": {
      "term": "幹葉図",
      "reading": "みきはず",
      "alt": [
        "幹葉表示"
      ]
    },
    "en": {
      "term": "stem-and-leaf plot",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "stemplot",
          "register": "written",
          "note": "話し言葉には出てこない。書き言葉はすべて OpenStax Introductory Statistics（stem plot を含む）。"
        },
        {
          "term": "stem-and-leaf graph",
          "register": "written",
          "note": "書き言葉に少しだけ出てくる（すべて OpenStax Introductory Statistics）。"
        }
      ],
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Intro Statistics",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データの上の位を幹、下の位を葉として並べ、分布の形ともとの値を同時に示す表。",
    "definition_en": "A display that splits each value into a stem (the leading digits) and a leaf (the last digit), showing the shape of the distribution while keeping the data values.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a stem-and-leaf plot, 7 | 3 means 73.",
        "ja": "幹葉図では 7 | 3 は 73 のこと。",
        "register": "spoken"
      },
      {
        "en": "Make a stemplot of the test scores and describe the shape of the distribution.",
        "ja": "テストの得点の幹葉図をかき、分布の形を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉は stem-and-leaf plot（すべて Khan Academy）、書き言葉は stemplot と stem-and-leaf plot が同じくらいで並ぶ。"
    ],
    "related": [
      "histogram",
      "distribution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "幹葉表示",
        "en": "Stem-and-leaf display"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "stem-and-leaf plot": 10
      },
      "written": {
        "stem-and-leaf plot": 9,
        "stemplot": 14,
        "stem-and-leaf graph": 3
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "step-function",
    "ja": {
      "term": "階段関数",
      "reading": "かいだんかんすう",
      "alt": []
    },
    "en": {
      "term": "step function",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1"
      ]
    },
    "definition_ja": "区間ごとに一定の値をとり、区間の区切りで値がとぶ関数。グラフが階段の形になり、「1 時間ごとに 200 円」のような料金を表すのに使う。",
    "definition_en": "A function that is constant on each interval and jumps to a new value at the break points, so its graph looks like a set of stairs.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Parking is $3 for each hour or part of an hour, so the cost is a step function: it stays flat, then jumps up at every hour.",
        "ja": "駐車料金は 1 時間ごと（1 時間未満も 1 時間）に 3 ドルだから、料金は階段関数。しばらく同じ値で、1 時間ごとに跳ね上がる。",
        "register": "spoken"
      },
      {
        "en": "A parking garage charges $2 for each hour or part of an hour, for up to 4 hours. Write the cost as a step function and graph it.",
        "ja": "ある駐車場の料金は、1 時間ごと（1 時間未満も 1 時間とする）に 2 ドルで、最大 4 時間までとする。料金を階段関数で表し、グラフをかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "最大整数関数 ⌊x⌋（日本のガウス記号、エントリ floor-function）は step function の代表。",
      "大学の微分方程式の講義では、t = 0 で 0 から 1 に跳ぶ unit step function（Heaviside function）の意味でも step function と言う。",
      "step function は区分的関数（エントリ piecewise-function）の一種で、グラフの各段の端には、含む点（●）と含まない点（○）をかき分ける。"
    ],
    "related": [
      "floor-function",
      "ceiling-function",
      "piecewise-function",
      "jump-discontinuity",
      "function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "階段関数",
        "en": "Step function"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "step function": 22
      },
      "written": {
        "step function": 4
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "yt:3blue1brown",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "stokes-theorem",
    "ja": {
      "term": "ストークスの定理",
      "reading": "すとーくすのていり",
      "alt": []
    },
    "en": {
      "term": "Stokes' theorem",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "曲面 S の境界の閉曲線 C に沿った線積分を、S 上の回転の面積分で表す定理。",
    "definition_en": "The theorem that equates a line integral around the boundary curve C of a surface S with the surface integral of the curl over S.",
    "latex": "\\oint_C \\mathbf{F} \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times \\mathbf{F}) \\cdot d\\mathbf{S}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Stokes' theorem is Green's theorem lifted off the plane.",
        "ja": "ストークスの定理は、グリーンの定理を平面から持ち上げたもの。",
        "register": "spoken"
      },
      {
        "en": "Use Stokes' theorem to evaluate ∬_S curl F · dS for F = ⟨−y, x, z⟩ over the hemisphere x² + y² + z² = 1, z ≥ 0.",
        "ja": "ストークスの定理を用いて、F = ⟨−y, x, z⟩、S を半球面 x² + y² + z² = 1、z ≥ 0 とするときの ∬_S curl F · dS を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "曲面の向きと境界の向きを右手の規則（right-hand rule）で合わせる。"
    ],
    "related": [
      "greens-theorem",
      "curl-vector",
      "surface-integral",
      "line-integral",
      "divergence-theorem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "Stokes' theorem": 33
      },
      "written": {
        "Stokes' theorem": 96
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "straight-angle",
    "ja": {
      "term": "平角",
      "reading": "へいかく",
      "alt": []
    },
    "en": {
      "term": "straight angle",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "頂点をはさんで 2 辺が一直線になる角。大きさは 180° である。",
    "definition_en": "An angle whose two sides point in opposite directions and form a straight line. It measures 180°.",
    "latex": "180^\\circ",
    "spoken_en": "one hundred eighty degrees",
    "examples": [
      {
        "en": "A straight angle is 180 degrees, so if this part is 65, the other part has to be 115.",
        "ja": "平角は 180° だから、こっちが 65° なら、もう一方は 115° になる。",
        "register": "spoken"
      },
      {
        "en": "Point B lies on line AC, so ∠ABC is a straight angle. If m∠ABD = (2x)° and m∠DBC = (x + 30)°, find the value of x.",
        "ja": "点 B は直線 AC 上にあるので、∠ABC は平角である。∠ABD = 2x°, ∠DBC = (x + 30)° のとき、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "平角を 1 本の半直線で 2 つに分けた 2 角の組は linear pair（一直線をなす角）と呼ぶ。"
    ],
    "related": [
      "angle",
      "linear-pair",
      "reflex-angle",
      "supplementary-angle"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 7)",
        "url": "https://im.kendallhunt.com/MS/students/2/index.html",
        "note": "glossary の見出し straight angle（Grade 7・Grade 8）。7.1 Relationships of Angles ほか"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "straight angle": 13
      },
      "sources": [
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "straightedge",
    "ja": {
      "term": "定規",
      "reading": "じょうぎ",
      "alt": []
    },
    "en": {
      "term": "ruler",
      "alt": [
        "straightedge"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「定規」は、目盛りで長さを測る定規にも、作図で直線を引くだけの定規にも使う。英語では目盛りのある定規（ものさしも）は ruler、作図で目盛りを使わずに直線を引く定規は straightedge と言い分け、Geometry の作図は compass and straightedge と言う。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直線を引いたり、目盛りで長さを測ったりする道具。作図では目盛りを使わず、直線を引くためだけに使う。",
    "definition_en": "A ruler is a tool with marks for measuring lengths and drawing straight lines. A straightedge is used only to draw straight lines, as in compass-and-straightedge constructions.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "In a construction, the straightedge is only for drawing lines, so don't use the marks on your ruler.",
        "ja": "作図では定規は線を引くためだけに使うから、定規の目盛りは使わないでね。",
        "register": "spoken"
      },
      {
        "en": "Using only a compass and a straightedge, construct the perpendicular bisector of segment AB.",
        "ja": "コンパスと定規だけを使って、線分 AB の垂直二等分線を作図せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "目盛りで長さを測る定規は ruler。日本語の「ものさし」も ruler。",
      "作図の問題文にある compass and straightedge の straightedge は、目盛りを使わずに直線を引くための定規のこと（エントリ construction）。"
    ],
    "related": [
      "construction",
      "compass",
      "construct",
      "ruler-postulate",
      "measure"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "定規",
        "en": "Ruler"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "ruler !postulate": 14
      },
      "written": {
        "ruler !postulate": 20
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "stratified-sampling",
    "ja": {
      "term": "層化抽出",
      "reading": "そうかちゅうしゅつ",
      "alt": [
        "層化抽出法",
        "層別抽出"
      ]
    },
    "en": {
      "term": "stratified sampling",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "母集団をいくつかの似たものどうしの層に分け、各層から無作為に標本を取り出す方法。",
    "definition_en": "Dividing the population into groups of similar members (strata) and taking a random sample from each group.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Split the school by grade, then randomly pick students from each grade: that's stratified sampling.",
        "ja": "学年ごとに分けて各学年から無作為に選ぶ。それが層化抽出。",
        "register": "spoken"
      },
      {
        "en": "Describe how to take a stratified sample of 40 students from the four grades.",
        "ja": "4 学年から 40 人の層化抽出の標本をとる方法を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "層は strata（単数 stratum）。"
    ],
    "related": [
      "cluster-sampling",
      "systematic-sampling",
      "random-sampling",
      "sampling"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "層化抽出法",
        "en": "Stratified sampling"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "stratified sampling": 3
      },
      "written": {
        "stratified sampling": 15
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "stretch-vertically",
    "ja": {
      "term": "縦に伸縮する",
      "reading": "たてにしんしゅくする",
      "alt": [
        "y 軸方向に拡大・縮小する"
      ]
    },
    "en": {
      "term": "stretch … vertically",
      "alt": [
        "stretch vertically",
        "vertically stretch"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "英語の stretch は a > 1 の拡大だけを指し、0 < a < 1 の縮小は compress（vertical compression）と言う。OpenStax Algebra and Trigonometry・Precalculus・Calculus Volume 1 と IM Algebra 2（5.8 Scaling the Outputs）は vertical stretch と vertical compression を分ける。日本語の「縦に伸縮する」「y 軸方向に拡大・縮小する」は両方を指す（高等学校学習指導要領解説（数学C 平面上の曲線と複素数平面）は、中心が原点で半径 a の円を「y 軸方向に b/a 倍して」楕円の標準形を導く、と拡大も縮小も一つの言い方で書く）。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "グラフの各点の y 座標を a 倍して、y 軸方向に拡大（a > 1）または縮小（0 < a < 1）する。y = f(x) は y = af(x) になる。",
    "definition_en": "To multiply every y-coordinate of a graph by a number a > 1, so that the graph is stretched vertically; y = f(x) becomes y = af(x). For 0 < a < 1 the graph is compressed vertically instead.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Multiplying by 2 on the outside stretches the graph vertically by a factor of 2.",
        "ja": "外側に 2 を掛けると、グラフは縦に 2 倍に伸びる。",
        "register": "spoken"
      },
      {
        "en": "Describe how the graph of y = 3 sin x is obtained from the graph of y = sin x.",
        "ja": "y = 3 sin x のグラフは、y = sin x のグラフをどのように変形したものか説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「y 軸方向に 3 倍に拡大」は vertically stretched by a factor of 3。縮小は compress（vertical compression）。",
      "OpenStax Algebra and Trigonometry は変換の名前として vertical stretch（40 件）・vertical compression（11 件）と書く。用例コーパスでは名詞の形のほうが stretch … vertically より多い。"
    ],
    "related": [
      "compress-horizontally",
      "amplitude"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 2 5.8 Scaling the Outputs: stretch と compress を分ける"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学C 平面上の曲線と複素数平面: 中心が原点で半径 a の円を y 軸方向に b/a 倍して楕円の標準形を導く"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "vertically stretch": 1,
        "stretch … vertically": 4
      },
      "written": {
        "stretch … vertically": 19,
        "vertically stretch": 10
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "strong-induction",
    "ja": {
      "term": "累積帰納法",
      "reading": "るいせききのうほう",
      "alt": [
        "強帰納法",
        "強い数学的帰納法"
      ]
    },
    "en": {
      "term": "strong induction",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "数学的帰納法の一種で、n = 1, 2, …, k のすべてで成り立つと仮定して、n = k + 1 でも成り立つことを示す証明法。直前の 1 つだけでなく、それより前のすべての場合を仮定に使える。",
    "definition_en": "A form of induction in which you assume the statement holds for every case from the base case up to k, and use that to prove it for k + 1. It helps when the k + 1 case depends on cases earlier than k.",
    "latex": "P(1) \\land \\cdots \\land P(k) \\implies P(k+1)",
    "spoken_en": "P of 1 and dot dot dot and P of k implies P of k plus 1",
    "examples": [
      {
        "en": "To show every integer greater than one has a prime factorization, knowing just the case k isn't enough, so we'll use strong induction.",
        "ja": "1 より大きいすべての整数が素因数分解できることを示すには、n = k の場合だけ仮定しても足りない。だから累積帰納法を使おう。",
        "register": "spoken"
      },
      {
        "en": "Use strong induction to prove that every integer n ≥ 12 can be written as 4a + 5b for some nonnegative integers a and b.",
        "ja": "累積帰納法を用いて、12 以上のすべての整数 n は、0 以上の整数 a, b を用いて 4a + 5b と表せることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数B の「n = k, k + 1 のとき成り立つと仮定して n = k + 2 を示す」形の帰納法は、strong induction の特別な場合にあたる。",
      "base case が 1 つでは足りないことがある。上の 4a + 5b の例では 12〜15 の 4 つを先に確かめる。",
      "用例コーパスでは話し言葉はすべて MIT OCW、書き言葉はすべて MIT の講義ノート。Levin は 4.6 Strong Induction ほかで 33 件使う。"
    ],
    "related": [
      "proof-by-induction",
      "well-ordering-principle",
      "structural-induction",
      "base-case",
      "inductive-hypothesis"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "strong induction": 15
      },
      "written": {
        "strong induction": 38
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "structural-induction",
    "ja": {
      "term": "構造帰納法",
      "reading": "こうぞうきのうほう",
      "alt": []
    },
    "en": {
      "term": "structural induction",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "再帰的に定義された集合（木や文字列、式など）のすべての要素がある性質をもつことを示す帰納法。基本の要素で成り立ち、新しい要素を作る各規則で性質が引き継がれることを示す。",
    "definition_en": "A form of induction for recursively defined sets such as trees, strings, or expressions. You show that the property holds for the basic elements and is preserved by each rule that builds new elements.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Strings are defined recursively, so we'll use structural induction: check the empty string, then show the property survives when you add one more character.",
        "ja": "文字列は再帰的に定義されているから、構造帰納法を使おう。空の文字列で確かめて、1 文字付け足しても性質が保たれることを示す。",
        "register": "spoken"
      },
      {
        "en": "Prove by structural induction that every full binary tree with n leaves has n − 1 internal vertices.",
        "ja": "構造帰納法を用いて、葉が n 個の全二分木の内部頂点はちょうど n − 1 個であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "自然数についての普通の帰納法は、「0 と、n から n + 1 を作る規則」で定義された集合に対する構造帰納法の特別な場合とみなせる。",
      "日本の高校数学では扱わず、大学の離散数学や情報科学で出てくる。",
      "用例コーパスで structural induction は書き言葉（MIT の講義ノート）にだけ出てきて、話し言葉には出てこない。"
    ],
    "related": [
      "proof-by-induction",
      "recursive-definition",
      "strong-induction",
      "tree"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "structural induction": 32
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "subscript",
    "ja": {
      "term": "添字",
      "reading": "そえじ",
      "alt": []
    },
    "en": {
      "term": "subscript",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "notation",
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "文字の右下に小さく書いて、何番目かなどを区別する数や文字。aₙ の n。",
    "definition_en": "A small number or letter written low to the right of a symbol to tell items apart, like the n in aₙ.",
    "latex": "a_n",
    "spoken_en": "a sub n",
    "examples": [
      {
        "en": "The subscript just tells you which term you're on.",
        "ja": "添字は何番目の項かを表しているだけ。",
        "register": "spoken"
      },
      {
        "en": "In the expression Σₖ₌₁ⁿ aₖ, the subscript k is called the index of summation.",
        "ja": "Σₖ₌₁ⁿ aₖ の添字 k を和の添字という。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "aₙ は a sub n と読む。Σ の k は index（of summation）とも呼ぶ。",
      "用例コーパスの書き言葉は半分ほどが統計（OpenStax Introductory Statistics）の添字。"
    ],
    "related": [
      "sequence",
      "nth-term",
      "shift-the-index",
      "summation-notation"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "添え字",
        "en": "Subscript and superscript"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences（subscript n、index variable）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "subscript": 57
      },
      "written": {
        "subscript": 46
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "subset",
    "ja": {
      "term": "部分集合",
      "reading": "ぶぶんしゅうごう",
      "alt": []
    },
    "en": {
      "term": "subset",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "集合 B の要素だけでできている集合 A のこと。A のどの要素も B の要素であるとき、A を B の部分集合という。",
    "definition_en": "A set A is a subset of a set B when nothing in A lies outside B: each member of A belongs to B as well.",
    "latex": "A \\subset B",
    "spoken_en": "A is a subset of B",
    "examples": [
      {
        "en": "So A is a subset of B, because everything in A is also in B.",
        "ja": "だから A は B の部分集合だ。A に入っているものは全部 B にも入っているからね。",
        "register": "spoken"
      },
      {
        "en": "List all subsets of {1, 2, 3}.",
        "ja": "集合 {1, 2, 3} の部分集合をすべて書き並べよ。",
        "register": "written"
      },
      {
        "en": "If A ⊂ B, then A is contained in B, and B contains A.",
        "ja": "A ⊂ B のとき、A は B に含まれる、または B は A を含むという。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「含む」「含まれる」は contain ／ is contained in。A ⊂ B は A is contained in B、B ⊃ A は B contains A と読む。",
      "センター試験（平成26年度 数学Ⅰ・数学Ａ）は「集合 X が集合 Y の部分集合であるとき X ⊂ Y と表す」と定め、A = B の場合も ⊂ で書く。日本語版 Wikipedia「部分集合」は ⊂ を真部分集合（A ⊆ B かつ A ≠ B）に使い、記法は統一されていないと書く。英語の本も分かれ、Levin（1.5）は ⊆ を部分集合・⊂ を真部分集合に使い、OpenStax Calculus は D ⊆ R と R ⊂ ℝ² の両方を書く。",
      "要素と部分集合を混同しない。1 ∈ A（1 is an element of A）と {1} ⊂ A（{1} is a subset of A）は別の主張。"
    ],
    "related": [
      "set",
      "element",
      "empty-set",
      "universal-set",
      "venn-diagram"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "部分集合",
        "en": "Subset"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "平成 26 年度 数学I・A: 集合 X が集合 Y の部分集合であるとき X ⊂ Y と表す"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「部分集合」",
        "url": "https://ja.wikipedia.org/wiki/%E9%83%A8%E5%88%86%E9%9B%86%E5%90%88",
        "note": "⊆ と真部分集合 ⊂。記法は統一されていない"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.5: ⊆（subset）と ⊂（proper subset）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "subset": 117
      },
      "written": {
        "subset": 179
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "subspace",
    "ja": {
      "term": "部分空間",
      "reading": "ぶぶんくうかん",
      "alt": [
        "線形部分空間"
      ]
    },
    "en": {
      "term": "subspace",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "ベクトル空間の部分集合で、0 を含み、和と実数倍について閉じているもの。それ自身もベクトル空間になる。",
    "definition_en": "A subset of a vector space that contains 0 and is closed under addition and scalar multiplication; it is a vector space itself.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A line through the origin is a subspace; a line that misses the origin isn't.",
        "ja": "原点を通る直線は部分空間だが、原点を通らない直線は違う。",
        "register": "spoken"
      },
      {
        "en": "Determine whether the set is a subspace of ℝ³.",
        "ja": "その集合が ℝ³ の部分空間かどうか判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉はすべて MIT 18.06。"
    ],
    "related": [
      "vector-space",
      "span",
      "null-space",
      "column-space"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "5.1 Subspaces and Spanning"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "subspace": 304
      },
      "sources": [
        "mit-18.06"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "substitute-new-variable",
    "ja": {
      "term": "置き換える",
      "reading": "おきかえる",
      "alt": [
        "置換する",
        "置き換え"
      ]
    },
    "en": {
      "term": "make the substitution",
      "alt": [],
      "variants": [
        {
          "term": "make a substitution",
          "register": "spoken",
          "note": "話し言葉では make the substitution と同じくらい使うが、Professor Leonard と MIT OCW に多い。書き言葉では make the substitution のほうが多い。"
        },
        {
          "term": "do a substitution",
          "register": "spoken",
          "note": "話し言葉は Professor Leonard が中心。書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "「x + 1 を A と置き換える」のような置き換えは、英語では Let A = x + 1. と宣言する形か、make the substitution（話し言葉では make a substitution とも）と言う。代入する（plug in）にも substitute を使うので、英語では両者の区別が文脈頼みになる。",
    "domains": [
      "algebra",
      "calculus"
    ],
    "level": {
      "jp": [
        "中3",
        "数I",
        "数III"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "AP Calculus AB",
        "Calculus I",
        "Algebra 2"
      ]
    },
    "definition_ja": "式の一部をひとまとまりとみて、新しい文字で表し直すこと。",
    "definition_en": "To rename part of an expression with a new letter so the expression becomes simpler.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's make a substitution: call x plus one A, and the whole thing factors like a quadratic.",
        "ja": "置き換えよう。x + 1 を A とすると、全体が 2 次式のように因数分解できる。",
        "register": "spoken"
      },
      {
        "en": "Let A = x + 1. Then (x + 1)² − 3(x + 1) + 2 = A² − 3A + 2 = (A − 1)(A − 2).",
        "ja": "x + 1 = A とおくと、(x + 1)² − 3(x + 1) + 2 = A² − 3A + 2 = (A − 1)(A − 2)。",
        "register": "written"
      },
      {
        "en": "To solve x⁴ − 5x² + 4 = 0, make the substitution t = x² and solve t² − 5t + 4 = 0 first.",
        "ja": "x⁴ − 5x² + 4 = 0 を解くには、x² を t と置き換えて、まず t² − 5t + 4 = 0 を解く。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「x + 1 = A とおく」は式が先だが、英語の答案では Let A = x + 1. と新しい文字を先に書く。"
    ],
    "related": [
      "substitute",
      "let-u-equal",
      "integration-by-substitution"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "make the substitution": 32,
        "make a substitution": 33,
        "do a substitution": 21
      },
      "written": {
        "make the substitution": 19,
        "make a substitution": 7
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "substitute",
    "ja": {
      "term": "代入する",
      "reading": "だいにゅうする",
      "alt": [
        "代入"
      ]
    },
    "en": {
      "term": "plug in",
      "alt": [],
      "variants": [
        {
          "term": "plug in",
          "register": "spoken",
          "note": "話し言葉で最も多い。大学の講義（MIT OCW）と YouTube の講義・解説で使われ、高校向けの Khan Academy ではほとんど出てこない。教科書の文章では少ない（書き言葉の大半は MIT の講義ノート）。"
        },
        {
          "term": "substitute",
          "register": "both",
          "note": "話し言葉では plug in より少ない。高校向けの Khan Academy はほぼこちらだけを使う。大学の講義（MIT OCW）でも使うが plug in のほうが多い。書き言葉（OpenStax・MIT の講義ノート）ではほぼこれだけ。書き言葉ではほぼこれだけなので、答案にはこちらを使う。"
        },
        {
          "term": "substitute back",
          "register": "spoken",
          "note": "求めた値や置き換えた式を、もとの式に戻して代入すること。書き言葉より話し言葉でよく使う。"
        },
        {
          "term": "sub in",
          "register": "spoken",
          "note": "略式。頻度は低い（話し言葉に少しあるだけで、すべて Professor Leonard）。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "classroom"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "文字に具体的な値や別の式を当てはめること。",
    "definition_en": "To put a value or another expression in place of a variable.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Plug x = 2 into the equation and see what you get.",
        "ja": "x=2 を方程式に代入してみて。",
        "register": "spoken"
      },
      {
        "en": "Substituting t = 2x gives a quadratic in t.",
        "ja": "t=2x を代入すると t の二次方程式になる。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "plug in x = 2",
        "ja": "x=2 を代入する"
      },
      {
        "en": "substitute u for 2x",
        "ja": "2x を u とおく"
      }
    ],
    "pitfalls": [
      "大学の講義や YouTube の解説では plug in がずっと多い。ただし高校向けの Khan Academy は substitute を使うので、どちらも耳にする。教科書と書き言葉（OpenStax・MIT の講義ノート）では substitute なので、答案にはこちらを使う。",
      "substitute A for B は「B の代わりに A を入れる」。順序を逆にしやすい。"
    ],
    "related": [
      "move-term-to-other-side",
      "substitute-new-variable"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-10",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "plug in": 1405,
        "substitute": 420,
        "substitute back": 42,
        "sub in": 2
      },
      "written": {
        "substitute": 1329,
        "substitute back": 9,
        "plug in": 35
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "substituting-values",
    "ja": {
      "term": "数値代入法",
      "reading": "すうちだいにゅうほう",
      "alt": []
    },
    "en": {
      "term": "plug in convenient values",
      "alt": [
        "substitute convenient values"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。手順として plug in convenient values of x（x に都合のよい値を代入する）と説明する。求めた値が恒等式の条件を満たすか確かめる（逆の確認）ことも英語では手順の一部として言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "恒等式の未知の係数を、x に適当な数値を代入してできる方程式から決める方法。",
    "definition_en": "Finding the unknown constants in an identity by plugging in convenient values of x and solving the resulting equations.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Plug in x = 1 so the B term drops out, and you get A right away.",
        "ja": "x = 1 を代入すると B の項が消えて、すぐ A が出る。",
        "register": "spoken"
      },
      {
        "en": "By substituting suitable values of x, find A and B such that 1/((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2).",
        "ja": "x に適当な値を代入して、1/((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2) となる A, B を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "substitute values（値を代入する）だけでは公式に値を入れる一般の言い方になる（用例コーパスの書き言葉ではすべてその意味）。数値代入法の意味では convenient（都合のよい）を付けて言う。",
      "数値代入法で求めた係数は、必要条件から出したものなので、恒等式になっているか確かめる（Check that it works for all x.）。"
    ],
    "related": [
      "identity",
      "compare-coefficients",
      "partial-fraction-decomposition"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "substitution-in-a-definite-integral",
    "ja": {
      "term": "定積分の置換積分",
      "reading": "ていせきぶんのちかんせきぶん",
      "alt": []
    },
    "en": {
      "term": "substitution with definite integrals",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。OpenStax Calculus が示すのは、区間を u の値に書き換える方法と、いったん不定積分を求めて x に戻し元の区間で計算する方法の 2 つ。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "定積分で置換積分を使うこと。置換に合わせて積分区間も新しい変数の範囲に書き換える。",
    "definition_en": "Using substitution in a definite integral, with the limits of integration changed to match the new variable.",
    "latex": "\\int_0^1 2x(x^2+1)^3\\,dx = \\int_1^2 u^3\\,du",
    "spoken_en": "the integral from zero to one of two x times the quantity x squared plus one, cubed, dx equals the integral from one to two of u cubed du",
    "examples": [
      {
        "en": "If you change the limits to u-values, you never have to go back to x.",
        "ja": "区間を u の値に変えておけば、x に戻さなくていい。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "区間を x の値のまま、u の式に上端・下端を代入してしまうのがよくある誤り。"
    ],
    "related": [
      "integration-by-substitution",
      "interval-of-integration",
      "change-the-limits-of-integration"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "substitution with definite integrals": 2
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 2 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "substitution-method",
    "ja": {
      "term": "代入法",
      "reading": "だいにゅうほう",
      "alt": []
    },
    "en": {
      "term": "substitution method",
      "alt": [
        "solve by substitution"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "連立方程式の一方の式を y = …（または x = …）の形に変形し、それをもう一方の式に代入して文字を 1 つ消す解き方。",
    "definition_en": "A way to solve a system of equations: solve one equation for one variable, then substitute that expression into the other equation.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The first equation is already solved for y, so the substitution method is the way to go here.",
        "ja": "1 つ目の式がもう y = … の形になっているから、ここは代入法がいいね。",
        "register": "spoken"
      },
      {
        "en": "Solve the system y = 2x − 1 and 3x + y = 9 using the substitution method.",
        "ja": "代入法を用いて、連立方程式 y = 2x − 1、3x + y = 9 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "積分の置換積分（エントリ integration-by-substitution）も英語では substitution と言う。連立方程式の代入法とは別のもの。",
      "加減法は elimination（エントリ elimination）。2 つの名前を組で覚える。",
      "代入する式は、かっこでくくって入れる。3x + y = 9 に y = 2x − 1 を入れると 3x + (2x − 1) = 9。"
    ],
    "related": [
      "substitute",
      "elimination",
      "system-of-linear-equations",
      "linear-equation-in-two-variables",
      "integration-by-substitution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "substitution method": 7,
        "solve by substitution": 2
      },
      "written": {
        "substitution method": 11,
        "solve by substitution": 3
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-10-09"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "substitution-property",
    "ja": {
      "term": "代入の性質",
      "reading": "だいにゅうのせいしつ",
      "alt": []
    },
    "en": {
      "term": "substitution property",
      "alt": [
        "substitution property of equality"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「代入の性質」は学習指導要領解説に無い、本プロジェクトの訳語。日本では「等しいものを代入してよい」を名前のある性質として挙げない。 用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。",
    "domains": [
      "proof",
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "a = b ならば、式や文の中の a を b に置き換えても成り立つという性質。",
    "definition_en": "The property that if a = b, then b can replace a in any equation or statement without changing whether it is true.",
    "latex": "a = b,\\ a + c = 10 \\implies b + c = 10",
    "spoken_en": "if a equals b and a plus c equals 10, then b plus c equals 10",
    "examples": [
      {
        "en": "We know angle A is 50 degrees, so by the substitution property we can plug 50 in for angle A in the equation.",
        "ja": "∠A は 50° とわかっているので、代入の性質により、式の ∠A に 50 を入れてよい。",
        "register": "spoken"
      },
      {
        "en": "Given m∠1 + m∠2 = 180 and m∠1 = m∠3, show that m∠3 + m∠2 = 180. State the property you used.",
        "ja": "∠1 + ∠2 = 180°、∠1 = ∠3 のとき、∠3 + ∠2 = 180° となることを示せ。また、使った性質の名前を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞の「代入する」は substitute（話し言葉では plug in）。substitution property は証明の理由として書く性質の名前。",
      "日本の証明では「∠1 = ∠3 を代入して」と書くだけで、理由に名前を付けない。",
      "用例コーパスでは話し言葉にわずかに出てくるだけ（The Organic Chemistry Tutor）で、書き言葉と IM には出てこない。"
    ],
    "related": [
      "transitive-property",
      "substitute",
      "reflexive-property"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "substitution property": 1
      },
      "sources": [
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 1 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "subtended-by",
    "ja": {
      "term": "弧に対する",
      "reading": "こにたいする",
      "alt": []
    },
    "en": {
      "term": "subtended by",
      "alt": [
        "subtend"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「弧 AB に対する円周角」の「に対する」はふつうの助詞。CK-12 Geometry 6.14 Inscribed Angles in Circles は角の側から intercepted arc（角の内部にある弧）・the angle that intercepts arc AB と言い、subtend は使わない（IM も 0 件）。subtend は OpenStax Algebra and Trigonometry・Precalculus の Angles の節（弧度法）と Calculus、Khan Academy の弧度法の講義に出てくる動詞で、「（弧や辺が、ある角を）張る・対する」の意味。the inscribed angle subtended by arc AB は本プロジェクトの書き方の例。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2",
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "円周角や中心角が、ある弧（または線分）の両端を通る 2 辺でできていることを表す言い方。「弧 AB に対する円周角」の「に対する」に当たる。",
    "definition_en": "Said of an angle whose sides pass through the two endpoints of an arc or a segment: the angle is subtended by that arc or segment. With the arc as the subject, you say the arc subtends the angle.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The central angle is 70, so the inscribed angle subtended by arc AB is half of that, 35 degrees.",
        "ja": "中心角が 70° だから、弧 AB に対する円周角はその半分の 35° だ。",
        "register": "spoken"
      },
      {
        "en": "In circle O, ∠BOC = 110°, and A is a point on major arc BC. Find ∠BAC, the inscribed angle subtended by arc BC.",
        "ja": "円 O で ∠BOC = 110° であり、点 A は長い方の弧 BC 上にある。弧 BC に対する円周角 ∠BAC の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "主語の向きに注意。弧を主語にすると能動で arc AB subtends ∠APB（弧 AB が ∠APB を張る）、角を主語にすると受け身で ∠APB is subtended by arc AB（∠APB は弧 AB に対する角）。OpenStax（Angles の節）は the arc subtended by the angle と逆の向きにも書くので、主語がどちらかは文ごとに確かめる。",
      "subtend は弧だけでなく、辺（線分）がある点に対して角を張るときにも使う（∠APB is subtended by segment AB）。",
      "角の側から弧を言う言い方はエントリ intercepted-arc。subtend とは主語と目的語が入れかわる。"
    ],
    "related": [
      "inscribed-angle",
      "central-angle",
      "arc",
      "intercepted-arc",
      "inscribed-angle-theorem"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.14 Inscribed Angles in Circles: intercepted arc（角の内部の弧）。subtend は 0 件"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "subtend は 0 件（intercepted arc を使う）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "subtended by": 14
      },
      "written": {
        "subtended by": 8
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "subtract",
    "ja": {
      "term": "引く",
      "reading": "ひく",
      "alt": []
    },
    "en": {
      "term": "subtract",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある数や式から別の数や式を取り去って、その差を求める。",
    "definition_en": "To take one number or expression away from another to find the difference.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Subtract 4 from both sides, so 2x + 4 = 10 becomes 2x = 6.",
        "ja": "両辺から 4 を引くと、2x + 4 = 10 は 2x = 6 になる。",
        "register": "spoken"
      },
      {
        "en": "Subtract 2x − 5 from 7x + 1.",
        "ja": "7x + 1 から 2x − 5 を引け。",
        "register": "written"
      },
      {
        "en": "Careful: subtract 9 from 2 means 2 minus 9, which is negative 7.",
        "ja": "気をつけて。「2 から 9 を引く」は 2 − 9 で、−7 だよ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "subtract A from B は B − A（B から A を引く）で、語順が日本語と逆。subtract 3 from 10 は 10 − 3。",
      "take away（7 take away 2）は OpenStax Prealgebra が引き算の説明に使う言い方。式の − は minus と読む。",
      "式を引くときは、引く式にかっこを付けて、そのすべての項の符号を変える（(7x + 1) − (2x − 5) = 5x + 6）。"
    ],
    "related": [
      "subtraction",
      "add",
      "the-four-operations",
      "move-term-to-other-side",
      "parentheses"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "subtract": 2556
      },
      "written": {
        "subtract": 1272
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "subtraction",
    "ja": {
      "term": "減法",
      "reading": "げんぽう",
      "alt": []
    },
    "en": {
      "term": "subtraction",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "ある数から別の数を引く計算（引き算）。結果を差といい、正負の数の減法は、ひく数の符号を変えて加法に直せる。",
    "definition_en": "Taking one number away from another; the answer is the difference. To subtract a number you can add its opposite instead.",
    "latex": "a - b = a + (-b)",
    "spoken_en": "a minus b equals a plus negative b",
    "examples": [
      {
        "en": "Subtraction with negatives gets easier if you add the opposite: 3 minus negative 5 is 3 plus 5, which is 8.",
        "ja": "負の数の減法は、ひく数の符号を変えて足すと楽になる。3 − (−5) は 3 + 5 で 8。",
        "register": "spoken"
      },
      {
        "en": "Rewrite the subtraction as an addition, and then find the value of (−7) − (−2).",
        "ja": "減法を加法に直して、(−7) − (−2) を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "記号 − は minus と読む（7 minus 3）。演算の名前は subtraction、動詞は subtract（エントリ subtract）、答えの「差」は difference。",
      "(−7) − (−2) のように − が並ぶときは、引き算の − を minus、負の符号の − を negative と読み分ける（negative seven minus negative two）。",
      "subtract A from B は B − A。「10 から 3 を引く」は subtract 3 from 10 で、数の出てくる順が式 10 − 3 と逆になる。"
    ],
    "related": [
      "subtract",
      "addition",
      "opposite",
      "positive-and-negative-numbers",
      "the-four-operations"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "減法",
        "en": "Subtraction"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "subtraction": 175
      },
      "written": {
        "subtraction": 358
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sufficient-condition",
    "ja": {
      "term": "十分条件",
      "reading": "じゅうぶんじょうけん",
      "alt": []
    },
    "en": {
      "term": "sufficient condition",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Calculus I",
        "Discrete Math"
      ]
    },
    "definition_ja": "「p ならば q」が真のとき、p を q であるための十分条件という。p が成り立てば q も必ず成り立つ。",
    "definition_en": "If \"if p, then q\" is true, p is a sufficient condition for q: whenever p holds, q holds as well.",
    "latex": "p \\Rightarrow q",
    "spoken_en": "p implies q, so p is sufficient for q",
    "examples": [
      {
        "en": "Being differentiable at a point is a sufficient condition for being continuous there, but the converse fails.",
        "ja": "ある点で微分可能であることは、そこで連続であるための十分条件だが、逆は成り立たない。",
        "register": "written"
      },
      {
        "en": "x = 2 is a sufficient condition for x² = 4. If x is two, x squared has to be four.",
        "ja": "x = 2 は x² = 4 であるための十分条件。x が 2 なら x² は 4 になるしかない。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "向きに注意。p ⇒ q が真のとき、矢印の出発点 p が十分条件、行き先 q が必要条件。"
    ],
    "related": [
      "necessary-condition",
      "necessary-and-sufficient-condition",
      "condition",
      "differentiable",
      "continuous"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "偏微分と smoothness: sufficient condition"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.2 Implications: sufficient condition（5 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "sufficient condition": 6
      },
      "sources": [
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 sufficient condition（本文 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sum-of-a-geometric-sequence",
    "ja": {
      "term": "等比数列の和",
      "reading": "とうひすうれつのわ",
      "alt": []
    },
    "en": {
      "term": "finite geometric series",
      "alt": [
        "sum of a geometric sequence"
      ],
      "uk": null,
      "register": "spoken",
      "variants": [
        {
          "term": "geometric sum",
          "register": "written",
          "note": "MIT の講義ノート（離散数学）と Levin（4.4）の言い方。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語では項を足したもの自体を series と呼ぶ。有限個の項の和であることを言うときは finite geometric series（話し言葉はほとんどが Khan Academy）。geometric series は無限等比級数にも使う（infinite-geometric-series を参照）。OpenStax Algebra and Trigonometry（13.4）は等比数列の項の和を geometric series と呼び、有限の和は the sum of the first n terms of a geometric series と書く。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "等比数列の初項から第 n 項までの和。公比 r ≠ 1 のとき a₁(1 − rⁿ)/(1 − r)。",
    "definition_en": "Adding up a geometric sequence from the first term through the nth; when r ≠ 1 the total is a₁(1 − rⁿ)/(1 − r).",
    "latex": "S_n = \\frac{a_1(1 - r^n)}{1 - r} \\quad (r \\ne 1)",
    "spoken_en": "S sub n equals a sub one times one minus r to the n, over one minus r",
    "examples": [
      {
        "en": "Multiply the finite geometric series by r and subtract — almost everything cancels.",
        "ja": "等比数列の和に r を掛けて引くと、ほとんどの項が消える。",
        "register": "spoken"
      },
      {
        "en": "Find the sum of the first 8 terms of the geometric sequence 1, 3, 9, 27, ….",
        "ja": "等比数列 1, 3, 9, 27, … の初項から第 8 項までの和を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "r = 1 のときは公式が使えない（和は n·a₁）。"
    ],
    "related": [
      "geometric-sequence",
      "common-ratio",
      "infinite-geometric-series",
      "sum-of-an-arithmetic-sequence",
      "partial-sum"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.4 Series and Their Notations（geometric series、the sum of the first n terms of a geometric series）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 2.25 Summing Up・2.26 Using the Sum（sum of a geometric sequence）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.4 Exponential Sequences（geometric sum）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "geometric sum": 3,
        "finite geometric series": 10,
        "sum of a geometric sequence": 2
      },
      "written": {
        "geometric sum": 13,
        "sum of a geometric sequence": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sum-of-an-arithmetic-sequence",
    "ja": {
      "term": "等差数列の和",
      "reading": "とうさすうれつのわ",
      "alt": []
    },
    "en": {
      "term": "arithmetic series",
      "alt": [
        "sum of an arithmetic sequence"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語では項を足したもの自体を series と呼び、等差数列の和を arithmetic series と言う（OpenStax Algebra and Trigonometry）。「数列の和」のまま sum of an arithmetic sequence と言うこともあるが少ない。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "等差数列の初項から第 n 項までの和。初項と末項の平均に項数を掛けたものになる。",
    "definition_en": "Adding up an arithmetic sequence from the first term through the nth: the result is the number of terms times the average of the first and last terms.",
    "latex": "S_n = \\frac{n(a_1 + a_n)}{2}",
    "spoken_en": "S sub n equals n times the quantity a sub one plus a sub n, all over two",
    "examples": [
      {
        "en": "Pair the first term with the last term — every pair adds up to the same thing.",
        "ja": "初項と末項を組にすると、どの組も同じ和になる。",
        "register": "spoken"
      },
      {
        "en": "Find the sum of the arithmetic series 2 + 5 + 8 + ⋯ + 62.",
        "ja": "等差数列の和 2 + 5 + 8 + ⋯ + 62 を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "series は単数・複数とも series。an arithmetic series。"
    ],
    "related": [
      "arithmetic-sequence",
      "sum-of-a-geometric-sequence",
      "partial-sum",
      "last-term",
      "find-the-sum"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.4 Series and Their Notations（arithmetic series）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "arithmetic series": 3,
        "sum of an arithmetic sequence": 2
      },
      "written": {
        "arithmetic series": 32,
        "sum of an arithmetic sequence": 1
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sum-of-cubes",
    "ja": {
      "term": "立方の和",
      "reading": "りっぽうのわ",
      "alt": []
    },
    "en": {
      "term": "sum of cubes",
      "alt": [
        "sum of two cubes"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "a³ + b³ の形の式。(a + b)(a² − ab + b²) と因数分解できる。",
    "definition_en": "An expression of the form a³ + b³, which factors as (a + b)(a² − ab + b²).",
    "latex": "a^3 + b^3 = (a + b)(a^2 - ab + b^2)",
    "spoken_en": "a cubed plus b cubed equals the quantity a plus b, times the quantity a squared minus a b plus b squared",
    "examples": [
      {
        "en": "x cubed plus 8 is a sum of cubes, since 8 is 2 cubed.",
        "ja": "x³ + 8 は立方の和だね。8 は 2 の 3 乗だから。",
        "register": "spoken"
      },
      {
        "en": "Factor the sum of cubes 27x³ + 1.",
        "ja": "立方の和 27x³ + 1 を因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "2 つ目の因数 a² − ab + b² は、実数の範囲ではそれ以上因数分解できない。(a + b)² の展開と取り違えて真ん中を −2ab にしないこと。",
      "立方の差（エントリ difference-of-cubes）とは符号の並びが逆になる。",
      "英語の cube は 3 乗（エントリ cube）にも立方体（エントリ cube-solid）にも使う。ここでは 3 乗のこと。"
    ],
    "related": [
      "difference-of-cubes",
      "difference-of-squares",
      "factoring",
      "cube",
      "polynomial"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "sum of cubes": 2,
        "sum of two cubes": 5
      },
      "written": {
        "sum of cubes": 23,
        "sum of two cubes": 3
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sum-of-random-variables",
    "ja": {
      "term": "確率変数の和",
      "reading": "かくりつへんすうのわ",
      "alt": []
    },
    "en": {
      "term": "sum of random variables",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "2 つ以上の確率変数を足した確率変数。期待値はいつも和になり、独立なときは分散も和になる。",
    "definition_en": "A random variable formed by adding two or more random variables; its expected value is always the sum, and its variance is the sum when they are independent.",
    "latex": "E(X + Y) = E(X) + E(Y)",
    "spoken_en": null,
    "examples": [
      {
        "en": "Even for a difference, the variances add, as long as the variables are independent.",
        "ja": "差であっても、変数が独立なら分散は足し算になる。",
        "register": "spoken"
      },
      {
        "en": "X and Y are independent with SD(X) = 3 and SD(Y) = 4. Find the standard deviation of X + Y.",
        "ja": "X と Y は独立で、SD(X) = 3, SD(Y) = 4 である。X + Y の標準偏差を求めよ。",
        "register": "written"
      },
      {
        "en": "For any sum of random variables the means add, but the variances add only when the variables are independent.",
        "ja": "確率変数の和では平均はいつも足せるが、分散が足せるのは独立なときだけ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "標準偏差はそのまま足さない。独立なら分散を足してから平方根をとる。"
    ],
    "related": [
      "independent-random-variables",
      "expected-value",
      "variance",
      "linearity-of-expectation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "7.2 The Central Limit Theorem for Sums・巻末 Practice Tests（sums of random variables）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Independence_(probability_theory)",
        "note": "節 Expectation, covariance, variance, and correlation（独立なら分散が和になる）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学II (8) 統計的な推測の発展・拡充の内容（確率変数の和、和の期待値、互いに独立であるときの分散の性質）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "sum of random variables": 4
      },
      "written": {
        "sum of random variables": 8
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 8 件（10 未満））。見出しは OpenStax の呼び方 sum of random variables（本文 2 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sum-of-the-exterior-angles",
    "ja": {
      "term": "外角の和",
      "reading": "がいかくのわ",
      "alt": []
    },
    "en": {
      "term": "exterior angle sum",
      "alt": [
        "sum of the exterior angles",
        "exterior angle sum theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "多角形の各頂点で外角を 1 つずつとり、それらをすべて足した大きさ。何角形でも 360° になる。",
    "definition_en": "The total of the exterior angles of a polygon, taking one exterior angle at each vertex. No matter how many sides the polygon has, the exterior angle sum is 360°.",
    "latex": "\\text{exterior angle sum} = 360^\\circ",
    "spoken_en": "the exterior angle sum equals 360 degrees",
    "examples": [
      {
        "en": "No matter how many sides the polygon has, the exterior angle sum is always 360 degrees.",
        "ja": "多角形の辺がいくつあっても、外角の和はいつも 360°。",
        "register": "spoken"
      },
      {
        "en": "Each exterior angle of a regular polygon measures 24°. Use the exterior angle sum to find how many sides the polygon has.",
        "ja": "1 つの外角が 24° である正多角形は正何角形か。外角の和を用いて求めよ。",
        "register": "written"
      },
      {
        "en": "By the Exterior Angle Sum Theorem, the exterior angles of a convex polygon, one at each vertex, add up to 360°.",
        "ja": "外角の和の定理より、凸多角形の外角（各頂点で 1 つずつ）の和は 360° である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "内角の和（sum of the interior angles、エントリ sum-of-the-interior-angles）は辺の数で変わるが、外角の和は変わらない。",
      "外角は各頂点で 1 つずつとる。1 つの頂点では、どちらの辺を延ばすかで外角が 2 つできる（対頂角で等しい）が、両方は足さない。",
      "CK-12 Geometry は外角の和が 360° であることを Exterior Angle Sum Theorem と呼ぶ（4.18 Exterior Angles and Theorems・5.28 Exterior Angles in Convex Polygons、10 件）。"
    ],
    "related": [
      "exterior-angle",
      "sum-of-the-interior-angles",
      "polygon",
      "regular-polygon",
      "interior-angle"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.18 Exterior Angles and Theorems・5.28 Exterior Angles in Convex Polygons"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 exterior angle sum（CK-12 Geometry 4.18 Exterior Angles and Theorems・CK-12 Geometry 5.28 Exterior Angles in Convex Polygons・計 8 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sum-of-the-interior-angles",
    "ja": {
      "term": "内角の和",
      "reading": "ないかくのわ",
      "alt": []
    },
    "en": {
      "term": "sum of the interior angles",
      "alt": [
        "interior angle sum",
        "polygon angle sum"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "多角形の内角をすべて足した大きさ。n 角形の内角の和は 180°(n − 2) で、1 つの頂点から引いた対角線で n − 2 個の三角形に分けられることから求められる。",
    "definition_en": "The total of all the interior angles of a polygon. For a polygon with n sides it is 180°(n − 2), because the diagonals from one vertex split the polygon into n − 2 triangles.",
    "latex": "180^\\circ (n - 2)",
    "spoken_en": "180 degrees times the quantity n minus 2",
    "examples": [
      {
        "en": "A pentagon splits into three triangles, so the sum of the interior angles is 3 times 180, or 540 degrees.",
        "ja": "五角形は三角形 3 つに分けられるから、内角の和は 180° × 3 で 540°。",
        "register": "spoken"
      },
      {
        "en": "The sum of the interior angles of a polygon is 1,260°. How many sides does the polygon have?",
        "ja": "内角の和が 1260° である多角形は何角形か。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「n 角形」は英語では an n-gon、または a polygon with n sides と、辺の数で言う。「何角形か」は How many sides does the polygon have? と聞く。",
      "外角の和（exterior angle sum、エントリ sum-of-the-exterior-angles）は何角形でも 360° で、内角の和と違って n によらない。",
      "三角形の内角の和 180° はエントリ angle-sum-of-a-triangle。公式 180°(n − 2) に n = 3 を入れたものになっている。"
    ],
    "related": [
      "interior-angle",
      "sum-of-the-exterior-angles",
      "angle-sum-of-a-triangle",
      "polygon",
      "diagonal"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.27 Interior Angles in Convex Polygons・5.26 Determine Missing Angle Measures"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sum of the interior angles": 8
      },
      "written": {
        "sum of the interior angles": 1
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 8 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは CK-12 の呼び方 sum of the interior angles（CK-12 Geometry 5.27 Interior Angles in Convex Polygons・CK-12 Geometry 5.26 Determine Missing Angle Measures・CK-12 Geometry 4.1 Classify Triangles・計 15 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "sum-of-the-series",
    "ja": {
      "term": "級数の和",
      "reading": "きゅうすうのわ",
      "alt": []
    },
    "en": {
      "term": "sum of the series",
      "alt": [],
      "variants": [
        {
          "term": "sum of an infinite series",
          "register": "written",
          "note": "少数の書き方。話し言葉には出てこない。書き言葉は OpenStax Algebra and Trigonometry と OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "sequences"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "無限級数が収束するときの、部分和の極限の値。",
    "definition_en": "The limit of the partial sums of a convergent series.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If the partial sums approach 2, the sum of the series is 2.",
        "ja": "部分和が 2 に近づくなら、級数の和は 2。",
        "register": "spoken"
      },
      {
        "en": "Find the sum of the series ∑_{n=0}^∞ (1/3)ⁿ.",
        "ja": "級数 ∑_{n=0}^∞ (1/3)ⁿ の和を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "infinite-series",
      "convergence",
      "partial-sum",
      "infinite-geometric-series"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "sum of the series": 4
      },
      "written": {
        "sum of the series": 12,
        "sum of an infinite series": 9
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sum-to-product-formulas",
    "ja": {
      "term": "和積公式",
      "reading": "わせきこうしき",
      "alt": [
        "和を積に直す公式"
      ]
    },
    "en": {
      "term": "sum-to-product formulas",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "sin A + sin B などの和・差を、三角関数の積で表す公式。",
    "definition_en": "The formulas that rewrite sums or differences such as sin A + sin B as products of trigonometric functions.",
    "latex": "\\sin A + \\sin B = 2\\sin\\frac{A + B}{2}\\cos\\frac{A - B}{2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Use sum-to-product to turn sin 3x + sin x into a product, then set each factor equal to zero.",
        "ja": "和積公式で sin 3x + sin x を積にして、それぞれの因数を 0 とおく。",
        "register": "spoken"
      },
      {
        "en": "Solve sin 3x + sin x = 0 for 0 ≤ x < π using a sum-to-product formula.",
        "ja": "和積公式を用いて、0 ≦ x < π のとき sin 3x + sin x = 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉はすべて OpenStax Algebra and Trigonometry。話し言葉には出てこない。"
    ],
    "related": [
      "product-to-sum-formulas",
      "angle-addition-formulas"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Sum-to-Product and Product-to-Sum Formulas（sum-to-product formulas）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学Ⅱ 三角関数の発展・拡充: 三角関数の和を積に直すなどの公式"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角関数の公式の一覧」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0%E3%81%AE%E5%85%AC%E5%BC%8F%E3%81%AE%E4%B8%80%E8%A6%A7",
        "note": "節「和積公式と積和公式」"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "sum-to-product formulas": 11
      },
      "sources": [
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "sum",
    "ja": {
      "term": "和",
      "reading": "わ",
      "alt": []
    },
    "en": {
      "term": "sum",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "足し算の答え。たとえば 3 と 4 の和は 7 である。",
    "definition_en": "What you get when you add two or more numbers or expressions together. For example, the sum of 3 and 4 is 7.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "What's the sum of negative 5 and 8? It's 3.",
        "ja": "−5 と 8 の和は？ 3 だね。",
        "register": "spoken"
      },
      {
        "en": "The sum of a number and 12 is 30. Find the number.",
        "ja": "ある数と 12 の和は 30 である。ある数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "sum と some（いくつかの）は発音が同じ（Merriam-Webster はどちらも ˈsəm）。聞き取りでは文脈で区別する。",
      "数列の和も sum（エントリ sum-of-an-arithmetic-sequence・find-the-sum）。Σ を使った書き方は summation notation（エントリ summation-notation）。"
    ],
    "related": [
      "addition",
      "add",
      "difference",
      "product",
      "find-the-sum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「sum」",
        "url": "https://www.merriam-webster.com/dictionary/sum",
        "note": "発音 ˈsəm。some も ˈsəm（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the sum of | a sum of": 783
      },
      "written": {
        "the sum of | a sum of": 1063
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "summation-formulas",
    "ja": {
      "term": "Σ の公式",
      "reading": "しぐまのこうしき",
      "alt": []
    },
    "en": {
      "term": "sums and powers of integers",
      "alt": [
        "summation formulas"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "notation"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "1 から n までの k、k²、k³ の和を n の式で表す公式。",
    "definition_en": "Formulas that give the sums of k, k², and k³ for k = 1 to n in terms of n.",
    "latex": "\\sum_{k=1}^{n} k = \\frac{n(n+1)}{2}, \\quad \\sum_{k=1}^{n} k^2 = \\frac{n(n+1)(2n+1)}{6}",
    "spoken_en": "the sum from k equals one to n of k is n times n plus one over two",
    "examples": [
      {
        "en": "Split the sum and use the formula for the sum of k squared.",
        "ja": "和を分けて、k² の和の公式を使おう。",
        "register": "spoken"
      },
      {
        "en": "Use the formulas for sums and powers of integers to evaluate Σₖ₌₁ⁿ (3k² − 2k).",
        "ja": "Σ の公式を用いて Σₖ₌₁ⁿ (3k² − 2k) を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 1 は Σi・Σi²・Σi³ の公式を Rule: Sums and Powers of Integers（5.1 Approximating Areas）としてまとめ、5.2（The Definite Integral）では the summation formula for Σi² のようにも書く。",
      "sum formula は、OpenStax Algebra and Trigonometry（9.2 Sum and Difference Identities）では三角関数の加法定理（the sum formula for sine）、Intermediate Algebra（12.2・12.3）では等差・等比数列の和の公式を指す。"
    ],
    "related": [
      "summation-notation",
      "partial-sum",
      "find-the-sum",
      "grouped-sequence"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "5.1 Approximating Areas（Rule: Sums and Powers of Integers）・5.2 The Definite Integral（the summation formula）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "9.2 Sum and Difference Identities（the sum formula for sine ／ cosine）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.2 Arithmetic Sequences・12.3 Geometric Sequences and Series（the sum formula）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "summation formulas": 1
      },
      "written": {
        "sums and powers of integers": 5,
        "summation formulas": 4
      },
      "sources": [
        "mit-18.06",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 sums and powers of integers（本文 5 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "summation-notation",
    "ja": {
      "term": "和の記号",
      "reading": "わのきごう",
      "alt": [
        "シグマ"
      ]
    },
    "en": {
      "term": "sigma notation",
      "alt": [],
      "variants": [
        {
          "term": "summation notation",
          "register": "both",
          "note": "書き言葉で最も多く、OpenStax Algebra and Trigonometry と OpenStax Intermediate Algebra が使う。AP の CED もこの言い方（topic 6.3）。話し言葉では sigma notation より少なく、ほとんどが MIT OCW。"
        },
        {
          "term": "sigma notation",
          "register": "both",
          "note": "話し言葉で最も多く（Khan Academy が中心）、書き言葉では OpenStax Calculus が使う。書き言葉では summation notation より少ない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences",
      "calculus",
      "notation"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus I"
      ]
    },
    "definition_ja": "Σ を使って和を簡潔に書く記法。Σ の下と上に、添字の始まりと終わりの値を書く。",
    "definition_en": "A compact way to write a sum with the symbol Σ, with the first and last values of the index written below and above it.",
    "latex": "\\sum_{i=1}^{n} i^2",
    "spoken_en": "the sum from i equals one to n of i squared",
    "examples": [
      {
        "en": "Let's write this sum in sigma notation so we don't have to list every term.",
        "ja": "全部の項を書かなくていいように、この和を Σ で書こう。",
        "register": "spoken"
      },
      {
        "en": "In summation notation, the Riemann sum is Σᵢ₌₁ⁿ f(xᵢ)Δx.",
        "ja": "Σ 記号を使うと、リーマン和は Σᵢ₌₁ⁿ f(xᵢ)Δx と書ける。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "in sigma notation",
        "ja": "Σ を使って"
      },
      {
        "en": "write … in sigma notation",
        "ja": "Σ で表す"
      }
    ],
    "pitfalls": [
      "センター試験（数学Ⅱ・数学Ｂ）の数列の問題は Σ_{k=1}^{n} のように添字に k を使い、OpenStax Precalculus（11.4 Series and Their Notations）も k を使う。OpenStax Calculus Volume 1 のリーマン和・定積分の節では i を使う（k・j より多い）。文字が違っても添字の役割は同じで、この i は虚数単位ではない。",
      "Σ は sigma と読み、式全体は the sum from i equals one to n of … と読む。"
    ],
    "related": [
      "riemann-sum",
      "limit-of-a-riemann-sum",
      "summation-formulas",
      "find-the-sum",
      "shift-the-index",
      "subscript"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "総和",
        "en": "Summation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.3（summation notation。sigma notation の語は使わない）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成30年度 大学入試センター試験 本試験 数学Ⅱ・数学Ｂ（問題）",
        "url": "https://web.archive.org/web/20190601/https://www.dnc.ac.jp/albums/abm.php?f=abm00033144.pdf&n=h30+suugaku2+02suugaku2B.pdf",
        "note": "第3問: Σ の添字 k"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "11.4 Series and Their Notations: Σ の添字 k（pitfalls[0]）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "sigma notation": 43,
        "summation notation": 10
      },
      "written": {
        "sigma notation": 23,
        "summation notation": 61
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "supplementary-angle-identity",
    "ja": {
      "term": "180° − θ の三角比",
      "reading": "ひゃくはちじゅうどまいなすしーたのさんかくひ",
      "alt": [
        "補角公式"
      ]
    },
    "en": {
      "term": "supplementary angle identity",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（supplementary angle identity は用例コーパス・参照に出てこず、cofunction identity にならった本プロジェクトの説明的な呼び方）。OpenStax Algebra and Trigonometry は 180° − θ の三角比を reference angle（|180° − t|）で鋭角の三角比に直す。高等学校学習指導要領解説（数学I）は cos A = −cos(180° − A) の形で公式を示すだけで名前を付けない。見出しの「180° − θ の三角比」は学習指導要領解説に無い（日本語版 Wikipedia「三角関数」は補角公式）。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "definition_ja": "θ の補角 180° − θ の三角比を θ の三角比で表す公式。sin(180° − θ) = sin θ、cos(180° − θ) = −cos θ、tan(180° − θ) = −tan θ。",
    "definition_en": "The identities that express the trigonometric ratios of 180° − θ in terms of those of θ: sin(180° − θ) = sin θ, cos(180° − θ) = −cos θ, and tan(180° − θ) = −tan θ.",
    "latex": "\\sin(180^\\circ - \\theta) = \\sin\\theta,\\quad \\cos(180^\\circ - \\theta) = -\\cos\\theta,\\quad \\tan(180^\\circ - \\theta) = -\\tan\\theta",
    "spoken_en": "sine of one eighty minus theta equals sine theta",
    "examples": [
      {
        "en": "Sine of 180 minus theta is the same as sine theta, but cosine and tangent flip sign.",
        "ja": "sin(180° − θ) は sin θ と同じだが、cos と tan は符号が変わる。",
        "register": "spoken"
      },
      {
        "en": "Use the supplementary angle identities to write sin 150° and cos 150° in terms of the trigonometric ratios of 30°.",
        "ja": "180° − θ の三角比を用いて、sin 150° と cos 150° を 30° の三角比で表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "補角（和が 180°）は supplementary angles、余角（和が 90°）は complementary angles。complementary を complimentary と綴らない。",
      "鈍角の三角比を鋭角の三角比に直すときに使う。cos と tan だけ符号が変わるのは、鈍角では x 座標が負になるから。"
    ],
    "related": [
      "cofunction-identity",
      "unit-circle",
      "reference-angle",
      "trigonometric-ratio",
      "table-of-trigonometric-ratios"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Unit Circle: reference angle |180° − t|（reference angle 84 件）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 鈍角の三角比 cos A = −cos(180° − A)（名前は無い）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0",
        "note": "節「負角・余角・補角公式」: 補角公式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "supplementary-angle",
    "ja": {
      "term": "補角",
      "reading": "ほかく",
      "alt": []
    },
    "en": {
      "term": "supplementary angles",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1",
        "Pre-Algebra"
      ]
    },
    "definition_ja": "和が 180° になる 2 つの角の、一方から見た他方の角。θ の補角は 180° − θ。",
    "definition_en": "Two angles are supplementary when their measures add up to 180°; the supplement of an angle θ is 180° − θ.",
    "latex": "\\alpha + \\beta = 180^\\circ",
    "spoken_en": null,
    "examples": [
      {
        "en": "These two are supplementary angles, so x is just 180 minus 65, which is 115.",
        "ja": "この 2 つは補角どうしだから、x は 180 − 65 で 115 だ。",
        "register": "spoken"
      },
      {
        "en": "Angles on a straight line are supplementary, so they add up to 180 degrees.",
        "ja": "一直線上に並んだ 2 つの角は互いに補角だから、足すと 180° になる。",
        "register": "spoken"
      },
      {
        "en": "Find the supplement of a 40° angle.",
        "ja": "40° の角の補角を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語版 Wikipedia「角度」は補角を 1 つの角（合わせて平角になる角）として定義する。OpenStax Prealgebra は 2 角の関係を形容詞で the angles are supplementary と書き、1 つの角には the supplement of … を使う（「40° の補角」は the supplement of 40°）。",
      "「補」の字につられて complementary（余角）と取り違えやすい。supplementary は 180°、complementary は 90°。"
    ],
    "related": [
      "complementary-angle",
      "angle",
      "adjacent-angles",
      "linear-pair"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「角度」",
        "url": "https://ja.wikipedia.org/wiki/%E8%A7%92%E5%BA%A6",
        "note": "補角（合わせて平角となる角）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 7（glossary: supplementary）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "supplementary": 70
      },
      "written": {
        "supplementary": 39
      },
      "sources": [
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "surface-area-of-revolution",
    "ja": {
      "term": "回転面の面積",
      "reading": "かいてんめんのめんせき",
      "alt": []
    },
    "en": {
      "term": "area of a surface of revolution",
      "alt": [
        "surface area of revolution"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線を軸のまわりに回転させてできる曲面の面積。x 軸のまわりなら 2π∫ₐᵇ f(x)√(1 + {f′(x)}²) dx で求める。",
    "definition_en": "The area of the surface swept out by revolving a curve about an axis; about the x-axis it is 2π ∫ₐᵇ f(x) √(1 + (f′(x))²) dx.",
    "latex": "S = 2\\pi \\int_a^b f(x)\\sqrt{1 + \\bigl(f'(x)\\bigr)^2}\\,dx",
    "spoken_en": "S equals two pi times the integral from a to b of f of x times the square root of one plus f prime of x squared, dx",
    "examples": [
      {
        "en": "Surface area uses the arc length piece, not just dx, so don't drop the square root.",
        "ja": "回転面の面積は dx ではなく曲線の長さの部分を使うから、ルートを落とさないように。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（数学III）は回転体の体積の公式を扱うが、回転面の面積の公式は出てこない（課題学習〈積分法〉の例に、球の表面積を定積分を利用して導くことが挙がる）。"
    ],
    "related": [
      "arc-length-of-a-curve",
      "volume-of-a-solid-of-revolution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "回転面",
        "en": "Surface of revolution",
        "note": "見出しのうち「回転面」の部分の対応"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III: 回転体の体積の公式を扱う。回転面の面積の公式は無い（課題学習〈積分法〉の例に球の表面積を定積分で導くこと）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "surface area of revolution": 2
      },
      "written": {
        "area of a surface of revolution": 8,
        "surface area of revolution": 1
      },
      "sources": [
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 area of a surface of revolution（本文 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "surface-area",
    "ja": {
      "term": "表面積",
      "reading": "ひょうめんせき",
      "alt": []
    },
    "en": {
      "term": "surface area",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "立体の表面全体の面積で、すべての面の面積の和。角柱・角錐・円柱・円錐では、展開図の面積と等しい。",
    "definition_en": "The total area of all the surfaces of a solid, including its bases. For a prism, pyramid, cylinder, or cone, it equals the area of the solid's net.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Surface area is like wrapping paper: how much paper would it take to cover the whole box?",
        "ja": "表面積は包み紙みたいなもの。箱全体を包むのに紙がどれだけいるか、ってこと。",
        "register": "spoken"
      },
      {
        "en": "Find the surface area of a rectangular prism that is 5 cm long, 4 cm wide, and 3 cm high.",
        "ja": "縦 5 cm、横 4 cm、高さ 3 cm の直方体の表面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "表面積は面積なので、単位は cm²（square centimeters）など。体積の cm³ と取り違えない。",
      "表面積 = 側面積 + 底面積。角柱・円柱は底面が 2 つあるので底面積を 2 倍し、角錐・円錐は底面が 1 つなので 1 つ分だけ足す。",
      "surface area は、微積分で曲線を軸のまわりに回転させてできる曲面の面積（エントリ surface-area-of-revolution）にも使う。"
    ],
    "related": [
      "lateral-area",
      "area-of-the-base",
      "net",
      "area",
      "surface-area-of-revolution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "表面積",
        "en": "Surface area"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "surface area": 236
      },
      "written": {
        "surface area": 246
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "surface-integral",
    "ja": {
      "term": "面積分",
      "reading": "めんせきぶん",
      "alt": []
    },
    "en": {
      "term": "surface integral",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "関数やベクトル場を曲面上で積分したもの。ベクトル場なら曲面を通る流束（flux）を表す。",
    "definition_en": "The integral of a function or a vector field over a surface; for a vector field it gives the flux through the surface.",
    "latex": "\\iint_S \\mathbf{F} \\cdot d\\mathbf{S}",
    "spoken_en": "the surface integral over S of F dot d S",
    "examples": [
      {
        "en": "For a surface integral, parametrize the surface with two parameters.",
        "ja": "面積分では、曲面を 2 つのパラメータで表そう。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the surface integral ∬_S z dS, where S is the part of the plane x + y + z = 1 in the first octant.",
        "ja": "S が平面 x + y + z = 1 の第 1 象限の部分のとき、面積分 ∬_S z dS を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉はすべて MIT OCW（18.02）。"
    ],
    "related": [
      "divergence-theorem",
      "stokes-theorem",
      "line-integral",
      "double-integral"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "面積分",
        "en": "Surface integral"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "surface integral": 50
      },
      "written": {
        "surface integral": 120
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "surplus-and-shortage",
    "ja": {
      "term": "過不足",
      "reading": "かぶそく",
      "alt": [
        "過不足算"
      ]
    },
    "en": {
      "term": "surplus and shortage",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。日本語版 Wikipedia「過不足算」は、ものを分けたときの過不足の情報 2 つから人数と全体の個数を求める算数の文章題とする。見出しの surplus and shortage は本プロジェクトの説明的な訳。英語で説明するときは「余る」を left over、「足りない」を short（2 short）と言って、状況をそのまま文にする。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "物を何人かに分けるとき、1 人あたりの数を変えると余ったり足りなくなったりする 2 通りの分け方から、人数や物の個数を求める問題。例えば「1 人 3 個ずつ配ると 5 個余り、4 個ずつ配ると 2 個足りない」から人数を求める。",
    "definition_en": "A type of word problem where sharing things one way leaves some left over and sharing another way leaves you short, and you find the number of people or things. For example, 3 each with 5 left over and 4 each with 2 short gives 3x + 5 = 4x − 2.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For a surplus and shortage problem like this, write the total two ways: 3x + 5 with 5 left over, and 4x − 2 when you're 2 short.",
        "ja": "こういう過不足の問題では、全体の個数を 2 通りに表そう。5 個余るなら 3x + 5、2 個足りないなら 4x − 2。",
        "register": "spoken"
      },
      {
        "en": "Solve the surplus and shortage problem: if each student gets 3 pencils, 5 are left over, and if each student gets 4, we are 2 pencils short. Find the number of students.",
        "ja": "次の過不足の問題を解け。生徒 1 人に鉛筆を 3 本ずつ配ると 5 本余り、4 本ずつ配ると 2 本足りない。生徒の人数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「余る」は left over（5 are left over）、「足りない」は short（2 short、we're 2 short）と言う。",
      "「余る」は割り算の余り（remainder、エントリ remainder）と同じ漢字だが、配った後に残る数は left over と言う。"
    ],
    "related": [
      "word-problem",
      "write-an-equation",
      "linear-equation",
      "remainder"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「過不足算」",
        "url": "https://ja.wikipedia.org/wiki/%E9%81%8E%E4%B8%8D%E8%B6%B3%E7%AE%97",
        "note": "概要: 算数の文章題の一種。ものを分けたときの過不足の情報 2 つから人数と全体の個数を求める（mapping_note・ja.alt）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "survey",
    "ja": {
      "term": "調査する",
      "reading": "ちょうさする",
      "alt": [
        "調査",
        "質問票",
        "アンケート"
      ]
    },
    "en": {
      "term": "survey",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Intro Statistics",
        "AP Statistics"
      ]
    },
    "definition_ja": "人や物について、質問や測定をして情報を集める。",
    "definition_en": "To collect information about people or things by asking questions or measuring.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "They surveyed 100 seniors about what they want in the new cafeteria.",
        "ja": "新しい食堂に何を望むか、3 年生 100 人に調査した。",
        "register": "spoken"
      },
      {
        "en": "A researcher surveys 400 randomly selected adults. Identify the population and the sample.",
        "ja": "研究者が無作為に選んだ成人 400 人を調査した。母集団と標本を答えよ。",
        "register": "written"
      },
      {
        "en": "The survey had ten questions, and only half the people sent it back.",
        "ja": "調査票には質問が 10 問あったが、返送したのは半分だけだった。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "名詞の survey は「調査」のほか「質問票・アンケート（調査票）」も指す。questionnaire も用例コーパスに出てくる（OpenStax Introductory Statistics・Khan Academy など）が、survey より少ない。"
    ],
    "related": [
      "sample-survey",
      "census"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "survey": 81
      },
      "written": {
        "survey": 429
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "symmetric-distribution",
    "ja": {
      "term": "左右対称な分布",
      "reading": "さゆうたいしょうなぶんぷ",
      "alt": []
    },
    "en": {
      "term": "symmetrical distribution",
      "alt": [
        "symmetric distribution"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "ヒストグラムを中央で折ると、左右がほぼ重なる形の分布。平均値と中央値がほぼ等しくなる。",
    "definition_en": "A distribution whose graph looks about the same on both sides of its center, like a mirror image. Its mean and median are about equal.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "See how the left half matches the right half? That's a symmetrical distribution, so the mean and the median are pretty much the same.",
        "ja": "左半分と右半分がそろっているでしょう。これは左右対称な分布だから、平均値と中央値はほぼ同じだよ。",
        "register": "spoken"
      },
      {
        "en": "The histogram shows a symmetrical distribution centered at 70. Estimate the mean and the median.",
        "ja": "ヒストグラムは 70 を中心とする左右対称な分布を表している。平均値と中央値を推定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は symmetrical と書くが、授業では symmetric distribution とも言う。どちらも「対称な」で意味は同じ。",
      "左右対称でない分布は skewed（エントリ be-skewed）。skewed right は右に裾が長く、山は左に寄る。山のある側ではなく、裾の伸びる側で左右を言う。",
      "左右対称な分布は正規分布（エントリ normal-distribution）に限らない。山が 2 つあっても左右がそろっていれば symmetrical。"
    ],
    "related": [
      "histogram",
      "be-skewed",
      "median",
      "arithmetic-mean",
      "normal-distribution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "symmetric distribution": 8
      },
      "written": {
        "symmetrical distribution": 8,
        "symmetric distribution": 2
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "symmetric-expression",
    "ja": {
      "term": "対称式",
      "reading": "たいしょうしき",
      "alt": [
        "対称多項式"
      ]
    },
    "en": {
      "term": "symmetric polynomial",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（symmetric polynomial ／ symmetric expression は CED・OpenStax・IM・CK-12 とも 0 件）。用例コーパスにも出てこない。見出しは英語版 Wikipedia の記事名「Symmetric polynomial」（ja 記事「対称式」の langlink 先。冒頭で symmetric polynomial を太字で定義する）。英語で説明するときは、名前の代わりに swapping x and y doesn't change it と性質を言う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": []
    },
    "definition_ja": "文字を入れ替えても元の式と同じになる多項式。x² + y² や x³ + y³ は x と y を入れ替えても変わらない。",
    "definition_en": "A polynomial that stays exactly the same when its variables are interchanged, such as x² + y² or x³ + y³.",
    "latex": "x^2 + y^2 = (x + y)^2 - 2xy",
    "spoken_en": "x squared plus y squared equals the quantity x plus y, squared, minus two x y",
    "examples": [
      {
        "en": "x² + y² is a symmetric polynomial: swap x and y and nothing changes, so we can write it using just x + y and xy.",
        "ja": "x² + y² は対称式だ。x と y を入れ替えても変わらないので、x + y と xy だけで表せる。",
        "register": "spoken"
      },
      {
        "en": "If x + y = 3 and xy = 1, find the value of the symmetric polynomial x³ + y³.",
        "ja": "x + y = 3, xy = 1 のとき、対称式 x³ + y³ の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "対称式はすべて基本対称式 x + y と xy で表せる（x³ + y³ = (x + y)³ − 3xy(x + y)）。英語で手順を言うなら rewrite it in terms of x + y and xy。",
      "形容詞の symmetric だけでは、グラフの対称性（symmetric about the y-axis）の意味にも取られる。"
    ],
    "related": [
      "elementary-symmetric-polynomial",
      "alternating-expression",
      "relationship-between-roots-and-coefficients",
      "polynomial"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "対称式",
        "en": "Symmetric polynomial"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II (1) いろいろな式: 解と係数の関係の指導で基本対称式・対称式の値（2 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Wikipedia の記事名 symmetric polynomial（英語版「Symmetric polynomial」、ja の langlink 先）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "symmetric-property",
    "ja": {
      "term": "対称律",
      "reading": "たいしょうりつ",
      "alt": []
    },
    "en": {
      "term": "symmetric property",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof",
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "a = b ならば b = a となる性質。合同についても、AB ≅ CD ならば CD ≅ AB が成り立つ。",
    "definition_en": "The property that if a = b, then b = a. The same holds for congruence: if AB ≅ CD, then CD ≅ AB.",
    "latex": "a = b \\implies b = a",
    "spoken_en": "if a equals b, then b equals a",
    "examples": [
      {
        "en": "We ended up with 5 equals x, and by the symmetric property we can flip it around to x equals 5.",
        "ja": "5 = x になったので、対称律で左右を入れ替えて x = 5 と書ける。",
        "register": "spoken"
      },
      {
        "en": "Name the property that justifies rewriting 12 = 3x as 3x = 12.",
        "ja": "12 = 3x を 3x = 12 と書き直してよい根拠となる性質の名前を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "symmetric だけだと、図形やグラフの対称（エントリ symmetric）の意味になる。性質の名前は symmetric property と property まで言う。",
      "対称式（symmetric expression）とも別のもの。",
      "用例コーパスにも参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）にも symmetric property of equality・symmetric property of congruence は出てこない。K12 LibreTexts の CK-12 Geometry には等式の性質の節（2.13・2.14）が無い。"
    ],
    "related": [
      "transitive-property",
      "reflexive-property",
      "symmetric",
      "symmetric-expression"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。symmetric property of equality ／ of congruence の形で数え直しても決まらず、そのまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "symmetric",
    "ja": {
      "term": "対称な",
      "reading": "たいしょうな",
      "alt": [
        "対称である",
        "線対称である"
      ]
    },
    "en": {
      "term": "symmetric",
      "alt": [],
      "variants": [
        {
          "term": "symmetric",
          "register": "both",
          "note": "symmetric about の形で数えた。話し言葉は大半が Professor Leonard、書き言葉は OpenStax Calculus と OpenStax Algebra and Trigonometry が中心。"
        },
        {
          "term": "symmetric with respect to",
          "register": "written",
          "note": "書き言葉（ほぼ OpenStax Algebra and Trigonometry）の言い方。話し言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "図形が、ある直線（または点）に関して折り返し（または 180° 回転し）ても重なる状態。2 点についても言う。",
    "definition_en": "Describing a figure (or a pair of points) that maps onto itself (onto each other) under a reflection across a line or a half-turn about a point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The graph of an even function is symmetric about the y-axis.",
        "ja": "偶関数のグラフは y 軸に関して対称。",
        "register": "spoken"
      },
      {
        "en": "Show that the points A and B are symmetric with respect to the line y = x.",
        "ja": "2 点 A, B が直線 y = x に関して対称であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "symmetric は symmetric matrix（対称行列）にも使う。書き言葉では symmetric with respect to もよく使う。",
      "「点対称」は symmetric about the origin（原点に関して）や symmetric about a point。"
    ],
    "related": [
      "reflection",
      "even-function",
      "odd-function"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "symmetric about": 37,
        "symmetric with respect to": 1
      },
      "written": {
        "symmetric about": 62,
        "symmetric with respect to": 39
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "synthetic-division",
    "ja": {
      "term": "組立除法",
      "reading": "くみたてじょほう",
      "alt": []
    },
    "en": {
      "term": "synthetic division",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "整式を 1 次式 x − k で割るとき、係数だけを並べて商と余りを求める簡便な方法。",
    "definition_en": "A shortcut for dividing a polynomial by x − k that works with the coefficients only.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's use synthetic division with k = 2: bring down the 1, multiply, add.",
        "ja": "k = 2 で組立除法を使おう。1 を下ろして、掛けて、足す。",
        "register": "spoken"
      },
      {
        "en": "Use synthetic division to divide x³ − 4x² + x + 6 by x − 3.",
        "ja": "組立除法を用いて、x³ − 4x² + x + 6 を x − 3 で割れ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry・Precalculus（Dividing Polynomials）と Intermediate Algebra は synthetic division を使う（51・43・25 件）。IM Algebra 2 は long division で割る（synthetic division は 0 件）。学習指導要領解説（数学II）には組立除法が出てこない（日本語版 Wikipedia「数学 (教科)」は数学II の高次方程式に挙げる）。"
    ],
    "related": [
      "polynomial-division",
      "remainder-theorem",
      "factor-theorem"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "組立除法",
        "en": "Synthetic division"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Dividing Polynomials（synthetic division）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "Dividing Polynomials（synthetic division）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Divide Polynomials（synthetic division）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 2.12・2.13 Polynomial Division（long division）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II いろいろな式: 組立除法は出てこない"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学II 高次方程式: 組立除法"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "synthetic division": 74
      },
      "written": {
        "synthetic division": 75
      },
      "sources": [
        "khan-algebra",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "system-of-inequalities",
    "ja": {
      "term": "連立不等式",
      "reading": "れんりつふとうしき",
      "alt": []
    },
    "en": {
      "term": "compound inequality",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語の compound inequality は 2 つの不等式を and または or でつないだもの。and でつないだもの（両方を満たす x、つまり共通範囲を答える）が日本の連立不等式にあたり、or でつないだもの（どちらかを満たす x）は連立不等式ではない。a < x < b の 1 行の形も、and の compound inequality をまとめて書いたもの。用例コーパスでは compound inequality は話し言葉より書き言葉に多く、書き言葉は大半が OpenStax Intermediate Algebra。日本語版 Wikipedia「不等式」は連立不等式を「同時に成り立つ不等式の組」と書く。OpenStax Intermediate Algebra は a < x < b の 1 行の形を double inequality とも呼ぶ（8 件）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 つ以上の不等式を組にして、それらを同時に満たす x の値の範囲を求めるもの。解は各不等式の解の共通範囲になる。",
    "definition_en": "Two or more inequalities in the same variable that must hold at the same time; the solution is the overlap of the solutions of the individual inequalities.",
    "latex": "\\begin{cases} 2x + 1 > 3 \\\\ x - 4 \\le 1 \\end{cases}",
    "spoken_en": "two x plus one is greater than three, and x minus four is less than or equal to one",
    "examples": [
      {
        "en": "This is an \"and\" compound inequality, so the answer is where both are true, the overlap on the number line.",
        "ja": "これは「かつ」の連立不等式なので、答えは両方が成り立つところ、つまり数直線上の共通範囲だ。",
        "register": "spoken"
      },
      {
        "en": "Solve the compound inequality 2x + 1 > 3 and x − 4 ≤ 1.",
        "ja": "連立不等式 2x + 1 > 3, x − 4 ≦ 1 を解け。",
        "register": "written"
      },
      {
        "en": "Solve the compound inequality −3 < 2x + 1 ≤ 7, and graph the solution set on a number line.",
        "ja": "連立不等式 −3 < 2x + 1 ≦ 7 を解き、解を数直線上に図示せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "or でつないだ compound inequality（x < −2 or x > 3）は連立不等式ではない。答えは 2 つの範囲を合わせたもの。",
      "共通範囲が無いときは no solution（解なし）。数直線に 2 つの解を描いて重なりを確かめるとよい。",
      "system of inequalities は、OpenStax では 2 変数の不等式の組をグラフで解く節（Elementary Algebra の Graphing Systems of Linear Inequalities ほか）の言い方で、高等学校学習指導要領解説（数学II 図形と方程式）の「不等式の表す領域」（慣習差 inequality-terms-scope、エントリ system-of-linear-inequalities）に当たる（書き言葉では OpenStax Elementary Algebra と OpenStax Algebra and Trigonometry に出てくるが、話し言葉にはほとんど出てこない）。1 変数の連立不等式の意味だけを数える形がないので、候補表現には数えない。"
    ],
    "related": [
      "linear-inequality",
      "solution-set",
      "number-line",
      "properties-of-inequalities",
      "system-of-linear-inequalities"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II 図形と方程式: 不等式の表す領域"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「不等式」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%8D%E7%AD%89%E5%BC%8F",
        "note": "連立不等式（同時に成り立つ不等式の組）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "compound inequality": 13
      },
      "written": {
        "compound inequality": 75
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "system-of-linear-equations",
    "ja": {
      "term": "連立一次方程式",
      "reading": "れんりついちじほうていしき",
      "alt": [
        "線形方程式系",
        "線型方程式系",
        "連立 1 次方程式",
        "連立方程式"
      ]
    },
    "en": {
      "term": "system of linear equations",
      "alt": [],
      "variants": [
        {
          "term": "linear system",
          "register": "spoken",
          "note": "話し言葉では system of linear equations より多いが、3Blue1Brown と MIT OCW（18.03 の微分方程式の系を含む）に偏る。書き言葉では system of linear equations のほうがずっと多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "中2",
        "数C",
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "1 次方程式をいくつか組にしたもの。中 2 の連立方程式（2 つの文字の一次方程式 2 つ）もその 1 つで、係数を並べた行列 A と定数のベクトル b を使って Ax = b とも書ける。",
    "definition_en": "A set of linear equations taken together; with the coefficient matrix A and the constant vector b it can be written Ax = b.",
    "latex": "A\\vec{x} = \\vec{b}",
    "spoken_en": null,
    "examples": [
      {
        "en": "A linear system can have no solution, exactly one, or infinitely many.",
        "ja": "連立一次方程式の解は、なし・ちょうど 1 つ・無数のどれかになる。",
        "register": "spoken"
      },
      {
        "en": "Write the system of linear equations as a matrix equation Ax = b.",
        "ja": "連立一次方程式を行列の方程式 Ax = b の形で表せ。",
        "register": "written"
      },
      {
        "en": "Solve the system of linear equations x + y = 10 and x − y = 4.",
        "ja": "連立方程式 x + y = 10、x − y = 4 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "中学の連立方程式は、1 次の式だけなら system of linear equations、一般には system of equations とも言う。system of equations は 1 次でない連立も含むので、この語の候補にしなかった。",
      "用例コーパスの話し言葉の linear system には、MIT 18.03 の線形微分方程式の系も混ざる。"
    ],
    "related": [
      "augmented-matrix",
      "gaussian-elimination",
      "homogeneous-system",
      "matrix"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "線型方程式系",
        "en": "System of linear equations"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "1.1 Solutions and Elementary Operations"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "system of linear equations": 23,
        "linear system": 41
      },
      "written": {
        "linear system": 40,
        "system of linear equations": 153
      },
      "sources": [
        "khan-middle",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "system-of-linear-inequalities",
    "ja": {
      "term": "2 変数の連立不等式",
      "reading": "にへんすうのれんりつふとうしき",
      "alt": [
        "連立不等式の表す領域",
        "連立不等式のグラフ"
      ]
    },
    "en": {
      "term": "system of linear inequalities",
      "alt": [],
      "variants": [
        {
          "term": "system of inequalities",
          "register": "written",
          "note": "linear を省いた言い方。教科書（OpenStax）では system of linear inequalities と同じくらい使う。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本の数I の「連立不等式」は 1 変数（英語の compound inequality、エントリ system-of-inequalities）。2 変数の連立不等式は数II の「不等式の表す領域」で扱う。米国の Algebra 1 の教科書（OpenStax Elementary Algebra）は、2 変数の一次不等式の組のグラフとして扱う。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "x と y の一次不等式を 2 つ以上組にしたもの。解はすべての不等式を同時に満たす点 (x, y) の集まりで、座標平面の領域として図示する。",
    "definition_en": "Two or more linear inequalities in x and y taken together. Its solution is the set of all points (x, y) that make every inequality true, shown as a shaded region of the coordinate plane.",
    "latex": "\\begin{cases} y > x + 1 \\\\ y \\le -2x + 4 \\end{cases}",
    "spoken_en": "y is greater than x plus one, and y is less than or equal to negative two x plus four",
    "examples": [
      {
        "en": "For a system of linear inequalities, graph both boundary lines, shade each one, and the answer is where the shading overlaps.",
        "ja": "連立不等式では、境界線を 2 本ともかいてそれぞれ塗る。塗った部分が重なるところが答えだよ。",
        "register": "spoken"
      },
      {
        "en": "Graph the system of linear inequalities y > x + 1 and y ≤ −2x + 4, and determine whether the point (0, 3) is a solution.",
        "ja": "連立不等式 y > x + 1、y ≦ −2x + 4 の表す領域を図示し、点 (0, 3) がその領域に含まれるかどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "境界線は、等号を含まない不等号（<、>）なら点線（dashed line）、含む不等号（≦、≧）なら実線（solid line）でかき、解の部分に色を塗る（shade）。",
      "日本の数I の「連立不等式」は 1 変数の不等式の組で、英語では compound inequality と言う（エントリ system-of-inequalities）。このエントリは 2 変数の不等式の組で、解は数の範囲ではなく平面の領域（エントリ region）。"
    ],
    "related": [
      "system-of-inequalities",
      "linear-inequality",
      "region",
      "feasible-region",
      "system-of-linear-equations"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "system of inequalities": 2
      },
      "written": {
        "system of linear inequalities": 28,
        "system of inequalities": 27
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-elemalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "system-of-recurrences",
    "ja": {
      "term": "連立漸化式",
      "reading": "れんりつぜんかしき",
      "alt": []
    },
    "en": {
      "term": "system of recurrences",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。Nicholson の Linear Algebra with Applications（3.3）は、2 つ以上の量を互いの前の値で表す漸化式を行列で vₖ₊₁ = Avₖ と書き、この条件を matrix recurrence、ベクトルの列を linear dynamical system と呼ぶ（線形の場合で、行列の対角化で解く）。日本語版 Wikipedia「数列」も、連立線型漸化式を係数行列 A を使って同じ形に書く。",
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "2 つ以上の数列の次の項を、それらの数列の前の項で表す漸化式の組。",
    "definition_en": "A set of recurrences that give the next terms of two or more sequences in terms of the earlier terms of all of them.",
    "latex": "a_{n+1} = a_n + 2b_n, \\quad b_{n+1} = a_n + b_n",
    "spoken_en": null,
    "examples": [
      {
        "en": "Add and subtract the two recurrences to decouple them.",
        "ja": "2 つの漸化式を足したり引いたりして、ばらばらにしよう。",
        "register": "spoken"
      },
      {
        "en": "The sequences {aₙ} and {bₙ} satisfy the system of recurrences aₙ₊₁ = 3aₙ + bₙ, bₙ₊₁ = aₙ + 3bₙ with a₁ = b₁ = 1. Find aₙ.",
        "ja": "数列 {aₙ}, {bₙ} は a₁ = b₁ = 1 と連立漸化式 aₙ₊₁ = 3aₙ + bₙ、bₙ₊₁ = aₙ + 3bₙ を満たす。aₙ を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "recurrence-relation",
      "solve-the-recurrence",
      "characteristic-equation"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "3.3 Diagonalization and Eigenvalues（matrix recurrence・linear dynamical system、Example 3.3.1）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数列」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%88%97",
        "note": "節「連立線型漸化式」: 2 つの数列の連立漸化式を係数行列で書く（mapping_note）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_(%E6%95%99%E7%A7%91)",
        "note": "新課程の学習内容: 数学B の漸化式に「二つの数列の漸化式」（level.jp 数B）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "system-of-three-equations",
    "ja": {
      "term": "三元連立一次方程式",
      "reading": "さんげんれんりついちじほうていしき",
      "alt": [
        "3 元 1 次連立方程式"
      ]
    },
    "en": {
      "term": "system of three equations",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "3 つの未知数を含む 1 次方程式 3 つを組にしたもの。",
    "definition_en": "A set of three linear equations in three unknowns.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Eliminate the same variable from two different pairs, and you're down to two equations.",
        "ja": "2 組の式から同じ文字を消すと、式が 2 つになる。",
        "register": "spoken"
      },
      {
        "en": "Solve the system of three equations in three variables: x + y + z = 2, x − y + 2z = 5, 2x + y − z = −1.",
        "ja": "連立方程式 x + y + z = 2, x − y + 2z = 5, 2x + y − z = −1 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は systems of three equations in three variables と書く（書き言葉はほとんどがこの本）。"
    ],
    "related": [
      "gaussian-elimination",
      "cramers-rule",
      "augmented-matrix"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "system of three equations": 3
      },
      "written": {
        "system of three equations": 32
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "systematic-sampling",
    "ja": {
      "term": "系統抽出",
      "reading": "けいとうちゅうしゅつ",
      "alt": [
        "系統抽出法"
      ]
    },
    "en": {
      "term": "systematic sampling",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "名簿の最初の 1 人を無作為に選び、そこから一定の間隔ごとに標本を取り出す方法。",
    "definition_en": "Choosing a random starting point in a list and then taking every kth member.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Start at a random name and take every tenth one after that.",
        "ja": "無作為に選んだ名前から始めて、そのあと 10 人おきに選ぶ。",
        "register": "spoken"
      },
      {
        "en": "Describe how to select a systematic sample of 50 from a list of 1,000 names.",
        "ja": "1,000 人の名簿から 50 人の系統抽出の標本を選ぶ方法を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "stratified-sampling",
      "cluster-sampling",
      "sampling"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "systematic sampling": 12
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "t-distribution",
    "ja": {
      "term": "t 分布",
      "reading": "てぃーぶんぷ",
      "alt": [
        "スチューデントの t 分布"
      ]
    },
    "en": {
      "term": "t-distribution",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "母標準偏差が分からないとき、標本平均を標本標準偏差で標準化した値が従う分布。標準正規分布に似た山形で、裾が厚く、自由度が大きいほど標準正規分布に近づく。",
    "definition_en": "The distribution of a sample mean standardized with the sample standard deviation; it is bell-shaped like the standard normal but has heavier tails, and it approaches the standard normal as the degrees of freedom grow.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "With only eight data points, we use the t-distribution instead of the z-table.",
        "ja": "データが 8 個しかないので、z 表ではなく t 分布を使う。",
        "register": "spoken"
      },
      {
        "en": "Find the critical value of the t-distribution with 14 degrees of freedom for a 95% confidence interval.",
        "ja": "95% 信頼区間のための、自由度 14 の t 分布の臨界値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は Student's t-distribution とも書く。見出しの t-distribution を中に含むので候補には置かなかった（含む関係）。",
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics。話し言葉にはほとんど出てこない。"
    ],
    "related": [
      "degrees-of-freedom",
      "standard-normal-distribution",
      "confidence-interval",
      "one-sample-t-test"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 4.2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "t-distribution": 3
      },
      "written": {
        "t-distribution": 65
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "t-test-for-the-slope",
    "ja": {
      "term": "傾きの t 検定",
      "reading": "かたむきのてぃーけんてい",
      "alt": []
    },
    "en": {
      "term": "t-test for the slope",
      "alt": [
        "test for the slope"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "AP Statistics の CED 2026 年版には出てこない（回帰は単元 5 で記述だけを扱い、傾きの推測はない）。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "回帰直線の傾きをもとに、母集団の回帰直線の傾きが 0（2 変数に直線的な関係がない）かどうかを t 分布で検定する方法。",
    "definition_en": "A t-test, based on the slope of the sample regression line, of whether the slope of the population regression line is zero.",
    "latex": "t = \\frac{b - 0}{SE_b}",
    "spoken_en": null,
    "examples": [
      {
        "en": "If the slope could really be zero, there's no linear relationship, and that's what the t-test for the slope checks.",
        "ja": "傾きが本当は 0 かもしれないなら直線的な関係はない。それを確かめるのが傾きの t 検定だ。",
        "register": "spoken"
      },
      {
        "en": "Conduct a t-test for the slope of the regression line at α = 0.05.",
        "ja": "α = 0.05 で回帰直線の傾きの t 検定を行え。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は傾きではなく相関係数の検定として扱う（節の名前 Testing the Significance of the Correlation Coefficient）。"
    ],
    "related": [
      "regression-line",
      "correlation-coefficient",
      "t-distribution"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Student%27s_t-test#Slope_of_a_regression_line",
        "note": "記事「Student's t-test」の節 Slope of a regression line（回帰直線の傾き β について帰無仮説 β = β₀ を t = (β̂ − β₀)/SE、自由度 n − 2 の t 分布で検定する。記事名 t-test と節の見出しで見出しの語になる。数学カテゴリから 4 段）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "table-of-trigonometric-ratios",
    "ja": {
      "term": "三角比の表",
      "reading": "さんかくひのひょう",
      "alt": [
        "三角比表"
      ]
    },
    "en": {
      "term": "right triangle table",
      "alt": [
        "trigonometric table",
        "trigonometry table",
        "table of trigonometric ratios"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "IM Geometry（4.6 Working with Trigonometric Ratios・4.9）は right triangle table と呼び、trigonometry table とも言う（1 件）。英語版 Wikipedia の記事名は Trigonometric table。用例コーパスにはどの言い方も出てこない。高等学校学習指導要領解説（数学I 内容の取扱い）は「三角比表」を積極的に利用すると書き、共通テスト（令和 4・6 年度 数学I 第2問）は問題冊子に「三角比の表」を付ける。IM の表は自分たちで測って作る直角三角形の比の表で、日本の三角比の表（0° から 90° の値の表）とは作りが違う。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "0° から 90° までの角について、sin・cos・tan の値の近似値をまとめた表。",
    "definition_en": "A table listing approximate values of sine, cosine, and tangent for angles from 0° to 90°.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Look it up in the right triangle table: tan 25° is about 0.4663.",
        "ja": "三角比の表で tan 25° を引いてごらん。0.4663 くらいだ。",
        "register": "spoken"
      },
      {
        "en": "Using the table of trigonometric ratios, find the acute angle θ such that sin θ = 0.6018.",
        "ja": "三角比の表を用いて、sin θ = 0.6018 となる鋭角 θ を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "表の値は小数第 4 位までの近似値。等号ではなく ≈ で書く（sin 25° ≈ 0.4226）。",
      "表は 0° から 90° まで。鈍角は sin(180° − θ) = sin θ、cos(180° − θ) = −cos θ で鋭角に直してから読む。"
    ],
    "related": [
      "trigonometric-ratio",
      "sine",
      "cosine",
      "tangent",
      "supplementary-angle-identity"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "4.6 Working with Trigonometric Ratios・4.9 Using Trigonometric Ratios to Find Angles: right triangle table（6 件）、trigonometry table（1 件）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Trigonometric_table",
        "note": "記事「Trigonometric table」（数学カテゴリから 4 段。冒頭で太字）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 内容の取扱い: 三角比表を積極的に利用する（2 件）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "令和 4・6 年度 数学I 第2問: 問題冊子の三角比の表"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 right triangle table（Geometry 4.6 Working with Trigonometric Ratios・Geometry 4.4 Ratios in Right Triangles・Geometry 4.9 Using Trigonometric Ratios to Find Angles・計 6 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "table-of-values",
    "ja": {
      "term": "対応表",
      "reading": "たいおうひょう",
      "alt": [
        "表"
      ]
    },
    "en": {
      "term": "table of values",
      "alt": [
        "input-output table",
        "x-y table"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "x のいくつかの値と、それに対応する y の値を並べて書いた表。x と y の変わり方を調べたり、グラフをかくための点を選んだりするのに使う。",
    "definition_en": "A table that lists several values of x together with the matching values of y. It helps you see how y changes as x changes and gives you points to plot for a graph.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's make a table of values for y = 2x + 1: when x is 0, y is 1, and when x is 1, y is 3.",
        "ja": "y = 2x + 1 の対応表を作ろう。x が 0 のとき y は 1、x が 1 のとき y は 3 だね。",
        "register": "spoken"
      },
      {
        "en": "Complete the table of values below for y = 12/x.",
        "ja": "y = 12/x について、下の対応表を完成させよ。",
        "register": "written"
      },
      {
        "en": "In this input-output table, the rule is multiply by 3, so an input of 4 gives an output of 12.",
        "ja": "この表のきまりは「3 をかける」だから、4 を入れると 12 が出てくる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "input-output table は、関数を「入れる数（input）」から「出てくる数（output）」が決まるものとして見た表。x が input、y が output に当たる。",
      "ja.alt の「表」をそのまま table と言うと、度数分布表（frequency table、エントリ frequency-table）など統計の表と区別がつかない。x と y の値を並べた表は table of values。"
    ],
    "related": [
      "function",
      "correspond",
      "graph",
      "direct-proportion",
      "inverse-proportion"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "table of values": 9
      },
      "written": {
        "table of values": 76
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "openstax-precalculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "take-the-average",
    "ja": {
      "term": "平均をとる",
      "reading": "へいきんをとる",
      "alt": []
    },
    "en": {
      "term": "calculate the mean",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "take the average",
          "register": "spoken",
          "note": "take the average of の形で数えた。話し言葉は Khan Academy と MIT OCW が中心で、書き言葉にはほとんど出てこない。"
        },
        {
          "term": "find the mean",
          "register": "both",
          "note": "話し言葉・書き言葉とも使い、書き言葉は OpenStax Prealgebra と OpenStax Introductory Statistics。"
        },
        {
          "term": "calculate the mean",
          "register": "both",
          "note": "話し言葉・書き言葉とも使う。話し言葉は大半が Khan Academy、書き言葉は大半が OpenStax Introductory Statistics。"
        },
        {
          "term": "find the average",
          "register": "spoken",
          "note": "find the average of の形で数えた（find the average rate of change ／ value を除くため）。話し言葉・書き言葉とも少ない。"
        }
      ],
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "いくつかの値の和を個数で割って、平均値を求める。",
    "definition_en": "To add up the values and divide by how many there are.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Take the average of the two middle numbers to get the median.",
        "ja": "真ん中の 2 つの数の平均をとると中央値になる。",
        "register": "spoken"
      },
      {
        "en": "Calculate the mean of the five test scores.",
        "ja": "5 回のテストの得点の平均値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは、話し言葉は take the average・calculate the mean・find the mean が並び、書き言葉は find the mean と calculate the mean が並ぶ（書き言葉は OpenStax Prealgebra と OpenStax Introductory Statistics）。",
      "find the average rate of change（平均変化率を求める）や find the average value（関数の平均値を求める）は別の意味。"
    ],
    "related": [
      "arithmetic-mean"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "take the average of": 22,
        "find the average of": 6,
        "calculate the mean": 15,
        "find the mean": 15
      },
      "written": {
        "find the average of": 7,
        "calculate the mean": 32,
        "find the mean": 41,
        "take the average of": 2
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "take-the-limit",
    "ja": {
      "term": "極限をとる",
      "reading": "きょくげんをとる",
      "alt": []
    },
    "en": {
      "term": "take the limit",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "変数を近づけたときに式がどんな値に近づくかを考え、lim をつけた式にする。",
    "definition_en": "To let a variable approach a value and find what an expression approaches.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Now take the limit as h goes to zero.",
        "ja": "ここで h → 0 の極限をとる。",
        "register": "spoken"
      },
      {
        "en": "Taking the limit of both sides as n → ∞ gives L = 1/2.",
        "ja": "両辺の n → ∞ の極限をとると L = 1/2 となる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "近づけ方は as ... で続ける（take the limit as h goes to 0）。"
    ],
    "related": [
      "limit",
      "as-x-approaches-a"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "take the limit": 157
      },
      "written": {
        "take the limit": 50
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "take-the-log-of-both-sides",
    "ja": {
      "term": "対数をとる",
      "reading": "たいすうをとる",
      "alt": [
        "両辺の対数をとる"
      ]
    },
    "en": {
      "term": "take the log",
      "alt": [],
      "variants": [
        {
          "term": "take the natural log",
          "register": "spoken",
          "note": "底 e の対数をとるとき。話し言葉は大半が Khan Academy。書き言葉では少ない。"
        },
        {
          "term": "take the log",
          "register": "spoken",
          "note": "話し言葉は 3Blue1Brown と MIT OCW が中心。書き言葉では少ない。"
        },
        {
          "term": "take the natural logarithm",
          "register": "written",
          "note": "書き言葉の言い方（OpenStax Calculus Volume 1 3.9 の対数微分法の Take the natural logarithm of both sides、OpenStax Algebra and Trigonometry 6.6）。"
        },
        {
          "term": "take the logarithm",
          "register": "written",
          "note": "書き言葉の言い方（OpenStax Algebra and Trigonometry 6.6 の taking the logarithm of each side、Intermediate Algebra 10.5）。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "等式（不等式）の両辺の対数を考えて、指数を係数に下ろす。",
    "definition_en": "To apply a logarithm to each side of an equation (or inequality) so that exponents come down as coefficients.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Take the log of both sides, and the x comes down in front.",
        "ja": "両辺の対数をとると、x が前に下りてくる。",
        "register": "spoken"
      },
      {
        "en": "Solve 3^x = 20 by taking the logarithm of both sides. Give the answer to three decimal places.",
        "ja": "両辺の対数をとって 3ˣ = 20 を解け。答えは小数第 3 位まで求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "書き言葉（OpenStax Algebra and Trigonometry・Calculus）は log を略さず、take the logarithm of each side ／ take the natural logarithm of both sides と書く。",
      "底を 10 にするか e にするかは自由（OpenStax Intermediate Algebra（10.5）は、どちらをとっても同じ結果になると書く）。OpenStax Algebra and Trigonometry（6.6）の手順は、底 10 の項があれば常用対数、なければ自然対数をとる。"
    ],
    "related": [
      "logarithm",
      "natural-logarithm",
      "exponential-equation",
      "logarithmic-differentiation",
      "undo-the-log"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.6 Exponential and Logarithmic Equations（take the logarithm of each side）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "10.5 Solve Exponential and Logarithmic Equations（take the logarithm）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "3.9 Derivatives of Exponential and Logarithmic Functions（Take the natural logarithm of both sides）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "take the log": 12,
        "take the logarithm": 7,
        "take the natural log": 21,
        "take the natural logarithm": 2
      },
      "written": {
        "take the natural logarithm": 13,
        "take the logarithm": 11,
        "take the log": 8,
        "take the natural log": 7
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "take-the-partial-derivative",
    "ja": {
      "term": "偏微分する",
      "reading": "へんびぶんする",
      "alt": []
    },
    "en": {
      "term": "take the partial derivative",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "多変数関数を、1 つの変数について（ほかを定数とみなして）微分する。",
    "definition_en": "To differentiate a function of several variables with respect to one variable, treating the others as constants.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Take the partial derivative with respect to y — x just comes along for the ride.",
        "ja": "y について偏微分しよう。x はそのままついてくるだけ。",
        "register": "spoken"
      },
      {
        "en": "Find ∂f/∂x for f(x, y) = e^{xy} sin y.",
        "ja": "f(x, y) = e^{xy} sin y を x について偏微分せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では derivative を省いて take the partial とも言う。"
    ],
    "related": [
      "partial-derivative",
      "differentiate"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "take the partial derivative": 26
      },
      "written": {
        "take the partial derivative": 2
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "tally",
    "ja": {
      "term": "集計する",
      "reading": "しゅうけいする",
      "alt": []
    },
    "en": {
      "term": "tally",
      "alt": [],
      "uk": null
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "集めたデータを数え、種類や階級ごとの個数にまとめる。",
    "definition_en": "To count the data and total them up by category or class.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Tally the results first, then turn the tallies into frequencies.",
        "ja": "まず結果を集計して、それを度数にする。",
        "register": "spoken"
      },
      {
        "en": "Tally the responses and make a frequency table.",
        "ja": "回答を集計し、度数分布表を作れ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "集計したものを表にしたのが度数分布表（frequency table）。"
    ],
    "related": [
      "frequency-table",
      "frequency"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "tally sheet（名詞）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "動詞の tally（数え上げて記録する）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "tally": 2
      },
      "written": {
        "tally": 7
      },
      "sources": [
        "openstax-algtrig",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは OpenStax の呼び方 tally（本文 7 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "tangent-chord-theorem",
    "ja": {
      "term": "接弦定理",
      "reading": "せつげんていり",
      "alt": [
        "接線と弦のなす角"
      ]
    },
    "en": {
      "term": "chord/tangent angle theorem",
      "alt": [
        "tangent-chord theorem",
        "tangent-chord angle"
      ]
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "CK-12 Geometry はこの定理を Chord/Tangent Angle Theorem と呼び、接線と弦のなす角はその角の内部にある弧の中心角の半分（half the measure of the intercepted arc）と述べる（6.16 Angles On and Inside a Circle、5 件）。見出しは CK-12 の呼び方。用例コーパスには chord/tangent angle theorem・tangent-chord theorem・tangent-chord angle が話・書とも出てこない。日本の接弦定理の形（接線と弦のなす角は、その角の内部にある弧に対する円周角に等しい）は、the angle between a tangent and a chord equals the inscribed angle on the other side of the chord のように文で言う（本プロジェクトの書き方の例）。英語版 Wikipedia「Circumcircle」は同じ定理を alternate segment theorem と呼ぶ（地域は書いていない）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円の接線と、接点を通る弦のなす角は、その角の内部にある弧に対する円周角に等しいという定理。",
    "definition_en": "The angle between a tangent to a circle and a chord drawn from the point of tangency equals any inscribed angle that intercepts the arc inside that angle; equivalently, it is half the measure of that arc.",
    "latex": "\\angle BAT = \\angle ACB",
    "spoken_en": "angle BAT equals angle ACB",
    "examples": [
      {
        "en": "The tangent-chord angle is half the intercepted arc, so if the arc is 100°, the angle is 50°.",
        "ja": "接線と弦のなす角は、その角の内部の弧の中心角の半分なので、弧が 100° なら角は 50°。",
        "register": "spoken"
      },
      {
        "en": "Line AT is tangent to the circle at A, and C is a point on the circle on the other side of chord AB. If ∠BAT = 65°, find ∠ACB.",
        "ja": "直線 AT は点 A で円に接し、点 C は弦 AB に関して T と反対側の円周上にある。∠BAT = 65° のとき、∠ACB を求めよ。",
        "register": "written"
      },
      {
        "en": "By the chord/tangent angle theorem, ∠ACB = ∠BAT = 65°.",
        "ja": "接弦定理より ∠ACB = ∠BAT = 65°。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "等しくなるのは、接線と弦のなす角の内部にある弧に対する円周角。反対側の弧に対する円周角と取り違えない。"
    ],
    "related": [
      "inscribed-angle",
      "chord",
      "tangent-line",
      "point-of-tangency",
      "power-of-a-point"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Circumcircle",
        "note": "記事「Circumcircle」: 接線と弦のなす角が反対側の弧の円周角に等しいことを alternate segment theorem と呼ぶ"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.16 Angles On and Inside a Circle（Chord/Tangent Angle Theorem、5 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 chord/tangent angle theorem（CK-12 Geometry 6.16 Angles On and Inside a Circle・計 5 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "tangent-line",
    "ja": {
      "term": "接線",
      "reading": "せっせん",
      "alt": []
    },
    "en": {
      "term": "tangent line",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数II",
        "数III"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "曲線（円など）と 1 点で接する直線。その点を接点という。",
    "definition_en": "A line that touches a curve, such as a circle, at a point without crossing it there; that point is the point of tangency.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The tangent line to a circle is perpendicular to the radius at the point of tangency.",
        "ja": "円の接線は、接点を通る半径に垂直。",
        "register": "spoken"
      },
      {
        "en": "Find the equation of the tangent line to the circle x² + y² = 25 at the point (3, 4).",
        "ja": "円 x² + y² = 25 上の点 (3, 4) における接線の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "tangent line to the circle",
        "ja": "円の接線"
      }
    ],
    "pitfalls": [
      "円 x² + y² = r² 上の点 (x₁, y₁) における接線は x₁x + y₁y = r²。",
      "tangent だけでは三角比の tan とまぎれるので、線の意味では tangent line と言う。「接する」は is tangent to。"
    ],
    "related": [
      "point-of-tangency",
      "slope-of-the-tangent-line",
      "normal-line",
      "find-the-equation-of-the-tangent-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "接線",
        "en": "Tangent"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "tangent line": 701
      },
      "written": {
        "tangent line": 322
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "tangent-plane",
    "ja": {
      "term": "接平面",
      "reading": "せっへいめん",
      "alt": []
    },
    "en": {
      "term": "tangent plane",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "曲面上の点で曲面に接する平面。z = f(x, y) では z − z₀ = f_x(x − x₀) + f_y(y − y₀)。",
    "definition_en": "The plane that touches a surface at a point; for z = f(x, y) it is z − z₀ = f_x(x − x₀) + f_y(y − y₀).",
    "latex": "z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)",
    "spoken_en": null,
    "examples": [
      {
        "en": "The tangent plane is the 3D version of the tangent line.",
        "ja": "接平面は、接線の 3 次元版。",
        "register": "spoken"
      },
      {
        "en": "Find an equation of the tangent plane to z = x² + y² at the point (1, 2, 5).",
        "ja": "z = x² + y² の点 (1, 2, 5) における接平面の方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "partial-derivative",
      "equation-of-a-plane",
      "linear-approximation",
      "total-differential",
      "gradient"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "tangent plane": 18
      },
      "written": {
        "tangent plane": 74
      },
      "sources": [
        "mit-18.02",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "tangent-problem",
    "ja": {
      "term": "接線問題",
      "reading": "せっせんもんだい",
      "alt": []
    },
    "en": {
      "term": "tangent problem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「接線問題」は学習指導要領解説に無い（日本語版 Wikipedia「解析学」は「曲線の接線問題」と書く）。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "曲線上の点で接線の傾きを求めるという問題。米国の教科書（OpenStax Calculus）では微分の導入に使う。",
    "definition_en": "The problem of how steep a curve is at a given point — the slope of its tangent line there — used to introduce the derivative.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The tangent problem leads to derivatives, and the area problem leads to integrals.",
        "ja": "接線問題から微分が、面積問題から積分が生まれる。",
        "register": "spoken"
      },
      {
        "en": "The tangent problem asks for the slope of the line tangent to y = x² at (1, 1).",
        "ja": "接線問題とは、たとえば点 (1, 1) における y = x² の接線の傾きを求める問題である。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "area-problem",
      "slope-of-the-tangent-line",
      "derivative"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「解析学」",
        "url": "https://ja.wikipedia.org/wiki/%E8%A7%A3%E6%9E%90%E5%AD%A6",
        "note": "ライプニッツが曲線の接線問題を解くのに dy/dx を考案した"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「接線問題」は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "tangent problem": 5
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは OpenStax の呼び方 tangent problem（本文 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "tangent-segments-are-equal",
    "ja": {
      "term": "接線の長さは等しい",
      "reading": "せっせんのながさはひとしい",
      "alt": []
    },
    "en": {
      "term": "tangent segments are equal",
      "alt": [
        "two tangents theorem"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "CK-12 Geometry は tangent segments と呼ぶ（6.18 Tangent Lines ほか、3 件）。用例コーパスには tangent segments がほとんど出てこない（話し言葉の blackpenredpen だけで、書き言葉には出てこない）。CK-12 Geometry はこの性質を Two Tangents Theorem と呼び、if two tangent segments are drawn to one circle from the same external point, then they are congruent と文で述べる（6.18 Tangent Lines）。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "円の外部の点 P から円に 2 本の接線を引き、接点を A, B とすると PA = PB が成り立つという性質。",
    "definition_en": "If two tangent lines are drawn to a circle from a point P outside it, touching the circle at A and B, then PA = PB.",
    "latex": "PA = PB",
    "spoken_en": null,
    "examples": [
      {
        "en": "PA and PB are tangent segments from the same point, so they have to be equal.",
        "ja": "PA と PB は同じ点から引いた接線だから、長さは等しくなるはずだ。",
        "register": "spoken"
      },
      {
        "en": "From a point P outside a circle, tangent lines touch the circle at A and B. Prove that PA = PB.",
        "ja": "円の外部の点 P から引いた 2 本の接線が円と A, B で接している。PA = PB を証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「接線の長さ」は、外部の点から接点までの線分の長さのこと。英語の tangent line は限りなく伸びる直線なので、長さを言うときは tangent segment か PA のように線分で言う。",
      "内接円の問題では、各頂点から接点までの 2 つの長さが等しいことを使って辺の長さを分ける。"
    ],
    "related": [
      "tangent-line",
      "inscribed-circle",
      "incenter",
      "power-of-a-point"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.18 Tangent Lines・6.2 Identify Circle Components"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "tangent segments": 1
      },
      "sources": [
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 tangent segments（CK-12 Geometry 6.18 Tangent Lines・CK-12 Geometry 6.2 Identify Circle Components・計 3 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "tangent",
    "ja": {
      "term": "正接",
      "reading": "せいせつ",
      "alt": [
        "タンジェント",
        "tan"
      ]
    },
    "en": {
      "term": "tangent",
      "alt": [
        "tan"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "角 θ について sin θ / cos θ で定める三角関数 tan θ。直角三角形では（対辺）/（隣辺）。",
    "definition_en": "The trigonometric function tan θ = sin θ / cos θ; in a right triangle, opposite over adjacent.",
    "latex": "\\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta}",
    "spoken_en": "tangent theta equals sine theta over cosine theta",
    "examples": [
      {
        "en": "The tangent of 45 degrees is 1.",
        "ja": "45° の正接（tan 45°）は 1。",
        "register": "spoken"
      },
      {
        "en": "Given that sin θ = 3/5 and θ is in the second quadrant, find tan θ.",
        "ja": "sin θ = 3/5 で θ が第 2 象限の角のとき、tan θ の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "tan θ は tangent theta とも tan theta とも読む（用例コーパスの話し言葉では tangent of の形が多く、tan of は少ない）。",
      "tangent だけでは接線（tangent line）とまぎれる。",
      "1 + tan²θ = 1/cos²θ のような相互関係は trigonometric identities（三角比の相互関係）を参照。"
    ],
    "related": [
      "trigonometric-identities",
      "tangent-line",
      "trigonometric-function",
      "sine",
      "cosine",
      "trigonometric-ratio"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "tangent of": 246,
        "tan of": 9
      },
      "written": {
        "tangent of": 26
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "tautology",
    "ja": {
      "term": "トートロジー",
      "reading": "とーとろじー",
      "alt": [
        "恒真式"
      ]
    },
    "en": {
      "term": "tautology",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "もとになる命題の真偽がどうであっても、常に真になる命題（論理式）のこと。",
    "definition_en": "A compound statement that is true for every possible combination of truth values of its parts.",
    "latex": "p \\lor \\lnot p",
    "spoken_en": "p or not p",
    "examples": [
      {
        "en": "Every row in the last column came out true, so this statement is a tautology.",
        "ja": "最後の列がどの行も真になったから、この命題はトートロジーだ。",
        "register": "spoken"
      },
      {
        "en": "Use a truth table to show that (p ∧ q) → p is a tautology.",
        "ja": "真理値表を用いて、(p ∧ q) → p が恒真式であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日常英語の tautology は「同じことを言い換えて繰り返すだけの言い方（同語反復）」の意味で使う。",
      "逆に、常に偽になる命題は contradiction（矛盾式）と呼ぶ。",
      "用例コーパスには話・書とも出てこない。CK-12 Geometry では 2.10 Negative Statements に 6 件、Levin の本では 6 件。"
    ],
    "related": [
      "truth-table",
      "propositional-logic",
      "contradiction",
      "logical-connective"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.10 Negative Statements"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 tautology（CK-12 Geometry 2.10 Negative Statements・計 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "taylor-polynomial",
    "ja": {
      "term": "テイラー多項式",
      "reading": "ていらーたこうしき",
      "alt": []
    },
    "en": {
      "term": "Taylor polynomial",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "関数 f を点 a のまわりで近似する n 次多項式。a での 0 階から n 階までの微分係数が f と一致する。",
    "definition_en": "The degree-n polynomial that approximates f near a, matching f and its first n derivatives at a.",
    "latex": "P_n(x) = \\sum_{k=0}^{n} \\frac{f^{(k)}(a)}{k!}(x-a)^k",
    "spoken_en": "P sub n of x equals the sum from k equals zero to n of the kth derivative of f at a over k factorial times x minus a to the k",
    "examples": [
      {
        "en": "The more terms you keep in the Taylor polynomial, the better the approximation near a.",
        "ja": "テイラー多項式の項を増やすほど、a の近くの近似がよくなる。",
        "register": "spoken"
      },
      {
        "en": "Find the third-degree Taylor polynomial for f(x) = ln x centered at x = 1.",
        "ja": "f(x) = ln x の x = 1 を中心とする 3 次のテイラー多項式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a = 0 のときは Maclaurin polynomial とも呼ぶ。「〜を中心とする」は centered at x = a。"
    ],
    "related": [
      "taylor-series",
      "maclaurin-series",
      "lagrange-error-bound",
      "taylors-theorem",
      "linear-approximation"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.11（Taylor polynomial）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "Taylor polynomial": 23
      },
      "written": {
        "Taylor polynomial": 46
      },
      "sources": [
        "khan-ap-calc",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "taylor-series",
    "ja": {
      "term": "テイラー級数",
      "reading": "ていらーきゅうすう",
      "alt": [
        "テイラー展開"
      ]
    },
    "en": {
      "term": "Taylor series",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "関数 f の点 a での各階の微分係数からつくるべき級数 Σf⁽ⁿ⁾(a)(x − a)ⁿ/n!。",
    "definition_en": "The power series Σf⁽ⁿ⁾(a)(x − a)ⁿ/n! built from the derivatives of f at a.",
    "latex": "f(x) = \\sum_{n=0}^{\\infty} \\frac{f^{(n)}(a)}{n!}(x-a)^n",
    "spoken_en": "f of x equals the sum from n equals zero to infinity of the nth derivative of f at a over n factorial times x minus a to the n",
    "examples": [
      {
        "en": "The Taylor series is just the Taylor polynomial that never stops.",
        "ja": "テイラー級数は、どこまでも続くテイラー多項式。",
        "register": "spoken"
      },
      {
        "en": "Find the Taylor series for f(x) = e^x centered at x = 2.",
        "ja": "f(x) = e^x の x = 2 を中心とするテイラー級数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「テイラー展開する」は expand … in a Taylor series ／ find the Taylor series of … と言う。"
    ],
    "related": [
      "maclaurin-series",
      "taylor-polynomial",
      "power-series",
      "taylors-theorem",
      "binomial-series"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "テイラー展開",
        "en": "Taylor series"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.14（Taylor series）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "Taylor series": 140
      },
      "written": {
        "Taylor series": 82
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "taylors-theorem",
    "ja": {
      "term": "テイラーの定理",
      "reading": "ていらーのていり",
      "alt": []
    },
    "en": {
      "term": "Taylor's theorem",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "関数を n 次テイラー多項式と剰余項 Rₙ(x) の和で表し、剰余項を (n + 1) 階導関数で表す定理。",
    "definition_en": "The theorem that writes a function as its nth Taylor polynomial plus a remainder Rₙ(x), expressed with the (n + 1)st derivative.",
    "latex": "f(x) = P_n(x) + R_n(x), \\quad R_n(x) = \\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Taylor's theorem tells you exactly what the leftover piece looks like.",
        "ja": "テイラーの定理は、残りの部分がどんな形かを正確に教えてくれる。",
        "register": "spoken"
      },
      {
        "en": "Use Taylor's theorem with remainder to estimate the error in approximating sin 0.2 by 0.2.",
        "ja": "剰余項つきのテイラーの定理を用いて、sin 0.2 を 0.2 で近似したときの誤差を評価せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは話し言葉には出てこず、書き言葉はすべて OpenStax Calculus。AP では剰余の評価を Lagrange error bound と呼ぶ（lagrange-error-bound を参照）。"
    ],
    "related": [
      "taylor-polynomial",
      "lagrange-error-bound",
      "taylor-series"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "テイラーの定理",
        "en": "Taylor's theorem"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "Taylor's theorem": 14
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "telescoping-series",
    "ja": {
      "term": "望遠鏡級数",
      "reading": "ぼうえんきょうきゅうすう",
      "alt": []
    },
    "en": {
      "term": "telescoping series",
      "alt": [
        "telescoping sum"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「望遠鏡級数」は学習指導要領解説に無い（日本語版 Wikipedia「畳み込み級数」は telescoping series の別名として挙げる）。日本の数B では Σ1/(k(k + 1)) を 1/k − 1/(k + 1) に分けて途中を消す方法として扱い、級数に名前をつけない。英語では消えていく様子を望遠鏡の筒にたとえて telescoping と呼ぶ。",
    "domains": [
      "sequences",
      "calculus"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "項を差の形に分けると、隣り合う項が打ち消し合って最初と最後の部分だけが残る級数。",
    "definition_en": "A series whose terms, written as differences, cancel in pairs so that only the first and last pieces of each partial sum remain.",
    "latex": "\\sum_{k=1}^{n} \\left(\\frac{1}{k} - \\frac{1}{k+1}\\right) = 1 - \\frac{1}{n+1}",
    "spoken_en": "the sum from k equals one to n of one over k minus one over k plus one equals one minus one over n plus one",
    "examples": [
      {
        "en": "Split it with partial fractions and watch the middle terms cancel — it telescopes.",
        "ja": "部分分数に分けると、途中の項が消えていく。望遠鏡のように縮む。",
        "register": "spoken"
      },
      {
        "en": "Show that Σ 1/(n(n + 1)) is a telescoping series and find its sum.",
        "ja": "Σ 1/(n(n + 1)) が望遠鏡級数であることを示し、その和を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞 telescope（途中が消える）も使う。"
    ],
    "related": [
      "partial-sum",
      "partial-fraction-decomposition",
      "sum-of-the-series"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「畳み込み級数」",
        "url": "https://ja.wikipedia.org/wiki/%E7%95%B3%E3%81%BF%E8%BE%BC%E3%81%BF%E7%B4%9A%E6%95%B0",
        "note": "英: telescoping series; 望遠鏡級数"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "telescoping series": 1
      },
      "written": {
        "telescoping series": 15,
        "telescoping sum": 3
      },
      "sources": [
        "openstax-calculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "term-by-term-differentiation",
    "ja": {
      "term": "級数の項別微分",
      "reading": "きゅうすうのこうべつびぶん",
      "alt": [
        "項別微分"
      ]
    },
    "en": {
      "term": "term-by-term differentiation",
      "alt": [
        "differentiate term-by-term"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "べき級数の各項を微分して、もとの関数の導関数のべき級数を得ること。収束半径は変わらない。",
    "definition_en": "Differentiating a power series one term at a time to get a power series for the derivative; the radius of convergence stays the same.",
    "latex": "\\frac{d}{dx}\\sum c_n x^n = \\sum n c_n x^{n-1}",
    "spoken_en": "d d x of the sum of c sub n x to the n equals the sum of n c sub n x to the n minus one",
    "examples": [
      {
        "en": "Differentiate the geometric series term by term and you get the series for 1/(1 − x)².",
        "ja": "等比級数を項別微分すると、1/(1 − x)² の級数が得られる。",
        "register": "spoken"
      },
      {
        "en": "Use term-by-term differentiation to find a power series for 1/(1 + x)².",
        "ja": "項別微分を用いて 1/(1 + x)² のべき級数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "項別積分は term-by-term integration。"
    ],
    "related": [
      "power-series",
      "radius-of-convergence",
      "taylor-series"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 10.13・10.15（term-by-term differentiation）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "differentiate term-by-term": 2
      },
      "written": {
        "term-by-term differentiation": 7,
        "differentiate term-by-term": 5
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満）。見出しは CED の呼び方 term-by-term differentiation（topic 10.13・topic 10.15）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "term",
    "ja": {
      "term": "項",
      "reading": "こう",
      "alt": []
    },
    "en": {
      "term": "term",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "式を + と − で区切ったひとつひとつの部分で、符号も含めて考える（3x − 5 の項は 3x と −5）。数列に並ぶひとつひとつの数も項という。",
    "definition_en": "One of the parts of an expression that are added or subtracted, taken together with its sign; the terms of 3x − 5 are 3x and −5. Each number in a sequence is also called a term.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "How many terms does 4x² − x + 7 have? Three: 4x squared, negative x, and 7.",
        "ja": "4x² − x + 7 の項はいくつ？ 4x²、−x、7 の 3 つだね。",
        "register": "spoken"
      },
      {
        "en": "Find the 10th term of the sequence 3, 7, 11, 15, …",
        "ja": "数列 3, 7, 11, 15, … の第 10 項を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "項は符号も含めて見る。3x − 5 の項は 3x と 5 ではなく、3x と −5。",
      "in terms of は「項」とは別の意味の熟語で、「〜を使って（表す）」。write y in terms of x は「y を x の式で表す」。",
      "項（term）は + と − で区切った部分、因数（factor）は掛け算で区切った部分。3x の 3 と x は因数で、項ではない。"
    ],
    "related": [
      "like-terms",
      "constant-term",
      "coefficient",
      "algebraic-expression",
      "first-term"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!in term": 4510
      },
      "written": {
        "!in term": 4086
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "terminal-point",
    "ja": {
      "term": "終点",
      "reading": "しゅうてん",
      "alt": []
    },
    "en": {
      "term": "terminal point",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "書き言葉は terminal point（OpenStax）。話し言葉では矢印の先の意味で the tip とも言うが、用例コーパスの tip は「こつ」などの意味も多いので数えていない。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "有向線分（ベクトル）の到着点。矢印の先。",
    "definition_en": "The end point of a directed line segment (vector): the tip of the arrow.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The terminal point is where the arrow ends.",
        "ja": "終点は矢印が終わるところ。",
        "register": "spoken"
      },
      {
        "en": "Find the terminal point of the vector ⟨3, −2⟩ if its initial point is (1, 5).",
        "ja": "始点が (1, 5) のとき、ベクトル ⟨3, −2⟩ の終点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "initial-point",
      "directed-segment",
      "vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "terminal point": 2
      },
      "written": {
        "terminal point": 141
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "terminal-side",
    "ja": {
      "term": "動径",
      "reading": "どうけい",
      "alt": []
    },
    "en": {
      "term": "terminal side",
      "alt": [
        "terminal ray"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "角を、始線を回転させてつくるとみたときの、回転したあとの半直線。",
    "definition_en": "The ray where an angle ends when the angle is made by rotating the initial side.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The terminal side of 210 degrees is in the third quadrant.",
        "ja": "210° の動径は第 3 象限にある。",
        "register": "spoken"
      },
      {
        "en": "Find sin θ and cos θ if the terminal side of θ passes through the point (−3, 4).",
        "ja": "θ の動径が点 (−3, 4) を通るとき、sin θ と cos θ の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry（7.1 Angles）は、固定した半直線を initial side、回転させた半直線を terminal side と呼ぶ。「動径 OP の表す角」は the angle whose terminal side is OP。",
      "日本語版 Wikipedia「半径」（「動径」はここへの転送）と「極座標系」は、極座標の r（原点からの距離）も動径と呼ぶ。この r は OpenStax Calculus Volume 2（7.3 Polar Coordinates）では radial coordinate で、terminal side ではない（エントリ polar-coordinates）。"
    ],
    "related": [
      "initial-side",
      "general-angle",
      "reference-angle",
      "coterminal-angle",
      "quadrantal-angle",
      "polar-coordinates"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles（initial side、terminal side）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 6.9（the terminal ray of the angle）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和4年度 大学入学共通テスト 本試験 数学Ⅱ（問題）",
        "url": "https://web.archive.org/web/20230601/https://www.dnc.ac.jp/albums/abm.php?d=67&f=abm00000831.pdf&n=R4%E5%95%8F%E9%A1%8C_%E5%95%8F%E9%A1%8C_%E6%95%B0%E5%AD%A6%E2%85%A1.pdf",
        "note": "第3問: β の動径は第…象限にあり、角の動径は x 軸の正の部分を始線とする"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「半径」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8D%8A%E5%BE%84",
        "note": "「動径」はここへの転送。極座標の r（座標系の動径）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「極座標系」",
        "url": "https://ja.wikipedia.org/wiki/%E6%A5%B5%E5%BA%A7%E6%A8%99%E7%B3%BB",
        "note": "1 個の動径 r と n − 1 個の偏角"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "7.3 Polar Coordinates（radial coordinate、the terminal ray of the angle）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "terminal side": 44,
        "terminal ray": 7
      },
      "written": {
        "terminal side": 52,
        "terminal ray": 2
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "terminating-decimal",
    "ja": {
      "term": "有限小数",
      "reading": "ゆうげんしょうすう",
      "alt": []
    },
    "en": {
      "term": "terminating decimal",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中3",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "小数点以下の数字が、ある位で終わる小数。0.25 や 1.375 などで、有限小数は分数で表せる数（有理数）である。",
    "definition_en": "A decimal whose digits after the decimal point stop after a certain place, such as 0.25 or 1.375. Every terminating decimal is a rational number.",
    "latex": "\\frac{3}{8} = 0.375",
    "spoken_en": "three eighths equals zero point three seven five",
    "examples": [
      {
        "en": "Three eighths is a terminating decimal: divide it out and you get 0.375, and then it just stops.",
        "ja": "8 分の 3 は有限小数。割り算をすると 0.375 になって、そこで終わる。",
        "register": "spoken"
      },
      {
        "en": "Select all of the following fractions that can be written as a terminating decimal: 3/8, 5/12, 7/20.",
        "ja": "次の分数のうち、有限小数で表せるものをすべて選べ。3/8、5/12、7/20",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "terminate は「終わる」という意味の動詞。terminating decimal は「桁がそこで終わる小数」という言い方で、「有限」に当たる語は入っていない。",
      "有理数を小数で表すと、有限小数か循環小数（repeating decimal、エントリ repeating-decimal）のどちらかになる。有限小数でない小数は無限小数（エントリ non-terminating-decimal）。",
      "既約分数が有限小数になるのは、分母の素因数が 2 と 5 だけのとき（3/8 = 0.375、7/20 = 0.35）。5/12 のように分母に 3 などの素因数があると循環小数になる。"
    ],
    "related": [
      "decimal",
      "repeating-decimal",
      "non-terminating-decimal",
      "rational-number",
      "prime-factorization"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "1.1 Real Numbers"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "terminating decimal": 1
      },
      "written": {
        "terminating decimal": 5
      },
      "sources": [
        "khan-middle",
        "openstax-algtrig"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 5 件（10 未満））。見出しは CK-12 の呼び方 terminating decimal（CK-12 Algebra 1.1 Real Numbers・CK-12 Algebra 2.4.3 Multi-Step Equations with Fractions・CK-12 Algebra 2.4.4 Multi-Step Equations with Decimals・計 8 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "tessellation",
    "ja": {
      "term": "平面充填",
      "reading": "へいめんじゅうてん",
      "alt": [
        "敷き詰め",
        "テセレーション"
      ]
    },
    "en": {
      "term": "tessellation",
      "alt": [
        "tiling"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "1 種類または何種類かの図形を、すき間も重なりもなく平面に敷き詰めること。また、そうして敷き詰めた模様。",
    "definition_en": "A pattern of one or more shapes that covers a plane with no gaps and no overlaps. The word also names the act of covering the plane this way.",
    "latex": "6 \\cdot 60^\\circ = 360^\\circ",
    "spoken_en": "6 times 60 degrees equals 360 degrees",
    "examples": [
      {
        "en": "Six equilateral triangles fit around a point with no gaps, because six times 60 is 360, so they make a tessellation.",
        "ja": "正三角形は 1 点のまわりに 6 つすき間なく集まる。60° の 6 倍が 360° だからだ。だから平面を敷き詰められる。",
        "register": "spoken"
      },
      {
        "en": "Explain why regular pentagons cannot form a tessellation of the plane.",
        "ja": "正五角形だけでは平面をすき間なく敷き詰められない理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "動詞は tessellate（敷き詰める）。Does this shape tessellate? のように、図形を主語にして使う。",
      "正多角形 1 種類で敷き詰められるのは正三角形・正方形・正六角形だけ。1 つの頂点に集まる角の和が 360° になるかで説明する。",
      "用例コーパスには話・書とも出てこない。IM では 45 件（Grade 8 9.1 Tessellations of the Plane 13 ほか）、CK-12 Geometry では 8.19 Tessellations に 12 件。"
    ],
    "related": [
      "rigid-motion",
      "rotational-symmetry"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "Grade 8 glossary・Grade 8 9.1 Tessellations of the Plane"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.19 Tessellations"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは IM の呼び方 tessellation（Grade 8 glossary・Geometry glossary・Grade 8 9.1 Tessellations of the Plane・計 45 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "test-for-homogeneity",
    "ja": {
      "term": "同質性の検定",
      "reading": "どうしつせいのけんてい",
      "alt": [
        "一様性の検定"
      ]
    },
    "en": {
      "term": "test for homogeneity",
      "alt": [
        "test of homogeneity"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "2 つ以上の母集団で、1 つのカテゴリ変数の分布が同じかどうかを χ² で検定する方法。",
    "definition_en": "A chi-square test of whether the distribution of one categorical variable is the same in two or more populations.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We sampled each school separately, so this is homogeneity, not independence.",
        "ja": "学校ごとに別々に標本をとったので、独立性ではなく同質性の検定だ。",
        "register": "spoken"
      },
      {
        "en": "Perform a test for homogeneity to compare the distributions of blood types in the three regions.",
        "ja": "同質性の検定を行い、3 地域の血液型の分布を比べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "計算は独立性の検定と同じだが、データの集め方（母集団ごとに標本をとる）が違う。",
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics（test for homogeneity と test of homogeneity の両方を使い、test for homogeneity のほうが多い）。"
    ],
    "related": [
      "chi-square-test",
      "test-for-independence"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.14・3.15"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "test for homogeneity": 15,
        "test of homogeneity": 7
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "test-for-independence",
    "ja": {
      "term": "独立性の検定",
      "reading": "どくりつせいのけんてい",
      "alt": []
    },
    "en": {
      "term": "test of independence",
      "alt": [
        "test for independence"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "1 つの母集団について 2 つのカテゴリ変数が独立かどうかを、分割表と χ² で検定する方法。",
    "definition_en": "A chi-square test, done on a two-way table from one population, of whether two categorical variables are independent.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "One sample, two variables: that's a test for independence.",
        "ja": "標本が 1 つで変数が 2 つなら、独立性の検定だ。",
        "register": "spoken"
      },
      {
        "en": "Use a test of independence to decide whether gender and preferred subject are related.",
        "ja": "独立性の検定を用いて、性別と好きな教科に関係があるか判断せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics は test of independence（節の名前）、AP Statistics の CED（2026 年版）は test for independence（topic 3.14 ほか）と書く。",
      "chi-square test for independence とも言う。見出しの言い方を中に含むので候補には置かなかった。",
      "母集団が 2 つ以上で 1 つの変数の分布を比べるときは同質性の検定（test for homogeneity）。"
    ],
    "related": [
      "chi-square-test",
      "test-for-homogeneity",
      "expected-count"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.14・3.15"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "test for independence": 1
      },
      "written": {
        "test of independence": 38,
        "test for independence": 7
      },
      "sources": [
        "mit-18.06",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "test-of-a-single-variance",
    "ja": {
      "term": "分散の検定",
      "reading": "ぶんさんのけんてい",
      "alt": []
    },
    "en": {
      "term": "test of a single variance",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "1 つの母集団の分散（標準偏差）が、ある値に等しいかどうかをカイ二乗分布を使って検定する方法。",
    "definition_en": "A chi-square test of whether the variance (or standard deviation) of one population equals a given value.",
    "latex": "\\chi^2 = \\frac{(n-1)s^2}{\\sigma_0^2}",
    "spoken_en": null,
    "examples": [
      {
        "en": "For a claim about the variance, the test statistic is chi-square, not t.",
        "ja": "分散についての主張なら、検定統計量は t ではなくカイ二乗だ。",
        "register": "spoken"
      },
      {
        "en": "Perform a test of a single variance to see whether the machine's variability exceeds the standard.",
        "ja": "分散の検定を行い、機械のばらつきが基準を超えているか調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語は「母分散の検定」「分散の検定」と言い、「1 つの」は付けない。",
      "用例コーパスの書き言葉はすべて OpenStax Introductory Statistics（節の名前 Test of a Single Variance）。"
    ],
    "related": [
      "variance",
      "chi-square-test",
      "hypothesis-testing"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "test of a single variance": 11
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "test-point",
    "ja": {
      "term": "テスト点",
      "reading": "てすとてん",
      "alt": [
        "代表点"
      ]
    },
    "en": {
      "term": "test point",
      "alt": [],
      "variants": [
        {
          "term": "test value",
          "register": "spoken",
          "note": "授業の言い方（多くは Khan Academy）。書き言葉にはほとんど出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「テスト点」は学習指導要領解説に無い、本プロジェクトの訳語。学習指導要領解説には、区間の代表の値を代入して符号を調べる手順の名前が出てこない。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus I"
      ]
    },
    "definition_ja": "不等式や導関数の符号を調べるとき、各区間から 1 つ選んで代入する値。",
    "definition_en": "A value picked from each interval and plugged in to find the sign of an expression, an inequality, or a derivative there.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Pick a test value in each interval — zero is usually the easiest.",
        "ja": "各区間から 1 つ値を選んで試す。0 が一番楽なことが多い。",
        "register": "spoken"
      },
      {
        "en": "Choose a test point in each interval to determine where f′(x) = x² − 4 is positive.",
        "ja": "各区間から代表の点を選び、f′(x) = x² − 4 が正となる区間を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "書き言葉の test point は大半が OpenStax Calculus で、OpenStax Elementary Algebra にも出てくる。OpenStax Calculus Volume 1 では導関数の符号を調べる点。話し言葉では test value のほうが多く、大半が Khan Academy。"
    ],
    "related": [
      "sign-chart-inequality",
      "polynomial-inequality",
      "first-derivative-test"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「テスト点」は出てこない・区間の代表の値を代入して符号を調べる手順の名前が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "test value": 10,
        "test point": 2
      },
      "written": {
        "test point": 52,
        "test value": 3
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "test-statistic",
    "ja": {
      "term": "検定統計量",
      "reading": "けんていとうけいりょう",
      "alt": []
    },
    "en": {
      "term": "test statistic",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "仮説検定で、標本のデータから計算し、帰無仮説のもとでの分布と比べる値。z 値や t 値など。",
    "definition_en": "In a hypothesis test, the value computed from the sample data and compared with its distribution under the null hypothesis, such as a z or t value.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Compute the test statistic and see how far out in the tail it lands.",
        "ja": "検定統計量を計算して、分布の裾のどのくらい先にあるかを見る。",
        "register": "spoken"
      },
      {
        "en": "Calculate the test statistic and the p-value.",
        "ja": "検定統計量と p 値を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "p-value",
      "critical-value",
      "hypothesis-testing",
      "z-score"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "検定統計量",
        "en": "Test statistic"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "test statistic": 25
      },
      "written": {
        "test statistic": 154
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "test",
    "ja": {
      "term": "検定する",
      "reading": "けんていする",
      "alt": []
    },
    "en": {
      "term": "test",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語の「検定する」は仮説検定に限った語だが、英語の test は「試す・テスト」一般の語で、検定の意味では test the claim ／ conduct a hypothesis test のように目的語で示す。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "仮説検定を行い、データから仮説（主張）を棄却するかどうかを判断する。",
    "definition_en": "To carry out a hypothesis test, deciding from the data whether to reject a hypothesis or claim.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's test the claim that the average wait time is more than 10 minutes.",
        "ja": "平均待ち時間が 10 分より長いという主張を検定しよう。",
        "register": "spoken"
      },
      {
        "en": "Test the claim at the 0.05 significance level.",
        "ja": "有意水準 0.05 でその主張を検定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "hypothesis-testing",
      "null-hypothesis",
      "reject"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "test the claim | test the hypothesis | conduct a hypothesis test | perform a hypothesis test | do a hypothesis test": 10
      },
      "written": {
        "test the claim | test the hypothesis | conduct a hypothesis test | perform a hypothesis test | do a hypothesis test": 93
      },
      "sources": [
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "tetrahedron",
    "ja": {
      "term": "四面体",
      "reading": "しめんたい",
      "alt": [
        "三角錐"
      ]
    },
    "en": {
      "term": "tetrahedron",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数I",
        "数A"
      ],
      "us": [
        "Geometry",
        "Calculus III"
      ]
    },
    "definition_ja": "4 つの三角形の面で囲まれた立体。三角錐と同じ図形で、4 つの面がすべて正三角形のものを正四面体という。",
    "definition_en": "A solid bounded by four triangular faces, the same shape as a triangular pyramid; when all four faces are equilateral triangles it is a regular tetrahedron.",
    "latex": "V = \\frac{\\sqrt{2}}{12}a^3",
    "spoken_en": "the volume of a regular tetrahedron with edge a is root two over twelve times a cubed",
    "examples": [
      {
        "en": "A tetrahedron has four faces, six edges, and four vertices.",
        "ja": "四面体には面が 4 つ、辺が 6 本、頂点が 4 つある。",
        "register": "spoken"
      },
      {
        "en": "Find the volume of a regular tetrahedron whose edges all have length 6.",
        "ja": "1 辺の長さが 6 の正四面体の体積を求めよ。",
        "register": "written"
      },
      {
        "en": "For the volume of the tetrahedron, take one third of the base area times the height.",
        "ja": "四面体の体積は、底面積 × 高さ ÷ 3 で求める。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "正四面体は regular tetrahedron。強勢は後ろから 2 つ目の音節（-he-、「ヒー」。Merriam-Webster: ˌte-trə-ˈhē-drən）。複数形は tetrahedrons または tetrahedra（Merriam-Webster）。",
      "正四面体の体積では、頂点から底面に下ろした垂線の足が底面の正三角形の重心（外心）になることを使って高さを出す。"
    ],
    "related": [
      "foot-of-the-perpendicular",
      "scalar-triple-product",
      "triangle",
      "law-of-cosines",
      "eulers-formula-for-polyhedra"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「tetrahedron」",
        "url": "https://www.merriam-webster.com/dictionary/tetrahedron",
        "note": "発音 ˌte-trə-ˈhē-drən、複数形 tetrahedrons または tetrahedra（2026-09-26 に確かめた）"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "三角錐",
        "en": "Tetrahedron"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 図形: 正四面体の模型・展開図"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "tetrahedron": 19
      },
      "written": {
        "tetrahedron": 25
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "the-five-centers-of-a-triangle",
    "ja": {
      "term": "五心",
      "reading": "ごしん",
      "alt": []
    },
    "en": {
      "term": "triangle centers",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語に 5 つをまとめる言い方はなく、centroid・circumcenter・incenter・orthocenter・excenter と個別に言う。まとめて言うなら triangle centers（三角形に決まる点の総称で、五心に限らない。英語版 Wikipedia の記事名 Triangle center）。米国の参照では CK-12 Geometry が circumcenter・incenter・centroid・orthocenter を名前で扱い（circumcenter 13 件・incenter 17 件・centroid 21 件・orthocenter 4 件）、IM Geometry は circumcenter（23 件）と incenter（29 件）を扱う。excenter はどちらにも出てこない。日本語版 Wikipedia「五心」に英語版へのリンクは無く、記事が挙げる英語名 five centroids of triangle は英語版の記事名にも参照にも出てこない。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "definition_ja": "三角形の外心・内心・重心・垂心・傍心の 5 つの点をまとめた呼び名。どれも三角形に関わる 3 本の直線が 1 点で交わる点。",
    "definition_en": "The Japanese name for five special points of a triangle taken together: the circumcenter, incenter, centroid, orthocenter, and excenter. Each is a point where three lines related to the triangle meet.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "These triangle centers all come from three lines meeting at one point: medians give the centroid, perpendicular bisectors the circumcenter, angle bisectors the incenter, altitudes the orthocenter, and one interior and two exterior angle bisectors an excenter.",
        "ja": "五心はどれも 3 本の直線が 1 点で交わる点。中線なら重心、垂直二等分線なら外心、角の二等分線なら内心、垂線なら垂心、内角の二等分線 1 本と外角の二等分線 2 本なら傍心。",
        "register": "spoken"
      },
      {
        "en": "Show that in an equilateral triangle, four of the triangle centers (the centroid, circumcenter, incenter, and orthocenter) coincide.",
        "ja": "正三角形では五心のうち重心・外心・内心・垂心が一致することを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスと参照には、五心の意味の the five centers は出てこない。英語で五心を言うときは 5 つの名前（circumcenter, incenter, centroid, orthocenter, excenter）を並べるか、まとめて the triangle centers と言う。ただし英語版 Wikipedia「Triangle center」は、傍心は一般には triangle center にならないと書く（節 Excenters）。",
      "正三角形では外心・内心・重心・垂心が 1 点に重なる。"
    ],
    "related": [
      "circumcenter",
      "incenter",
      "centroid",
      "orthocenter",
      "excenter",
      "concurrent"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Triangle_center",
        "note": "記事「Triangle center」の冒頭と節 Formal definition で triangle center を太字で定義（三角形に決まる点の総称で五心に限らない。数学カテゴリから 5 段: 本文の定義を出典に使える 5 段以内で、見出しを記事名で決める 4 段の外。ja 記事「五心」に英語版へのリンクは無い）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「五心」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BA%94%E5%BF%83",
        "note": "外心・内心・重心・垂心・傍心の総称。「三角形の五心」はこの記事へのリダイレクト。英語版へのリンクは無い"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "circumcenter 13 件・incenter 17 件・centroid 21 件・orthocenter 4 件・excenter 0 件（5 つをまとめた名前は無い）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 7.5 Triangles in Circles・7.6 A Special Point・7.7 Circles in Triangles（circumcenter 23 件・incenter 29 件。centroid・orthocenter・excenter は 0 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-26。docs/DECISIONS.md「Phase 5 監査（セッション 2）」2: en.term を triangle centers、mapping near）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "the-four-operations",
    "ja": {
      "term": "四則",
      "reading": "しそく",
      "alt": [
        "四則演算"
      ]
    },
    "en": {
      "term": "arithmetic operations",
      "alt": [
        "four operations",
        "basic operations"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "「四則」に当たる 1 語は用例コーパスと参照に出てこない。4 つを addition, subtraction, multiplication, and division と並べて言うか、the (four) arithmetic operations とまとめて言う。",
    "domains": [
      "arithmetic"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "加法・減法・乗法・除法の 4 つの計算をまとめていう言葉。",
    "definition_en": "Addition, subtraction, multiplication, and division, taken together as the basic operations of arithmetic.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This one has all four arithmetic operations, so do the multiplication and division before the addition and subtraction.",
        "ja": "この式には四則が全部入っているから、加減より先に乗除を計算しよう。",
        "register": "spoken"
      },
      {
        "en": "Find the value of 18 − 12 ÷ (−3) × 2 + 1, which uses all four arithmetic operations.",
        "ja": "四則の混じった式 18 − 12 ÷ (−3) × 2 + 1 の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "四則の混じった計算の順序（かっこ → 累乗 → 乗除 → 加減）は、英語で order of operations と言う。",
      "「乗除」「加減」にも英語の 1 語はなく、multiplication and division ／ addition and subtraction と並べて言う。"
    ],
    "related": [
      "addition",
      "subtraction",
      "multiplication",
      "division",
      "compute"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "arithmetic operations": 11,
        "basic operations": 1,
        "four operations": 4
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "the-limit-does-not-exist",
    "ja": {
      "term": "極限が存在しない",
      "reading": "きょくげんがそんざいしない",
      "alt": []
    },
    "en": {
      "term": "the limit does not exist",
      "alt": [],
      "variants": [
        {
          "term": "the limit does not exist",
          "register": "both",
          "note": "話し言葉でも書き言葉でも最も多い。話し言葉は NancyPi と Khan Academy が中心、書き言葉は大半が OpenStax Calculus。"
        },
        {
          "term": "DNE",
          "register": "written",
          "note": "does not exist の略。答案や表で使う。話し言葉では少なく（すべて NancyPi）、書き言葉は OpenStax Calculus が中心。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "変数を近づけたとき、式の値が 1 つの有限の値に近づかない。",
    "definition_en": "Said when an expression does not approach a single finite value as the variable approaches its target.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The left and right limits are different, so the limit does not exist.",
        "ja": "左側極限と右側極限が違うので、極限は存在しない。",
        "register": "spoken"
      },
      {
        "en": "Since the one-sided limits differ, lim_{x→0} |x|/x does not exist (DNE).",
        "ja": "片側極限が一致しないので、lim_{x→0} |x|/x は存在しない。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus は does not exist を DNE と略して書く（AP の CED は略さずに does not exist と書く）。",
      "口頭では doesn't exist とも言う（用例コーパスの話し言葉にも the limit doesn't exist が出てくる）。"
    ],
    "related": [
      "limit",
      "one-sided-limit",
      "right-hand-limit",
      "left-hand-limit",
      "diverge-to-positive-infinity"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 1.3 ほか（does not exist）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the limit does not exist": 15,
        "DNE": 6
      },
      "written": {
        "the limit does not exist": 24,
        "DNE": 19
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "theorem",
    "ja": {
      "term": "定理",
      "reading": "ていり",
      "alt": []
    },
    "en": {
      "term": "theorem",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "証明によって正しいことが確かめられた事柄のうち、大切でよく使うもの。一度証明した定理は、ほかの証明の根拠に使える。",
    "definition_en": "A statement that has been proved true using definitions, postulates, and theorems that were already proved. Once a theorem is proved, you can use it as a reason in other proofs.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Once we prove this, it's a theorem, and you can use it as a reason in any proof after that.",
        "ja": "これを証明したら定理になるから、その後はどの証明でも根拠として使っていいよ。",
        "register": "spoken"
      },
      {
        "en": "Prove the following theorem: The base angles of an isosceles triangle are equal.",
        "ja": "次の定理を証明せよ。「二等辺三角形の底角は等しい。」",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "定理（theorem）は証明して正しいとわかった事柄。証明せずに正しいと認める事柄は公準・公理で、postulate ／ axiom と呼ぶ（エントリ postulate、axiom）。",
      "「〜の定理より」は by the … theorem（二等辺三角形の定理なら by the base angles theorem、エントリ isosceles-triangle-theorem）。定理の逆（converse、エントリ converse）は正しいとは限らないので、使うなら別に証明する。",
      "theorem の th は think の th と同じ音。"
    ],
    "related": [
      "postulate",
      "axiom",
      "proof",
      "lemma",
      "converse"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "定理",
        "en": "Theorem"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "theorem": 1309
      },
      "written": {
        "theorem": 1891
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "three-perpendiculars-theorem",
    "ja": {
      "term": "三垂線の定理",
      "reading": "さんすいせんのていり",
      "alt": []
    },
    "en": {
      "term": "three perpendiculars theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（three perpendiculars theorem は用例コーパスにも CED・OpenStax・Nicholson・Levin にも出てこない）。英語で説明するときは、「PH が平面に垂直で、HO が直線 ℓ に垂直だから PO も ℓ に垂直」のように、垂直の関係を順に述べる。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "definition_ja": "平面 α 上の直線 ℓ、ℓ 上の点 O、α 上にない点 P と α 上の点 H について、PH ⊥ α かつ OH ⊥ ℓ ならば PO ⊥ ℓ である、という定理。PH ⊥ α かつ PO ⊥ ℓ ならば OH ⊥ ℓ などの形も含む。",
    "definition_en": "For a line ℓ in a plane α, a point O on ℓ, a point P off α, and a point H in α: if PH ⊥ α and OH ⊥ ℓ, then PO ⊥ ℓ; related forms such as PH ⊥ α and PO ⊥ ℓ implying OH ⊥ ℓ are included.",
    "latex": "PH \\perp \\alpha,\\ OH \\perp \\ell \\implies PO \\perp \\ell",
    "spoken_en": "if PH is perpendicular to plane alpha and OH is perpendicular to line ell, then PO is perpendicular to ell",
    "examples": [
      {
        "en": "PH is perpendicular to the plane and HO is perpendicular to the line, so by the three perpendiculars theorem, PO is perpendicular to the line too.",
        "ja": "PH は平面に垂直で、HO は直線に垂直だから、三垂線の定理より PO も直線に垂直になる。",
        "register": "spoken"
      },
      {
        "en": "Point H is the foot of the perpendicular from P to plane α, and O is the foot of the perpendicular from H to line ℓ in α. Use the three perpendiculars theorem to show that PO ⊥ ℓ.",
        "ja": "点 P から平面 α に下ろした垂線の足を H、H から α 上の直線 ℓ に下ろした垂線の足を O とする。三垂線の定理を用いて PO ⊥ ℓ を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "答案では定理名に頼らず、直線と平面の垂直の条件（平面上の交わる 2 直線と垂直）から PO ⊥ ℓ を導く形で書く。"
    ],
    "related": [
      "line-perpendicular-to-a-plane",
      "foot-of-the-perpendicular",
      "perpendicular-lines",
      "distance-from-a-point-to-a-line"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「代数・幾何」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E3%83%BB%E5%B9%BE%E4%BD%95",
        "note": "旧課程の内容の取扱い: 三垂線の定理を導く程度の内容"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping none なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": true
  },
  {
    "id": "tick-mark",
    "ja": {
      "term": "数直線上の目盛り",
      "reading": "すうちょくせんじょうのめもり",
      "alt": [
        "目盛り"
      ]
    },
    "en": {
      "term": "hash mark",
      "alt": [
        "tick mark"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "definition_ja": "数直線や座標軸の上に、等しい間隔でつける短い線。目盛り。",
    "definition_en": "A short line drawn across a number line or an axis to mark equally spaced values.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each hash mark on this number line is worth 5, so this one right here is 15.",
        "ja": "この数直線の目盛りは 1 つが 5 だから、ここの目盛りは 15 だね。",
        "register": "spoken"
      },
      {
        "en": "Label each tick mark on the number line, counting by 0.5 from 0 to 3.",
        "ja": "数直線の各目盛りに、0 から 3 まで 0.5 ずつ数を書き入れよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "授業では hash mark と呼ぶことが多く、教科書（OpenStax Introductory Statistics）は tick mark と書く。どちらも同じ目盛りのこと。",
      "同じ hash mark ／ tick mark を、幾何では辺の長さが等しいことを示す印（等しい辺につける小さな線）にも使う（エントリ congruent）。数直線の目盛りとは別の使い方。"
    ],
    "related": [
      "number-line",
      "coordinate-axes",
      "x-axis",
      "coordinate-plane",
      "congruent"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "hash mark": 41,
        "tick mark": 7
      },
      "written": {
        "tick mark": 3,
        "hash mark": 1
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "time-series",
    "ja": {
      "term": "時系列",
      "reading": "じけいれつ",
      "alt": [
        "時系列データ"
      ]
    },
    "en": {
      "term": "time series",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "時間の順に並べて記録したデータ。",
    "definition_en": "Data recorded in order of time.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "With a time series, the order of the data matters, so plot it against time.",
        "ja": "時系列では順番に意味があるので、時間に対してグラフをかく。",
        "register": "spoken"
      },
      {
        "en": "Make a time series graph of the monthly sales and describe any trend.",
        "ja": "月ごとの売上高の時系列グラフをかき、傾向を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "時系列のグラフは time series graph（OpenStax Introductory Statistics、16 件）。"
    ],
    "related": [
      "moving-average",
      "trend",
      "line-graph"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "時系列",
        "en": "Time series"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "time series": 17
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "times",
    "ja": {
      "term": "〜倍",
      "reading": "ばい",
      "alt": []
    },
    "en": {
      "term": "times",
      "alt": [],
      "variants": [
        {
          "term": "twice as",
          "register": "both",
          "note": "2 倍を言う形。話・書とも three times as many などの times より多い。"
        },
        {
          "term": "times",
          "register": "both",
          "note": "three times as many ／ as long ／ larger などの形。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "「〜倍」は、× の読み（3 times 4）と同じ times を数の後に置いて言う。「A は B の 3 倍」は A is three times B で、数や量を比べるときは three times as many as B ／ three times as long as B の形になる。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "ある量が、もとにする量のいくつ分に当たるかを表す言い方。b が a の k 倍のとき、b = ka となる。",
    "definition_en": "Used after a number to compare two quantities by multiplication: if b is k times a, then b = ka.",
    "latex": "b = ka",
    "spoken_en": "b equals k times a",
    "examples": [
      {
        "en": "If Mia has 4 stickers and Jake has three times as many, Jake has 12, and twice as many would be 8.",
        "ja": "ミアのシールが 4 枚で、ジェイクがその 3 倍持っているなら、ジェイクは 12 枚。2 倍なら 8 枚だね。",
        "register": "spoken"
      },
      {
        "en": "Find the width of a rectangle whose length is three times its width and whose perimeter is 48 cm.",
        "ja": "横の長さが縦の長さの 3 倍で、周の長さが 48 cm の長方形がある。縦の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「3 倍する」は動詞で multiply by 3 ／ triple と言う。",
      "2 倍は two times のほか twice（twice as long as）とも言う。半分（0.5 倍）は half、1.5 倍は 1.5 times。",
      "「B の 3 倍」は three times B、three times as many as B の順で、「B の」に当たる語が times の後ろに来る。"
    ],
    "related": [
      "multiply",
      "multiplication",
      "ratio",
      "percent",
      "fraction"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "twice as": 80,
        "times as many | times as much | times as large | times as long | times as big | times larger | times bigger | times greater": 51
      },
      "written": {
        "twice as": 42,
        "times as many | times as much | times as large | times as long | times as big | times larger | times bigger | times greater": 35
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "toss",
    "ja": {
      "term": "投げる",
      "reading": "なげる",
      "alt": []
    },
    "en": {
      "term": "toss",
      "alt": [],
      "variants": [
        {
          "term": "flip",
          "register": "both",
          "note": "コインを投げるときの言い方。話し言葉では toss より多い（Khan Academy の中学の講義に多い）。IM もこの言い方。"
        },
        {
          "term": "roll",
          "register": "both",
          "note": "さいころを投げるときの言い方（roll a die）。話・書とも使う。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "日本語は硬貨もさいころも「投げる」と言うが、英語では硬貨は toss ／ flip（toss a coin、flip a coin）、さいころは roll（roll a die）と言い分ける。IM は flip a coin の言い方が多く、OpenStax は toss a coin が多い。",
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "確率の試行で、硬貨やさいころを投げて、どの面が出るかを見る。",
    "definition_en": "To throw a coin into the air so that it lands heads or tails. For a die, the verb is roll.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Toss the coin first, then roll the die. How many outcomes are there in all?",
        "ja": "まず硬貨を投げて、次にさいころを投げる。結果は全部で何通りある？",
        "register": "spoken"
      },
      {
        "en": "A coin is tossed and a die is rolled. Find the probability of getting heads and an even number.",
        "ja": "1 枚の硬貨と 1 個のさいころを投げる。硬貨は表が出て、さいころは偶数の目が出る確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「投げる」を throw と直訳しない。硬貨は toss ／ flip（toss a coin、flip a coin）、さいころは roll（roll a die、roll two dice。エントリ die）。",
      "toss は名詞でも使い、a coin toss は硬貨を 1 回投げること。「表が出る」は land heads up ／ come up heads（エントリ coin）。"
    ],
    "related": [
      "coin",
      "die",
      "trial",
      "probability",
      "equally-likely"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "flip a coin（12 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "toss a coin | tosses a coin | coin toss | toss the coin": 11,
        "flip a coin | flips a coin | coin flip | flip the coin": 19,
        "roll a die | roll the die | roll two dice | rolls a die | roll a number cube": 18
      },
      "written": {
        "flip a coin | flips a coin | coin flip | flip the coin": 10,
        "toss a coin | tosses a coin | coin toss | toss the coin": 30,
        "roll a die | roll the die | roll two dice | rolls a die | roll a number cube": 11
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "total-differential",
    "ja": {
      "term": "全微分",
      "reading": "ぜんびぶん",
      "alt": []
    },
    "en": {
      "term": "total differential",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "多変数関数の値の微小な変化を、各変数の微分 dx、dy と偏導関数で表したもの dz = f_x dx + f_y dy。",
    "definition_en": "The expression dz = f_x dx + f_y dy that gives the small change in a function of several variables from small changes in each variable.",
    "latex": "dz = \\frac{\\partial f}{\\partial x}dx + \\frac{\\partial f}{\\partial y}dy",
    "spoken_en": "d z equals partial f partial x d x plus partial f partial y d y",
    "examples": [
      {
        "en": "Use the total differential to estimate how much the volume changes.",
        "ja": "全微分を使って、体積がどれだけ変わるかを見積もろう。",
        "register": "spoken"
      },
      {
        "en": "Find the total differential of z = x²y + ln y.",
        "ja": "z = x²y + ln y の全微分を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉は大半が MIT OCW（18.02）。"
    ],
    "related": [
      "partial-derivative",
      "differential",
      "tangent-plane",
      "multivariable-chain-rule"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "全微分",
        "en": "Derivative (multivariable calculus)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "total differential": 41
      },
      "written": {
        "total differential": 12
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "total-distance-traveled",
    "ja": {
      "term": "移動距離",
      "reading": "いどうきょり",
      "alt": []
    },
    "en": {
      "term": "total distance traveled",
      "alt": [],
      "uk": "total distance travelled",
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "動く点が実際に動いた道のりの合計。速度の絶対値の定積分 ∫ₐᵇ |v(t)|dt で求まる。",
    "definition_en": "The total length of the path an object covers, ∫ₐᵇ |v(t)| dt, counting motion in either direction as positive.",
    "latex": "\\int_a^b |v(t)|\\,dt",
    "spoken_en": "the integral from a to b of the absolute value of v of t, dt",
    "examples": [
      {
        "en": "For total distance traveled, split the integral wherever the velocity changes sign.",
        "ja": "移動距離は、速度の符号が変わるところで積分を分ける。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "数IIIの「道のり」に当たる。"
    ],
    "related": [
      "displacement",
      "riemann-sum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 積分法: 点 P の道のり"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/American_and_British_English_spelling_differences#Doubled_in_British_English",
        "note": "記事「American and British English spelling differences」の節 Doubled in British English（英国では語末の l を重ねる。travelled ／ traveled）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "total distance traveled": 9
      },
      "written": {
        "total distance traveled": 19
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "transcendental-function",
    "ja": {
      "term": "超越関数",
      "reading": "ちょうえつかんすう",
      "alt": []
    },
    "en": {
      "term": "transcendental function",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "definition_ja": "代数関数でない関数。指数関数・対数関数・三角関数など。",
    "definition_en": "A function that is not algebraic, such as exponential, logarithmic, and trigonometric functions.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Exponential and trig functions are transcendental functions.",
        "ja": "指数関数や三角関数は超越関数だ。",
        "register": "spoken"
      },
      {
        "en": "Classify f(x) = eˣ + x² as an algebraic or a transcendental function.",
        "ja": "f(x) = eˣ + x² が代数関数か超越関数かを答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 1 は、関数の種類を紹介する節（Basic Classes of Functions）で algebraic functions と transcendental functions に分ける。"
    ],
    "related": [
      "algebraic-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "超越関数",
        "en": "Transcendental function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "transcendental function": 8
      },
      "written": {
        "transcendental function": 6
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 8 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは OpenStax の呼び方 transcendental function（本文 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "transformation-of-a-variable",
    "ja": {
      "term": "変量の変換",
      "reading": "へんりょうのへんかん",
      "alt": [
        "データの変換"
      ]
    },
    "en": {
      "term": "transforming data",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスには transforming data ／ transformation of data が出てこない）。「すべての値に定数を足す（add a constant）・掛ける（multiply by a constant）と平均値・標準偏差がどう変わるか」と操作で説明する（AP Statistics の CED topic 1.7 は changing units of measurement と言う）。見出しの「変量の変換」は学習指導要領解説に無く、センター試験（平成 31 年度 数学I・A 第2問）と共通テスト（令和 5 年度 数学I・A）は「(x_i − x̄)/s と変換した」「千円単位に変換する」と動詞で言う。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics"
      ]
    },
    "definition_ja": "変量 x の各値を u = (x − a)/c のような式で別の変量に置き換えること。平均値や分散がどう変わるかを調べる。対数をとるなど、散布図の点を直線に近づけるための変換もある。",
    "definition_en": "Replacing each value of a variable x by a new value such as u = (x − a)/c and seeing how the mean and variance change; also a transformation such as taking logarithms to straighten a scatter plot.",
    "latex": "u = \\frac{x - a}{c}",
    "spoken_en": null,
    "examples": [
      {
        "en": "If you transform the data by adding 10 to every score, the mean goes up by 10 but the spread stays the same.",
        "ja": "全員の得点に 10 を足すデータの変換では、平均値は 10 上がるが散らばりは変わらない。",
        "register": "spoken"
      },
      {
        "en": "Let u = (x − 50)/10. Find the mean and variance of x from those of u.",
        "ja": "u = (x − 50)/10 とする。u の平均値と分散から x の平均値と分散を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "定数を足すと平均値だけが変わり（shift）、定数を掛けると平均値も標準偏差も変わる（scale）。",
      "散布図が曲線状のとき、y の対数をとるなどして点を直線に近づける変換も変量の変換の一つ。この意味の transforming data も、用例コーパスと参照（CED・OpenStax・IM・CK-12）には出てこない。"
    ],
    "related": [
      "arithmetic-mean",
      "variance",
      "standard-deviation"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.7（changing units of measurement affects the values of the calculated statistics）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "センター試験 平成 31 年度 数学I・A 第2問・共通テスト 令和 5 年度 数学I・A: 変量を変換した（動詞）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I データの分析: 「変量の変換」は 0 件（見出しの日本語は解説に出てこない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "transformations-of-functions",
    "ja": {
      "term": "グラフの移動",
      "reading": "ぐらふのいどう",
      "alt": [
        "関数の変換"
      ]
    },
    "en": {
      "term": "transformations of the graph",
      "alt": [
        "transformations of functions",
        "function transformations"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本では「平行移動」「対称移動」を別々に学び、まとめた名前はあまり使わない。OpenStax Algebra and Trigonometry は、基本の関数（toolkit functions、エントリ parent-function）の transformations としてまとめて扱う。OpenStax Algebra and Trigonometry は Transformation of Functions の節で扱う。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "関数 y = f(x) のグラフの平行移動・対称移動・拡大縮小をまとめた言い方。y = f(x − h) + k は右に h、上に k の平行移動。",
    "definition_en": "The ways a graph of y = f(x) can be changed: shifts (translations), reflections, stretches, and compressions. For example, y = f(x − h) + k shifts the graph h units right and k units up.",
    "latex": "y = f(x - h) + k",
    "spoken_en": "y equals f of x minus h, plus k",
    "examples": [
      {
        "en": "What transformations of the graph take y = x² to y = (x − 2)² + 3? It shifts right 2 and up 3.",
        "ja": "y = x² のグラフをどう移動すると y = (x − 2)² + 3 のグラフになるかな。右に 2、上に 3 平行移動だ。",
        "register": "spoken"
      },
      {
        "en": "Describe the transformations of the graph of f(x) = |x| that produce the graph of g(x) = −|x + 1|.",
        "ja": "f(x) = |x| のグラフをどのように移動すると g(x) = −|x + 1| のグラフになるかを説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "y = f(x − h) は右に h の平行移動。x − h のマイナスにつられて左と読まない。",
      "英語の transformations には平行移動（shift、エントリ translation）と対称移動（reflection、エントリ reflect）のほか、拡大縮小（stretch、compression、エントリ stretch-vertically）も入る。",
      "y = −f(x) は x 軸に関する対称移動。"
    ],
    "related": [
      "translation",
      "reflect",
      "stretch-vertically",
      "compress-horizontally",
      "absolute-value-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "function transformations": 1,
        "transformations of functions": 1
      },
      "written": {
        "transformations of functions": 5,
        "function transformations": 8,
        "transformations of the graph": 9
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 2 件（10 未満） ／ 書: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満）。見出しは OpenStax の呼び方 transformations of the graph（本文 9 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "transitive-property",
    "ja": {
      "term": "推移律",
      "reading": "すいいりつ",
      "alt": []
    },
    "en": {
      "term": "transitive property",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof",
      "logic"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "a = b かつ b = c ならば a = c となる性質。合同や不等号 < についても同じ形の性質が成り立つ。",
    "definition_en": "The property that if a = b and b = c, then a = c. The same rule holds for congruence and for inequalities such as <.",
    "latex": "a = b,\\ b = c \\implies a = c",
    "spoken_en": "if a equals b and b equals c, then a equals c",
    "examples": [
      {
        "en": "Angle 1 is congruent to angle 2, and angle 2 is congruent to angle 3, so by the transitive property, angle 1 is congruent to angle 3.",
        "ja": "∠1 ≅ ∠2 で、∠2 ≅ ∠3 だから、推移律より ∠1 ≅ ∠3。",
        "register": "spoken"
      },
      {
        "en": "Given: ∠1 ≅ ∠2 and ∠2 ≅ ∠3. Name the property that justifies the statement ∠1 ≅ ∠3.",
        "ja": "∠1 = ∠2、∠2 = ∠3 のとき、∠1 = ∠3 と言える根拠となる性質の名前を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本では推移律は大学（同値関係・順序関係）の語で、中学・高校の証明では「∠1 = ∠2、∠2 = ∠3 より」と理由に名前を付けずに書く。",
      "米国の Geometry の証明では、等式なら transitive property of equality、合同なら transitive property of congruence と対象を付けて書く。",
      "用例コーパスでは話し言葉にほとんど出てこず、書き言葉には出てこない。IM では 0 件、CK-12 Geometry では 5 件（2.15 Proofs- Angle Pairs and Segments）。"
    ],
    "related": [
      "reflexive-property",
      "symmetric-property",
      "substitution-property"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.15 Proofs- Angle Pairs and Segments"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "transitive property": 3
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 transitive property（CK-12 Geometry 2.15 Proofs- Angle Pairs and Segments・計 5 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "translate",
    "ja": {
      "term": "平行に移動する",
      "reading": "へいこうにいどうする",
      "alt": []
    },
    "en": {
      "term": "translate",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "verb",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "図形のすべての点を、同じ向きに同じ距離だけ動かす。形も大きさも向きも変わらない。",
    "definition_en": "To slide a figure so that every point moves the same distance in the same direction, without turning or flipping it.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's translate the triangle 4 units to the left. Every vertex slides over 4, and the shape doesn't change.",
        "ja": "三角形を左に 4 だけ平行移動しよう。どの頂点も 4 ずつずれるだけで、形は変わらない。",
        "register": "spoken"
      },
      {
        "en": "Triangle ABC has vertices A(1, 2), B(4, 2), and C(1, 5). Translate the triangle 3 units right and 2 units down, and give the coordinates of the vertices of the image.",
        "ja": "A(1, 2)、B(4, 2)、C(1, 5) を頂点とする △ABC を、右に 3、下に 2 だけ平行移動する。移した三角形の頂点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "translate は日常語では「翻訳する」。数学でも、文章題の文を式に直すことを translate と言う（translate the sentence into an equation）。図形を動かす意味かどうかは目的語（the triangle、the point など）で見分ける。",
      "動かす量は 3 units to the right、2 units down のように units と向きの語で言う（日本語の「x 軸方向に 3、y 軸方向に −2」に当たる）。",
      "名詞は translation（エントリ translation）で、平行移動という移動そのものを指す。"
    ],
    "related": [
      "translation",
      "reflect",
      "rotate",
      "rigid-motion",
      "write-an-equation"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 1.12 ほか"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "translate the graph | translated … units | translate … units | translate the figure | translate the triangle | translate the point | translated up | translated down | translated left | translated right | translate it": 15
      },
      "written": {
        "translate the graph | translated … units | translate … units | translate the figure | translate the triangle | translate the point | translated up | translated down | translated left | translated right | translate it": 8
      },
      "sources": [
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "translation",
    "ja": {
      "term": "平行移動",
      "reading": "へいこういどう",
      "alt": []
    },
    "en": {
      "term": "translation",
      "alt": [],
      "variants": [
        {
          "term": "shift",
          "register": "both",
          "note": "関数のグラフの平行移動の言い方。用例コーパス（講義・OpenStax）では vertical ／ horizontal shift、shifted … units などの形が、話し言葉・書き言葉とも translation の形よりずっと多い。"
        }
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "図形やグラフ上のすべての点を、同じ向きに同じ距離だけ動かす移動。点 (x, y) は (x + a, y + b) に移る。",
    "definition_en": "A transformation that moves every point of a figure or graph the same distance in the same direction, sending (x, y) to (x + a, y + b).",
    "latex": "(x, y) \\mapsto (x + a, y + b)",
    "spoken_en": "x comma y maps to x plus a comma y plus b",
    "examples": [
      {
        "en": "So this is a horizontal shift: the whole parabola moves 3 units to the right.",
        "ja": "これは x 軸方向の平行移動で、放物線全体が右に 3 動く。",
        "register": "spoken"
      },
      {
        "en": "The graph of y = (x − 3)² + 2 is the graph of y = x² shifted 3 units to the right and 2 units up.",
        "ja": "y = (x − 3)² + 2 のグラフは、y = x² のグラフを x 軸方向に 3、y 軸方向に 2 だけ平行移動したものである。",
        "register": "written"
      },
      {
        "en": "Under the translation (x, y) → (x + 4, y − 1), find the image of triangle ABC.",
        "ja": "平行移動 (x, y) → (x + 4, y − 1) による三角形 ABC の像を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の「x 軸方向に p、y 軸方向に q だけ平行移動」は、英語では向きで言う（shift 3 units to the right and 2 units up）。y = f(x) のグラフの平行移動は y = f(x − p) + q で、x − p の符号と右向きの対応に注意。",
      "関数のグラフと図形で言い方が分かれる。関数のグラフの平行移動は、用例コーパス（講義・OpenStax Algebra and Trigonometry ほか）では shift（vertical ／ horizontal shift・shifted … units など）が話し言葉・書き言葉とも translation の形よりずっと多い。図形の平行移動は、IM の Grade 8 と Geometry で translation（65 件・129 件。両方の glossary の見出し）、動詞は translate（40 件・95 件）で、shift は 2 件・4 件しかない。IM Algebra 2 はグラフにも translation ／ translate を使う（28 件・40 件、shift は 11 件）。答案で図形を動かすときは translation、グラフを「右に 3」と言うときは shift が無難。",
      "平行移動ベクトル（どの向きにどれだけ動かすか）は translation vector で、⟨a, b⟩ のように成分で書く。"
    ],
    "related": [
      "reflection",
      "rotation",
      "phase-shift",
      "composition-of-transformations",
      "axis-of-symmetry"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "平行移動",
        "en": "Translation (geometry)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8) / IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Grade 8・Geometry の glossary の見出し translation。Grade 8 1 章・Geometry 1 章（rigid transformations）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "vertical translation | horizontal translation | translation of the graph | translations of the graph | translation of the parabola | translation of the function | translated … units": 2,
        "vertical shift | horizontal shift | shifted … units | shift … units": 96
      },
      "written": {
        "vertical shift | horizontal shift | shifted … units | shift … units": 248,
        "vertical translation | horizontal translation | translation of the graph | translations of the graph | translation of the parabola | translation of the function | translated … units": 11
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "transpose",
    "ja": {
      "term": "転置",
      "reading": "てんち",
      "alt": [
        "転置行列"
      ]
    },
    "en": {
      "term": "transpose",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "行列の行と列を入れ替えること。また、入れ替えてできた行列。A の転置行列を Aᵀ と書く。",
    "definition_en": "Swapping the rows and columns of a matrix, or the matrix that results; the transpose of A is written Aᵀ.",
    "latex": "(A^{T})_{ij} = A_{ji}",
    "spoken_en": null,
    "examples": [
      {
        "en": "A transpose A is always square and symmetric.",
        "ja": "A の転置と A の積はいつも正方で対称だ。",
        "register": "spoken"
      },
      {
        "en": "Find the transpose of the matrix and verify that (AB)ᵀ = Bᵀ Aᵀ.",
        "ja": "行列の転置を求め、(AB)ᵀ = Bᵀ Aᵀ を確かめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Aᵀ は A transpose と読む。英語は名詞・動詞とも transpose（transpose the matrix）。",
      "用例コーパスの話し言葉はほとんどが MIT 18.06。"
    ],
    "related": [
      "matrix",
      "matrix-multiplication",
      "square-matrix"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "2.1 Matrix Addition, Scalar Multiplication, and Transposition"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "transpose": 752
      },
      "written": {
        "transpose": 1
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "transversal",
    "ja": {
      "term": "横断線",
      "reading": "おうだんせん",
      "alt": []
    },
    "en": {
      "term": "transversal",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 本以上の直線と、それぞれ異なる点で交わる直線。同位角や錯角は、この直線に対する位置で決まる。",
    "definition_en": "A line that crosses two or more other lines at different points. Pairs such as corresponding angles and alternate interior angles are named by their position relative to it.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Line t is our transversal here. It cuts across both parallel lines, so we get eight angles to work with.",
        "ja": "ここでは直線 t が横断線。平行な 2 直線の両方を横切るから、角が 8 つできる。",
        "register": "spoken"
      },
      {
        "en": "Lines ℓ and m are parallel, and line t is a transversal. One of the angles formed measures 65°. Find the measures of the other seven angles.",
        "ja": "直線 ℓ と m は平行で、直線 t はこの 2 直線と交わっている。できる角の 1 つが 65° のとき、残り 7 つの角の大きさを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の教科書は交わる直線に名前を付けず「2 直線に 1 直線が交わるとき」と言う。英語の授業や問題文では、この直線を transversal と名前で呼ぶ。",
      "同位角は corresponding angles、錯角は alternate interior angles。どちらも transversal に対する位置で決まる名前。",
      "用例コーパスで transversal は話し言葉だけに出てくる（Khan Academy の中学の講義が最も多く、次が The Organic Chemistry Tutor）。"
    ],
    "related": [
      "parallel-lines",
      "alternate-interior-angles-theorem",
      "corresponding-angles-postulate",
      "alternate-exterior-angles",
      "same-side-exterior-angles"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "横断線",
        "en": "Transversal (geometry)"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math (Grade 8)",
        "url": "https://im.kendallhunt.com/MS/students/3/index.html",
        "note": "glossary の見出し transversal（Grade 8）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "transversal": 61
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "transverse-axis",
    "ja": {
      "term": "主軸",
      "reading": "しゅじく",
      "alt": []
    },
    "en": {
      "term": "transverse axis",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus II"
      ]
    },
    "definition_ja": "双曲線の 2 つの焦点を通る軸。2 つの頂点を結ぶ線分の長さは 2a。",
    "definition_en": "The axis of a hyperbola through its two foci; the segment joining the two vertices has length 2a.",
    "latex": "\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1",
    "spoken_en": null,
    "examples": [
      {
        "en": "For this hyperbola the transverse axis is horizontal, since x squared comes first.",
        "ja": "この双曲線は x² が先に来るから、主軸は横向き。",
        "register": "spoken"
      },
      {
        "en": "Find the length of the transverse axis of the hyperbola 9x² − 4y² = 36.",
        "ja": "双曲線 9x² − 4y² = 36 の主軸の長さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では少なく、書き言葉（大半が OpenStax Algebra and Trigonometry）の語。",
      "主軸に垂直な軸は conjugate axis（共役軸）。"
    ],
    "related": [
      "major-axis",
      "focus",
      "conic-section"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "transverse axis": 7
      },
      "written": {
        "transverse axis": 64
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trapezoid",
    "ja": {
      "term": "台形",
      "reading": "だいけい",
      "alt": []
    },
    "en": {
      "term": "trapezoid",
      "alt": [],
      "uk": "trapezium",
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「台形」と共通テスト（令和3年度 第1日程 数学Ⅱ・数学Ｂ 第5問）は、少なくとも一組の対辺が平行な四角形を台形とする（平行四辺形も台形に入る）。米国の教材では台形の定義が分かれる。CK-12 Geometry は平行な辺がちょうど 1 組（exactly one pair）の四角形として平行四辺形を含めず、IM Geometry は少なくとも 1 組（at least one pair）として平行四辺形も台形に含める。英国では trapezium と呼ぶ。",
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "向かい合う 1 組の辺が平行な四角形。平行な 2 辺を上底・下底という。",
    "definition_en": "A quadrilateral with a pair of parallel opposite sides, called the bases. Some textbooks require exactly one such pair, while others allow at least one.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A trapezoid has a pair of parallel sides, and those two sides are called the bases.",
        "ja": "台形には平行な辺が 1 組あって、その 2 辺を底（上底と下底）と呼ぶよ。",
        "register": "spoken"
      },
      {
        "en": "The bases of a trapezoid are 5 cm and 9 cm long, and its height is 4 cm. Find the area of the trapezoid.",
        "ja": "上底が 5 cm、下底が 9 cm、高さが 4 cm の台形の面積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "平行な 2 辺は、上底も下底もどちらも base（2 本で bases）。平行でない 2 辺は legs と呼び、直角三角形の直角をはさむ 2 辺（エントリ legs）と同じ語。",
      "「平行四辺形は台形か」の答えは教材の定義による。「ちょうど 1 組」の定義なら台形ではなく、「少なくとも 1 組」の定義なら台形でもある（エントリ parallelogram）。",
      "英国では trapezium と呼ぶ。"
    ],
    "related": [
      "parallelogram",
      "quadrilateral",
      "parallel",
      "area",
      "height"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "台形",
        "en": "Trapezoid"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Trapezoid",
        "note": "記事「Trapezoid」の冒頭: 北米英語の trapezoid、英国英語の trapezium"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "5.7 Quadrilateral Classification・5.13 Trapezoids: exactly one pair of parallel sides"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 6.17 Lines in Triangles の練習問題（Unit 6 Lesson 14 から）: at least one pair of parallel sides"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「台形」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8F%B0%E5%BD%A2",
        "note": "少なくとも一組の対辺が互いに平行な四角形"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和3年度 大学入学共通テスト 第1日程 数学Ⅱ・数学Ｂ（問題）",
        "url": "https://web.archive.org/web/20220601/https://www.dnc.ac.jp/albums/abm.php?d=88&f=abm00001266.pdf&n=R3_%E6%95%B0%E5%AD%A6%E2%85%A1%E3%83%BB%E6%95%B0%E5%AD%A6%EF%BC%A2%EF%BC%88%EF%BC%91%E6%9C%8816%E6%97%A5%E3%83%BB17%E6%97%A5%EF%BC%89%E3%81%AE%E5%95%8F%E9%A1%8C.pdf",
        "note": "第5問: ただし，少なくとも一組の対辺が平行な四角形を台形という"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "trapezoid": 154
      },
      "written": {
        "trapezoid": 94
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-prealgebra",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trapezoidal-rule",
    "ja": {
      "term": "台形公式",
      "reading": "だいけいこうしき",
      "alt": [
        "台形則"
      ]
    },
    "en": {
      "term": "trapezoidal sum",
      "alt": [],
      "variants": [
        {
          "term": "trapezoidal rule",
          "register": "written",
          "note": "大学の教科書の書き言葉で、数値積分の方法の名前として使う。話し言葉では少なく（すべて MIT OCW）、書き言葉は大半が OpenStax Calculus。"
        },
        {
          "term": "trapezoid rule",
          "register": "spoken",
          "note": "MIT OCW の言い方。話し言葉はすべて MIT OCW で、書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II"
      ]
    },
    "definition_ja": "各小区間で曲線を直線で結び、できた台形の面積の和で定積分を近似する公式。",
    "definition_en": "A method that approximates a definite integral by the sum of the areas of trapezoids under the curve.",
    "latex": "T_n = \\frac{\\Delta x}{2}\\bigl(f(x_0) + 2f(x_1) + \\cdots + 2f(x_{n-1}) + f(x_n)\\bigr)",
    "spoken_en": "T sub n equals delta x over two times the quantity f of x sub zero plus two f of x sub one, and so on, plus two f of x sub n minus one plus f of x sub n",
    "examples": [
      {
        "en": "The trapezoidal sum is just the average of the left and right Riemann sums.",
        "ja": "台形公式の値は、左と右のリーマン和の平均にすぎない。",
        "register": "spoken"
      },
      {
        "en": "Use the trapezoidal rule with n = 4 to approximate ∫₀² x³ dx.",
        "ja": "n = 4 の台形公式で ∫₀² x³ dx の近似値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Riemann sum の一つとして習うときは trapezoidal sum（AP の CED topic 6.2。話し言葉はすべて Khan Academy）、数値積分の方法としては trapezoidal rule と呼ぶ。",
      "trapezoid（台形）は米国の言い方。英国では trapezium と言う。"
    ],
    "related": [
      "numerical-integration",
      "simpsons-rule",
      "left-riemann-sum",
      "right-riemann-sum"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "台形公式",
        "en": "Trapezoidal rule"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 6.2（trapezoidal sums）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Trapezoid",
        "note": "記事「Trapezoid」の冒頭: 北米英語の trapezoid、英国英語の trapezium"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学II (7) 積分法（発展・拡充）: 面積を台形公式やシンプソンの公式で求める"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "trapezoidal rule": 9,
        "trapezoid rule": 10,
        "trapezoidal sum": 16
      },
      "written": {
        "trapezoidal rule": 61
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "treatment",
    "ja": {
      "term": "処理",
      "reading": "しょり",
      "alt": [
        "処理条件"
      ]
    },
    "en": {
      "term": "treatment",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "実験で、対象に与える条件（薬の種類や量など）。",
    "definition_en": "In an experiment, a condition applied to the subjects, such as a kind or amount of a drug.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Randomly assign the subjects to the two treatments.",
        "ja": "被験者を 2 つの処理に無作為に割り当てる。",
        "register": "spoken"
      },
      {
        "en": "Identify the treatments and the response variable in the experiment.",
        "ja": "実験の処理と応答変数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「処理」はデータの処理の意味でも使うが、英語の treatment は実験の処理（与える条件）。医療の「治療」も treatment。"
    ],
    "related": [
      "control-group",
      "placebo",
      "completely-randomized-design",
      "blinding"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "treatment": 52
      },
      "written": {
        "treatment": 72
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-calculus",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "tree-diagram",
    "ja": {
      "term": "樹形図",
      "reading": "じゅけいず",
      "alt": []
    },
    "en": {
      "term": "tree diagram",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Integrated Math 2",
        "Intro Statistics"
      ]
    },
    "definition_ja": "起こりうる場合を、枝分かれする線で順にすべて書き出した図。",
    "definition_en": "A diagram that lists all the possible outcomes by branching lines, one stage at a time.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Draw a tree diagram so you don't miss any of the outcomes.",
        "ja": "場合を見落とさないように樹形図をかこう。",
        "register": "spoken"
      },
      {
        "en": "Use a tree diagram to find the probability of getting two heads in three tosses of a coin.",
        "ja": "樹形図を使って、硬貨を 3 回投げて表が 2 回出る確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "条件付き確率では、枝に確率を書き込み、枝に沿って掛け合わせる。"
    ],
    "related": [
      "probability",
      "sample-space",
      "number-of-possible-outcomes",
      "coin"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "tree diagram": 4
      },
      "written": {
        "tree diagram": 93
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "tree",
    "ja": {
      "term": "木（グラフ）",
      "reading": "き",
      "alt": []
    },
    "en": {
      "term": "tree",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフ理論で、閉路をもたない連結なグラフ。頂点が n 個なら辺は n − 1 本。",
    "definition_en": "In graph theory, a connected graph with no cycles; a tree with n vertices has n − 1 edges.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A tree is connected but has no loops, so there's exactly one path between any two vertices.",
        "ja": "木は連結だがループがないので、どの 2 頂点の間にも道がちょうど 1 本ある。",
        "register": "spoken"
      },
      {
        "en": "Show that a tree with n vertices has exactly n − 1 edges.",
        "ja": "頂点が n 個の木はちょうど n − 1 本の辺をもつことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a tree（木）と a tree diagram（樹形図）は別の語。",
      "すべての頂点を含む木は spanning tree（全域木）。"
    ],
    "related": [
      "vertex-graph",
      "tree-diagram",
      "eulerian-path"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.2 Trees"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "a tree !diagram": 60
      },
      "written": {
        "a tree !diagram": 36
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-introstats",
        "openstax-prealgebra"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trend",
    "ja": {
      "term": "傾向",
      "reading": "けいこう",
      "alt": []
    },
    "en": {
      "term": "trend",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中1",
        "数I",
        "数B"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "AP Statistics",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "データ全体に見られる増え方・減り方などのおおまかな向き。",
    "definition_en": "The general direction, such as increasing or decreasing, that the data follow as a whole.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The points aren't on a line, but there's a clear upward trend.",
        "ja": "点は直線上にはないが、はっきりとした上向きの傾向がある。",
        "register": "spoken"
      },
      {
        "en": "Describe the trend shown in the scatter plot.",
        "ja": "散布図に表れている傾向を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "upward trend",
        "ja": "増加の傾向"
      },
      {
        "en": "downward trend",
        "ja": "減少の傾向"
      }
    ],
    "pitfalls": [],
    "related": [
      "data"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "trend": 73
      },
      "written": {
        "trend": 61
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.03",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trial",
    "ja": {
      "term": "試行",
      "reading": "しこう",
      "alt": []
    },
    "en": {
      "term": "trial",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「試行」は、標本空間や事象を考える実験そのもの（学習指導要領解説（数学B）の「2枚の硬貨を投げる試行についての標本空間」、センター試験 平成 30 年度 数学I・数学A 第 3 問の「大小 2 個のさいころを同時に投げる試行」）にも、その 1 回 1 回（共通テスト 令和 6 年度 数学I・数学A 第 3 問の「n 回の試行で」「n 回目の試行で初めて」）にも使う。日本語版 Wikipedia「試行 (確率論)」も英名に trial と experiment を並べる。AP Statistics の CED（topic 2.3）は outcome を one trial of a random process の結果と書き、IM（Grade 7 8.6・8.7）・OpenStax Introductory Statistics も trial を 1 回 1 回に使う。実験そのものは、CED は random process、IM（Grade 7）は chance experiment、OpenStax Introductory Statistics は experiment と呼ぶ。",
    "domains": [
      "combinatorics",
      "probability"
    ],
    "level": {
      "jp": [
        "中1",
        "数A"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "Integrated Math 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "同じ条件のもとで繰り返すことができ、結果が偶然によって決まる実験や観測。繰り返すときの 1 回 1 回も試行と呼ぶ。",
    "definition_en": "One run of an experiment that can be repeated under the same conditions and whose result depends on chance.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each flip of the coin is one trial, so ten flips means ten trials.",
        "ja": "硬貨を 1 回投げるのが 1 回の試行だから、10 回投げれば 10 回の試行だ。",
        "register": "spoken"
      },
      {
        "en": "A trial consists of rolling two dice and recording the sum of the numbers shown.",
        "ja": "2 個のさいころを投げ、出た目の和を記録することを 1 回の試行とする。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉は大半が OpenStax Introductory Statistics。",
      "「試行錯誤」は trial and error で、確率の試行とは別の言い方。"
    ],
    "related": [
      "repeated-trials",
      "elementary-event",
      "sample-space",
      "probability",
      "experiment",
      "toss"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（experiment、trial）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles（an experiment or trial）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.3（one trial of a random process）・2.10"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 7 8.2・8.3（chance experiment）・8.6・8.7（trial）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学B 統計的な推測: 2枚の硬貨を投げる試行についての標本空間"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成30年度 大学入試センター試験 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://web.archive.org/web/20190601/https://www.dnc.ac.jp/albums/abm.php?f=abm00033141.pdf&n=h30+suugaku1+02suugaku1A.pdf",
        "note": "第3問: 大小2個のさいころを同時に投げる試行、試行を2回繰り返す"
      },
      {
        "type": "reference",
        "title": "大学入試センター 令和6年度 大学入学共通テスト 本試験 数学Ⅰ・数学Ａ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?d=666&f=abm00004685.pdf&n=2024_or_19_sugaku1A.pdf",
        "note": "第3問: n回の試行、n回目の試行で初めて"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「試行 (確率論)」",
        "url": "https://ja.wikipedia.org/wiki/%E8%A9%A6%E8%A1%8C_%28%E7%A2%BA%E7%8E%87%E8%AB%96%29",
        "note": "英: trial, experiment"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "trial": 332
      },
      "written": {
        "trial": 187
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "triangle-inequality",
    "ja": {
      "term": "三角形の成立条件",
      "reading": "さんかくけいのせいりつじょうけん",
      "alt": [
        "三角不等式",
        "三角形が成立するための条件"
      ]
    },
    "en": {
      "term": "triangle inequality",
      "alt": [
        "triangle inequality theorem"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "algebra"
    ],
    "level": {
      "jp": [
        "数A",
        "数II"
      ],
      "us": [
        "Geometry",
        "Calculus I",
        "Calculus III"
      ]
    },
    "definition_ja": "三角形の 2 辺の長さの和は他の 1 辺より大きいという不等式。絶対値では |a + b| ≦ |a| + |b| の形になる。",
    "definition_en": "The inequality saying any side of a triangle is shorter than the sum of the other two; for absolute values, |a + b| ≤ |a| + |b|.",
    "latex": "|a + b| \\le |a| + |b|",
    "spoken_en": "the absolute value of a plus b is less than or equal to the absolute value of a plus the absolute value of b",
    "examples": [
      {
        "en": "By the triangle inequality, the sum of the two shorter sides has to be bigger than the longest side.",
        "ja": "三角形の成立条件から、短い 2 辺の和は一番長い辺より大きくないといけない。",
        "register": "spoken"
      },
      {
        "en": "Prove that |a + b| ≤ |a| + |b| for all real numbers a and b.",
        "ja": "すべての実数 a, b について |a + b| ≦ |a| + |b| を証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語版 Wikipedia「三角不等式」は、三角比（三角関数）を含む不等式（trigonometric-inequality）も三角不等式と呼ぶ場合があると注意する。英語の triangle inequality は、三角形の辺の不等式と、それを絶対値・ベクトルの大きさで書いたもの（|a + b| ≦ |a| + |b|、OpenStax Calculus Volume 3（2.1）の ‖v + w‖ ≦ ‖v‖ + ‖w‖）で、参照に三角関数を含む不等式の意味は出てこない。",
      "学習指導要領解説（数学II）は、絶対値の性質などを用いて不等式が成り立つことを証明することを挙げる。OpenStax Calculus Volume 1（2.5 The Precise Definition of a Limit）は |a + b| ≤ |a| + |b| を the triangle inequality と呼ぶ。"
    ],
    "related": [
      "trigonometric-inequality",
      "side-angle-inequality",
      "absolute-value"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "2.5 The Precise Definition of a Limit（the triangle inequality）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.1 Vectors in the Plane（triangle inequality）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.26 Triangle Inequality Theorem・4.29 Pythagorean Theorem to Classify Triangles"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角不等式」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E4%B8%8D%E7%AD%89%E5%BC%8F",
        "note": "冒頭: 三角比を含む不等式を三角不等式と呼ぶ場合がある"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A 図形の性質: 三角形が成立するための条件、数学II: 絶対値の性質を用いた不等式の証明"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "!sas triangle inequality": 4
      },
      "written": {
        "!sas triangle inequality": 6
      },
      "sources": [
        "khan-middle",
        "openstax-calculus",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 4 件（10 未満） ／ 書: 判断不能: 重み付け後 6 件（10 未満））。見出しは CK-12 の呼び方 !sas triangle inequality（CK-12 Geometry 4.26 Triangle Inequality Theorem・CK-12 Geometry 4.29 Pythagorean Theorem to Classify Triangles・計 12 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "triangle-proportionality-theorem",
    "ja": {
      "term": "三角形と比の定理",
      "reading": "さんかくけいとひのていり",
      "alt": [
        "平行線と線分の比"
      ]
    },
    "en": {
      "term": "triangle proportionality theorem",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "三角形の 1 辺に平行な直線が他の 2 辺と交わるとき、その 2 辺を同じ比に分けるという定理。逆も成り立つ。",
    "definition_en": "When a line runs parallel to one side of a triangle and meets the other two sides, it splits those two sides in the same ratio. The converse holds as well.",
    "latex": "DE \\parallel BC \\implies \\frac{AD}{DB} = \\frac{AE}{EC}",
    "spoken_en": "If DE is parallel to BC, then AD over DB equals AE over EC.",
    "examples": [
      {
        "en": "This segment is parallel to the base, so by the triangle proportionality theorem it cuts the other two sides in the same ratio. That gives x over 6 equals 4 over 8.",
        "ja": "この線分は底辺に平行だから、三角形と比の定理より残りの 2 辺を同じ比に分ける。よって x/6 = 4/8。",
        "register": "spoken"
      },
      {
        "en": "In △ABC, point D lies on AB and point E lies on AC, with DE ∥ BC. If AD = 6, DB = 4, and AE = 9, find EC.",
        "ja": "△ABC で、点 D, E はそれぞれ辺 AB, AC 上にあり、DE ∥ BC である。AD = 6, DB = 4, AE = 9 のとき、EC を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "DE ∥ BC のとき AD : DB = AE : EC だが、DE : BC は AD : DB ではなく AD : AB に等しい。平行な辺の比を部分の比で立てない。",
      "用例コーパスには triangle proportionality theorem が出てこない。IM Geometry 3.5・3.11 Splitting Triangle Sides with Dilation はこの性質を扱うが、定理に名前を付けない。",
      "英語版 Wikipedia はこの仲間の定理を記事「Intercept theorem」で扱い、冒頭で side splitter theorem・basic proportionality theorem という別名も挙げる。",
      "中学校学習指導要領解説はこの内容を「平行線と線分の比についての性質」と呼び、「三角形と比の定理」は出てこない（日本語版 Wikipedia「六円定理」はこの名前で使う）。"
    ],
    "related": [
      "similar-triangles",
      "parallel-lines",
      "point-of-internal-division"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Geometry 3.11 Splitting Triangle Sides with Dilation, Part 2: 平行線が 2 辺を同じ比に分けること（定理の名前は無い）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Intercept_theorem",
        "note": "記事「Intercept theorem」（数学カテゴリから 4 段）: 冒頭で side splitter theorem・basic proportionality theorem とも"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 図形: 平行線と線分の比についての性質（「三角形と比の定理」は 0 件）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「六円定理」",
        "url": "https://ja.wikipedia.org/wiki/%E5%85%AD%E5%86%86%E5%AE%9A%E7%90%86",
        "note": "本文で「三角形と比の定理より」（pitfalls[3]）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。そのまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "triangle",
    "ja": {
      "term": "三角形",
      "reading": "さんかくけい",
      "alt": []
    },
    "en": {
      "term": "triangle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "中1",
        "数A",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Precalculus",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "同じ直線上にない 3 点を線分で結んでできる図形。3 つの内角の和は 180° になる。",
    "definition_en": "A polygon with three sides, formed by joining three points that do not lie on one line; its three interior angles add up to 180°.",
    "latex": "\\angle A + \\angle B + \\angle C = 180^\\circ",
    "spoken_en": null,
    "examples": [
      {
        "en": "The three angles of a triangle always add up to 180 degrees, so the missing one is 65.",
        "ja": "三角形の 3 つの角の和はいつも 180° だから、残りの角は 65° だ。",
        "register": "spoken"
      },
      {
        "en": "Classify the triangle with side lengths 5, 7, and 9 as acute, right, or obtuse.",
        "ja": "3 辺の長さが 5, 7, 9 の三角形は、鋭角三角形・直角三角形・鈍角三角形のどれか。",
        "register": "written"
      },
      {
        "en": "Triangle ABC satisfies a cos A = b cos B. What kind of triangle is ABC?",
        "ja": "△ABC が a cos A = b cos B を満たすとき、△ABC はどのような三角形か。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "right triangle",
        "ja": "直角三角形"
      },
      {
        "en": "classify the triangle",
        "ja": "三角形を分類する"
      }
    ],
    "pitfalls": [
      "三角形の分類は、角で acute ／ right ／ obtuse triangle（鋭角・直角・鈍角）、辺で scalene ／ isosceles ／ equilateral triangle（不等辺・二等辺・正三角形）。",
      "「三角形の形状を決定せよ」を shape of the triangle と直訳するより、What kind of triangle is ABC? や acute, right, or obtuse? のように何三角形かを問う形にすると意味がはっきりする。余弦定理で最大辺の 2 乗と他の 2 辺の 2 乗の和を比べるのは同じ。",
      "△ABC は英語でも triangle ABC と読む。"
    ],
    "related": [
      "angle",
      "vertex",
      "isosceles-triangle",
      "congruent",
      "solving-triangles"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "三角形",
        "en": "Triangle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "triangle": 1479
      },
      "written": {
        "triangle": 1132
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "triangular-prism",
    "ja": {
      "term": "三角柱",
      "reading": "さんかくちゅう",
      "alt": []
    },
    "en": {
      "term": "triangular prism",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "definition_ja": "底面が三角形の角柱。面は 5 つ（底面 2 つと側面 3 つ）、辺は 9 本、頂点は 6 つある。",
    "definition_en": "A prism whose two bases are triangles. It has 5 faces, 9 edges, and 6 vertices.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A triangular prism has two triangles for bases and three rectangles on the sides, so that's five faces.",
        "ja": "三角柱は、三角形の底面が 2 つと、側面の長方形が 3 つ。だから面は 5 つだね。",
        "register": "spoken"
      },
      {
        "en": "Draw a net for a triangular prism whose bases are equilateral triangles.",
        "ja": "底面が正三角形の三角柱の展開図をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "三角柱の高さ（height of the prism）は 2 つの底面の間の距離で、底面の三角形の高さ（height of the triangle）とは別。1 つの問題に両方が出てくるので区別する。",
      "角柱は prism、角錐は pyramid。三角錐は triangular pyramid（エントリ tetrahedron）で、三角柱とは別の立体。",
      "三角柱が横に倒れた図では、両端の三角形が底面（base）で、下に接している長方形は側面（lateral face）。"
    ],
    "related": [
      "prism",
      "rectangular-prism",
      "lateral-face",
      "net",
      "volume"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "三角柱",
        "en": "Triangular prism"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "triangular prism": 16
      },
      "sources": [
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trigonometric-equation",
    "ja": {
      "term": "三角方程式",
      "reading": "さんかくほうていしき",
      "alt": []
    },
    "en": {
      "term": "trigonometric equation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "角 θ の三角関数を含む方程式。範囲を指定して、またはすべての解（一般解）を求める。",
    "definition_en": "An equation involving trigonometric functions of an angle θ, solved on a given interval or for all solutions.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Find all the angles in one period first, then add 2π n for the general solution.",
        "ja": "まず 1 周期の中の角を全部見つけて、一般解なら 2πn を足す。",
        "register": "spoken"
      },
      {
        "en": "Solve the trigonometric equation 2 cos²θ − cos θ − 1 = 0 for 0 ≤ θ < 2π.",
        "ja": "0 ≦ θ < 2π のとき、2 cos²θ − cos θ − 1 = 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「0 ≦ θ < 2π の範囲で」は on the interval [0, 2π) または for 0 ≤ θ < 2π。",
      "用例コーパスでは書き言葉（OpenStax Algebra and Trigonometry・Precalculus）に出てくるが、話し言葉には出てこない。"
    ],
    "related": [
      "trigonometric-inequality",
      "general-solution",
      "trigonometric-function",
      "solve"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Solving Trigonometric Equations（trigonometric equation、all possible solutions）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学II の内容: 三角方程式と三角不等式"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "trigonometric equation": 60
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trigonometric-function",
    "ja": {
      "term": "三角関数",
      "reading": "さんかくかんすう",
      "alt": []
    },
    "en": {
      "term": "trigonometric function",
      "alt": [],
      "variants": [
        {
          "term": "trig function",
          "register": "spoken",
          "note": "話し言葉の短い言い方。話し言葉に多く、書き言葉では少ない。"
        },
        {
          "term": "trigonometric function",
          "register": "both",
          "note": "書き言葉に多く、OpenStax Algebra and Trigonometry と OpenStax Calculus が中心。話し言葉でも使うが、trig function のほうが多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "一般角 θ に対して sin θ, cos θ, tan θ などを対応させる関数。単位円上の点の座標で定める。",
    "definition_en": "A function such as sin θ, cos θ, or tan θ that assigns a value to an angle θ of any size; sin θ and cos θ are the y- and x-coordinates where the terminal side of θ crosses the unit circle.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "All the trig functions are periodic, so their graphs repeat.",
        "ja": "三角関数はどれも周期関数なので、グラフが繰り返す。",
        "register": "spoken"
      },
      {
        "en": "Find the maximum and minimum values of the trigonometric function y = sin²θ + 2 cos θ for 0 ≤ θ < 2π.",
        "ja": "0 ≦ θ < 2π のとき、関数 y = sin²θ + 2 cos θ の最大値と最小値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は sin・cos・tan に加えて csc（cosecant）・sec（secant）・cot（cotangent）を定義して使う（節 The Other Trigonometric Functions）。高等学校学習指導要領の〔用語・記号〕は sin・cos・tan だけを挙げ、解説に sec・csc・cot は出てこない。",
      "OpenStax Algebra and Trigonometry は三角関数のグラフを、章 Periodic Functions の節 Graphs of the Sine and Cosine Functions・Graphs of the Other Trigonometric Functions で扱う。"
    ],
    "related": [
      "tangent",
      "trigonometric-identities",
      "period",
      "amplitude",
      "inverse-trigonometric-function",
      "periodic-function"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "三角関数",
        "en": "Trigonometric functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "The Other Trigonometric Functions・Graphs of the Sine and Cosine Functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）〔用語・記号〕",
        "note": "数I 図形と計量: sin、cos、tan"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "sec・csc・cot（正割・余割・余接）は出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "trig function": 174,
        "trigonometric function": 67
      },
      "written": {
        "trigonometric function": 346,
        "trig function": 10
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trigonometric-identities",
    "ja": {
      "term": "三角比の相互関係",
      "reading": "さんかくひのそうごかんけい",
      "alt": [
        "三角関数の相互関係"
      ]
    },
    "en": {
      "term": "trigonometric identities",
      "alt": [],
      "variants": [
        {
          "term": "trig identities",
          "register": "spoken",
          "note": "話し言葉の短い言い方。話し言葉に多く（MIT OCW が中心）、書き言葉では少ない。"
        },
        {
          "term": "trigonometric identities",
          "register": "both",
          "note": "話し言葉・書き言葉とも使う。書き言葉では trig identities よりずっと多く、OpenStax Algebra and Trigonometry と OpenStax Calculus が中心。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "高等学校学習指導要領解説（数学I）の「三角比の基本的な相互関係」は、sin A = cos(90° − A)・cos A = sin(90° − A)、tan A = sin A / cos A、sin²A + cos²A = 1、1 + tan²A = 1/cos²A と、180° − A の三角比の関係を含む（数学II は「三角関数の相互関係」）。英語の trigonometric identities は、OpenStax Algebra and Trigonometry の章 Trigonometric Identities and Equations のように加法定理（sum and difference formulas）・倍角の公式（double-angle formulas）まで含む三角関数の恒等式全体で、範囲が広い。相互関係の式は Pythagorean identities・quotient identities・reciprocal identities と呼び分ける。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "同じ角 θ の sin θ, cos θ, tan θ の間に成り立つ関係式。1 つの値からほかの値を求めるのに使う。",
    "definition_en": "The relations among sin θ, cos θ, and tan θ for the same angle, used to find the other values from one of them.",
    "latex": "\\sin^2\\theta + \\cos^2\\theta = 1,\\quad \\tan\\theta = \\frac{\\sin\\theta}{\\cos\\theta},\\quad 1 + \\tan^2\\theta = \\frac{1}{\\cos^2\\theta}",
    "spoken_en": null,
    "examples": [
      {
        "en": "Use the Pythagorean identity to get cosine from sine, then divide to get tangent.",
        "ja": "ピタゴラスの関係式で sin から cos を出し、割って tan を出す。",
        "register": "spoken"
      },
      {
        "en": "Use trigonometric identities to find cos θ and tan θ, given sin θ = 1/3 and 90° < θ < 180°.",
        "ja": "sin θ = 1/3, 90° < θ < 180° のとき、cos θ と tan θ の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "1 + tan²θ = 1/cos²θ は米国の教科書では 1 + tan²θ = sec²θ と secant で書く（OpenStax Algebra and Trigonometry）。学習指導要領解説には sec（正割）が出てこない。"
    ],
    "related": [
      "tangent",
      "trigonometric-function",
      "pythagorean-identity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 三角比の基本的な相互関係（sin A = cos(90° − A)、1 + tan²A = 1/cos²A ほか。正割・sec は 0 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "trig identities": 77,
        "trigonometric identities": 30
      },
      "written": {
        "trigonometric identities": 60,
        "trig identities": 8
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trigonometric-inequality",
    "ja": {
      "term": "三角不等式",
      "reading": "さんかくふとうしき",
      "alt": []
    },
    "en": {
      "term": "trigonometric inequality",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（trigonometric inequality は用例コーパスにも OpenStax にも出てこない）。参照には、三角関数を含む不等式を解く問題そのものが出てこない（OpenStax Algebra and Trigonometry（7.3 Unit Circle）は、sin t と cos t の符号から t が決める単位円上の点の象限を答えさせるだけ）。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": []
    },
    "definition_ja": "角 θ の三角関数を含む不等式。単位円やグラフを使って θ の範囲を求める。",
    "definition_en": "An inequality involving trigonometric functions of an angle θ, solved for a range of θ using the unit circle or a graph.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Look at the unit circle: where is the y-coordinate bigger than one half?",
        "ja": "単位円を見よう。y 座標が 1/2 より大きいのはどこ？",
        "register": "spoken"
      },
      {
        "en": "Solve sin θ > 1/2 for 0 ≤ θ < 2π.",
        "ja": "0 ≦ θ < 2π のとき、不等式 sin θ > 1/2 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語版 Wikipedia「三角不等式」は、三角不等式を |a + b| ≦ |a| + |b|（英語の triangle inequality。エントリ triangle-inequality）とし、三角比を含む不等式も三角不等式と呼ぶ場合があるので注意が要ると書く。"
    ],
    "related": [
      "trigonometric-equation",
      "triangle-inequality"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角不等式」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E4%B8%8D%E7%AD%89%E5%BC%8F",
        "note": "三角不等式は triangle inequality。三角比を含む不等式も三角不等式（英: trigonometric inequalities）と呼ぶ場合がある"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「数学 (教科)」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B0%E5%AD%A6_%28%E6%95%99%E7%A7%91%29",
        "note": "数学II の内容: 三角方程式と三角不等式"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.3 Unit Circle（sin t と cos t の符号から象限を答える問題）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "trigonometric-integrals",
    "ja": {
      "term": "三角関数の積の積分",
      "reading": "さんかくかんすうのせきのせきぶん",
      "alt": [
        "三角関数の積分"
      ]
    },
    "en": {
      "term": "trigonometric integrals",
      "alt": [],
      "variants": [
        {
          "term": "trig integrals",
          "register": "spoken",
          "note": "略した言い方で、話し言葉の主見出し（MIT OCW が中心）。書き言葉にはほとんど出てこない（MIT の講義ノートだけ）。"
        },
        {
          "term": "trigonometric integrals",
          "register": "written",
          "note": "節の名前（OpenStax Calculus Volume 2）。話し言葉では少なく、書き言葉は OpenStax Calculus が中心。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I",
        "Calculus II"
      ]
    },
    "definition_ja": "三角関数の不定積分・定積分。基本は ∫sin x dx = −cos x + C、∫cos x dx = sin x + C で、sin³x cos²x や sin 3x cos 2x のような積・べきは、積和の公式や次数を下げる公式を使って計算する。",
    "definition_en": "Integrals of trigonometric functions: the basic ∫ sin x dx = −cos x + C and ∫ cos x dx = sin x + C, and products and powers such as sin³x cos²x, handled with product-to-sum and power-reducing identities.",
    "latex": "\\int \\sin^2 x\\,dx = \\frac{x}{2} - \\frac{\\sin 2x}{4} + C",
    "spoken_en": "the integral of sine squared x dx equals x over two minus sine two x over four, plus C",
    "examples": [
      {
        "en": "The integral of sine is negative cosine, so watch that sign.",
        "ja": "sin の積分は −cos。符号に気をつけて。",
        "register": "spoken"
      },
      {
        "en": "With an odd power of sine, save one sine for du and turn the rest into cosines.",
        "ja": "sin の奇数乗なら、sin を 1 つ du 用に残して、残りを cos に直す。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "∫(1/cos²x)dx = tan x + C は、OpenStax Calculus では sec²x（secant squared x）を使って ∫sec²x dx = tan x + C と書く（学習指導要領解説には sec が出てこない）。",
      "積和の公式は product-to-sum formulas、次数を下げる公式は power-reducing formulas（half-angle formulas とも）と言う。",
      "OpenStax Calculus Volume 2 の節 Trigonometric Integrals は三角関数の積・べきの積分を扱い、sin x の積分のような基本の積分は Volume 1 で扱う。"
    ],
    "related": [
      "trigonometric-substitution",
      "integration-formulas",
      "integration-by-substitution",
      "integrals-giving-inverse-trig-functions"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "sec が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "trig integrals": 36,
        "trigonometric integrals": 7
      },
      "written": {
        "trigonometric integrals": 16,
        "trig integrals": 2
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trigonometric-ratio",
    "ja": {
      "term": "三角比",
      "reading": "さんかくひ",
      "alt": []
    },
    "en": {
      "term": "trigonometric ratio",
      "alt": [
        "trig ratio"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "英語の trigonometric ratio は直角三角形の辺の比で、IM Geometry（用語集・4.6・4.9）は鋭角についてだけ使う。鈍角の sin・cos・tan は、OpenStax Precalculus・Algebra and Trigonometry では単位円で定める三角関数（trigonometric functions、エントリ trigonometric-function）の値として扱う。IM Algebra 2（6.6 The Pythagorean Identity (Part 2)）と OpenStax Calculus（Volume 2 3.3 Trigonometric Substitution、Volume 3 2.1 Vectors in the Plane）は、一般の角 θ の sin θ・cos θ・tan θ の値も trigonometric ratios と呼ぶ。定義として直角三角形の鋭角の比を言うのは IM Geometry・CK-12 Geometry。",
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "直角三角形の辺の比で定める sin θ, cos θ, tan θ の総称。鈍角を含む 0° ≦ θ ≦ 180° では、原点を中心とする半円上の点の座標で定める。",
    "definition_en": "In a right triangle, the sine, cosine, and tangent of either acute angle, given as ratios of its sides: opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent.",
    "latex": "\\sin\\theta = \\frac{\\text{opposite}}{\\text{hypotenuse}},\\quad \\cos\\theta = \\frac{\\text{adjacent}}{\\text{hypotenuse}},\\quad \\tan\\theta = \\frac{\\text{opposite}}{\\text{adjacent}}",
    "spoken_en": "sine theta is opposite over hypotenuse, cosine theta is adjacent over hypotenuse, tangent theta is opposite over adjacent",
    "examples": [
      {
        "en": "The trig ratios only depend on the angle, not on how big the triangle is.",
        "ja": "三角比は角だけで決まり、三角形の大きさにはよらない。",
        "register": "spoken"
      },
      {
        "en": "Find sin 150°, cos 150°, and tan 150°.",
        "ja": "150° の三角比の値を求めよ。",
        "register": "written"
      },
      {
        "en": "In right triangle ABC with ∠C = 90°, express BC in terms of AB and a trigonometric ratio of ∠A.",
        "ja": "∠C = 90° の直角三角形 ABC において、BC を AB と ∠A の三角比を用いて表せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "鈍角では cos と tan が負になる。sin(180° − θ) = sin θ、cos(180° − θ) = −cos θ、tan(180° − θ) = −tan θ。",
      "英語の覚え方 SOH-CAH-TOA は、Sine = Opposite/Hypotenuse、Cosine = Adjacent/Hypotenuse、Tangent = Opposite/Adjacent の頭文字。",
      "高等学校学習指導要領解説では、数学I の三角比を直角三角形の鋭角で定めてから鈍角（0° ≦ θ ≦ 180°）まで広げ、数学II で一般角に広げたものを三角関数（trigonometric function）と呼ぶ。"
    ],
    "related": [
      "sine",
      "cosine",
      "tangent",
      "adjacent-side",
      "trigonometric-function",
      "sohcahtoa"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "3.3 Trigonometric Substitution（the trigonometric ratios produced from the reference triangle。mapping_note）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.1 Vectors in the Plane（Use trigonometric ratios。mapping_note）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry glossary・Geometry 4.6 Working with Trigonometric Ratios・Geometry 4.9 Using Trigonometric Ratios to Find Angles"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 鋭角の三角比・鈍角の三角比。数学II 三角関数: 一般角"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "5.2 Unit Circle: Sine and Cosine Functions（単位円で定める sine・cosine。mapping_note）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.3 Unit Circle（mapping_note）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 6.6 The Pythagorean Identity (Part 2)（the two missing trigonometric ratios。mapping_note）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "用語欄 Trigonometric Ratios（sides and angles of right triangles。mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "trig ratio": 1
      },
      "written": {
        "trigonometric ratio": 4
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは IM の呼び方 trigonometric ratio（Geometry glossary・Geometry 4.6 Working with Trigonometric Ratios・Geometry 4.9 Using Trigonometric Ratios to Find Angles・計 8 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "trigonometric-substitution",
    "ja": {
      "term": "三角置換",
      "reading": "さんかくちかん",
      "alt": []
    },
    "en": {
      "term": "trigonometric substitution",
      "alt": [],
      "variants": [
        {
          "term": "trig substitution",
          "register": "spoken",
          "note": "話し言葉で最も多い（MIT OCW が中心）。書き言葉にはほとんど出てこない（MIT の講義ノートだけ）。"
        },
        {
          "term": "trig sub",
          "register": "spoken",
          "note": "さらに略した言い方。NancyPi に多い。書き言葉には出てこない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学III）は x = a sin θ の置き換えを置換積分法に含め、別の名前を付けない。OpenStax Calculus Volume 2 は Trigonometric Substitution の節で体系的に扱い、不定積分の答えを x に戻すために直角三角形（reference triangle）を描く。",
    "domains": [
      "calculus",
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "definition_ja": "√(a² − x²) などを含む積分で、x = a sin θ、x = a tan θ、x = a sec θ のように x を三角関数で置き換えて根号を外す方法。",
    "definition_en": "A method for integrals with √(a² − x²), √(a² + x²) or √(x² − a²): replace x with a sin θ, a tan θ or a sec θ to remove the root.",
    "latex": "\\int \\sqrt{1-x^2}\\,dx, \\quad x = \\sin\\theta",
    "spoken_en": "the integral of the square root of one minus x squared dx, with x equals sine theta",
    "examples": [
      {
        "en": "Square root of one minus x squared? That's a trig sub: let x be sine theta.",
        "ja": "√(1 − x²)？ それは三角置換。x = sin θ とおく。",
        "register": "spoken"
      },
      {
        "en": "Using the trigonometric substitution x = 3 tan θ, we have dx = 3 sec² θ dθ.",
        "ja": "三角置換 x = 3 tan θ を用いると dx = 3 sec²θ dθ となる。",
        "register": "written"
      },
      {
        "en": "For the square root of two x plus one, let u be two x plus one, and it turns into a power.",
        "ja": "√(2x + 1) は u = 2x + 1 とおけば、べき乗の形になる。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "1/cos²θ は OpenStax では sec²θ と書く（学習指導要領解説には sec が出てこない）。",
      "AP Calculus の CED（2020 年版）には trigonometric substitution の topic がない。BC の積分の技法は topic 6.11（部分積分）・6.12（部分分数）・6.13（広義積分）。",
      "√ を含む関数（日本の「無理関数」）は、OpenStax と AP の CED では irrational function と呼ばず、radical function ／ square root function と呼ぶ。√(ax + b) は u-substitution、√(a² − x²) などは trigonometric substitution で扱う。"
    ],
    "related": [
      "integration-by-substitution",
      "integrals-giving-inverse-trig-functions"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "topic なし（Unit 6 の積分の技法は 6.9–6.14。三角置換は含まれない）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 積分法: 置換積分法は ax + b = t、x = a sin θ と置き換えるもの"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "trig substitution": 79,
        "trig sub": 27,
        "trigonometric substitution": 7
      },
      "written": {
        "trigonometric substitution": 22,
        "trig substitution": 3
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "trinomial",
    "ja": {
      "term": "三項式",
      "reading": "さんこうしき",
      "alt": []
    },
    "en": {
      "term": "trinomial",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "項が 3 つの多項式。x² + 5x + 6 のような式。",
    "definition_en": "A polynomial with exactly three terms, such as x² + 5x + 6.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For this trinomial, x squared plus 5x plus 6, we need two numbers that multiply to 6 and add to 5.",
        "ja": "この三項式 x² + 5x + 6 では、掛けて 6、足して 5 になる 2 つの数を探そう。",
        "register": "spoken"
      },
      {
        "en": "Factor the trinomial x² − 7x + 12.",
        "ja": "三項式 x² − 7x + 12 を因数分解せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "tri- は「3 つの」の意味。項が 1 つなら monomial（エントリ monomial）、2 つなら binomial（エントリ binomial）。",
      "OpenStax Elementary Algebra は x² + bx + c ／ ax² + bx + c の因数分解を Factor Trinomials of the Form … の節で扱う（IM Algebra 1 は trinomial を使わず、Rewriting Quadratic Expressions in Factored Form）。",
      "x² + 6x + 9 = (x + 3)² のように、2 乗の形に因数分解できる三項式は perfect square trinomial（完全平方式、エントリ perfect-square-trinomial）と呼ぶ。"
    ],
    "related": [
      "binomial",
      "monomial",
      "polynomial",
      "factoring",
      "perfect-square-trinomial"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "三項式",
        "en": "Trinomial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 7.6〜7.10 Rewriting Quadratic Expressions in Factored Form（trinomial は 0 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "trinomial": 49
      },
      "written": {
        "trinomial": 431
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "triple-angle-formulas",
    "ja": {
      "term": "3 倍角の公式",
      "reading": "さんばいかくのこうしき",
      "alt": [
        "三倍角の公式"
      ]
    },
    "en": {
      "term": "triple-angle formulas",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "definition_ja": "sin 3α = 3 sin α − 4 sin³α、cos 3α = 4 cos³α − 3 cos α の公式。",
    "definition_en": "The formulas sin 3α = 3 sin α − 4 sin³α and cos 3α = 4 cos³α − 3 cos α.",
    "latex": "\\sin 3\\alpha = 3\\sin\\alpha - 4\\sin^3\\alpha,\\quad \\cos 3\\alpha = 4\\cos^3\\alpha - 3\\cos\\alpha",
    "spoken_en": null,
    "examples": [
      {
        "en": "Write 3θ as 2θ plus θ and expand — that's where the triple-angle formula comes from.",
        "ja": "3θ を 2θ + θ として展開する。3 倍角の公式はそこから出てくる。",
        "register": "spoken"
      },
      {
        "en": "Derive the formula for cos 3α from the addition and double-angle formulas.",
        "ja": "加法定理と 2 倍角の公式から cos 3α の公式を導け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスと参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）には triple-angle formula の名前が出てこない。OpenStax Algebra and Trigonometry・Precalculus（節 Double-Angle, Half-Angle, and Reduction Formulas の練習問題）は、sin(3x) = 3 sin x cos²x − sin³x を sin(x + 2x) と書いて加法定理と 2 倍角の公式で示させる（cos²x = 1 − sin²x で書き直すと 3 sin x − 4 sin³x）。"
    ],
    "related": [
      "double-angle-formulas",
      "angle-addition-formulas"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/List_of_trigonometric_identities",
        "note": "記事「List of trigonometric identities」の節 Triple-angle formulae（節の見出し。数学カテゴリから 2 段）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Double-Angle, Half-Angle, and Reduction Formulas（練習問題: sin(3x)、cos(3x)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "Double-Angle, Half-Angle, and Reduction Formulas（練習問題）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「三角関数の公式の一覧」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%89%E8%A7%92%E9%96%A2%E6%95%B0%E3%81%AE%E5%85%AC%E5%BC%8F%E3%81%AE%E4%B8%80%E8%A6%A7",
        "note": "節「倍角・三倍角・半角の公式」（三倍角の公式）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学II (4) 三角関数 イ(ア): 三角関数の加法定理から新たな性質（2 倍角の公式・三角関数の合成）を導く（3 倍角の公式は出てこない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "triple-integral",
    "ja": {
      "term": "三重積分",
      "reading": "さんじゅうせきぶん",
      "alt": []
    },
    "en": {
      "term": "triple integral",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "3 変数関数を空間の領域で積分したもの。",
    "definition_en": "The integral of a function of three variables over a solid region of space.",
    "latex": "\\iiint_E f(x, y, z)\\,dV",
    "spoken_en": "the triple integral over E of f of x, y, z d V",
    "examples": [
      {
        "en": "If the integrand is 1, the triple integral just gives you the volume.",
        "ja": "被積分関数が 1 なら、三重積分は体積になる。",
        "register": "spoken"
      },
      {
        "en": "Evaluate the triple integral ∭_E z dV, where E is the box [0, 1] × [0, 2] × [0, 3].",
        "ja": "E が直方体 [0, 1] × [0, 2] × [0, 3] のとき、三重積分 ∭_E z dV を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "double-integral",
      "cylindrical-coordinates",
      "spherical-coordinates",
      "change-of-variables"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "triple integral": 95
      },
      "written": {
        "triple integral": 165
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "true",
    "ja": {
      "term": "真",
      "reading": "しん",
      "alt": []
    },
    "en": {
      "term": "true",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "adjective",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "命題が正しいこと。真理値表では T と書く。",
    "definition_en": "Describes a statement that is correct; in a truth table it is marked T.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Is that true for every x? Let's check x = 0.",
        "ja": "それってすべての x で真かな。x = 0 で確かめてみよう。",
        "register": "spoken"
      },
      {
        "en": "Show that the statement is true for all positive integers n.",
        "ja": "この命題がすべての正の整数 n について真であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「真である」は英語では is true。「真」を名詞のまま訳さず、The statement is true. のように形容詞で使う。",
      "命題や式の正しさは true、答えや計算の正しさは correct で言い分ける（本プロジェクトの書き方の案内）。"
    ],
    "related": [
      "false",
      "truth-value",
      "proposition",
      "counterexample"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "true": 1874
      },
      "written": {
        "true": 949
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "truth-table",
    "ja": {
      "term": "真理値表",
      "reading": "しんりちひょう",
      "alt": []
    },
    "en": {
      "term": "truth table",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "もとになる命題の真偽のすべての組み合わせを行に並べ、それぞれについて全体の命題の真偽を書いた表のこと。",
    "definition_en": "A table that lists every combination of truth values for the component statements and shows the truth value of the whole statement in each row.",
    "latex": "\\begin{array}{cc|c} p & q & p \\land q \\\\ \\hline T & T & T \\\\ T & F & F \\\\ F & T & F \\\\ F & F & F \\end{array}",
    "spoken_en": "p, q, p and q: true, true, true; true, false, false; false, true, false; false, false, false",
    "examples": [
      {
        "en": "With two statements you need four rows in your truth table, and with three statements you need eight.",
        "ja": "命題が 2 つなら真理値表は 4 行、3 つなら 8 行必要だよ。",
        "register": "spoken"
      },
      {
        "en": "Make a truth table for p ∨ ¬q.",
        "ja": "p ∨ ¬q の真理値表を作れ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数I では命題の真偽を集合の包含関係（ベン図）で調べ、真理値表は使わない。",
      "表の T ／ F は true ／ false と読む。行の数はもとの命題が n 個なら 2ⁿ 行。",
      "用例コーパスで truth table は話し言葉では少なく（MIT OCW）、書き言葉はすべて MIT の講義ノート。CK-12 Geometry では 35 件（2.8 Truth Tables ほか）、Levin の本では 74 件。"
    ],
    "related": [
      "truth-value",
      "logical-connective",
      "tautology",
      "biconditional"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "真理値表",
        "en": "Truth table"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.8 Truth Tables"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学I (1) 数と式（発展・拡充）: 真理値表を用いて命題「p→q」の否定が「p∧（¬q）」であること"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "truth table": 3
      },
      "written": {
        "truth table": 20
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "truth-value",
    "ja": {
      "term": "真偽",
      "reading": "しんぎ",
      "alt": [
        "真理値"
      ]
    },
    "en": {
      "term": "truth value",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "命題が真であるか偽であるかということ。",
    "definition_en": "The value, true or false, that a statement takes.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Before we prove anything, let's figure out the truth value of each statement.",
        "ja": "証明に入る前に、それぞれの命題の真偽を確かめよう。",
        "register": "spoken"
      },
      {
        "en": "Determine whether each statement is true or false. If a statement is false, give a counterexample.",
        "ja": "次の命題の真偽を調べよ。偽であるものは反例を挙げよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "問題文は truth value を使わず、OpenStax Calculus・Algebra and Trigonometry・CK-12 Geometry とも Determine whether the statement is true or false. と書く。truth value は CK-12 Geometry・Levin の説明の文に出る。",
      "Levin の離散数学の本も truth value と呼ぶ（1.1 Mathematical Statements ほか）。"
    ],
    "related": [
      "true",
      "false",
      "proposition",
      "counterexample"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1 Mathematical Statements"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.9 And and Or Statements・2.10 Negative Statements・2.8 Truth Tables"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "determine whether the statement is true or false（7 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Determine whether the statement is true or false（2 件）"
      },
      {
        "type": "wikipedia-langlink",
        "ja": "真理値",
        "en": "Truth value"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "truth value": 3
      },
      "sources": [
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは CK-12 の呼び方 truth value（CK-12 Geometry 2.9 And and Or Statements・CK-12 Geometry 2.10 Negative Statements・CK-12 Geometry 2.8 Truth Tables・計 14 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "turning-points",
    "ja": {
      "term": "極値をとる点",
      "reading": "きょくちをとるてん",
      "alt": [
        "極大・極小の点"
      ]
    },
    "en": {
      "term": "turning point",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本では数II・数III で微分して極大・極小を求め、グラフの形から極値をとる点の個数を数える言い方はしない。米国の教科書（OpenStax Algebra and Trigonometry）は微分を使わず、多項式関数の次数から turning points の個数の上限を読む。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "グラフが増加から減少、または減少から増加に変わる点。微分を使う場面では極大・極小の点に当たる。",
    "definition_en": "A point where a graph changes from increasing to decreasing, or from decreasing to increasing.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This cubic goes up, comes down, then goes back up, so it has two turning points.",
        "ja": "この 3 次関数は上がって、下がって、また上がるから、極値をとる点が 2 つある。",
        "register": "spoken"
      },
      {
        "en": "What is the greatest possible number of turning points of the graph of f(x) = x⁵ − 4x³ + x?",
        "ja": "f(x) = x⁵ − 4x³ + x のグラフについて、極値をとる点の個数として考えられる最大の数を答えよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Algebra and Trigonometry は、n 次の多項式関数のグラフの turning points は多くても n − 1 個と書く。上限なので、それより少ないこともある。",
      "微積分では同じ点を local maximum ／ local minimum（エントリ local-extremum）と言う。",
      "turning point はふつうの英語では「転機」の意味でも使う。"
    ],
    "related": [
      "local-extremum",
      "local-maximum",
      "local-minimum",
      "polynomial-function",
      "increasing-and-decreasing"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "turning point": 6
      },
      "written": {
        "turning point": 64
      },
      "sources": [
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "two-column-proof",
    "ja": {
      "term": "二段組みの証明",
      "reading": "にだんぐみのしょうめい",
      "alt": []
    },
    "en": {
      "term": "two-column proof",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「二段組みの証明」は学習指導要領解説に無い、本プロジェクトの訳語。日本の中学・高校の証明は文章で書き、「statements（主張）」と「reasons（理由）」を 2 列に並べる答案の形式はない。CK-12 Geometry はこの形式を two-column proof と呼ぶ（4.15 ASA and AAS ほか、9 件）。",
    "domains": [
      "proof",
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "definition_ja": "左の列に主張（statements）を、右の列にその理由（reasons）を 1 行ずつ対応させて並べる証明の書き方。",
    "definition_en": "A proof laid out as a table, with numbered statements in the left column and the reason for each statement in the right column.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's set this up as a two-column proof: statements on the left, reasons on the right.",
        "ja": "これを二段組みの証明で書こう。左に主張、右に理由だ。",
        "register": "spoken"
      },
      {
        "en": "Points A, B, C, and D lie on a line in that order, and AB = CD. Write a two-column proof that AC = BD.",
        "ja": "点 A, B, C, D はこの順に一直線上にあり、AB = CD である。AC = BD であることを二段組みの証明で示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "1 行目の理由は Given（仮定）、最後の行は Prove: に書かれた結論になる。日本の答案の「仮定より」は、この形式では理由の列に Given と書く。",
      "理由の列には定義・公準・定理の名前（Definition of midpoint、Reflexive Property など）を書く。",
      "two-column proof は用例コーパスにも IM にも出てこない（IM Geometry はこの形式の名前を使わない）。"
    ],
    "related": [
      "paragraph-proof",
      "flowchart-proof",
      "proof",
      "hypothesis"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.15 ASA and AAS・2.15 Proofs- Angle Pairs and Segments・3.8 Angles and Perpendicular Lines"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 two-column proof（CK-12 Geometry 4.15 ASA and AAS・CK-12 Geometry 2.15 Proofs- Angle Pairs and Segments・CK-12 Geometry 3.8 Angles and Perpendicular Lines・計 9 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "two-proportion-z-test",
    "ja": {
      "term": "2 標本比率の検定",
      "reading": "にひょうほんひりつのけんてい",
      "alt": []
    },
    "en": {
      "term": "test of two proportions",
      "alt": [
        "two-proportion z-test"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「2 標本比率の検定」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "2 つの母集団の比率が等しいかどうかを、2 つの標本比率の差から z 検定で調べる方法。",
    "definition_en": "A z-test of whether two population proportions are equal, based on the difference between two sample proportions.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We're comparing two groups' proportions, so it's a two-proportion z-test.",
        "ja": "2 つの群の比率を比べるので、2 標本比率の z 検定。",
        "register": "spoken"
      },
      {
        "en": "Perform a test of two proportions to decide whether the rates differ.",
        "ja": "2 標本比率の検定を行い、割合に差があるか判断せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "two-proportion z-test（2 標本比率の z 検定）とも呼ぶが、用例コーパスには出てこない。OpenStax Introductory Statistics は test of two proportions（3 件）と書く。"
    ],
    "related": [
      "population-proportion",
      "sample-proportion",
      "hypothesis-testing"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "written": {
        "test of two proportions": 3
      },
      "sources": [
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 3 件（10 未満））。見出しは OpenStax の呼び方 test of two proportions（本文 3 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "two-sample-t-test",
    "ja": {
      "term": "2 標本 t 検定",
      "reading": "にひょうほんてぃーけんてい",
      "alt": []
    },
    "en": {
      "term": "two-sample t-test",
      "alt": [
        "t-test for the difference of two means"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "独立な 2 つの標本の平均をもとに、2 つの母平均が等しいかどうかを t 分布で検定する方法。",
    "definition_en": "A t-test of whether two population means are equal, based on the means of two independent samples.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The two groups are independent, so we run a two-sample t-test, not a paired one.",
        "ja": "2 つのグループは独立なので、対応のある検定ではなく 2 標本 t 検定を行う。",
        "register": "spoken"
      },
      {
        "en": "Use a two-sample t-test to decide whether the mean scores of the two classes differ.",
        "ja": "2 標本 t 検定を用いて、2 クラスの平均点が異なるかどうか判断せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスにも OpenStax Introductory Statistics にも出てこない。OpenStax は two population means with unknown standard deviations（節の名前）として扱う。"
    ],
    "related": [
      "one-sample-t-test",
      "paired-t-test",
      "t-distribution",
      "two-proportion-z-test"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 4.9・4.10"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは AP Statistics の CED の呼び方 two-sample t-test（topic 4.9・topic 4.10）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "two-tailed-test",
    "ja": {
      "term": "両側検定",
      "reading": "りょうがわけんてい",
      "alt": []
    },
    "en": {
      "term": "two-tailed test",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "対立仮説が「μ ≠ μ₀」のように両方向の違いを考え、棄却域を分布の両側にとる検定。",
    "definition_en": "A test whose alternative hypothesis allows a difference in either direction (such as μ ≠ μ₀), with the rejection region split between both tails.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's two-tailed, so we double the one-tail area to get the p-value.",
        "ja": "両側検定なので、片側の面積を 2 倍して p 値にする。",
        "register": "spoken"
      },
      {
        "en": "Perform a two-tailed test at the 1% significance level.",
        "ja": "有意水準 1% で両側検定を行え。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "形容詞で The test is two-tailed. とも言う（OpenStax Introductory Statistics、two-tailed 単独を含めると 41 件）。"
    ],
    "related": [
      "one-tailed-test",
      "hypothesis-testing",
      "alternative-hypothesis"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "two-tailed test": 3
      },
      "written": {
        "two-tailed test": 22
      },
      "sources": [
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "two-variable-data",
    "ja": {
      "term": "二変量データ",
      "reading": "にへんりょうでーた",
      "alt": []
    },
    "en": {
      "term": "bivariate data",
      "alt": [
        "two-variable data"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Integrated Math 1"
      ]
    },
    "definition_ja": "1 つの対象について、2 つの変数の値を組にして記録したデータ（身長と体重など）。",
    "definition_en": "Data that pair the values of two variables for each individual, such as each student's height and weight.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Height and weight for each kid is bivariate data, so let's put it on a scatter plot.",
        "ja": "一人ひとりの身長と体重は二変量データだから、散布図にしてみよう。",
        "register": "spoken"
      },
      {
        "en": "The table shows bivariate data on hours studied and test scores for a group of students. Make a scatter plot and describe the association.",
        "ja": "表は、ある生徒の集団の学習時間とテストの得点の二変量データである。散布図をかき、関連を述べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の数I では「2 つの変量のデータ」として散布図・相関係数を学ぶ。英語の bivariate の bi- は「2 つの」の意味で、two-variable data とも言う。",
      "量的変数どうしは散布図（エントリ scatter-plot）と相関・回帰直線で、質的変数どうしは二元分割表（two-way table）で表す。",
      "変数が 1 つだけのデータは one-variable data（エントリ one-variable-data）。"
    ],
    "related": [
      "one-variable-data",
      "scatter-plot",
      "correlation-coefficient",
      "regression-line",
      "contingency-table"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 5.4"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "bivariate data": 5
      },
      "written": {
        "bivariate data": 7
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 7 件（10 未満））。見出しは AP Statistics の CED の呼び方 bivariate data（topic 5.4）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "type-i-error",
    "ja": {
      "term": "第一種の過誤",
      "reading": "だいいっしゅのかご",
      "alt": [
        "第 1 種の誤り"
      ]
    },
    "en": {
      "term": "Type I error",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "帰無仮説が正しいのに、それを棄却してしまう誤り。その確率は有意水準 α。",
    "definition_en": "Rejecting the null hypothesis when it is actually true; its probability is the significance level α.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A Type I error is a false alarm: you think something happened when it didn't.",
        "ja": "第一種の過誤は誤報のようなもの。何も起きていないのに起きたと思う。",
        "register": "spoken"
      },
      {
        "en": "Describe a Type I error in the context of this problem.",
        "ja": "この問題の文脈で第一種の過誤を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Type I は type one と読む。"
    ],
    "related": [
      "type-ii-error",
      "significance-level",
      "null-hypothesis"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "第一種過誤と第二種過誤",
        "en": "Type I and type II errors"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "Type I error": 33
      },
      "written": {
        "Type I error": 43
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "type-ii-error",
    "ja": {
      "term": "第二種の過誤",
      "reading": "だいにしゅのかご",
      "alt": [
        "第 2 種の誤り"
      ]
    },
    "en": {
      "term": "Type II error",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "帰無仮説が誤っているのに、それを棄却しない誤り。その確率を β と書く。",
    "definition_en": "Failing to reject the null hypothesis when it is actually false; its probability is written β.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A Type II error is a miss: something really happened, but the test didn't catch it.",
        "ja": "第二種の過誤は見逃し。本当に何か起きたのに、検定がとらえられなかった。",
        "register": "spoken"
      },
      {
        "en": "What is the consequence of a Type II error in this situation?",
        "ja": "この状況で第二種の過誤が起きるとどうなるか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Type II は type two と読む。1 − β を検定の power（検出力）と言う。"
    ],
    "related": [
      "type-i-error",
      "null-hypothesis"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "第一種過誤と第二種過誤",
        "en": "Type I and type II errors"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "Type II error": 40
      },
      "written": {
        "Type II error": 55
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "unbiased-estimator",
    "ja": {
      "term": "不偏推定量",
      "reading": "ふへんすいていりょう",
      "alt": []
    },
    "en": {
      "term": "unbiased estimator",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "unbiased estimate",
          "register": "spoken",
          "note": "推定した値のほうを指す言い方。話し言葉だけに出てきて、すべて Khan Academy。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "その期待値が推定したい母数に等しくなる推定量。標本平均は母平均の不偏推定量。",
    "definition_en": "An estimator whose expected value equals the parameter it estimates; the sample mean is an unbiased estimator of the population mean.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We divide by n minus 1 so that the sample variance is an unbiased estimator of the population variance.",
        "ja": "標本分散が母分散の不偏推定量になるように、n − 1 で割る。",
        "register": "spoken"
      },
      {
        "en": "Show that the sample mean is an unbiased estimator of μ.",
        "ja": "標本平均が μ の不偏推定量であることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉では unbiased estimate のほうが unbiased estimator より多い（どちらも Khan Academy）。OpenStax Introductory Statistics には出てこない。"
    ],
    "related": [
      "unbiased",
      "estimation",
      "population-variance"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.1（推定量が unbiased であることの説明。unbiased estimator という連語は 0 件）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "unbiased estimate": 15,
        "unbiased estimator": 6
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "unbiased",
    "ja": {
      "term": "かたよりのない",
      "reading": "かたよりのない",
      "alt": [
        "偏りのない",
        "不偏"
      ]
    },
    "en": {
      "term": "unbiased",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "adjective",
    "mapping": "near",
    "mapping_note": "英語の unbiased は 3 つの意味にまたがる: 標本の選び方（中学校学習指導要領解説は「偏りなく抽出する」と書く）、推定量（日本語は「不偏」。エントリ unbiased-estimator。AP Statistics の CED topic 3.1）、硬貨などの公平さ（MIT OCW の講義ノート）。日本語の「かたよりのない」は 1 つ目に当たる。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "標本の選び方や推定の方法が、特定の結果に系統的に寄っていないこと。",
    "definition_en": "Not tending systematically toward particular results, said of a way of choosing a sample or of estimating.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you only count the people who volunteer to answer, the sample isn't unbiased, because the people who care most are overrepresented.",
        "ja": "自分から答えてくれた人だけを数えると、関心の強い人が多く入るので、かたよりのない標本にはならない。",
        "register": "spoken"
      },
      {
        "en": "Is the sampling method likely to produce an unbiased sample? Explain.",
        "ja": "この抽出方法でかたよりのない標本が得られるか。説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "表と裏が同じ確率で出る硬貨は、OpenStax Introductory Statistics では fair coin と書く（unbiased coin は用例コーパスでは MIT OCW の講義ノートだけ）。",
      "反対は biased（偏った）。"
    ],
    "related": [
      "is-biased",
      "unbiased-estimator",
      "random-sampling"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 3.1（unbiased）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 3 学年 標本調査: 偏りなく抽出するための代表的な方法として無作為抽出"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "unbiased sample | unbiased estimate | unbiased estimator | unbiased statistic": 33
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "undefined-terms",
    "ja": {
      "term": "無定義用語",
      "reading": "むていぎようご",
      "alt": [
        "点・直線・平面"
      ]
    },
    "en": {
      "term": "undefined terms",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "理論の出発点として、定義をせずに使う用語。ユークリッド幾何では点・直線・平面がそれにあたり、その性質は公理で決まる。",
    "definition_en": "The basic words that a theory uses without defining them; in Euclidean geometry these are point, line, and plane, whose behavior is fixed by the axioms.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Point, line, and plane are our undefined terms. We describe them, but we don't define them.",
        "ja": "点・直線・平面は無定義用語。説明はするけど、定義はしない。",
        "register": "spoken"
      },
      {
        "en": "Name the three undefined terms of geometry and use them to define a segment.",
        "ja": "幾何の 3 つの無定義用語を挙げ、それを使って線分を定義せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "an undefined term は「値が定まらない項」（0 で割る項など）の意味でも使う。",
      "ja.alt の「点・直線・平面」は無定義用語の中身。英語でも point, line, plane の順に並べて言える。"
    ],
    "related": [
      "axiom",
      "segment",
      "segment-addition-postulate",
      "angle-addition-postulate",
      "proof"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 中学の単元の前の修正」の 3）。そのまま。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "undercount",
    "ja": {
      "term": "数え落とす",
      "reading": "かぞえおとす",
      "alt": []
    },
    "en": {
      "term": "undercount",
      "alt": [],
      "uk": null
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（用例コーパスには undercount が話・書とも出てこない）。",
    "domains": [
      "combinatorics"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math"
      ]
    },
    "definition_ja": "数えるべき場合の一部を見落として、個数を実際より少なく数えること。",
    "definition_en": "To leave out some of the cases that should be counted, so the total comes out too small.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you only list the pairs where the first number is smaller, you'll undercount — you miss the doubles like (3, 3).",
        "ja": "1 つ目の数が小さい組だけを書き出すと数え落とす。(3, 3) のようなぞろ目が抜けるからね。",
        "register": "spoken"
      },
      {
        "en": "Draw a tree diagram so you don't miss a case.",
        "ja": "数え落とさないように樹形図をかこう。",
        "register": "spoken"
      },
      {
        "en": "To count the pairs with x ≤ y, listing only those with x < y undercounts: the pairs with x = y are missing.",
        "ja": "x ≦ y となる組を数えるのに x < y の組だけを書き出すと、x = y の組が抜けて数え落とす。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説（数学A）の「場合の数をもれなく重複なく数え上げる」は、英語では Levin（3.3）のように each case is counted exactly once と説明できる。"
    ],
    "related": [
      "double-count",
      "enumeration",
      "counting",
      "case-analysis"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学A (2) 場合の数と確率: 場合の数をもれなく重複なく数え上げる（数え上げの原則）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.3 Non-Disjoint Outcomes（counted exactly once）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-27"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "undo-the-log",
    "ja": {
      "term": "log を外す",
      "reading": "ろぐをはずす",
      "alt": []
    },
    "en": {
      "term": "exponentiate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "見出しの「log を外す」は日本側の資料には出てこない言い方で、log が消えるという結果で言う。OpenStax Calculus Volume 2（4.3・4.4）は、両辺を指数にする操作で exponentiate both sides と書き、OpenStax Algebra and Trigonometry（6.3）は convert … to exponential form ／ rewrite … in exponential form と書く。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "log_a M = p を M = aᵖ に直して、対数のない式にする。",
    "definition_en": "To turn log_a M = p into M = aᵖ, getting rid of the logarithm.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To get rid of the ln, exponentiate both sides: e to the left equals e to the right.",
        "ja": "ln を外すには両辺を e の肩にのせる。",
        "register": "spoken"
      },
      {
        "en": "Solve ln(2x − 1) = 3 by exponentiating both sides.",
        "ja": "両辺を e の肩にのせて（log を外して）ln(2x − 1) = 3 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "exponentiate は用例コーパスでは話し言葉・書き言葉とも使われ、話し言葉のほうが多い。eˣ を作る意味でも使う。"
    ],
    "related": [
      "rewrite-in-exponential-form",
      "logarithmic-equation",
      "take-the-log-of-both-sides"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "4.3 Separable Equations・4.4 The Logistic Equation（exponentiate both sides）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.3 Logarithmic Functions（convert … to exponential form、rewrite … in exponential form）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "exponentiate": 37
      },
      "written": {
        "exponentiate": 14
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "uniform-distribution",
    "ja": {
      "term": "一様分布",
      "reading": "いちようぶんぷ",
      "alt": []
    },
    "en": {
      "term": "uniform distribution",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "probability",
      "statistics"
    ],
    "level": {
      "jp": [
        "理数"
      ],
      "us": [
        "Intro Statistics"
      ]
    },
    "definition_ja": "ある範囲のどの値も同じように起こる確率分布。連続型では区間 [a, b] で確率密度が一定。",
    "definition_en": "A distribution in which every value in a range is equally likely; for a continuous variable on [a, b] the density is constant.",
    "latex": "f(x) = \\frac{1}{b - a},\\ a \\le x \\le b",
    "spoken_en": null,
    "examples": [
      {
        "en": "A uniform distribution looks like a flat rectangle.",
        "ja": "一様分布は平らな長方形の形をしている。",
        "register": "spoken"
      },
      {
        "en": "The waiting time is uniformly distributed between 0 and 15 minutes. Find the probability of waiting more than 10 minutes.",
        "ja": "待ち時間は 0 分から 15 分の一様分布に従う。10 分より長く待つ確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの書き言葉は大半が OpenStax Introductory Statistics、話し言葉は 3Blue1Brown が最も多い。"
    ],
    "related": [
      "probability-density-function",
      "exponential-distribution",
      "continuous-random-variable"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "理数数学II (8) 統計的な推測（発展・拡充）: 確率密度関数が 0 < x ≦ 1 で 1 の一様分布"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "uniform distribution": 14
      },
      "written": {
        "uniform distribution": 53
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-introstats",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "union-of-events",
    "ja": {
      "term": "和事象",
      "reading": "わじしょう",
      "alt": []
    },
    "en": {
      "term": "union of events",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "combinatorics",
      "probability"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Discrete Math",
        "Integrated Math 2",
        "Intro Statistics",
        "AP Statistics"
      ]
    },
    "definition_ja": "事象 A と事象 B の少なくとも一方が起こるという事象。A ∪ B と書く。",
    "definition_en": "The event that at least one of the events A and B happens, written A ∪ B.",
    "latex": "A \\cup B",
    "spoken_en": "A union B; in probability, often read as A or B",
    "examples": [
      {
        "en": "The union of two events is A or B, and that includes the case where both happen.",
        "ja": "和事象は A または B で、両方起こる場合も含む。",
        "register": "spoken"
      },
      {
        "en": "A card is drawn from a standard deck. Let A be the event of a heart and B the event of a face card. Find the probability of the union of events A and B.",
        "ja": "52 枚のトランプから 1 枚引く。ハートである事象を A、絵札である事象を B とする。和事象 A ∪ B の確率を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Introductory Statistics（3.1 Terminology）は A ∪ B を A OR B と書き（A Union B とも）、AP Statistics の CED（topic 2.7）は P(A union B) と書いて the probability of the union と呼ぶ。"
    ],
    "related": [
      "union",
      "intersection-of-events",
      "general-addition-rule",
      "mutually-exclusive-events",
      "complementary-event",
      "double-count"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（A OR B、A Union B）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 2.7（the union of two events、P(A union B)）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "union of two events | union of events": 1
      },
      "written": {
        "union of two events | union of events": 13
      },
      "sources": [
        "mit-6.042",
        "openstax-algtrig"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "union",
    "ja": {
      "term": "和集合",
      "reading": "わしゅうごう",
      "alt": []
    },
    "en": {
      "term": "union",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "集合 A と B の少なくとも一方に属する要素全体の集合。A ∪ B と書く。",
    "definition_en": "The set made up of everything in A together with everything in B (an element in both counts once), written A ∪ B.",
    "latex": "A \\cup B",
    "spoken_en": "A union B",
    "examples": [
      {
        "en": "A union B means anything that's in A or in B, or in both.",
        "ja": "A ∪ B（A と B の和集合）は、A か B の少なくとも一方に入っているもの全部だ。",
        "register": "spoken"
      },
      {
        "en": "Let A = {1, 3, 5} and B = {3, 4}. Find the union of A and B.",
        "ja": "A = {1, 3, 5}、B = {3, 4} のとき、A と B の和集合を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "A ∪ B は「A または B」で、両方に入る要素も含む。確率の和事象（union of events）にも同じ union を使う。",
      "日本語では「A かつ B」の共通部分 A ∩ B と取り違えやすい。英語では ∪ が union、∩ が intersection。"
    ],
    "related": [
      "set",
      "complement",
      "de-morgans-laws",
      "venn-diagram",
      "number-of-elements",
      "intersection"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "和集合",
        "en": "Union (set theory)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "union of": 16
      },
      "written": {
        "union of": 79
      },
      "sources": [
        "mit-18.01",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "uniqueness-proof",
    "ja": {
      "term": "一意性の証明",
      "reading": "いちいせいのしょうめい",
      "alt": []
    },
    "en": {
      "term": "proof of uniqueness",
      "alt": [
        "uniqueness proof"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（proof of uniqueness ／ uniqueness proof は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "proof"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "条件を満たすものが高々 1 つしかないことを示す証明のこと。ふつうは条件を満たすものを 2 つ x, y とおき、x = y を導く。",
    "definition_en": "A proof that at most one object satisfies a given condition. The usual approach is to assume that two objects x and y both satisfy it and then show that x = y.",
    "latex": "P(x) \\land P(y) \\Rightarrow x = y",
    "spoken_en": "P of x and P of y implies x equals y",
    "examples": [
      {
        "en": "For the proof of uniqueness, suppose two numbers both work, call them x and y, and show that x has to equal y.",
        "ja": "一意性の証明では、条件を満たす数が 2 つあるとして x, y とおき、x = y となることを示そう。",
        "register": "spoken"
      },
      {
        "en": "Let a be a real number. Give a proof of uniqueness for the real number b with a + b = 0: assume that a + b = 0 and a + c = 0, and show that b = c.",
        "ja": "実数 a に対し、a + b = 0 を満たす実数 b はただ 1 つであることを証明せよ。a + b = 0, a + c = 0 と仮定して b = c を示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「ただ 1 つ存在する」を示すには、存在の証明（existence proof）と一意性の証明の 2 つが要る。一意性だけでは、条件を満たすものが 1 つもない場合を除けない。",
      "記号 ∃! x は there exists a unique x と読む。「一意」は unique、「一意性」は uniqueness。",
      "用例コーパスには話・書とも出てこない。参照では Nicholson の 11.2 The Jordan Canonical Form に proof of uniqueness が 1 件あるだけ。"
    ],
    "related": [
      "existence-proof",
      "prove",
      "existential-quantifier"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "11.2 The Jordan Canonical Form"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは Nicholson の呼び方 proof of uniqueness（11.2 The Jordan Canonical Form）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "unit-circle",
    "ja": {
      "term": "単位円",
      "reading": "たんいえん",
      "alt": []
    },
    "en": {
      "term": "unit circle",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "trigonometry"
    ],
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "原点を中心とする半径 1 の円。円周上の点の x 座標を cos θ、y 座標を sin θ として三角比・三角関数を定める。",
    "definition_en": "The circle with center at the origin and radius 1. For the point on it at angle θ, the x-coordinate is cos θ and the y-coordinate is sin θ.",
    "latex": "x^2 + y^2 = 1",
    "spoken_en": "x squared plus y squared equals one",
    "examples": [
      {
        "en": "On the unit circle, the x-coordinate is cosine and the y-coordinate is sine, so at 120° you get negative one half and root three over two.",
        "ja": "単位円では x 座標が cos、y 座標が sin だから、120° なら −1/2 と √3/2 になる。",
        "register": "spoken"
      },
      {
        "en": "Use the unit circle to find sin 135° and cos 135°.",
        "ja": "単位円を用いて sin 135° と cos 135° の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "高等学校学習指導要領解説（数学I 図形と計量）は単位円を使わず、座標平面の第 1 象限で原点を端点とする長さ α の線分 OP と点 P の座標 (α cos θ, α sin θ) で三角比を鈍角まで拡張する。単位円を使うのは数学II の三角関数（共通テスト（令和 7 年度 数学II・B・C）は「単位円を用いて」と書く）。",
      "解説（数学I）の長さ α の線分 OP による定め方でも、単位円（α = 1）なら座標がそのまま (cos θ, sin θ) になる。tan θ は y/x で、θ = 90° では定まらない。"
    ],
    "related": [
      "trigonometric-ratio",
      "trigonometric-function",
      "sine",
      "cosine",
      "reference-angle"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "単位円",
        "en": "Unit circle"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 図形と計量: 鈍角の三角比。数学II 三角関数: 一般角"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "令和 7 年度 数学II・B・C: 単位円を用いて考える"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "unit circle": 302
      },
      "written": {
        "unit circle": 171
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "unit-normal-vector",
    "ja": {
      "term": "主法線ベクトル",
      "reading": "しゅほうせんべくとる",
      "alt": [
        "単位主法線ベクトル"
      ]
    },
    "en": {
      "term": "principal unit normal vector",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "単位接ベクトル T の微分の向きをもつ長さ 1 のベクトル N = T′/‖T′‖。曲線が曲がる側を向く。",
    "definition_en": "The unit vector N = T′/‖T′‖ in the direction of the derivative of the unit tangent vector; it points toward the side the curve bends.",
    "latex": "\\mathbf{N}(t) = \\frac{\\mathbf{T}'(t)}{\\|\\mathbf{T}'(t)\\|}",
    "spoken_en": "N of t equals T prime of t over the magnitude of T prime of t",
    "examples": [
      {
        "en": "The unit normal vector points toward the inside of the curve.",
        "ja": "主法線ベクトルは曲線の内側を向く。",
        "register": "spoken"
      },
      {
        "en": "Find the principal unit normal vector N(t) for r(t) = ⟨cos t, sin t, t⟩.",
        "ja": "r(t) = ⟨cos t, sin t, t⟩ の主法線ベクトル N(t) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "unit normal vector だけだと、曲面の単位法線ベクトルの意味にもなる。",
      "T × N を binormal vector（従法線ベクトル）B と呼ぶ。"
    ],
    "related": [
      "unit-tangent-vector",
      "curvature",
      "space-curve"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "principal unit normal vector": 15
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "unit-tangent-vector",
    "ja": {
      "term": "単位接ベクトル",
      "reading": "たんいせっべくとる",
      "alt": []
    },
    "en": {
      "term": "unit tangent vector",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "曲線の接線の向きをもつ長さ 1 のベクトル T(t) = r′(t)/‖r′(t)‖。",
    "definition_en": "The tangent vector of length 1, T(t) = r′(t)/‖r′(t)‖.",
    "latex": "\\mathbf{T}(t) = \\frac{\\mathbf{r}'(t)}{\\|\\mathbf{r}'(t)\\|}",
    "spoken_en": "T of t equals r prime of t over the magnitude of r prime of t",
    "examples": [
      {
        "en": "Divide the velocity by its length and you get the unit tangent vector.",
        "ja": "速度をその長さで割れば単位接ベクトルになる。",
        "register": "spoken"
      },
      {
        "en": "Find the unit tangent vector T(t) for r(t) = ⟨3 cos t, 3 sin t, 4t⟩.",
        "ja": "r(t) = ⟨3 cos t, 3 sin t, 4t⟩ の単位接ベクトル T(t) を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "unit-normal-vector",
      "curvature",
      "velocity-vector",
      "derivative-of-a-vector-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "unit tangent vector": 13
      },
      "written": {
        "unit tangent vector": 51
      },
      "sources": [
        "mit-18.02",
        "mit-notes",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "unit-vector",
    "ja": {
      "term": "単位ベクトル",
      "reading": "たんいべくとる",
      "alt": []
    },
    "en": {
      "term": "unit vector",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "大きさが 1 のベクトル。0 でないベクトル a と同じ向きの単位ベクトルは a/|a|。",
    "definition_en": "A vector of magnitude 1; the unit vector in the direction of a nonzero vector a is a/|a|.",
    "latex": "\\frac{\\vec{a}}{|\\vec{a}|}",
    "spoken_en": null,
    "examples": [
      {
        "en": "To get a unit vector, divide the vector by its own length.",
        "ja": "単位ベクトルを作るには、ベクトルを自分の長さで割る。",
        "register": "spoken"
      },
      {
        "en": "Find the unit vector in the direction of v = ⟨6, −8⟩.",
        "ja": "v = ⟨6, −8⟩ と同じ向きの単位ベクトルを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "x 軸・y 軸方向の単位ベクトル i, j は基本ベクトル（standard unit vectors）。"
    ],
    "related": [
      "magnitude",
      "standard-unit-vectors",
      "vector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "単位ベクトル",
        "en": "Unit vector"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "unit vector": 244
      },
      "written": {
        "unit vector": 159
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "unit",
    "ja": {
      "term": "単位",
      "reading": "たんい",
      "alt": []
    },
    "en": {
      "term": "unit",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "長さ・重さ・時間などの量を測るときの基準にする決まった量。cm、kg、秒など。",
    "definition_en": "A fixed amount used as a standard for measuring a quantity, such as a centimeter, a kilogram, or a second.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Your answer is 12, but 12 what? Always write the unit, like centimeters or meters.",
        "ja": "答えは 12。でも 12 何？ センチなのかメートルなのか、単位を必ず書こう。",
        "register": "spoken"
      },
      {
        "en": "A rectangle is 8 cm long and 5 cm wide. Find its perimeter, and include the unit in your answer.",
        "ja": "長さ 8 cm、幅 5 cm の長方形の周の長さを求めよ。答えには単位を付けること。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "面積を 1 辺 1 の正方形いくつ分で答えるときは square units（エントリ square-units）、体積を 1 辺 1 の立方体いくつ分で答えるときは cubic units（エントリ cubic-units）と言う。",
      "unit circle（単位円）・unit vector（単位ベクトル）の unit は「大きさ 1 の」という意味で、量を測る単位とは別の語（エントリ unit-circle、unit-vector）。"
    ],
    "related": [
      "square-units",
      "cubic-units",
      "quantity",
      "speed",
      "unit-circle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "units of measure | unit of measure | units of measurement | unit of measurement | same units | label the units | include units | in the units | units of length | unit of length": 48
      },
      "written": {
        "units of measure | unit of measure | units of measurement | unit of measurement | same units | label the units | include units | in the units | units of length | unit of length": 138
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "universal-quantifier",
    "ja": {
      "term": "全称量化子",
      "reading": "ぜんしょうりょうかし",
      "alt": [
        "全称記号"
      ]
    },
    "en": {
      "term": "universal quantifier",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "notation"
    ],
    "level": {
      "jp": [
        "数I",
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "「すべての x について」を表す量化子で、記号 ∀ で書く。∀x P(x) は、考えている範囲のどの x についても P(x) が真であることを表す。",
    "definition_en": "The quantifier \"for all,\" written ∀. The statement ∀x P(x) says that P(x) is true for every x in the domain.",
    "latex": "\\forall x \\in \\mathbb{R},\\ x^2 \\ge 0",
    "spoken_en": "for all x in the real numbers, x squared is greater than or equal to zero",
    "examples": [
      {
        "en": "The upside-down A is the universal quantifier, so read it as \"for all\" or \"for every.\"",
        "ja": "逆さまの A は全称記号だから、「すべての」と読もう。",
        "register": "spoken"
      },
      {
        "en": "Rewrite the statement \"Every integer is a rational number\" using the universal quantifier.",
        "ja": "命題「すべての整数は有理数である」を、全称記号を用いて書き直せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "式の中の ∀ は for all や for every と読む。",
      "¬∀x P(x) は ∃x ¬P(x) と同じ。数I の「すべての…」の否定が「ある…でない」になるのと同じ規則。",
      "用例コーパスで universal quantifier は書き言葉にほとんど出てこず（OpenStax Calculus と MIT の講義ノート）、話し言葉には出てこない。Levin の本では 8 件。"
    ],
    "related": [
      "quantifier",
      "existential-quantifier",
      "all",
      "predicate"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "全称記号",
        "en": "Universal quantification"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "universal quantifier": 2
      },
      "sources": [
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは OpenStax の呼び方 universal quantifier（本文 1 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "universal-set",
    "ja": {
      "term": "全体集合",
      "reading": "ぜんたいしゅうごう",
      "alt": []
    },
    "en": {
      "term": "universe",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "definition_ja": "考えている集合をすべて部分集合として含む、議論の土台になる集合。ふつう U で表す。",
    "definition_en": "The set of everything under consideration in a given problem, usually written U; every set in the problem is a subset of it.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's take the universe to be the integers from 1 to 10.",
        "ja": "全体集合を 1 から 10 までの整数としよう。",
        "register": "spoken"
      },
      {
        "en": "Let U = {1, 2, 3, …, 10} be the universal set, and let A = {2, 4, 6, 8, 10}. Find the complement of A.",
        "ja": "U = {1, 2, 3, …, 10} を全体集合とし、A = {2, 4, 6, 8, 10} とする。A の補集合を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「全体集合」は共通テストの数学I 第1問（令和 3〜8 年度ほか）が「全体集合 U を…とする」と書く語で、学習指導要領解説には出てこない。",
      "補集合は全体集合を決めてはじめて決まる。同じ A でも U が変われば補集合も変わる。",
      "universal set という言い方は米国の高校課程の参照（CED・OpenStax・IM・CK-12）に 0 件。CK-12 Geometry 2.10 Negative Statements はベン図の全体を the universe of all things と呼び、Levin は the universe 𝒰（0.2.2・5.1 Sets）、論理の文脈では universe of discourse（1.6）と言う。英語版 Wikipedia の同じ概念の記事は「Universe (mathematics)」で、「Universal set」はすべての集合を含む集合（通常の集合論には存在しない）という別の概念。"
    ],
    "related": [
      "set",
      "subset",
      "complement",
      "venn-diagram"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Universe_(mathematics)",
        "note": "記事「Universe (mathematics)」（数学カテゴリから 3 段。冒頭で universe を太字で定義）。「Universal set」は別の概念（WIKIPEDIA_NOT_SAME）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "0.2.2・5.1 Sets: the universe 𝒰。1.6: universe of discourse"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.10 Negative Statements: the universe of all things（ベン図の全体）"
      },
      {
        "type": "reference",
        "title": "大学入学共通テスト・大学入試センター試験 本試験 数学（問題）",
        "note": "数学I 第1問（令和 3〜8 年度、平成 26・30 年度）: 全体集合 U"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 集合と命題: 「全体集合」は 0 件"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the universe of all things | universe of discourse | universe is u | universe u | the universe be the set": 1
      },
      "sources": [
        "mit-6.042"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 the universe of all things | universe of discourse | universe is u | universe u | the universe be the set（CK-12 Geometry 2.10 Negative Statements・計 1 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "unknown",
    "ja": {
      "term": "未知数",
      "reading": "みちすう",
      "alt": []
    },
    "en": {
      "term": "unknown",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "方程式の中で、値がまだ分かっていない数。ふつう x などの文字で表し、方程式を解いてその値を求める。",
    "definition_en": "A number in an equation whose value you don't know yet, usually written as a letter such as x. Solving the equation means finding the value of the unknown.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We've got two equations and two unknowns, so we can find both x and y.",
        "ja": "式が 2 つで未知数も 2 つだから、x も y も求められるね。",
        "register": "spoken"
      },
      {
        "en": "In the equation 4x − 7 = 13, the unknown is x. Find its value.",
        "ja": "方程式 4x − 7 = 13 で、未知数は x である。その値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "unknown は形容詞「分からない」（an unknown number）としてもよく使う。名詞で「未知数」と言うときは the unknown、複数なら two unknowns。",
      "日本語では方程式の x を未知数、関数の x を変数と分けて呼ぶが、英語では方程式の x も variable と呼ぶことがある（isolate the variable、エントリ variable）。",
      "「2 つの未知数を含む 2 つの方程式」（連立方程式）は two equations and two unknowns と言う。"
    ],
    "related": [
      "variable",
      "equation",
      "solve",
      "solution",
      "system-of-linear-equations"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "the unknown !population | an unknown !population | two unknowns | three unknowns | unknown number | unknown value | unknown quantity": 94
      },
      "written": {
        "the unknown !population | an unknown !population | two unknowns | three unknowns | unknown number | unknown value | unknown quantity": 130
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "upper-limit",
    "ja": {
      "term": "上端",
      "reading": "じょうたん",
      "alt": []
    },
    "en": {
      "term": "upper limit",
      "alt": [
        "upper limit of integration"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "定積分 ∫ₐᵇ f(x)dx で、積分記号の上に書く b。",
    "definition_en": "The number b written at the top of the integral sign in ∫ₐᵇ f(x) dx.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Plug in the upper limit first, then subtract what you get at the lower limit.",
        "ja": "まず上端を代入して、そこから下端を代入した値を引く。",
        "register": "spoken"
      },
      {
        "en": "When the upper limit is the variable x, the integral defines a function of x.",
        "ja": "上端が変数 x のとき、定積分は x の関数を表す。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "話し言葉では upper bound（「上界」と同じ語）や the top、the top number とも言う。",
      "数列の和の Σ の上の数（upper limit of summation）や、統計の階級・信頼区間の上限も upper limit と言う。"
    ],
    "related": [
      "lower-limit",
      "interval-of-integration",
      "definite-integral",
      "accumulation-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "upper limit of integration | at the upper limit": 10
      },
      "written": {
        "upper limit of integration | at the upper limit": 4
      },
      "sources": [
        "mit-18.03",
        "openstax-calculus",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "value-of-a-ratio",
    "ja": {
      "term": "比の値",
      "reading": "ひのあたい",
      "alt": []
    },
    "en": {
      "term": "value of a ratio",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。比 a : b を 1 つの数 a/b（= a ÷ b）で表したものの名前は用例コーパスと参照に出てこず、OpenStax Prealgebra は「比を分数で書く」（write a ratio as a fraction）と説明する。2 つの比が等しい（equivalent ratios）かどうかは、この数が等しいかで確かめられる。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "比 a : b で、a を b で割った値 a/b。2 つの比は、比の値が等しいとき等しい。",
    "definition_en": "The number a/b that you get by dividing a by b in the ratio a : b. Two ratios are equivalent when they have the same value.",
    "latex": "a \\div b = \\frac{a}{b}",
    "spoken_en": "a divided by b equals a over b",
    "examples": [
      {
        "en": "Write 6 to 8 as a fraction: that's six eighths, which simplifies to three fourths.",
        "ja": "6 : 8 の比の値を分数で書くと 6/8 で、約分すると 3/4 だね。",
        "register": "spoken"
      },
      {
        "en": "Write each ratio as a fraction in simplest form, and decide whether 12 : 18 and 10 : 15 are equivalent ratios.",
        "ja": "それぞれの比の値を既約分数で表し、12 : 18 と 10 : 15 が等しい比かどうかを調べよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "a : b の比の値は a ÷ b = a/b で、前の数を後ろの数で割る。順番を入れかえた b : a の比の値は b/a で、別の数になる。",
      "比の値は比（a : b）ではなく 1 つの数で、3/4 や 0.75 のように分数や小数で書く。比そのものは ratio（エントリ ratio）。"
    ],
    "related": [
      "ratio",
      "proportion",
      "fraction",
      "division"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "write a ratio as a fraction"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "value of a ratio | value of the ratio": 7
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 7 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "value-of-the-expression",
    "ja": {
      "term": "式の値",
      "reading": "しきのあたい",
      "alt": []
    },
    "en": {
      "term": "value of the expression",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "式の中の文字に数を代入して計算した結果。x = 2 のとき、3x + 1 の値は 7。",
    "definition_en": "The number you get when you substitute given numbers for the variables in an expression and calculate. For example, when x = 2, the value of the expression 3x + 1 is 7.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Plug in 3 for x, and the value of the expression 2x + 5 is 11.",
        "ja": "x に 3 を代入すると、式 2x + 5 の値は 11 になるね。",
        "register": "spoken"
      },
      {
        "en": "Find the value of the expression x² − 4x when x = −1.",
        "ja": "x = −1 のとき、式 x² − 4x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「式の値を求めよ」は Evaluate the expression とも書く。evaluate 1 語で「値を求める」の意味になる（エントリ evaluate）。",
      "「x = 3 のとき」は when x = 3 または if x = 3。「x に 3 を代入する」は plug in 3 for x ／ substitute 3 for x（エントリ substitute）。"
    ],
    "related": [
      "evaluate",
      "substitute",
      "expression",
      "value-of-the-function",
      "substituting-values"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "value of the expression": 12
      },
      "written": {
        "value of the expression": 8
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-notes",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "value-of-the-function",
    "ja": {
      "term": "関数の値",
      "reading": "かんすうのあたい",
      "alt": []
    },
    "en": {
      "term": "value of the function",
      "alt": [],
      "uk": null,
      "variants": [
        {
          "term": "function value",
          "register": "both",
          "note": "OpenStax Algebra and Trigonometry と OpenStax Calculus に多い。話し言葉では value of the function より少ない。"
        },
        {
          "term": "output value",
          "register": "both",
          "note": "入力と出力の言い方。OpenStax Algebra and Trigonometry・Precalculus は各 180 件前後使い、IM・CK-12 Algebra も使う。用例コーパスの書き言葉では 3 つの言い方の中で最も多く、話し言葉では value of the function より少ない。"
        }
      ]
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "関数 f で、x に値 a を与えたときに対応して決まる y の値 f(a)。",
    "definition_en": "The output f(a) that a function f assigns to a given input a.",
    "latex": "f(a)",
    "spoken_en": "f of a",
    "examples": [
      {
        "en": "Plug in x = 2, and the value of the function there is 7.",
        "ja": "x = 2 を代入すると、そこでの関数の値は 7 になる。",
        "register": "spoken"
      },
      {
        "en": "Find the value of the function f(x) = 2x² − 3x at x = −1.",
        "ja": "関数 f(x) = 2x² − 3x の x = −1 における値を求めよ。",
        "register": "written"
      },
      {
        "en": "The table lists the function values f(0), f(1), and f(2).",
        "ja": "表は関数の値 f(0), f(1), f(2) を示している。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「x = −1 における値」は英語では at x = −1 で言う（the value of f at x = −1）。"
    ],
    "related": [
      "function-notation",
      "evaluate",
      "substituting-values",
      "maximum",
      "minimum"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "output value（178 件）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "output value（34 件）"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "output value（13 件）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "value of the function": 136,
        "function value": 34,
        "output value": 41
      },
      "written": {
        "function value": 94,
        "value of the function": 116,
        "output value": 185
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "variable-of-integration",
    "ja": {
      "term": "積分変数",
      "reading": "せきぶんへんすう",
      "alt": []
    },
    "en": {
      "term": "variable of integration",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "積分される関数の変数。dx の x、dt の t にあたる。",
    "definition_en": "The variable you integrate with respect to, such as the x in dx or the t in dt.",
    "latex": "\\int_0^1 t^2\\,dt = \\int_0^1 x^2\\,dx",
    "spoken_en": "the integral from zero to one of t squared dt equals the integral from zero to one of x squared dx",
    "examples": [
      {
        "en": "The t here is just the variable of integration. You could call it anything and the answer wouldn't change.",
        "ja": "この t はただの積分変数。どんな文字にしても答えは変わらない。",
        "register": "spoken"
      },
      {
        "en": "Changing the variable of integration from t to x does not change the value of the integral.",
        "ja": "積分変数を t から x に変えても、定積分の値は変わらない。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "積分変数を何の文字にしても値が変わらないことを言うとき、授業では dummy variable（見かけの変数）と呼ぶことがある。dummy variable は Σ の添え字にも使う広い語で、積分変数の別名ではない。統計の dummy variable（ダミー変数）とも別の意味。"
    ],
    "related": [
      "integrand",
      "accumulation-function",
      "definite-integral"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "variable of integration": 19
      },
      "written": {
        "variable of integration": 14
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "openstax-calculus",
        "yt:nancypi"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "variable",
    "ja": {
      "term": "変数",
      "reading": "へんすう",
      "alt": [
        "文字"
      ]
    },
    "en": {
      "term": "variable",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "中学校学習指導要領解説は、式の中の x や a を「文字」と呼ぶ（「文字を用いた式」）。英語では letter ではなく variable と言う（x is a variable）ので、「文字」も「変数」も英語では variable になる。",
    "domains": [
      "algebra",
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "いろいろな値をとることができる文字。式の中で、数の代わりに使う x や y など。",
    "definition_en": "A letter, such as x or y, that stands for a number that can change or that we don't know yet.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "We don't know the number yet, so let's call it x. That x is our variable.",
        "ja": "その数はまだわからないから x とおこう。この x が変数（文字）だ。",
        "register": "spoken"
      },
      {
        "en": "Define a variable for the unknown number, then write and solve an equation.",
        "ja": "求める数を文字で表して方程式をつくり、それを解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "式の中の「文字」は英語では variable と言う。「文字を使った式」は an expression with variables、「x について解く」は solve for x（エントリ solve-for）。",
      "英語の variable は、統計の「変量」も指す（エントリ statistical-variable）。式の中の変数か、データの項目（身長・点数など）かは文脈で読み分ける。"
    ],
    "related": [
      "algebraic-expression",
      "constant",
      "substitute",
      "solve-for",
      "statistical-variable"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "変数 (数学)",
        "en": "Variable (mathematics)"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A 数と式「文字を用いた式」"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "!random !independent !dependent !response !explanatory !categorical !quantitative !lurking !confounding !dummy !free !basic !indicator variable": 1586
      },
      "written": {
        "!random !independent !dependent !response !explanatory !categorical !quantitative !lurking !confounding !dummy !free !basic !indicator variable": 2642
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "variance",
    "ja": {
      "term": "分散",
      "reading": "ぶんさん",
      "alt": []
    },
    "en": {
      "term": "variance",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "definition_ja": "各データの偏差の 2 乗の平均値。データの散らばりの大きさを表す。",
    "definition_en": "The mean of the squared deviations of the data from their mean; a measure of spread.",
    "latex": "s^2 = \\frac{1}{n}\\sum_{i=1}^{n}(x_i - \\bar{x})^2",
    "spoken_en": "s squared equals one over n times the sum from i equals one to n of the quantity x sub i minus x bar, squared",
    "examples": [
      {
        "en": "The variance is in squared units, which is why we usually take the square root.",
        "ja": "分散は単位が 2 乗なので、ふつうは平方根をとる。",
        "register": "spoken"
      },
      {
        "en": "Compute the variance and the standard deviation of the data.",
        "ja": "データの分散と標準偏差を計算せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "センター試験 平成 30 年度 数学I の問題文は分散 s² を n で割る式で示し、日本語版 Wikipedia「分散 (確率論)」も n で割る分散（標本分散）と n − 1 で割る不偏分散を分ける。高等学校学習指導要領解説（数学I）は分散を、平均値との差に基づいてデータの散らばりの度合いを表す指標と書く。OpenStax Introductory Statistics（Measures of the Spread of the Data）と AP Statistics の CED（topic 1.7）の sample variance（s²）は n − 1 で割る。n で割るのは母集団の分散（population variance, σ²。エントリ population-variance）の式。"
    ],
    "related": [
      "standard-deviation",
      "deviation",
      "spread"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "分散 (確率論)",
        "en": "Variance"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I データの分析: 分散、標準偏差（平均値との差に基づく散らばりの指標）"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成30年度 大学入試センター試験 本試験 数学Ⅰ（問題）",
        "url": "https://www.dnc.ac.jp/albums/abm.php?f=abm00033140.pdf&n=h30+suugaku1+01suugaku1.pdf",
        "note": "分散 s² を n で割る式"
      },
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.7（sample variance）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "variance": 366
      },
      "written": {
        "variance": 250
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "mit-notes",
        "openstax-introstats",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector-addition",
    "ja": {
      "term": "ベクトルの加法",
      "reading": "べくとるのかほう",
      "alt": []
    },
    "en": {
      "term": "vector addition",
      "alt": [
        "add vectors"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "2 つのベクトル a, b から和 a + b を作ること。一方の終点に他方の始点を重ねて（三角形の法則）、または平行四辺形をつくって求める。",
    "definition_en": "Forming the sum a + b of two vectors, by placing the tail of one at the tip of the other (triangle method) or by completing a parallelogram.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To add vectors, put them tip to tail and draw the arrow from start to finish.",
        "ja": "ベクトルを足すには、先と尾をつないで、最初から最後まで矢印を引く。",
        "register": "spoken"
      },
      {
        "en": "Use vector addition to find a + b for a = ⟨2, 5⟩ and b = ⟨−1, 3⟩.",
        "ja": "a = ⟨2, 5⟩, b = ⟨−1, 3⟩ について、ベクトルの加法で a + b を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "和のベクトルそのものは resultant（合力・和ベクトル）とも言う。",
      "先と尾をつなぐ置き方を tip to tail と言う（用例コーパスの話し言葉ではすべて 3Blue1Brown）。",
      "用例コーパスの話し言葉は大半が 3Blue1Brown。"
    ],
    "related": [
      "vector-subtraction",
      "resultant",
      "vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "add vectors": 4,
        "vector addition": 14
      },
      "written": {
        "vector addition": 22,
        "add vectors": 5
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector-equation-of-a-circle",
    "ja": {
      "term": "円のベクトル方程式",
      "reading": "えんのべくとるほうていしき",
      "alt": []
    },
    "en": {
      "term": "vector equation of a circle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない。米国の教科書は円を (x − h)² + (y − k)² = r² の形で扱い、ベクトル方程式としては立てない。",
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "中心 C（位置ベクトル c）、半径 r の円上の点 P の位置ベクトル p が満たす式 |p − c| = r。",
    "definition_en": "The equation |p − c| = r satisfied by the position vector p of a point on the circle with center C (position vector c) and radius r.",
    "latex": "|\\vec{p} - \\vec{c}| = r",
    "spoken_en": null,
    "examples": [
      {
        "en": "The distance from P to the center has to be r, and that's the whole equation.",
        "ja": "P から中心までの距離が r、それが方程式のすべて。",
        "register": "spoken"
      },
      {
        "en": "Write the vector equation of the circle with center C and radius 3, and convert it to an equation in x and y.",
        "ja": "中心 C、半径 3 の円のベクトル方程式を書き、x, y の方程式に直せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector-equation",
      "equation-of-a-circle",
      "magnitude"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-no-fixed-expression",
        "note": "用例コーパスと参照（CED・OpenStax・IM・CK-12）には決まった言い方が出てこない（話 0 件 ／ 書 0 件）。mapping near なので、決まった言い方が出てこないと判定した（英語版 Wikipedia の数学記事の記事名にも候補が無い）。register は主張しない。人間レビューには回さない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector-equation-of-a-line",
    "ja": {
      "term": "直線のベクトル方程式",
      "reading": "ちょくせんのべくとるほうていしき",
      "alt": []
    },
    "en": {
      "term": "vector equation of a line",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "点 A を通り、方向ベクトル u に平行な直線上の点 P の位置ベクトル p を、p = a + tu（t は実数）と表した式。",
    "definition_en": "The equation p = a + tu (t real) for the position vector p of a point on the line through A parallel to the direction vector u.",
    "latex": "\\vec{p} = \\vec{a} + t\\vec{u}",
    "spoken_en": "p equals a plus t times u",
    "examples": [
      {
        "en": "As t runs through all real numbers, the point traces out the whole line.",
        "ja": "t がすべての実数を動くと、点は直線全体をえがく。",
        "register": "spoken"
      },
      {
        "en": "Write a vector equation of the line through A(2, −1) and B(5, 3).",
        "ja": "2 点 A(2, −1), B(5, 3) を通る直線のベクトル方程式を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "各成分に分けた式 x = 2 + 3t, y = −1 + 4t は parametric equations of the line（OpenStax Calculus Volume 3、for the line を含め 13 件）と言う。"
    ],
    "related": [
      "vector-equation",
      "direction-vector",
      "parameter"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "vector equation of a line": 4
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 4 件（10 未満））。見出しは OpenStax の呼び方 vector equation of a line（本文 4 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector-equation",
    "ja": {
      "term": "ベクトル方程式",
      "reading": "べくとるほうていしき",
      "alt": []
    },
    "en": {
      "term": "vector equation",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "図形上の点 P の位置ベクトル p が満たす条件を、ベクトルの式で表したもの。直線 p = a + tu など。",
    "definition_en": "An equation in terms of vectors satisfied by the position vector p of a point on a figure, such as p = a + tu for a line.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The vector equation says: start at A and move t times the direction vector.",
        "ja": "ベクトル方程式は、A から出発して方向ベクトルの t 倍だけ進む、ということ。",
        "register": "spoken"
      },
      {
        "en": "Find a vector equation of the line through (1, 2, 0) parallel to ⟨3, −1, 4⟩.",
        "ja": "点 (1, 2, 0) を通り ⟨3, −1, 4⟩ に平行な直線のベクトル方程式を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector-equation-of-a-line",
      "vector-equation-of-a-circle",
      "position-vector",
      "parameter"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "vector equation": 1
      },
      "written": {
        "vector equation": 16
      },
      "sources": [
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector-field",
    "ja": {
      "term": "ベクトル場",
      "reading": "べくとるば",
      "alt": []
    },
    "en": {
      "term": "vector field",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "definition_ja": "平面や空間の各点にベクトルを対応させたもの F(x, y) = ⟨P, Q⟩。",
    "definition_en": "An assignment of a vector to each point of the plane or space, F(x, y) = ⟨P, Q⟩.",
    "latex": "\\mathbf{F}(x, y) = \\langle P(x, y), Q(x, y) \\rangle",
    "spoken_en": null,
    "examples": [
      {
        "en": "Think of a vector field as wind: at every point there's an arrow.",
        "ja": "ベクトル場は風のようなもの。どの点にも矢印がある。",
        "register": "spoken"
      },
      {
        "en": "Sketch the vector field F(x, y) = ⟨−y, x⟩.",
        "ja": "ベクトル場 F(x, y) = ⟨−y, x⟩ をかけ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "line-integral",
      "conservative-vector-field",
      "curl-vector",
      "divergence-vector",
      "gradient"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ベクトル場",
        "en": "Vector field"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "vector field": 161
      },
      "written": {
        "vector field": 507
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector-projection",
    "ja": {
      "term": "正射影ベクトル",
      "reading": "せいしゃえいべくとる",
      "alt": [
        "ベクトル射影"
      ]
    },
    "en": {
      "term": "vector projection",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトル a を、ベクトル b の方向に正射影してできるベクトル。((a·b)/|b|²) b。",
    "definition_en": "The vector obtained by projecting a onto the direction of b: ((a·b)/|b|²) b.",
    "latex": "\\operatorname{proj}_{\\vec{b}}\\vec{a} = \\frac{\\vec{a}\\cdot\\vec{b}}{|\\vec{b}|^2}\\,\\vec{b}",
    "spoken_en": "the projection of a onto b equals a dot b over the magnitude of b squared, times b",
    "examples": [
      {
        "en": "The vector projection is the shadow of a on the line of b.",
        "ja": "正射影ベクトルは、a を b の方向の直線に落とした影。",
        "register": "spoken"
      },
      {
        "en": "Find the vector projection of a = (3, 4) onto b = (1, 0).",
        "ja": "a = (3, 4) の b = (1, 0) への正射影ベクトルを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Calculus Volume 3 は proj_b a と書き、長さだけのもの（符号つき）を scalar projection（comp_b a）と呼び分ける。",
      "用例コーパスでは書き言葉（OpenStax Calculus Volume 3）にだけ出てくる。"
    ],
    "related": [
      "orthogonal-projection",
      "dot-product",
      "vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "written": {
        "vector projection": 9
      },
      "sources": [
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 vector projection（本文 9 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector-space",
    "ja": {
      "term": "ベクトル空間",
      "reading": "べくとるくうかん",
      "alt": [
        "線形空間",
        "線型空間"
      ]
    },
    "en": {
      "term": "vector space",
      "alt": [
        "linear space"
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "definition_ja": "和と実数倍が定義され、それらが交換法則・分配法則などの決まった性質を満たす集合。その要素をベクトルという。",
    "definition_en": "A set with an addition and a scalar multiplication that satisfy a fixed list of rules (commutativity, distributivity and so on); its elements are called vectors.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Polynomials of degree at most two form a vector space, even though they don't look like arrows.",
        "ja": "2 次以下の多項式全体はベクトル空間になる。矢印には見えなくても。",
        "register": "spoken"
      },
      {
        "en": "Verify that the set of all 2 × 2 matrices is a vector space.",
        "ja": "2 × 2 行列全体がベクトル空間であることを確かめよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉は大半が MIT 18.06。"
    ],
    "related": [
      "subspace",
      "basis",
      "dimension",
      "vector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ベクトル空間",
        "en": "Vector space"
      },
      {
        "type": "reference",
        "title": "W. Keith Nicholson, Linear Algebra with Applications (Lyryx, 2021A)",
        "url": "https://openlibrary-repo.ecampusontario.ca/jspui/bitstream/123456789/897/2/Nicholson-OpenLAWA-2021A.pdf",
        "note": "6.1 Examples and Basic Properties"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "vector space": 90
      },
      "sources": [
        "mit-18.06",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector-subtraction",
    "ja": {
      "term": "ベクトルの減法",
      "reading": "べくとるのげんぽう",
      "alt": []
    },
    "en": {
      "term": "vector subtraction",
      "alt": [
        "subtract vectors"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "ベクトルの差 a − b を作ること。a − b = a + (−b)。始点をそろえると、b の終点から a の終点へのベクトルになる。",
    "definition_en": "Forming the difference a − b = a + (−b); with the tails together, it is the vector from the tip of b to the tip of a.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "a minus b goes from the tip of b to the tip of a.",
        "ja": "a − b は b の先から a の先へ向かう。",
        "register": "spoken"
      },
      {
        "en": "Find a − b and |a − b| for a = ⟨4, 1⟩ and b = ⟨1, −3⟩.",
        "ja": "a = ⟨4, 1⟩, b = ⟨1, −3⟩ について a − b と |a − b| を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "vector-addition",
      "opposite-vector",
      "vector"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "written": {
        "vector subtraction": 6,
        "subtract vectors": 5
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 首位が 3 倍に届かず、次点も 10 件未満）。見出しは OpenStax の呼び方 vector subtraction（本文 6 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector-valued-function",
    "ja": {
      "term": "ベクトル値関数",
      "reading": "べくとるちかんすう",
      "alt": [
        "ベクトル値函数"
      ]
    },
    "en": {
      "term": "vector-valued function",
      "alt": [],
      "variants": [
        {
          "term": "vector-valued function",
          "register": "both",
          "note": "教科書（OpenStax Calculus Volume 3）と AP の CED の言い方。話し言葉はすべて Khan Academy、書き言葉はほとんどが OpenStax Calculus。"
        },
        {
          "term": "vector function",
          "register": "spoken",
          "note": "Professor Leonard の言い方（話し言葉のほとんどが Professor Leonard）。書き言葉では vector-valued function のほうがずっと多い。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus III"
      ]
    },
    "definition_ja": "実数 t に対してベクトル r(t) を対応させる関数。成分ごとの関数で表される。",
    "definition_en": "A function that assigns a vector r(t) to each real number t, given by one function per component.",
    "latex": "\\mathbf{r}(t) = \\langle \\cos t, \\sin t, t \\rangle",
    "spoken_en": "r of t equals the vector cosine t, sine t, t",
    "examples": [
      {
        "en": "A vector-valued function is really just parametric equations bundled into one vector.",
        "ja": "ベクトル値関数は、媒介変数表示を 1 つのベクトルにまとめたもの。",
        "register": "spoken"
      },
      {
        "en": "Find the domain of the vector-valued function r(t) = ⟨√t, 1/(t − 2), ln t⟩.",
        "ja": "ベクトル値関数 r(t) = ⟨√t, 1/(t − 2), ln t⟩ の定義域を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "parametric-equations",
      "derivative-of-a-vector-function",
      "space-curve",
      "velocity-vector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ベクトル値函数",
        "en": "Vector-valued function"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 9.4–9.6（vector-valued functions）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "vector-valued function": 22,
        "vector function": 75
      },
      "written": {
        "vector-valued function": 190,
        "vector function": 12
      },
      "sources": [
        "khan-ap-calc",
        "mit-notes",
        "openstax-calculus",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vector",
    "ja": {
      "term": "ベクトル",
      "reading": "べくとる",
      "alt": []
    },
    "en": {
      "term": "vector",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "definition_ja": "大きさと向きをもつ量。有向線分で表し、位置を問わず大きさと向きが同じものは等しいとみなす。",
    "definition_en": "A quantity with both magnitude and direction, represented by a directed segment; two with the same magnitude and direction are equal regardless of position.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A vector has a magnitude and a direction — where you draw it doesn't matter.",
        "ja": "ベクトルには大きさと向きがある。どこにかくかは関係ない。",
        "register": "spoken"
      },
      {
        "en": "For the vectors a = (2, −1) and b = (1, 3), find 2a − b and its magnitude.",
        "ja": "a = (2, −1), b = (1, 3) のとき、2a − b とその大きさを求めよ。",
        "register": "written"
      },
      {
        "en": "The vector has magnitude 10 and makes a 30° angle with the horizontal, so its horizontal component is 10 cos 30°.",
        "ja": "このベクトルは大きさ 10 で水平方向と 30° をなすので、水平成分は 10 cos 30° だ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の教科書は矢印（→）を文字の上に書くが、米国の教科書は太字（v）か、手書きでは上の矢印 v⃗ を使う。",
      "大きさ |a| は magnitude（または length、norm）。",
      "空間のベクトル（空間ベクトル）も英語は vector（vectors in space ／ in three dimensions）。"
    ],
    "related": [
      "component-form",
      "dot-product",
      "standard-unit-vectors",
      "direction-angle",
      "vector-field",
      "magnitude",
      "unit-vector",
      "position-vector",
      "vector-addition",
      "scalar"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "vector": 3467
      },
      "written": {
        "vector": 2484
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "velocity-vector",
    "ja": {
      "term": "速度ベクトル",
      "reading": "そくどべくとる",
      "alt": []
    },
    "en": {
      "term": "velocity vector",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "vectors"
    ],
    "level": {
      "jp": [
        "数III",
        "大学"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus III"
      ]
    },
    "definition_ja": "平面や空間を動く点の位置ベクトル r(t) を t で微分したベクトル v(t) = r′(t)。大きさが速さ。",
    "definition_en": "The derivative v(t) = r′(t) of the position vector of a moving point; its length is the speed.",
    "latex": "\\mathbf{v}(t) = \\mathbf{r}'(t), \\quad \\text{speed} = \\|\\mathbf{v}(t)\\|",
    "spoken_en": "v of t equals r prime of t, and the speed is the magnitude of v of t",
    "examples": [
      {
        "en": "The velocity vector is tangent to the path, and its length is the speed.",
        "ja": "速度ベクトルは道筋に接していて、その長さが速さ。",
        "register": "spoken"
      },
      {
        "en": "A particle moves with position r(t) = ⟨t², 2t⟩. Find its velocity vector and speed at t = 1.",
        "ja": "点が位置 r(t) = ⟨t², 2t⟩ で動く。t = 1 における速度ベクトルと速さを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "加速度ベクトルは acceleration vector a(t) = v′(t)。",
      "velocity（向きを含む）と speed（大きさ）を区別する。"
    ],
    "related": [
      "velocity",
      "acceleration",
      "derivative-of-a-vector-function",
      "unit-tangent-vector",
      "vector-valued-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 9.6（velocity vector）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "velocity vector": 65
      },
      "written": {
        "velocity vector": 42
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-calculus",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "velocity",
    "ja": {
      "term": "速度",
      "reading": "そくど",
      "alt": []
    },
    "en": {
      "term": "velocity",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "位置を時刻で微分したもの。向き（符号）を含む。",
    "definition_en": "How fast position changes over time, with a sign (or direction) that shows which way the object moves.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Velocity is the derivative of position, and negative velocity means it's moving to the left.",
        "ja": "速度は位置の導関数で、速度が負なら左に動いている。",
        "register": "spoken"
      },
      {
        "en": "The position of a particle is s(t) = t³ − 6t². Find its velocity and acceleration at t = 2.",
        "ja": "点の位置が s(t) = t³ − 6t² のとき、t = 2 における速度と加速度を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "velocity は向きを含み、負にもなる。速さ（大きさ）は speed = |v|。"
    ],
    "related": [
      "acceleration",
      "instantaneous-velocity",
      "derivative",
      "velocity-vector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "速度",
        "en": "Velocity"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 4.2（velocity）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "velocity": 913
      },
      "written": {
        "velocity": 657
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "venn-diagram",
    "ja": {
      "term": "ベン図",
      "reading": "べんず",
      "alt": []
    },
    "en": {
      "term": "Venn diagram",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "logic",
      "set-theory"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math",
        "Intro Statistics"
      ]
    },
    "definition_ja": "集合を円などの閉じた図形で表し、共通部分・和集合・補集合の関係を目で見えるようにした図。全体集合を長方形で囲み、その中に集合を円などでかく。",
    "definition_en": "A diagram that shows sets as overlapping circles or other closed shapes, inside a rectangle that stands for the universal set, so that intersections, unions, and complements can be seen at a glance.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's draw a Venn diagram: one circle for A, one for B, and the overlap is everything in both.",
        "ja": "ベン図をかこう。A の円と B の円をかいて、重なった部分が両方に入っているものだ。",
        "register": "spoken"
      },
      {
        "en": "Draw a Venn diagram and shade the region that represents A ∩ B′.",
        "ja": "ベン図をかき、A ∩ B̄ を表す部分に斜線を引け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "Venn は人名なので大文字で書く（Venn diagram）。",
      "補集合の記号は高等学校学習指導要領解説（数学I）では Ā、OpenStax Introductory Statistics では A′（エントリ complement）。「斜線を引く」は shade。"
    ],
    "related": [
      "set",
      "union",
      "complement",
      "universal-set",
      "number-of-elements"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "ベン図",
        "en": "Venn diagram"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 集合と命題: 集合の記号 Ā（A の補集合）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "Venn diagram": 10
      },
      "written": {
        "Venn diagram": 24
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vertex-angle",
    "ja": {
      "term": "頂角",
      "reading": "ちょうかく",
      "alt": []
    },
    "en": {
      "term": "vertex angle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "二等辺三角形で、長さの等しい 2 辺にはさまれた角。頂角の二等分線は、底辺を垂直に二等分する。",
    "definition_en": "In an isosceles triangle, the angle formed by the two equal sides (the legs). The line that bisects this angle also cuts the base in half at a right angle.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you bisect the vertex angle, that line hits the base at a right angle and cuts it in half.",
        "ja": "頂角を二等分すると、その線は底辺と垂直に交わって、底辺をちょうど半分に分けるよ。",
        "register": "spoken"
      },
      {
        "en": "In isosceles triangle ABC with AB = AC, the bisector of the vertex angle A meets BC at D. Prove that BD = CD and AD ⊥ BC.",
        "ja": "AB = AC の二等辺三角形 ABC で、頂角 ∠A の二等分線が辺 BC と交わる点を D とする。BD = CD、AD ⊥ BC であることを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "三角形の 3 つの頂点はどれも vertex（エントリ vertex）だが、二等辺三角形で vertex angle と言えば、等しい 2 辺にはさまれた角（頂角）のこと。",
      "頂角と底角は vertex angle と base angles（エントリ base-angle）。頂角の二等分線は底辺の垂直二等分線にもなる（エントリ angle-bisector、perpendicular-bisector）。"
    ],
    "related": [
      "base-angle",
      "isosceles-triangle",
      "angle-bisector",
      "perpendicular-bisector",
      "vertex"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.4 Isosceles Triangles"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "sources": [],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 0 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは CK-12 の呼び方 vertex angle（CK-12 Geometry 4.4 Isosceles Triangles・CK-12 Geometry 5.16 Kites・CK-12 Geometry 5.21 Area of Regular and Irregular Polygons・計 15 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "vertex-form",
    "ja": {
      "term": "標準形",
      "reading": "ひょうじゅんけい",
      "alt": [
        "頂点形"
      ]
    },
    "en": {
      "term": "vertex form",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「二次関数」が「標準形」と呼ぶ y = a(x − p)² + q は英語の vertex form に当たる。高等学校学習指導要領解説（数学I）はこの形を y = a(x − p)² + q に変形して頂点 (p, q) に着目すると書き、名前は付けない（「標準形」は数学C の放物線・楕円の方程式に使う）。英語では頂点を (h, k) として y = a(x − h)² + k と書く。「標準形」を standard form と直訳すると食い違う: OpenStax Algebra and Trigonometry は二次関数の a(x − h)² + k を standard form（vertex form とも）と呼ぶ一方、二次方程式 ax² + bx + c = 0 を standard form と呼び、IM Algebra 1 は ax² + bx + c を standard form と呼ぶ。見出しの「標準形」は学習指導要領解説に無い（日本語版 Wikipedia「二次関数」にはある）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "二次関数を y = a(x − p)² + q の形に書いたもの。頂点 (p, q) と軸 x = p がそのまま読み取れる。",
    "definition_en": "A quadratic function written as y = a(x − h)² + k, which shows the vertex (h, k) and the axis of symmetry x = h directly.",
    "latex": "y = a(x - h)^2 + k",
    "spoken_en": "y equals a times the quantity x minus h squared, plus k",
    "examples": [
      {
        "en": "Let's complete the square to get this into vertex form.",
        "ja": "平方完成して標準形に直そう。",
        "register": "spoken"
      },
      {
        "en": "Once it's in vertex form, you can just read off the vertex: it's at (3, −2).",
        "ja": "標準形にすれば頂点がそのまま読める。頂点は (3, −2) だ。",
        "register": "spoken"
      },
      {
        "en": "Write f(x) = x² − 6x + 7 in vertex form and find the vertex of its graph.",
        "ja": "f(x) = x² − 6x + 7 を標準形に直し、グラフの頂点を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の「標準形」を standard form と直訳しない。英語の standard form は ax² + bx + c を指すことも a(x − h)² + k を指すこともある（OpenStax Algebra and Trigonometry は二次関数では後者、二次方程式では前者。IM Algebra 1 は前者）。どちらの形か式で確かめる。",
      "英語では頂点を (h, k) と書き、学習指導要領解説（数学I）や日本語版 Wikipedia「二次関数」の (p, q) と文字が違う。符号にも注意: y = 2(x + 1)² − 5 の頂点は (−1, −5)。"
    ],
    "related": [
      "vertex",
      "completing-the-square",
      "axis-of-symmetry",
      "standard-form",
      "factored-form"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「二次関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%BA%8C%E6%AC%A1%E9%96%A2%E6%95%B0",
        "note": "標準形 f(x) = a(x − p)² + q（vertex form）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 二次関数: y = a(x − p)² + q の形に変形し、頂点 (p, q) に着目（名前は無い）。標準形は数学C の放物線・楕円の方程式"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 1)",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "用語集: standard form (of a quadratic expression) は ax² + bx + c"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "vertex form": 35
      },
      "written": {
        "vertex form": 12
      },
      "sources": [
        "khan-algebra",
        "openstax-algtrig",
        "openstax-intalg",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vertex-graph",
    "ja": {
      "term": "頂点（グラフ）",
      "reading": "ちょうてん",
      "alt": [
        "点（グラフ理論）"
      ]
    },
    "en": {
      "term": "vertex",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "discrete"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "グラフ理論で、辺で結ばれる点。",
    "definition_en": "In graph theory, one of the points that the edges join.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Each vertex here is a person, and an edge means they know each other.",
        "ja": "ここでは各頂点が人を表し、辺は知り合いであることを表す。",
        "register": "spoken"
      },
      {
        "en": "Find the degree of each vertex of the graph.",
        "ja": "グラフの各頂点の次数を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形は vertices。MIT の講義ノートは node（節点）とも書く。",
      "図形や放物線の頂点も英語は vertex で、別のエントリ（頂点）にした。"
    ],
    "related": [
      "adjacency-matrix",
      "tree",
      "eulerian-path",
      "vertex",
      "eulers-formula-for-polyhedra"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "2.1 Problems and Definitions（vertex ／ vertices。本文 576 件）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "adjacent vertices | degree of a vertex | vertices u and v | set of vertices": 2
      },
      "written": {
        "adjacent vertices | degree of a vertex | vertices u and v | set of vertices": 13
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vertex",
    "ja": {
      "term": "頂点",
      "reading": "ちょうてん",
      "alt": []
    },
    "en": {
      "term": "vertex",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry",
      "functions"
    ],
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "角をつくる 2 辺の共通の端点、多角形・多面体の辺が集まる点。また、放物線と軸の交点。",
    "definition_en": "The common endpoint of the two sides of an angle, a corner point of a polygon or solid, or the point where a parabola meets its axis of symmetry.",
    "latex": "y = a(x - h)^2 + k",
    "spoken_en": "y equals a times the quantity x minus h squared, plus k; the vertex is at h comma k",
    "examples": [
      {
        "en": "The vertex of this parabola is at (2, −1), and it opens up, so that's the minimum.",
        "ja": "この放物線の頂点は (2, −1) で、下に凸だから、そこが最小値だね。",
        "register": "spoken"
      },
      {
        "en": "Find the vertex of the parabola y = x² − 4x + 3.",
        "ja": "放物線 y = x² − 4x + 3 の頂点を求めよ。",
        "register": "written"
      },
      {
        "en": "A triangle has three vertices, and the sum of its interior angles is 180°.",
        "ja": "三角形には頂点が 3 つあり、内角の和は 180° である。",
        "register": "written"
      }
    ],
    "collocations": [
      {
        "en": "vertex of the parabola",
        "ja": "放物線の頂点"
      }
    ],
    "pitfalls": [
      "複数形は vertices（ヴァーティシーズ）。三角形の頂点を指すときも the vertices of the triangle と複数形になる。",
      "グラフ理論（点と辺のグラフ）の頂点も vertex だが、意味が違うので別のエントリ（頂点（グラフ））にした。",
      "放物線の頂点は、平方完成して y = a(x − h)² + k の形にしたときの (h, k)。x 座標は −b/(2a) でも求められる。"
    ],
    "related": [
      "vertex-graph",
      "angle",
      "axis-of-symmetry",
      "completing-the-square",
      "triangle",
      "eulers-formula-for-polyhedra"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "頂点",
        "en": "Vertex (geometry)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "vertex": 537
      },
      "written": {
        "vertex": 963
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vertical-angles",
    "ja": {
      "term": "対頂角",
      "reading": "たいちょうかく",
      "alt": []
    },
    "en": {
      "term": "vertical angles",
      "alt": [],
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "2 本の直線が交わってできる 4 つの角のうち、向かい合っている 2 つの角。対頂角は等しい。",
    "definition_en": "A pair of opposite angles formed when two lines intersect. Vertical angles are always congruent.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Angles 1 and 3 are vertical angles, so if this one's 40 degrees, that one's 40 too.",
        "ja": "∠1 と ∠3 は対頂角だから、こっちが 40° なら、あっちも 40°。",
        "register": "spoken"
      },
      {
        "en": "Lines AB and CD intersect at point E, and ∠AEC and ∠BED are vertical angles. If m∠AEC = (3x + 12)° and m∠BED = (5x − 20)°, find x.",
        "ja": "直線 AB と CD が点 E で交わり、∠AEC と ∠BED は対頂角である。∠AEC = (3x + 12)°、∠BED = (5x − 20)° のとき、x の値を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "vertical は「垂直」（perpendicular、エントリ perpendicular）の意味ではなく、vertex（頂点）から来た語で、「頂点を共有する」という意味。対頂角をつくる 2 直線が垂直に交わっている必要はない。",
      "「対頂角は等しい」は vertical angles are congruent と言う（CK-12 Geometry 1.16 Vertical Angles）。角が等しいことに congruent（エントリ congruent）を使う。",
      "vertical angles は 2 つの角の組の名前で、∠1 and ∠3 are vertical angles のように 2 つの角を主語にして言う。英語版 Wikipedia「Angle」は opposite angles ／ vertically opposite angles とも呼ぶ。"
    ],
    "related": [
      "angle",
      "linear-pair",
      "adjacent-angles",
      "congruent",
      "perpendicular"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "対頂角",
        "en": "Vertical angles"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.16 Vertical Angles"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Angle",
        "note": "記事「Angle」: vertical angles, opposite angles or vertically opposite angles（地域は書いていない）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "vertical angles": 59
      },
      "sources": [
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vertical-asymptote",
    "ja": {
      "term": "垂直漸近線",
      "reading": "すいちょくぜんきんせん",
      "alt": []
    },
    "en": {
      "term": "vertical asymptote",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "グラフが限りなく近づく、x 軸に垂直な直線 x = a。分数関数では、約分した後に分母を 0 にする x の値のところにできる。",
    "definition_en": "A vertical line x = a that a graph gets closer and closer to as x approaches a, while the y-values grow without bound. For a rational function, it occurs where the denominator is zero after the fraction is simplified.",
    "latex": "x = a",
    "spoken_en": "x equals a",
    "examples": [
      {
        "en": "Set the denominator equal to zero and you get x equals 3, so there's a vertical asymptote at x = 3.",
        "ja": "分母 = 0 とおくと x = 3。だから x = 3 に垂直漸近線がある。",
        "register": "spoken"
      },
      {
        "en": "Find the vertical asymptote of f(x) = (x + 1)/(x − 4).",
        "ja": "f(x) = (x + 1)/(x − 4) の垂直漸近線を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "約分で消える因数のところは垂直漸近線ではなく、グラフの穴（hole、removable discontinuity。エントリ removable-discontinuity）になる。約分する前の分母を 0 にする値をそのまま答えないこと。",
      "日本では数III で漸近線（エントリ asymptote）として扱い、向きで名前を分けない。英語の見出しは向きで vertical ／ horizontal ／ slant（エントリ horizontal-asymptote、slant-asymptote）に分かれる。",
      "x = a では関数が定義されないので、グラフは垂直漸近線と交わらない。水平漸近線とは違う。"
    ],
    "related": [
      "asymptote",
      "horizontal-asymptote",
      "slant-asymptote",
      "rational-function",
      "removable-discontinuity"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "vertical asymptote": 130
      },
      "written": {
        "vertical asymptote": 155
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "openstax-precalculus",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vertical-line-test",
    "ja": {
      "term": "垂直線テスト",
      "reading": "すいちょくせんてすと",
      "alt": []
    },
    "en": {
      "term": "vertical line test",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "none",
    "mapping_note": "見出しの「垂直線テスト」は学習指導要領解説に無い、本プロジェクトの訳語。日本では「x の値を決めると y の値がただ 1 つに決まるとき y は x の関数」という定義で考え、グラフでの判定法に名前を付けない。OpenStax Algebra and Trigonometry・Intermediate Algebra は vertical line test の名前で扱う。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Algebra 2"
      ]
    },
    "definition_ja": "座標平面上のグラフに垂直な直線を引き、どの垂直線ともグラフが 2 点以上で交わらなければ、それは y が x の関数であるグラフだと判定する方法。",
    "definition_en": "A way to tell whether a graph shows y as a function of x: if no vertical line crosses the graph more than once, the graph passes and represents a function.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The circle fails the vertical line test: the line x = 0 hits it at y = 1 and at y = −1.",
        "ja": "この円は垂直線テストに通らない。直線 x = 0 と y = 1 と y = −1 の 2 点で交わるからね。",
        "register": "spoken"
      },
      {
        "en": "Use the vertical line test to determine whether the graph represents y as a function of x.",
        "ja": "垂直な直線とグラフの交わり方を調べて、このグラフが y を x の関数として表しているかどうかを判定せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「通る／通らない」は pass ／ fail the vertical line test と言う。円 x² + y² = 1 は通らない。",
      "ここでの vertical は「上下方向の」（x = a の形の直線）の意味。日本語の「垂直」が表す「直交する」は perpendicular（エントリ perpendicular）。",
      "1 対 1 かどうか（逆関数をもつか）を判定する horizontal line test（エントリ horizontal-line-test）とは別の判定法。"
    ],
    "related": [
      "function",
      "horizontal-line-test",
      "relation",
      "graph",
      "circle"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "vertical line test": 22
      },
      "written": {
        "vertical line test": 37
      },
      "sources": [
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "vertical-tangent",
    "ja": {
      "term": "垂直接線",
      "reading": "すいちょくせっせん",
      "alt": []
    },
    "en": {
      "term": "vertical tangent",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "曲線のある点での接線が y 軸に平行になるもの。傾きは定まらず（導関数の極限は ±∞）、その点で関数は微分可能でない。",
    "definition_en": "A tangent line that is vertical, so its slope is undefined; the function is not differentiable there.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The cube root of x has a vertical tangent at the origin.",
        "ja": "∛x のグラフは原点で接線が垂直になる。",
        "register": "spoken"
      },
      {
        "en": "Show that f(x) = ∛x has a vertical tangent at x = 0.",
        "ja": "f(x) = ∛x のグラフが x = 0 で垂直な接線をもつことを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説には、接線が y 軸に平行になる点の名前が出てこない。"
    ],
    "related": [
      "not-differentiable",
      "cusp",
      "corner-nondifferentiable",
      "slope-of-the-tangent-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "垂直接線",
        "en": "Vertical tangent"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "接線が y 軸に平行になる点の名前が出てこない"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "vertical tangent": 16
      },
      "written": {
        "vertical tangent": 17
      },
      "sources": [
        "khan-ap-calc",
        "mit-notes",
        "openstax-calculus",
        "openstax-precalculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "volume-by-cross-sections",
    "ja": {
      "term": "断面積による体積",
      "reading": "だんめんせきによるたいせき",
      "alt": []
    },
    "en": {
      "term": "slicing method",
      "alt": [
        "volumes with cross sections"
      ],
      "variants": [
        {
          "term": "slicing method",
          "register": "written",
          "note": "OpenStax Calculus の書き方。話し言葉には出てこず、書き言葉はすべて OpenStax Calculus。"
        },
        {
          "term": "method of slicing",
          "register": "written",
          "note": "話し言葉・書き言葉とも少ない。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「断面積による体積」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "立体の断面積を積分して体積を求める方法。断面積を S(x) とすると V = ∫ₐᵇ S(x)dx。",
    "definition_en": "Finding a volume by integrating the area of its cross sections: V = ∫ₐᵇ A(x) dx.",
    "latex": "V = \\int_a^b A(x)\\,dx",
    "spoken_en": "V equals the integral from a to b of A of x dx",
    "examples": [
      {
        "en": "The cross sections perpendicular to the x-axis are squares, so A of x is the side length squared.",
        "ja": "x 軸に垂直な断面が正方形だから、A(x) は 1 辺の長さの 2 乗。",
        "register": "spoken"
      },
      {
        "en": "Find the volume of a square pyramid with base edge 6 and height 9 by the slicing method.",
        "ja": "断面積を積分して、底面の 1 辺が 6、高さが 9 の正四角錐の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "AP の CED は、断面が正方形・長方形の場合（topic 8.7）と三角形・半円の場合（topic 8.8）を別の topic にしている。問題文では cross sections perpendicular to the x-axis are squares（x 軸に垂直な断面は正方形）のように断面の形を指定する。"
    ],
    "related": [
      "cross-sectional-area",
      "volume-of-a-solid-of-revolution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.7–8.8（volumes with cross sections）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "「断面積による体積」は出てこない（mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "method of slicing": 1
      },
      "written": {
        "slicing method": 21,
        "method of slicing": 4
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus"
      ],
      "counted": "2026-10-09"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "volume-of-a-solid-of-revolution",
    "ja": {
      "term": "回転体の体積",
      "reading": "かいてんたいのたいせき",
      "alt": []
    },
    "en": {
      "term": "volume of a solid of revolution",
      "alt": [],
      "variants": [
        {
          "term": "volume of a solid of revolution",
          "register": "written",
          "note": "書き言葉はすべて OpenStax Calculus。話し言葉にはほとんど出てこない（MIT OCW だけ）。"
        },
        {
          "term": "volume of revolution",
          "register": "both",
          "note": "略した言い方。話し言葉では少なく、書き言葉は大半が OpenStax Calculus。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus",
      "geometry"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "平面の図形を軸のまわりに 1 回転させてできる立体の体積。x 軸のまわりなら π∫ₐᵇ {f(x)}² dx で求める。",
    "definition_en": "The volume of the solid formed by revolving a region around an axis; around the x-axis it is π ∫ₐᵇ (f(x))² dx.",
    "latex": "V = \\pi \\int_a^b \\bigl(f(x)\\bigr)^2\\,dx",
    "spoken_en": "V equals pi times the integral from a to b of f of x, squared, dx",
    "examples": [
      {
        "en": "When you spin this region around the x-axis, every slice is a disk, so the volume is pi times the integral of f squared.",
        "ja": "この部分を x 軸のまわりに回すと断面はどれも円板だから、体積は π × (f の 2 乗の積分)。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "回転体そのものは solid of revolution。OpenStax Calculus は体積の求め方を disk method ／ washer method ／ shell method と方法の名前で呼び分ける（AP の CED は disc method ／ washer method）。"
    ],
    "related": [
      "cross-sectional-area",
      "volume-by-cross-sections",
      "disk-method",
      "washer-method",
      "shell-method",
      "revolve-around-the-x-axis",
      "axis-of-revolution",
      "surface-area-of-revolution"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "回転体",
        "en": "Solid of revolution"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.9–8.12（disc method ／ washer method）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "volume of revolution": 6,
        "volume of a solid of revolution": 1
      },
      "written": {
        "volume of a solid of revolution": 20,
        "volume of revolution": 10
      },
      "sources": [
        "mit-18.01",
        "mit-notes",
        "openstax-calculus",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "volume",
    "ja": {
      "term": "体積",
      "reading": "たいせき",
      "alt": []
    },
    "en": {
      "term": "volume",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "geometry"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "立体が空間の中で占める大きさ。1 辺が 1 cm の立方体がいくつ分かで表し、単位は cm³（立方センチメートル）などを使う。",
    "definition_en": "How much space a solid takes up, counted in cubic units, such as cubic centimeters (cm³).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the volume of a box, just multiply length times width times height.",
        "ja": "箱の体積を求めるには、縦 × 横 × 高さを計算すればいい。",
        "register": "spoken"
      },
      {
        "en": "Find the volume of a square pyramid whose base has a side length of 6 cm and whose height is 5 cm.",
        "ja": "底面が 1 辺 6 cm の正方形で、高さが 5 cm の正四角錐の体積を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "cm³ は cubic centimeters と読む。長さの単位がない問題では cubic units（エントリ cubic-units）を付けて答える。",
      "入れ物に入る量（容積）も英語では volume と言う（capacity とも言う）。",
      "音量の volume と同じ語。"
    ],
    "related": [
      "cubic-units",
      "surface-area",
      "area-of-the-base",
      "prism",
      "solid"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "体積",
        "en": "Volume"
      },
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "volume": 1046
      },
      "written": {
        "volume": 959
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "voluntary-response-bias",
    "ja": {
      "term": "自発的回答の偏り",
      "reading": "じはつてきかいとうのかたより",
      "alt": []
    },
    "en": {
      "term": "voluntary response bias",
      "alt": [
        "voluntary response sample"
      ],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "見出しの「自発的回答の偏り」は学習指導要領解説に無い、本プロジェクトの訳語。",
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics"
      ]
    },
    "definition_ja": "回答するかどうかを本人に任せた調査で、強い意見をもつ人ばかりが答えるために生じる偏り。",
    "definition_en": "The bias in a survey where people choose whether to respond, so that mostly those with strong opinions answer.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "An online poll is a voluntary response sample, so the people who care most are overrepresented.",
        "ja": "オンラインの投票は自発的回答の標本なので、関心の強い人が多くなりすぎる。",
        "register": "spoken"
      },
      {
        "en": "Explain why a call-in poll suffers from voluntary response bias.",
        "ja": "電話で意見を募る投票に自発的回答の偏りが生じる理由を説明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスでは voluntary response も voluntary response sample も話し言葉にわずかに出てくるだけ（すべて Khan Academy）。",
      "語のままの voluntary response は見出しと voluntary response sample の両方の中に現れるので候補にしない（含む関係）。"
    ],
    "related": [
      "convenience-sample",
      "nonresponse",
      "is-biased"
    ],
    "sources": [
      {
        "type": "reference",
        "title": "College Board, AP Statistics Course and Exam Description (Effective Fall 2026)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf",
        "note": "Topic 1.12（voluntary response bias）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "voluntary response sample": 1
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 0 件（10 未満））。見出しは AP Statistics の CED の呼び方 voluntary response bias（topic 1.12）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "washer-method",
    "ja": {
      "term": "ワッシャー法",
      "reading": "わっしゃーほう",
      "alt": []
    },
    "en": {
      "term": "washer method",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "学習指導要領解説（数学III）は、x 軸と曲線 y = f(x) の間の図形を x 軸のまわりに回してできる立体の体積を π∫{f(x)}²dx で求めることを挙げ、穴のある回転体の求め方の名前は出てこない。日本語版 Wikipedia「回転体」は、2 曲線 f(x), g(x) で囲まれた図形の回転体の体積 π∫|f(x)² − g(x)²|dx を円板法（円板分割法）の中で扱い、別の名前を付けない。OpenStax Calculus と AP の CED（topic 8.11・8.12）は、断面がワッシャー（座金。中央に穴のある円板）の形になる場合を disk method（CED は disc method と綴る）と分けて washer method と呼ぶ。見出しの「ワッシャー法」は学習指導要領解説にも日本語版 Wikipedia にも無い、本プロジェクトの訳語。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus II",
        "Calculus I"
      ]
    },
    "definition_ja": "穴のある回転体を、中央に穴のある薄い円板（ワッシャー）に分けて体積を求める方法。",
    "definition_en": "A way to get the volume of a solid of revolution that has a hole in it, by cutting it into washers (disks with a hole in the middle).",
    "latex": "V = \\pi \\int_a^b \\Bigl(\\bigl(R(x)\\bigr)^2 - \\bigl(r(x)\\bigr)^2\\Bigr)\\,dx",
    "spoken_en": "V equals pi times the integral from a to b of big R of x squared minus little r of x squared, dx",
    "examples": [
      {
        "en": "There's a gap between the region and the axis, so use the washer method.",
        "ja": "領域と回転軸の間にすき間があるので、ワッシャー法を使う。",
        "register": "spoken"
      },
      {
        "en": "It's big R squared minus little r squared, not the difference squared.",
        "ja": "大きい R の 2 乗から小さい r の 2 乗を引く。差を 2 乗するんじゃないよ。",
        "register": "spoken"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "(R − r)² としてしまうのがよくある誤り。英語でも outer radius squared minus inner radius squared と言って注意する。"
    ],
    "related": [
      "disk-method",
      "shell-method",
      "volume-of-a-solid-of-revolution",
      "axis-of-revolution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "6.2 Determining Volumes by Slicing（The Washer Method。mapping_note）"
      },
      {
        "type": "reference",
        "title": "College Board, AP Calculus AB and BC Course and Exam Description (Effective Fall 2020)",
        "url": "https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf",
        "note": "Topic 8.11–8.12（washer method）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学III 積分法: y = f(x) と x 軸の間の図形の回転体の体積 π∫{f(x)}²dx（穴のある回転体の求め方の名前・「ワッシャー法」は出てこない）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「回転体」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%9E%E8%BB%A2%E4%BD%93",
        "note": "節「円板法」: 2 曲線の間の図形の回転体の体積 π∫|f(x)² − g(x)²|dx（mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "washer method": 15
      },
      "written": {
        "washer method": 21
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "openstax-calculus",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "well-ordering-principle",
    "ja": {
      "term": "整列原理",
      "reading": "せいれつげんり",
      "alt": [
        "整列性",
        "自然数の整列性"
      ]
    },
    "en": {
      "term": "well-ordering principle",
      "alt": [],
      "uk": null
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": "米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない（well-ordering principle は CED・OpenStax・IM・CK-12 とも 0 件）。",
    "domains": [
      "proof",
      "number-theory"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": []
    },
    "definition_ja": "正の整数からなる空でない集合には、必ず最小の要素があるという性質。数学的帰納法と同値で、「最小の反例」をとる背理法の証明に使う。",
    "definition_en": "The principle that every nonempty set of positive integers has a smallest element. It is equivalent to induction and is used in proofs that pick a smallest counterexample.",
    "latex": "\\varnothing \\neq S \\subseteq \\mathbb{N} \\implies S \\text{ has a least element}",
    "spoken_en": "if S is a nonempty subset of the natural numbers, then S has a least element",
    "examples": [
      {
        "en": "Suppose there's a counterexample. By the well-ordering principle, there has to be a smallest one, so let's call it n.",
        "ja": "反例があると仮定しよう。整列原理により最小の反例があるはずだから、それを n とおく。",
        "register": "spoken"
      },
      {
        "en": "Use the well-ordering principle to prove that every integer n ≥ 2 has a prime factor.",
        "ja": "整列原理を用いて、2 以上のすべての整数 n は素因数をもつことを証明せよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "整数全体や正の実数では成り立たない。負の整数全体にも、区間 (0, 1) にも最小の要素は無い。",
      "英語の natural numbers が 0 を含むかは本によるが、整列原理はどちらでも成り立つ。"
    ],
    "related": [
      "proof-by-induction",
      "strong-induction",
      "proof-by-contradiction"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Well-ordering_principle",
        "note": "記事名 Well-ordering principle（en.term のリダイレクト先。見出しを決めた参照）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "well-ordering principle": 1
      },
      "written": {
        "well-ordering principle": 1
      },
      "sources": [
        "mit-6.042",
        "mit-notes"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 1 件（10 未満） ／ 書: 判断不能: 重み付け後 1 件（10 未満））。見出しは Wikipedia の記事名 well-ordering principle（英語版「Well-ordering principle」、en.term のリダイレクト先）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "which-one-to-differentiate",
    "ja": {
      "term": "どちらを微分するか",
      "reading": "どちらをびぶんするか",
      "alt": []
    },
    "en": {
      "term": "pick u",
      "alt": [],
      "variants": [
        {
          "term": "choose u",
          "register": "written",
          "note": "書き言葉ではこれだけで、すべて OpenStax Calculus。話し言葉では少ない。"
        }
      ],
      "uk": null,
      "register": "spoken"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語では「どちらを微分し、どちらを積分するか」と言うが、英語（OpenStax Calculus）では ∫u dv の u（微分する側）と dv（積分する側）を選ぶ、と言う。目安として LIATE（対数・逆三角・代数・三角・指数の順に u にする）という覚え方を教えることがある。",
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "definition_ja": "部分積分で、積の 2 つの因数のうちどちらを微分する側にするかという選択。",
    "definition_en": "In integration by parts, the choice of which factor to differentiate (u) and which to integrate (dv).",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Pick u to be the part that gets simpler when you differentiate it, so here u is x.",
        "ja": "微分すると簡単になるほうを u に選ぶ。ここでは u = x。",
        "register": "spoken"
      },
      {
        "en": "We choose u = x and dv = eˣ dx, so that du = dx and v = eˣ.",
        "ja": "u = x、dv = eˣ dx と選ぶと、du = dx、v = eˣ となる。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "選び方を間違えると、残る積分がかえって難しくなる。そのときは u と dv を入れ替えてやり直す。",
      "話し言葉の pick u は MIT OCW・NancyPi・patrickJMT・Professor Leonard に分かれていて、1 つのソースに頼らない。書き言葉の choose u は OpenStax Calculus だけが使う。"
    ],
    "related": [
      "integration-by-parts",
      "integrate-by-parts",
      "liate"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "pick u": 17,
        "choose u": 4
      },
      "written": {
        "choose u": 15
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "openstax-calculus",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "whisker",
    "ja": {
      "term": "ひげ",
      "reading": "ひげ",
      "alt": []
    },
    "en": {
      "term": "whisker",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "箱ひげ図で、箱の両端から最小値・最大値まで伸ばした線分。",
    "definition_en": "In a box plot, either of the two lines drawn out from the box to the smallest and the largest value.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The left whisker is longer, so the lower quarter of the data is more spread out.",
        "ja": "左のひげのほうが長いので、下の 4 分の 1 のデータのほうが散らばっている。",
        "register": "spoken"
      },
      {
        "en": "What do the whiskers of the box plot represent?",
        "ja": "箱ひげ図のひげは何を表しているか。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "箱の長さは四分位範囲（IQR）に当たる。"
    ],
    "related": [
      "box-plot"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "whisker": 17
      },
      "written": {
        "whisker": 8
      },
      "sources": [
        "khan-ap-stats",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "without-loss-of-generality",
    "ja": {
      "term": "一般性を失わずに",
      "reading": "いっぱんせいをうしなわずに",
      "alt": [
        "一般性を失うことなく"
      ]
    },
    "en": {
      "term": "without loss of generality",
      "alt": [
        "WLOG"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "proof"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "definition_ja": "いくつかの場合のうち 1 つを示せば残りも対称性などで同じように示せるとき、その 1 つの場合に絞って議論することを断る言い方。「一般性を失わずに a ≤ b としてよい」のように使う。",
    "definition_en": "A phrase used in a proof to restrict attention to one case, because the other cases follow in the same way, often by symmetry. For example, \"Without loss of generality, assume a ≤ b.\"",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The roles of x and y are symmetric here, so without loss of generality, let's say x is the bigger one.",
        "ja": "ここでは x と y の役割が対称なので、一般性を失わずに x のほうが大きいとしよう。",
        "register": "spoken"
      },
      {
        "en": "Let a and b be real numbers with a ≠ b. Without loss of generality, assume a < b.",
        "ja": "a, b を a ≠ b である実数とする。一般性を失わずに a < b としてよい。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "使えるのは、残りの場合が文字の入れ替えなどで同じ議論になるときだけ。場合ごとに議論が変わるなら、場合分け（proof by cases）で全部書く。",
      "書き言葉では略して WLOG と書くことがある。",
      "用例コーパスでは話・書とも少ない（話し言葉はすべて MIT OCW、書き言葉は OpenStax Calculus）。Levin の本でも 1.4 Proofs に 1 件。"
    ],
    "related": [
      "prove",
      "proof-by-cases"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Precise Definition of a Limit（見出しを決めた参照）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Surface Integrals（見出しを決めた参照）"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "without loss of generality": 5
      },
      "written": {
        "without loss of generality": 2
      },
      "sources": [
        "mit-6.042",
        "openstax-calculus"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 5 件（10 未満） ／ 書: 判断不能: 重み付け後 2 件（10 未満））。見出しは OpenStax の呼び方 without loss of generality（本文 2 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "word-problem",
    "ja": {
      "term": "文章題",
      "reading": "ぶんしょうだい",
      "alt": []
    },
    "en": {
      "term": "word problem",
      "alt": [
        "story problem"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "身近な場面などを文章で書き表した問題。文章から数量の関係を読み取り、式を立てて解く。",
    "definition_en": "A math problem that describes a situation in words instead of giving you an equation. You read it, write an equation or expression, and solve it.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "For this word problem, first decide what x stands for.",
        "ja": "この文章題は、まず x が何を表すかを決めよう。",
        "register": "spoken"
      },
      {
        "en": "For each word problem, write an equation and solve it.",
        "ja": "次の文章題について、それぞれ方程式をつくって解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「文章」を sentence と直訳した sentence problem ではなく、word problem と言う。word は「単語」ではなく「言葉で書かれた」の意味。story problem とも言う。",
      "文章題で「式を立てる」は write an equation ／ set up an equation（エントリ write-an-equation）。",
      "答えは問題が聞いている量で書く。x = 5 で終わらせず、単位をつけて「5 冊」（5 notebooks）のように答える。"
    ],
    "related": [
      "write-an-equation",
      "unknown",
      "equation",
      "variable",
      "represent"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "文章題",
        "en": "Word problem (mathematics education)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "word problem": 36
      },
      "written": {
        "word problem": 127
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "work-backwards",
    "ja": {
      "term": "逆算する",
      "reading": "ぎゃくさんする",
      "alt": [
        "逆演算で解く"
      ]
    },
    "en": {
      "term": "undo",
      "alt": [
        "work backwards",
        "inverse operations"
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "verb",
    "mapping": "near",
    "mapping_note": "「逆算する」に当たる 1 語の英語はない。1 つの操作を打ち消すのは undo（undo the addition by subtracting 3）、操作を最後から順にさかのぼるのは work backwards と言う。打ち消すための演算は inverse operations（逆演算）で、足し算には引き算、掛け算には割り算を使う（use inverse operations）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "definition_ja": "行った計算を、最後の操作から順に逆にたどって、もとの数を求めること。足し算は引き算で、掛け算は割り算で打ち消す。",
    "definition_en": "To undo an operation with its inverse operation, such as subtracting 3 to undo adding 3. Working backwards means undoing the steps in reverse order, starting with the last one.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "It's x plus 3 equals 10, so undo the plus 3 by subtracting 3 from both sides.",
        "ja": "x + 3 = 10 だから、両辺から 3 を引いて、3 を足したのを元に戻そう。",
        "register": "spoken"
      },
      {
        "en": "A number is multiplied by 4, and then 5 is added. The result is 33. Work backwards to find the number.",
        "ja": "ある数に 4 を掛けてから 5 を足すと 33 になった。逆算して、もとの数を求めよ。",
        "register": "written"
      },
      {
        "en": "Undo the addition first: 4x = 33 − 5 = 28, so x = 7.",
        "ja": "まず足し算を逆算する。4x = 33 − 5 = 28 より、x = 7。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "undo は「元に戻す」（パソコンの「元に戻す」も undo）。1 つの操作を打ち消すことを言い、操作を最後から順にさかのぼる全体は work backwards と言う。",
      "打ち消す順序は、行った操作の逆。4x + 5 = 33 では、掛け算より先に「5 を足す」を打ち消す。",
      "「逆演算」は inverse operations。逆関数（inverse function、エントリ inverse-function）や逆数（reciprocal、エントリ reciprocal）とは別の語。"
    ],
    "related": [
      "properties-of-equality",
      "both-sides",
      "move-term-to-other-side",
      "solve",
      "word-problem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "work backwards": 14,
        "undo": 101,
        "inverse operations": 4
      },
      "written": {
        "undo": 85,
        "inverse operations": 8,
        "work backwards": 6
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "work",
    "ja": {
      "term": "仕事",
      "reading": "しごと",
      "alt": []
    },
    "en": {
      "term": "work",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Calculus I",
        "Calculus II",
        "Calculus III"
      ]
    },
    "definition_ja": "力 F で物体を変位 d だけ動かしたときの F·d（力の変位方向の成分 × 距離）。",
    "definition_en": "The quantity F·d for a force F moving an object through a displacement d: the component of the force along the motion times the distance.",
    "latex": "W = \\vec{F}\\cdot\\vec{d}",
    "spoken_en": "W equals F dot d",
    "examples": [
      {
        "en": "Only the part of the force along the direction of motion does any work.",
        "ja": "仕事をするのは、力のうち動く向きの成分だけ。",
        "register": "spoken"
      },
      {
        "en": "A force F = (3, 4) N moves an object from (0, 0) to (5, 2) m. Find the work done.",
        "ja": "力 F = (3, 4)（N）が物体を (0, 0) から (5, 2)（m）まで動かすとき、した仕事を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "学習指導要領解説の数学C のベクトルの節には仕事が出てこない（解説で仕事が出てくるのは付録の学習指導要領 理科「物理基礎」など）。W = F·d は数学C の内積で計算できる。"
    ],
    "related": [
      "dot-product",
      "vector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "仕事 (物理学)",
        "en": "Work (physics)"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学C のベクトルに仕事は出てこない。付録の学習指導要領 理科「物理基礎」: 仕事と関連付けて運動エネルギーと位置エネルギーを理解する"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "work done": 4
      },
      "written": {
        "work done": 86
      },
      "sources": [
        "mit-18.01",
        "mit-6.042",
        "openstax-calculus",
        "openstax-introstats",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "write-an-equation",
    "ja": {
      "term": "式で表す",
      "reading": "しきであらわす",
      "alt": [
        "文字を用いて表す",
        "式を立てる",
        "式に表す"
      ]
    },
    "en": {
      "term": "write an equation",
      "alt": [
        "set up an equation"
      ],
      "variants": [
        {
          "term": "write an expression",
          "register": "spoken",
          "note": "等号を含まない式を書くときの言い方。話し言葉では write an equation に次いで多い（ほとんどが Khan Academy）。"
        }
      ],
      "uk": null,
      "register": "both"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「式で表す」は、等号を含まない式も、等式（方程式）や不等式も指す（中学校学習指導要領解説の第 1 学年 A(2) は、入館料の合計を a + 2b と表すことと、数量の相等関係・大小関係を等式・不等式に表すことを、同じ文字を用いた式の項目で扱う）が、英語は write an expression ／ write an equation ／ write an inequality と言い分ける（IM Algebra 1 2.2 の練習問題の指示は Write an expression, an equation, or an inequality）。",
    "domains": [
      "functions",
      "algebra"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "言葉で述べられた数量や数量の関係を、文字と数を使った式に書き表す。等号で結べば等式（方程式）、等号も不等号も含まなければ式になる。",
    "definition_en": "To translate a quantity or a relationship given in words into math, using variables and numbers. The result is an equation if it has an equal sign and an expression if it has neither an equal sign nor an inequality sign.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's write an equation for this one: if the number is x, then 2x plus 7 equals 25.",
        "ja": "これを式で表してみよう。ある数を x とすると、2x + 7 = 25 だね。",
        "register": "spoken"
      },
      {
        "en": "Write an equation for the following and solve it: when 8 is added to a number, the result is three times the number.",
        "ja": "ある数に 8 を加えると、もとの数の 3 倍になる。このことを方程式で表し、ある数を求めよ。",
        "register": "written"
      },
      {
        "en": "Write an expression for the total cost, in dollars, of x pens at 2 dollars each and one notebook at 5 dollars.",
        "ja": "1 本 2 ドルのペン x 本と、5 ドルのノート 1 冊の代金の合計（ドル）を表す式を書け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "等号を含むなら equation、含まないなら expression（エントリ expression）。「代金を表す式を書け」は write an expression、「方程式で表せ」は write an equation。不等号で結ぶ不等式は write an inequality（IM Algebra 1 2.2）。",
      "文章題で方程式をつくる「式を立てる」は write an equation ／ set up an equation。",
      "グラフや条件から直線・円などの式を決める「式を求める」は別のエントリ（find-the-equation）。"
    ],
    "related": [
      "equation",
      "expression",
      "word-problem",
      "represent",
      "find-the-equation"
    ],
    "sources": [
      {
        "type": "mext-translation"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "第 1 学年 A(2) 文字を用いた式: 入館料の合計 a + 2b、相等関係・大小関係を等式・不等式に表す（mapping_note）・数量の関係や法則などを文字を用いた式に表す（ja.alt 式に表す）・第 3 学年 数学的活動: 連続する数などを文字を用いて表す（ja.alt）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 1)",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 2.2 Writing Equations to Model Relationships (Part 1): Write an expression, an equation, or an inequality（mapping_note・pitfalls[0]）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "write an equation": 27,
        "write an expression": 12,
        "set up an equation": 9
      },
      "written": {
        "write an equation": 75,
        "set up an equation": 1,
        "write an expression": 21
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-10-09"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "write-dx-in-terms-of-du",
    "ja": {
      "term": "dx を du で表す",
      "reading": "でぃーえっくすをでぃーゆーであらわす",
      "alt": []
    },
    "en": {
      "term": "solve for dx",
      "alt": [
        "write dx in terms of du"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "calculus"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "definition_ja": "置換積分で、du = g′(x)dx の関係から dx を du を使った式に書き直すこと。",
    "definition_en": "In a substitution, to rewrite dx using du, from the relation du = g′(x) dx.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "du is two x dx, so solve for dx: dx is du over two x.",
        "ja": "du = 2x dx だから dx について解くと、dx = du/(2x)。",
        "register": "spoken"
      },
      {
        "en": "From du = 2x dx we get dx = du/(2x), and the x in the denominator cancels.",
        "ja": "du = 2x dx より dx = du/(2x) となり、分母の x は約分で消える。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "dx を解き出す（solve for dx）か、被積分関数の中の 2x dx をそのまま du に置き換えるかは教科書による。",
      "関連変化率の solve for dx dt（dx/dt を求める）とは別。"
    ],
    "related": [
      "integration-by-substitution",
      "let-u-equal"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "solve for dx !dt": 8
      },
      "sources": [
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "見出しは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 統計・ベクトルの単元 2 の前の修正」の、残った ③ を人間が決めた行）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "write-in-descending-order",
    "ja": {
      "term": "降べきの順に整理する",
      "reading": "こうべきのじゅんにせいりする",
      "alt": []
    },
    "en": {
      "term": "written in descending order",
      "alt": [
        "write … in descending order"
      ],
      "uk": null
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "OpenStax Algebra and Trigonometry（5.2・5.4・5.5）と Elementary Algebra（6.1・6.6・7.3）は written in descending order ／ in descending order of degrees と書く。OpenStax Elementary Algebra 6.1・Intermediate Algebra 5.1・Prealgebra 10.1 は、次数の高い順に並べた多項式を standard form（standard form of a polynomial）と定義し、IM Algebra 2 2.6 Different Forms も同じ（Algebra and Trigonometry 5.2 は general form）。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "多項式の項を、ある文字についての次数が高いものから低いものへ順に並べ直す。",
    "definition_en": "To arrange the terms of a polynomial from the highest power of a chosen variable down to the lowest.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Let's write it in descending order first, so the x cubed term goes in front.",
        "ja": "まず降べきの順に整理しよう。x³ の項が先頭に来る。",
        "register": "spoken"
      },
      {
        "en": "Write 3x − 5 + 2x³ − x² in descending order.",
        "ja": "3x − 5 + 2x³ − x² を降べきの順に整理せよ。",
        "register": "written"
      },
      {
        "en": "Write x² + 3xy − 2y² + x − 4y in descending order of powers of x, grouping the terms with the same power of x.",
        "ja": "x² + 3xy − 2y² + x − 4y を x について降べきの順に整理せよ。",
        "register": "written"
      },
      {
        "en": "A polynomial written in descending order has its leading term first.",
        "ja": "降べきの順に整理した多項式は、最高次の項が先頭に来る。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "「x について整理する」では x 以外の文字を係数とみなし、x の次数ごとにまとめる：x² + (3y + 1)x − (2y² + 4y)。",
      "日本語の「整理する」は同類項をまとめること（combine like terms）も含む。並べるだけの「降べきの順」と分けて考える。",
      "逆の、次数の低い順は昇べきの順（エントリ ascending-order）。"
    ],
    "related": [
      "ascending-order",
      "leading-term",
      "combine-like-terms",
      "arrange-by-the-variable-of-lowest-degree"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 2 2.6 Different Forms: standard form of a polynomial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "write … in descending order": 2,
        "written in descending order": 1
      },
      "written": {
        "written in descending order": 7,
        "write … in descending order": 2
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-reference-fallback",
        "note": "コーパスで決まらない（話: 判断不能: 重み付け後 3 件（10 未満） ／ 書: 判断不能: 重み付け後 9 件（10 未満））。見出しは OpenStax の呼び方 written in descending order（本文 7 件）。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "en_is_explanatory_translation": false
  },
  {
    "id": "write-out-the-first-few-terms",
    "ja": {
      "term": "項を書き出す",
      "reading": "こうをかきだす",
      "alt": []
    },
    "en": {
      "term": "write the first … terms",
      "alt": [
        "list the first … terms"
      ],
      "variants": [
        {
          "term": "write out the first few terms",
          "register": "written",
          "note": "OpenStax Calculus Volume 2（5.1・5.2）・Intermediate Algebra（12.3）と Levin（4.1）の言い方。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "sequences"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1",
        "Discrete Math",
        "Calculus II"
      ]
    },
    "definition_ja": "数列の初めのいくつかの項を具体的に書き並べる。一般項や漸化式から項を求めるときや、規則を見るときに行う。",
    "definition_en": "To write down the first several terms of a sequence explicitly, for example from its formula or to look for a pattern.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "If you're stuck, just write out the first few terms and look for a pattern.",
        "ja": "行き詰まったら、項を書き出して規則を探そう。",
        "register": "spoken"
      },
      {
        "en": "Write the first five terms of the sequence aₙ = (−1)ⁿ/n.",
        "ja": "数列 aₙ = (−1)ⁿ/n の初めの 5 項を書き出せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "sequence",
      "partial-sum"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences（write out the first few terms）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.1 Sequences and Their Notations（Write the first five terms of the sequence …）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.1 Sequences（Write the first few terms of a sequence）・12.3 Geometric Sequences and Series（write out the first few terms）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 1.5 Sequences are Functions・1.6 Representing Sequences（Write the first five terms of each sequence）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.1 Describing Sequences（writing out the first few terms）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "written": {
        "write out the first few terms": 6,
        "write the first … terms of the sequence | write the first … terms of a sequence | write the first … terms of each sequence | write the first few terms": 67,
        "list the first … terms": 8
      },
      "sources": [
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-intalg"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "write-with-the-same-base",
    "ja": {
      "term": "底をそろえる",
      "reading": "ていをそろえる",
      "alt": []
    },
    "en": {
      "term": "rewrite … with a common base",
      "alt": [],
      "variants": [
        {
          "term": "write … with the same base",
          "register": "written",
          "note": "OpenStax Intermediate Algebra（10.2）の手順は Write both sides of the equation with the same base と書く。with the same base は、指数法則の multiply powers with the same base（同じ底の累乗の積）にも使う。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "phrase",
    "mapping": "near",
    "mapping_note": "日本語の「底をそろえる」は底を目的語にするが、OpenStax Algebra and Trigonometry（6.6）は両辺を目的語にして rewrite each side with a common base（両辺を共通の底で書き直す）と書く（with a common base 10 件）。底を目的語にした言い方（match the bases など）は参照（CED・OpenStax・IM・CK-12）に出てこない。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "指数方程式・対数方程式で、両辺を同じ底の累乗（対数）で表す。",
    "definition_en": "In exponential or logarithmic equations, to write both sides as powers (or logarithms) of the same base.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Write both sides with the same base: 8 is 2 cubed and 4 is 2 squared.",
        "ja": "両辺の底をそろえよう。8 は 2³、4 は 2²。",
        "register": "spoken"
      },
      {
        "en": "Solve 8^(x+1) = 4^(2x) by rewriting both sides with a common base.",
        "ja": "底をそろえて 8^(x+1) = 4^(2x) を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [],
    "related": [
      "exponential-equation",
      "change-of-base-formula",
      "one-to-one-property"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "6.6 Exponential and Logarithmic Equations（rewrite each side with a common base）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "10.2 Evaluate and Graph Exponential Functions（Write both sides of the equation with the same base）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "with a common base": 2
      },
      "written": {
        "with a common base": 10,
        "as a power with the same base": 2
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "x-axis",
    "ja": {
      "term": "x 軸",
      "reading": "えっくすじく",
      "alt": []
    },
    "en": {
      "term": "x-axis",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "座標平面で、原点を通る横の数直線。右へ行くほど値が大きくなる。",
    "definition_en": "The horizontal number line on the coordinate plane. It crosses the y-axis at the origin, and its numbers increase to the right.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "Every point on the x-axis has a y-coordinate of 0.",
        "ja": "x 軸上の点は、どれも y 座標が 0 だよ。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinates of the point where the line y = −2x + 8 crosses the x-axis.",
        "ja": "直線 y = −2x + 8 と x 軸との交点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "複数形は axes で、発音はアクシーズ（Merriam-Webster: ˈak-ˌsēz）。斧（ax ／ axe、ˈaks）の複数形も axes と綴る（Merriam-Webster）。",
      "x 軸は直線 y = 0 のこと（x = 0 ではない）。",
      "x 軸との交点は x-intercept。答えは点 (4, 0) のことも、x 座標の 4 だけのこともある（エントリ x-intercept）。"
    ],
    "related": [
      "y-axis",
      "coordinate-axes",
      "x-intercept",
      "origin",
      "number-line"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「axis」",
        "url": "https://www.merriam-webster.com/dictionary/axis",
        "note": "発音 複数形 axes ˈak-ˌsēz。axe は ˈaks、複数形 axes（2026-09-26 に確かめた）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "x-axis": 685
      },
      "written": {
        "x-axis": 723
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "x-coordinate",
    "ja": {
      "term": "x 座標",
      "reading": "えっくすざひょう",
      "alt": []
    },
    "en": {
      "term": "x-coordinate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "点の座標 (x, y) の 1 つ目の数。点が原点から右（正）または左（負）にどれだけの位置にあるかを表す。",
    "definition_en": "In an ordered pair (x, y), the number written first. It tells how far right (positive) or left (negative) of the origin the point is.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The x-coordinate always comes first, so for (3, −2) you go right 3.",
        "ja": "x 座標がいつも先だから、(3, −2) なら右に 3 進むよ。",
        "register": "spoken"
      },
      {
        "en": "Point P lies on the line y = x + 2 and has an x-coordinate of 5. Find the coordinates of P.",
        "ja": "点 P は直線 y = x + 2 上にあり、x 座標は 5 である。点 P の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "点 (3, −2) の x 座標は 3、y 座標は −2。英語で点を読むときは three, negative two と言う。",
      "x 座標 1 つを指すので単数の coordinate。(x, y) の組は複数形 coordinates（エントリ coordinates）。"
    ],
    "related": [
      "coordinates",
      "y-coordinate",
      "x-axis",
      "coordinate-plane",
      "x-intercept"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "x-coordinate": 145
      },
      "written": {
        "x-coordinate": 87
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "x-intercept",
    "ja": {
      "term": "x 軸との交点",
      "reading": "えっくすじくとのこうてん",
      "alt": [
        "x 軸との共有点",
        "x 切片"
      ]
    },
    "en": {
      "term": "x-intercept",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語の「x 軸との交点」は x 軸と交わる点で、接する点も含めて言うときは「x 軸との共有点」と言う（高等学校学習指導要領解説（数学I）は二次方程式の解をグラフと x 軸との共有点の x 座標と捉える）。日本語版 Wikipedia「切片 (数学)」は、x 軸との交点を x 切片と呼び、一次関数では交点の x 座標を x 切片と呼ぶ。英語の x-intercept は接する点も含み（OpenStax Algebra and Trigonometry 5.3 は重解のところでグラフが touches the axis at the intercept と書く）、ふつうは点を指すが、x 座標の数だけを言うこともある（IM Algebra 1 の用語集は horizontal intercept（x-intercept）を (6, 0) のような点とし、x 座標だけを指すこともあると書く。OpenStax Precalculus 2.3 は数で言う）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1",
        "Algebra 2",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "グラフと x 軸の共有点（x 軸と交わる点・接する点）。y = 0 とおいて求める。",
    "definition_en": "A point where a graph crosses or touches the x-axis, found by setting y = 0; the word is also used for just the x-coordinate of that point.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To find the x-intercepts, set y equal to zero and solve for x.",
        "ja": "x 軸との交点を求めるには、y = 0 とおいて x について解く。",
        "register": "spoken"
      },
      {
        "en": "For what values of k does the graph of y = x² + kx + 4 have two x-intercepts?",
        "ja": "y = x² + kx + 4 のグラフが x 軸と異なる 2 点で交わるような k の値の範囲を求めよ。",
        "register": "written"
      },
      {
        "en": "The discriminant is negative, so this parabola has no x-intercepts.",
        "ja": "判別式が負だから、この放物線は x 軸と共有点をもたない。",
        "register": "spoken"
      }
    ],
    "collocations": [
      {
        "en": "find the x-intercepts",
        "ja": "x 軸との交点を求める"
      },
      {
        "en": "how many x-intercepts",
        "ja": "x 軸との共有点の個数"
      }
    ],
    "pitfalls": [
      "答えを点 (3, 0) で書くか、x 座標の 3 で書くかは問題文を見る。",
      "グラフが x 軸に接する点（重解）も x-intercept に数える。「x 軸との共有点」の訳も x-intercept でよく、その個数は判別式の符号で決まる。"
    ],
    "related": [
      "discriminant",
      "point-of-intersection",
      "coordinates",
      "axis-of-symmetry",
      "zeros-of-a-polynomial"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "5.3 Graphs of Polynomial Functions（touches the axis at the intercept。mapping_note）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "2.3 Modeling with Linear Functions（x-intercept を数で言う。mapping_note）"
      },
      {
        "type": "reference",
        "title": "高等学校学習指導要領（平成30年告示）解説 数学編 理数編",
        "url": "https://www.mext.go.jp/content/20260115-mxt_kyoiku02-100002620_04.pdf",
        "note": "数学I 二次関数: 二次方程式の解をグラフと x 軸との共有点の x 座標と捉える（mapping_note）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「切片 (数学)」",
        "url": "https://ja.wikipedia.org/wiki/%E5%88%87%E7%89%87_(%E6%95%B0%E5%AD%A6)",
        "note": "冒頭・1 次関数の場合: x 軸との交点を x 切片、一次関数では交点の x 座標を x 切片と呼ぶ（mapping_note）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Algebra 1)",
        "url": "https://im.kendallhunt.com/HS/students/1/index.html",
        "note": "Algebra 1 glossary（horizontal intercept: also called the x-intercept。mapping_note）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-10-09",
      "human": null
    },
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "x-intercept": 98
      },
      "written": {
        "x-intercept": 148
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "y-axis",
    "ja": {
      "term": "y 軸",
      "reading": "わいじく",
      "alt": []
    },
    "en": {
      "term": "y-axis",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "座標平面で、原点で x 軸と垂直に交わる縦の数直線。上へ行くほど値が大きくなる。",
    "definition_en": "The vertical number line on the coordinate plane. It crosses the x-axis at the origin, and its numbers increase going up.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "This point is right on the y-axis, so its x-coordinate is 0.",
        "ja": "この点はちょうど y 軸上にあるから、x 座標は 0 だね。",
        "register": "spoken"
      },
      {
        "en": "Find the coordinates of the point where the line y = 4x − 3 crosses the y-axis.",
        "ja": "直線 y = 4x − 3 と y 軸との交点の座標を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "y 軸は直線 x = 0 のこと（y = 0 ではない）。y 軸上の点は x 座標が 0。",
      "y 軸との交点は y-intercept。答えは点 (0, −3) のことも、y 座標の −3 だけのこともある。"
    ],
    "related": [
      "x-axis",
      "coordinate-axes",
      "y-coordinate",
      "origin",
      "coordinate-plane"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "y-axis": 348
      },
      "written": {
        "y-axis": 451
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "y-coordinate",
    "ja": {
      "term": "y 座標",
      "reading": "わいざひょう",
      "alt": []
    },
    "en": {
      "term": "y-coordinate",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "点の座標 (x, y) の 2 つ目の数。点が原点から上（正）または下（負）にどれだけの位置にあるかを表す。",
    "definition_en": "In an ordered pair (x, y), the number written second. It tells how far above (positive) or below (negative) the origin the point is.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The y-coordinate is negative 2, so from there you go down 2.",
        "ja": "y 座標が −2 だから、そこから下に 2 進むよ。",
        "register": "spoken"
      },
      {
        "en": "The point (4, k) lies on the line y = 3x − 5. Find its y-coordinate k.",
        "ja": "点 (4, k) は直線 y = 3x − 5 上にある。この点の y 座標 k を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "関数では、x = a のときの y の値が、グラフ上の点の y 座標になる（エントリ value-of-the-function）。",
      "x 軸上の点は y 座標が 0。x 軸との交点を求めるときは y = 0 とおく（エントリ x-intercept）。"
    ],
    "related": [
      "coordinates",
      "x-coordinate",
      "y-axis",
      "coordinate-plane",
      "value-of-the-function"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "y-coordinate": 106
      },
      "written": {
        "y-coordinate": 104
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.02",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "openstax-precalculus",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "y-intercept",
    "ja": {
      "term": "切片",
      "reading": "せっぺん",
      "alt": [
        "y 切片"
      ]
    },
    "en": {
      "term": "y-intercept",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "日本語版 Wikipedia「一次関数」は y = ax + b の b（直線と y 軸との交点の座標）を y 切片、あるいは単に切片と呼び、センター試験の問題文（平成 31 年度 数学Ⅰ）も直線について「切片が −15」と書く。OpenStax Algebra and Trigonometry は y を付けて y-intercept と言い、x 軸と交わる点は x-intercept（エントリ x-intercept）と言い分ける（回帰直線では OpenStax Introductory Statistics が単に the intercept とも言う）。",
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 1"
      ]
    },
    "definition_ja": "直線やグラフが y 軸と交わる点の y 座標。y = ax + b では b。",
    "definition_en": "The y-coordinate of the point where a line or graph crosses the y-axis. In y = mx + b, it is b.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "To get the y-intercept, set x equal to 0; that's the point where the graph hits the y-axis.",
        "ja": "切片を求めるには x に 0 を代入する。そこがグラフと y 軸との交点。",
        "register": "spoken"
      },
      {
        "en": "Find the slope and the y-intercept of the line 2x + y = 6.",
        "ja": "直線 2x + y = 6 の傾きと切片を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本語の「切片」は y 切片のことだが、英語では y を付けて y-intercept と言う。x 軸と交わる点は x-intercept（エントリ x-intercept）。",
      "y-intercept は x = 0 を、x-intercept は y = 0 を代入して求める。取り違えやすい。"
    ],
    "related": [
      "x-intercept",
      "slope",
      "linear-function",
      "y-axis",
      "equation-of-a-line"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "切片 (数学)",
        "en": "Y-intercept"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「一次関数」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%80%E6%AC%A1%E9%96%A2%E6%95%B0",
        "note": "b を y 切片、あるいは単に切片と呼ぶ"
      },
      {
        "type": "reference",
        "title": "大学入試センター 平成31年度 大学入試センター試験 本試験 数学Ⅰ（問題）",
        "url": "https://web.archive.org/web/20200601/https://www.dnc.ac.jp/albums/abm.php?f=abm00036452.pdf&n=k+18+suugaku1.pdf",
        "note": "直線の「切片が −15」"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "回帰直線の the intercept"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "y-intercept": 288
      },
      "written": {
        "y-intercept": 476
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-introstats",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "z-score",
    "ja": {
      "term": "z 値",
      "reading": "ぜっとち",
      "alt": [
        "z スコア"
      ]
    },
    "en": {
      "term": "z-score",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "statistics"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Algebra 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "値 x が平均から標準偏差いくつ分離れているかを表す数 z = (x − μ)/σ。",
    "definition_en": "The number of standard deviations a value x is from the mean, z = (x − μ)/σ.",
    "latex": "z = \\frac{x - \\mu}{\\sigma}",
    "spoken_en": "z equals x minus mu over sigma",
    "examples": [
      {
        "en": "A z-score of negative two means you're two standard deviations below the mean.",
        "ja": "z スコアが −2 とは、平均より標準偏差 2 つ分下ということ。",
        "register": "spoken"
      },
      {
        "en": "Find the z-score of a test score of 85 if the mean is 70 and the standard deviation is 6.",
        "ja": "平均 70、標準偏差 6 のとき、得点 85 の z スコアを求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "英語の standard score は z スコアのこと。日本の偏差値（hensachi）とは別。"
    ],
    "related": [
      "standardize",
      "standard-normal-distribution",
      "standard-score",
      "normal-distribution"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "z-score": 111
      },
      "written": {
        "z-score": 113
      },
      "sources": [
        "khan-ap-stats",
        "openstax-calculus",
        "openstax-introstats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "zero-exponent",
    "ja": {
      "term": "0 乗",
      "reading": "ぜろじょう",
      "alt": [
        "零乗"
      ]
    },
    "en": {
      "term": "zero exponent",
      "alt": [],
      "variants": [
        {
          "term": "to the zero power",
          "register": "spoken",
          "note": "「x の 0 乗」の読み（x to the zero power）。話し言葉に多く、大半が Khan Academy。書き言葉では zero exponent のほうがずっと多い。"
        }
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 2",
        "Precalculus",
        "Integrated Math 1",
        "Algebra 1"
      ]
    },
    "definition_ja": "a ≠ 0 のとき a⁰ = 1 と定めること。",
    "definition_en": "The rule a⁰ = 1 for a ≠ 0.",
    "latex": "a^0 = 1\\quad(a \\ne 0)",
    "spoken_en": "a to the zero power equals one",
    "examples": [
      {
        "en": "Anything to the zero power is 1, as long as it isn't zero.",
        "ja": "0 でないものの 0 乗は 1。",
        "register": "spoken"
      },
      {
        "en": "Use the zero exponent rule to simplify (3x²y)⁰.",
        "ja": "0 乗の定め方を使って (3x²y)⁰ を簡単にせよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "OpenStax Prealgebra（10.5）は a ≠ 0 のときだけ a⁰ = 1 とし、0⁰ の値は決めない。indeterminate（不定形）は極限の形 0⁰ の呼び方（OpenStax Calculus Volume 1 の 4.8）。べき級数では x = 0 でも x⁰ = 1 とする（OpenStax Calculus Volume 2 の 6.1）。"
    ],
    "related": [
      "negative-exponent",
      "laws-of-exponents"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "10.5 Integer Exponents and Scientific Notation（Zero Exponent Property）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "4.8 L’Hôpital’s Rule（indeterminate form 0⁰）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "6.1 Power Series and Functions（x⁰ = 1）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 7.4 Dividing Powers of 10（zero exponent）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "to the zero power": 92
      },
      "written": {
        "zero exponent": 61,
        "to the zero power": 13
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "openstax-prealgebra",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "zero-matrix",
    "ja": {
      "term": "零行列",
      "reading": "ぜろぎょうれつ",
      "alt": []
    },
    "en": {
      "term": "zero matrix",
      "alt": [],
      "uk": null,
      "register": "spoken"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "definition_ja": "成分がすべて 0 の行列。O と書く。",
    "definition_en": "A matrix all of whose entries are 0, written O.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "A matrix times the zero matrix is always the zero matrix.",
        "ja": "どの行列に零行列を掛けても零行列になる。",
        "register": "spoken"
      },
      {
        "en": "Find a nonzero 2 × 2 matrix A such that A² is the zero matrix.",
        "ja": "A² が零行列になる、零行列でない 2 × 2 行列 A を求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉は大半が MIT OCW 18.06（線形代数）。"
    ],
    "related": [
      "matrix",
      "identity-matrix"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "零行列",
        "en": "Zero matrix"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "zero matrix": 16
      },
      "written": {
        "zero matrix": 1
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "zero-product-property",
    "ja": {
      "term": "零積法則",
      "reading": "れいせきほうそく",
      "alt": [
        "零積の性質",
        "零因子の性質"
      ]
    },
    "en": {
      "term": "zero product property",
      "alt": [
        "zero-product property",
        "zero factor property"
      ],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "near",
    "mapping_note": "見出しの「零積法則」は学習指導要領解説に無い（日本語版 Wikipedia「整域」が零因子の非存在を零積法則と呼び、「整数の合同」は零積性質と書く）。中学校学習指導要領解説はこの性質に名前を付けず、「AB = 0 ならば、A = 0 または B = 0」と文で書く。米国の教科書 OpenStax Elementary Algebra は zero product property と名前を付け、因数分解で二次方程式を解くときの根拠として書く。OpenStax Algebra and Trigonometry・Precalculus は zero factor property と呼ぶ。",
    "domains": [
      "algebra"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Integrated Math 2"
      ]
    },
    "definition_ja": "2 つの数の積が 0 ならば、少なくとも一方は 0 であるという性質。AB = 0 ならば A = 0 または B = 0。",
    "definition_en": "When a product of two numbers equals 0, at least one factor is 0: ab = 0 forces a = 0 or b = 0.",
    "latex": "ab = 0 \\Rightarrow a = 0 \\text{ or } b = 0",
    "spoken_en": "if a times b equals zero, then a equals zero or b equals zero",
    "examples": [
      {
        "en": "The product is zero, so by the zero product property, either x − 4 is zero or x + 1 is zero.",
        "ja": "積が 0 だから、「AB = 0 ならば A = 0 または B = 0」で、x − 4 = 0 か x + 1 = 0 のどちらかだ。",
        "register": "spoken"
      },
      {
        "en": "Use the zero product property to solve (2x − 1)(x + 5) = 0.",
        "ja": "AB = 0 ならば A = 0 または B = 0 であることを用いて、方程式 (2x − 1)(x + 5) = 0 を解け。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "右辺が 0 のときだけ使える。(x − 1)(x − 2) = 6 から x − 1 = 6 や x − 2 = 6 とはできない（積が 6 になる 2 数の組は無数にある）。展開して x² − 3x − 4 = 0 と右辺を 0 にしてから因数分解する。",
      "「または」（or）は両方が 0 の場合も含む。A = 0 or B = 0 は「どちらか一方だけが 0」の意味ではない。",
      "「零因子」（zero divisor）は 0 でない元どうしをかけて 0 になるもの。実数に零因子が無いというのがこの性質の中身なので、日本語版 Wikipedia「整域」はこれを零積法則と呼ぶ。英語の名前は zero product property（OpenStax Elementary Algebra）と zero factor property（OpenStax Algebra and Trigonometry・Precalculus）。"
    ],
    "related": [
      "solving-by-factoring",
      "factoring",
      "quadratic-equation",
      "factored-form",
      "zeros-of-a-polynomial"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      },
      {
        "type": "reference",
        "title": "中学校学習指導要領（平成29年告示）解説 数学編",
        "url": "https://www.mext.go.jp/component/a_menu/education/micro_detail/__icsFiles/afieldfile/2019/03/18/1387018_004.pdf",
        "note": "中学校 第 3 学年 数と式（二次方程式）: 「AB=0ならば,A=0またはB=0」であることを用いる方法（性質の名前は無い。「零因子の性質」「零積の性質」は 0 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「整域」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B4%E5%9F%9F",
        "note": "零因子の非存在（零積法則）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「整数の合同」",
        "url": "https://ja.wikipedia.org/wiki/%E6%95%B4%E6%95%B0%E3%81%AE%E5%90%88%E5%90%8C",
        "note": "零積性質「積が 0 に等しいならばいずれかの数が 0 に等しい」"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-26",
      "human": null
    },
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "zero product property": 8
      },
      "written": {
        "zero-product property": 16,
        "zero product property": 80,
        "zero factor property": 3
      },
      "sources": [
        "openstax-algtrig",
        "openstax-elemalg",
        "openstax-intalg",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-26"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "zero-vector",
    "ja": {
      "term": "零ベクトル",
      "reading": "ぜろべくとる",
      "alt": []
    },
    "en": {
      "term": "zero vector",
      "alt": [],
      "uk": null,
      "register": "both"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "vectors",
      "linear-algebra"
    ],
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "definition_ja": "大きさが 0 のベクトル。0 と書き、向きは考えない。",
    "definition_en": "The vector of magnitude 0, written 0; it has no direction.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The only vector with length zero is the zero vector.",
        "ja": "長さが 0 のベクトルは零ベクトルだけ。",
        "register": "spoken"
      },
      {
        "en": "Show that a + (−a) is the zero vector.",
        "ja": "a + (−a) が零ベクトルであることを示せ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "用例コーパスの話し言葉は大半が MIT OCW（おもに線形代数）。"
    ],
    "related": [
      "vector",
      "opposite-vector"
    ],
    "sources": [
      {
        "type": "wikipedia-langlink",
        "ja": "零ベクトル",
        "en": "Zero vector"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-25",
    "evidence": {
      "spoken": {
        "zero vector": 108
      },
      "written": {
        "zero vector": 15
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-notes",
        "openstax-algtrig",
        "openstax-calculus",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  },
  {
    "id": "zeros-of-a-polynomial",
    "ja": {
      "term": "多項式の零点",
      "reading": "たこうしきのれいてん",
      "alt": []
    },
    "en": {
      "term": "zeros of a polynomial",
      "alt": [],
      "uk": null,
      "register": "written"
    },
    "pos": "noun",
    "mapping": "exact",
    "mapping_note": null,
    "domains": [
      "functions"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Integrated Math 2",
        "Integrated Math 3"
      ]
    },
    "definition_ja": "多項式 P(x) の値が 0 になる x。方程式 P(x) = 0 の解と同じ。",
    "definition_en": "The values of x where a polynomial P(x) equals 0: the solutions of P(x) = 0.",
    "latex": null,
    "spoken_en": null,
    "examples": [
      {
        "en": "The zeros of the polynomial are where the graph crosses or touches the x-axis.",
        "ja": "多項式の零点は、グラフが x 軸と交わる（接する）ところ。",
        "register": "spoken"
      },
      {
        "en": "Find all zeros of P(x) = x³ − 2x² + 4x − 8, including complex zeros.",
        "ja": "P(x) = x³ − 2x² + 4x − 8 の零点を、複素数も含めてすべて求めよ。",
        "register": "written"
      }
    ],
    "collocations": [],
    "pitfalls": [
      "日本の高校の教科書は「方程式 P(x) = 0 の解」として扱う。米国では zero（関数の値が 0 になる x）・root（方程式の解）・x-intercept（グラフの交点）を使い分ける。",
      "冠詞の違う zeros of the polynomial も書き言葉に少し出てくる。"
    ],
    "related": [
      "solution",
      "factor-theorem",
      "fundamental-theorem-of-algebra",
      "rational-root-theorem"
    ],
    "sources": [
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      }
    ],
    "confidence": "likely",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": null,
      "human": null
    },
    "updated": "2026-09-24",
    "evidence": {
      "spoken": {
        "zeros of a polynomial": 1
      },
      "written": {
        "zeros of a polynomial": 22
      },
      "sources": [
        "openstax-algtrig",
        "yt:organicchem"
      ],
      "counted": "2026-09-25"
    },
    "en_is_explanatory_translation": false
  }
]