[
  {
    "id": "a-sub-n",
    "latex": "a_n,\\ a_{n+1}",
    "spoken_en": [
      {
        "text": "a sub n",
        "register": "standard"
      }
    ],
    "spoken_ja": "a n",
    "name_en": "subscript",
    "name_ja": "添字",
    "category": "subscripts",
    "term_ref": "subscript",
    "related": [
      "sequence-starting-index"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "aₙ は a sub n と読む。sub を言わずに a n と読む人もいる。",
      "aₙ₊₁ は a sub n plus one。aₙ + 1 は a sub n, plus one と区切って読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Summation Notation（the sum of a sub i from i equals one to five）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "a sub *": 307
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "absolute-value-bars",
    "latex": "|x|",
    "spoken_en": [
      {
        "text": "the absolute value of x",
        "register": "standard"
      },
      {
        "text": "absolute value of x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "x の絶対値",
    "name_en": "absolute value bars",
    "name_ja": "絶対値記号",
    "category": "arithmetic",
    "term_ref": "absolute-value",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "|x| は the absolute value of x と読む。|−5| は the absolute value of negative five。",
      "縦線 | は条件付き確率（given）や集合の記法（such that）、割り切る（divides）にも使う。読みは使い方で変わる。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 7.6 Absolute Value of Numbers（the absolute value of）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the absolute value of *": 753,
        "!the absolute value of *": 136
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "alpha",
    "latex": "\\alpha",
    "spoken_en": [
      {
        "text": "alpha",
        "register": "standard"
      }
    ],
    "spoken_ja": "アルファ",
    "name_en": "alpha",
    "name_ja": "ギリシャ文字 α",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus",
        "AP Statistics"
      ]
    },
    "notes": [
      "α は alpha と読む。",
      "AP Statistics の CED（topic 3.7）は有意水準（significance level）を α で表す。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles・10.1 Non-right Triangles: Law of Sines（angle α (alpha)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "5.1 Angles（ギリシャ文字の表: α alpha）・8.1 Non-right Triangles: Law of Sines（angle α (alpha)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "付録 Mathematical Phrases, Symbols, and Formulas（α、alpha、probability of Type I error）"
      },
      {
        "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 3.7（significance level）・Topic 4.5・Topic 4.10"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Geometry 4.9・7.2・7.4（α）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "alpha": 514
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "angle-abc",
    "latex": "\\angle ABC",
    "spoken_en": [
      {
        "text": "angle ABC",
        "register": "standard"
      }
    ],
    "spoken_ja": "角 ABC",
    "name_en": "angle sign",
    "name_ja": "角の記号（∠ABC）",
    "category": "geometry",
    "term_ref": "angle",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "notes": [
      "∠ABC は angle ABC と読む。頂点の B が真ん中。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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: ∠"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.9 Measuring Angles（Say It: angle ABC）・1.7（the middle letter should be the vertex）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem（∠A）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "angle a b c | angle abc | angle d e f | angle def | angle x y z | angle xyz | angle p q r | angle pqr | angle a b d | angle abd | angle d b c | angle dbc | angle b a c | angle bac | angle a c b | angle acb": 85
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.02",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "figure-and-measure-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "angle-bracket-vector",
    "latex": "\\langle a, b \\rangle",
    "spoken_en": [
      {
        "text": "the vector a, b",
        "register": "standard"
      },
      {
        "text": "the vector with components a and b",
        "register": "spoken"
      }
    ],
    "spoken_ja": "成分表示 (a, b) ／ ベクトル a b",
    "name_en": "component form of a vector",
    "name_ja": "成分表示",
    "category": "vectors",
    "term_ref": "component-form",
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "notes": [
      "⟨a, b⟩ は the vector a, b と読み、the vector one, zero, minus one のように成分を続けて言う。the vector with components a and b とも言う。OpenStax Calculus Volume 3（2.1 Vectors in the Plane）は成分を山かっこ（angle brackets）でくくって書く。山かっこは読まない（用例コーパスと参照に angle bracket と読む例は出てこない）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.1 Vectors in the Plane（component form、angle brackets）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "8.8 Vectors（component form ⟨a, b⟩）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "the vector zero | the vector one | the vector two | the vector three | the vector four | the vector five | the vector six | the vector seven | the vector eight | the vector nine | the vector minus | the vector negative": 42,
        "with components *": 4
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "approaches-arrow",
    "latex": "x \\to a",
    "spoken_en": [
      {
        "text": "x approaches a",
        "register": "standard"
      },
      {
        "text": "x goes to a",
        "register": "standard"
      },
      {
        "text": "x tends to a",
        "register": "standard"
      }
    ],
    "spoken_ja": "x が a に近づく ／ x → a",
    "name_en": "approaches arrow",
    "name_ja": "近づく（→）",
    "category": "limits",
    "term_ref": "approaches",
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "x → a は x approaches a と読む。x goes to a とも言い、大学の講義（MIT OCW）では goes to が多い。",
      "→ は写像の矢印（f: A → B）や条件文（p → q）にも使い、そのときは approaches と読まない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "Basic Classes of Functions（f(x) approaches two as x goes to infinity）"
      },
      {
        "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"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1（P → Q is read if P then Q）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "as * goes to *": 396,
        "as * tends to *": 14,
        "as * approaches *": 1047
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "approximately-equal-sign",
    "latex": "a \\approx b",
    "spoken_en": [
      {
        "text": "a is approximately b",
        "register": "standard"
      },
      {
        "text": "a is approximately equal to b",
        "register": "spoken"
      }
    ],
    "spoken_ja": "a は b にほぼ等しい",
    "name_en": "approximately equal sign",
    "name_ja": "ほぼ等しい（≈）",
    "category": "arithmetic",
    "term_ref": "approximate-value",
    "level": {
      "jp": [
        "数B",
        "数III"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Calculus I"
      ]
    },
    "notes": [
      "π ≈ 3.14 は pi is approximately three point one four と読む。is approximately equal to と equal to まで言う読みもある。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Simplify and Use Square Roots（≈ is read approximately）"
      },
      {
        "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 微分法の応用（Δy ≒ f′(x)Δx）・数学B 統計的な推測（≒）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "is approximately !equal !normal !the !going": 151,
        "is approximately equal to * | approximately equals *": 49
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "approximately-equal-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "arc-ab",
    "latex": "\\overset{\\frown}{AB}",
    "spoken_en": [
      {
        "text": "arc AB",
        "register": "standard"
      },
      {
        "text": "the measure of arc AB",
        "register": "standard"
      }
    ],
    "spoken_ja": "弧 AB",
    "name_en": "arc notation",
    "name_ja": "弧の記号（AB⌒）",
    "category": "geometry",
    "term_ref": "arc",
    "related": [
      "arc-measure-notation"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry"
      ]
    },
    "notes": [
      "弧 AB（AB の上の弧の印）は arc AB と読む。",
      "度数を言うときは the measure of arc AB（CK-12 Geometry 6.9 は m を付けて書く。arc-measure-notation）。"
    ],
    "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（弧の記号 152 件、度数は m を付けて書く）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「円 (数学)」",
        "url": "https://ja.wikipedia.org/wiki/%E5%86%86_%28%E6%95%B0%E5%AD%A6%29",
        "note": "弧AB（⌒ を AB の上にかぶせて書く）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 3 フレーズの前の修正」の 2）。arc AB ／ the measure of arc AB。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "argument-arg-z",
    "latex": "\\arg z",
    "spoken_en": [
      {
        "text": "the argument of z",
        "register": "standard"
      },
      {
        "text": "arg z",
        "register": "standard"
      }
    ],
    "spoken_ja": "z の偏角 ／ アーグ z",
    "name_en": "argument of a complex number",
    "name_ja": "偏角（arg z）",
    "category": "complex-numbers",
    "term_ref": "argument-of-a-complex-number",
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "notes": [
      "arg z は the argument of z、短く arg z と読む。",
      "Nicholson の付録 A（複素数）は偏角を θ = arg z と書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "8.5 Polar Form of Complex Numbers（θ を argument と呼ぶ）"
      },
      {
        "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: θ = arg z"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the argument of z | the argument of a complex | the argument of w": 1
      },
      "sources": [
        "mit-18.01"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 3 フレーズの前の修正」の 2）。the argument of z ／ arg z。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "base-n-subscript",
    "latex": "1011_{(2)},\\ 1011_2",
    "spoken_en": [
      {
        "text": "one zero one one in binary",
        "register": "standard"
      },
      {
        "text": "one zero one one base two",
        "register": "standard"
      }
    ],
    "spoken_ja": "2 進法の 1011 ／ いち れい いち いち（2）",
    "name_en": "base-n numeral",
    "name_ja": "n 進法の表し方",
    "category": "number-theory",
    "term_ref": "base-n",
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "1011₂ は数字を 1 つずつ読んで one zero one one in binary と言う（one zero one one base two とも言う）。",
      "one thousand eleven とは読まない（10 進法の数ではない）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.2 Combining Outcomes の Practice Problems 3（decimal form (or base 10)、hexadecimal or base 16。level.us の Discrete Math）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "one in binary | zero in binary": 3
      },
      "sources": [
        "khan-middle"
      ],
      "counted": "2026-09-28"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた。one zero one one in binary ／ one zero one one base two（2026-09-28。docs/DECISIONS.md「Phase 5 監査（セッション 11）の前の決定」の 2）。register は主張しない。",
        "raised": "2026-09-28"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "because-sign",
    "latex": "\\because",
    "spoken_en": [
      {
        "text": "because",
        "register": "standard"
      },
      {
        "text": "since",
        "register": "spoken"
      }
    ],
    "spoken_ja": "なぜならば",
    "name_en": "because sign",
    "name_ja": "なぜならば（∵）",
    "category": "logic",
    "term_ref": null,
    "related": [
      "therefore-because-symbols"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": []
    },
    "notes": [
      "∵ は because か since と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Therefore_sign",
        "note": "節 Similar signs（∵ is the because sign）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「∵」",
        "url": "https://ja.wikipedia.org/wiki/%E2%88%B5",
        "note": "数学などで「なぜならば」と読む記号"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "because": 9247,
        "since": 1182
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "beta",
    "latex": "\\beta",
    "spoken_en": [
      {
        "text": "beta",
        "register": "standard"
      }
    ],
    "spoken_ja": "ベータ",
    "name_en": "beta",
    "name_ja": "ギリシャ文字 β",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "数I",
        "数II"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Intro Statistics"
      ]
    },
    "notes": [
      "β は beta と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles・10.1 Non-right Triangles: Law of Sines（angle β (beta)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "5.1 Angles（ギリシャ文字の表: β beta）・8.1 Non-right Triangles: Law of Sines（angle β (beta)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "9.2 Outcomes and the Type I and Type II Errors（denoted by the Greek letter beta, β）・付録 Mathematical Phrases, Symbols, and Formulas（β、beta）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Β」",
        "url": "https://ja.wikipedia.org/wiki/%CE%92",
        "note": "ベータ（英: beta）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Geometry 4.6・7.4・7.6（β で角を表す）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "beta": 167
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "binomial-distribution-b",
    "latex": "B(n, p)",
    "spoken_en": [
      {
        "text": "the binomial distribution with n trials and success probability p",
        "register": "standard"
      }
    ],
    "spoken_ja": "二項分布 B(n, p)",
    "name_en": "binomial distribution notation",
    "name_ja": "二項分布の記号（B(n, p)）",
    "category": "probability-statistics",
    "term_ref": "binomial-distribution",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "B(n, p) は the binomial distribution with n trials and success probability p と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "4.3 Binomial Distribution（X ~ B(n, p) の記法）"
      },
      {
        "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": "巻末の公式集 Formulas for AP Statistics（If X has a binomial distribution with parameters n and p）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-26。docs/DECISIONS.md「Phase 3 フレーズ 2 の前の修正」の 4）。the binomial distribution with n trials and success probability p。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "cardinality-notation",
    "latex": "n(A),\\ |A|",
    "spoken_en": [
      {
        "text": "the cardinality of A",
        "register": "standard"
      },
      {
        "text": "the number of elements in A",
        "register": "spoken"
      }
    ],
    "spoken_ja": "A の要素の個数 ／ n(A)",
    "name_en": "number of elements (cardinality)",
    "name_ja": "要素の個数",
    "category": "sets",
    "term_ref": "number-of-elements",
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Precalculus",
        "Discrete Math"
      ]
    },
    "notes": [
      "|A| も n(A) も the number of elements in A と読める。Levin（1.5・5.1）は |A| を the cardinality of A（the number of elements in A）と説明する。",
      "OpenStax Algebra and Trigonometry（13.7）は n(E) を the number of elements in E として確率を n(E)/n(S) と書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.5・5.1（|A| denotes the cardinality of A）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.7 Probability（n(E) = number of elements in E）"
      },
      {
        "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（n(A∪B) = n(A) + n(B) − n(A∩B)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "the cardinality of *": 57,
        "the number of elements in *": 4
      },
      "sources": [
        "mit-6.042"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "cartesian-product-cross",
    "latex": "A \\times B",
    "spoken_en": [
      {
        "text": "the Cartesian product of A and B",
        "register": "standard"
      }
    ],
    "spoken_ja": "A と B の直積",
    "name_en": "Cartesian product",
    "name_ja": "直積（×）",
    "category": "sets",
    "term_ref": "cartesian-product",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "A × B は the Cartesian product of A and B と読む。Levin は A × B をこの名前で定義する（5.1 Sets）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets ほか。A × B を the Cartesian product of A and B と書く（5 件。読みを決めた参照）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「×」",
        "url": "https://ja.wikipedia.org/wiki/%C3%97",
        "note": "集合の直積"
      }
    ],
    "confidence": "verified",
    "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: 対象外）。読みは Levin の呼び方 cartesian product（5.1 Sets・2.6 Relations and Graphs・3.7 Applications to Probability）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "ceiling-brackets",
    "latex": "\\lceil x \\rceil",
    "spoken_en": [
      {
        "text": "the ceiling of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "x の天井関数",
    "name_en": "ceiling function",
    "name_ja": "天井関数（⌈x⌉）",
    "category": "functions",
    "term_ref": "ceiling-function",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "notes": [
      "⌈x⌉ は the ceiling of x と読む。意味は the smallest integer greater than or equal to x（x 以上の最小の整数）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "極限の練習問題（f(x) is the ceiling 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": "天井関数"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "ceiling of *": 4
      },
      "sources": [
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 3 フレーズの前の修正」の 2）。the ceiling of x。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "chi-square-symbol",
    "latex": "\\chi^2",
    "spoken_en": [
      {
        "text": "chi-square",
        "register": "standard"
      }
    ],
    "spoken_ja": "カイ 2 乗",
    "name_en": "chi-square",
    "name_ja": "カイ 2 乗（χ²）",
    "category": "probability-statistics",
    "term_ref": "chi-square-test",
    "level": {
      "jp": [],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "χ² は chi-square と読む。chi は「カイ」と発音する（Merriam-Webster: ˈkī）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "Facts About the Chi-Square Distribution ほか本文 101 件。X ~ χ²_df の記法を the chi-square distribution と呼ぶ（読みを裏づける参照）"
      },
      {
        "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。chi-square（読みを決めた参照）"
      },
      {
        "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）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "chi-square | chi square": 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: 対象外）。読みは AP Statistics の CED の呼び方 chi-square | chi square（topic 3.14・topic 3.15）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-26",
      "human": null
    }
  },
  {
    "id": "circle-o",
    "latex": "\\odot O",
    "spoken_en": [
      {
        "text": "circle O",
        "register": "standard"
      }
    ],
    "spoken_ja": "円 O",
    "name_en": "circle notation",
    "name_ja": "円の記号（⊙O）",
    "category": "geometry",
    "term_ref": "circle",
    "level": {
      "jp": [],
      "us": [
        "Geometry"
      ]
    },
    "notes": [
      "⊙O は circle O と読む。"
    ],
    "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 3.1・7.13（circle O の形。IM は ⊙ を使わない）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "6.2 Identify Circle Components（We call this circle circle A, and it is labeled ⊙A。⊙ を定義して読む）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 7 3.2・3.3、Grade 8 1.12・2.2・5.13（circle C の形）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "circle o | circle p | circle c | circle q": 2
      },
      "sources": [
        "khan-algebra",
        "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 2 件（10 未満） ／ 書: 対象外）。読みは IM の呼び方 circle o | circle p | circle c | circle q（Grade 8 2.2 Circular Grid・Geometry 3.1 Scale Drawings・Geometry 7.13 Using Radians・計 11 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "closed-interval-brackets",
    "latex": "[a, b]",
    "spoken_en": [
      {
        "text": "the closed interval from a to b",
        "register": "standard"
      },
      {
        "text": "the closed interval a, b",
        "register": "spoken"
      }
    ],
    "spoken_ja": "閉区間 a b ／ a 以上 b 以下",
    "name_en": "closed interval",
    "name_ja": "閉区間",
    "category": "sets",
    "term_ref": "closed-interval",
    "related": [
      "interval-notation-vs-inequalities"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "[a, b] は the closed interval from a to b と読む（the closed interval a, b とも）。on the closed interval from zero to two のように使う。",
      "端点を含むので角かっこ（brackets）。"
    ],
    "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 1.16（closed interval）・Topic 5.5（closed interval）"
      },
      {
        "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（閉区間[a,b]において）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "3.6 Graphs of Functions（The domain is [−3, 3]。level.us の Algebra 2）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "1.2 Domain and Range（In interval notation, the domain is [1973, 2008]。level.us の Precalculus）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "4.3 Maxima and Minima・4.4 The Mean Value Theorem（the closed interval [a, b]。level.us の Calculus I）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "closed interval from * | closed interval, from *": 45,
        "closed interval a, b | closed interval a b | closed interval a and b | closed interval a to b": 4
      },
      "sources": [
        "khan-ap-calc",
        "yt:organicchem"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "combination-ncr",
    "latex": "{}_n\\mathrm{C}_r = \\binom{n}{r}",
    "spoken_en": [
      {
        "text": "n choose r",
        "register": "standard"
      }
    ],
    "spoken_ja": "エヌ シー アール",
    "name_en": "binomial coefficient (combination)",
    "name_ja": "組合せの数（ₙCᵣ）",
    "category": "probability-statistics",
    "term_ref": "combination",
    "related": [
      "combination-permutation-notation"
    ],
    "level": {
      "jp": [
        "数A",
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Discrete Math"
      ]
    },
    "notes": [
      "ₙCᵣ・C(n, r)・縦に並べた二項係数はどれも n choose r と読む（5 choose 2 のように数を入れて読む）。書き方の違いは combination-permutation-notation。",
      "二項定理の一般項 ₙCᵣ aⁿ⁻ʳbʳ でも同じ読み方をする。"
    ],
    "tts_text": "n choose r",
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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A 場合の数と確率: nCr"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.4 Binomial Theorem（We read (n r) as n choose r）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "11.5 Counting Principles・11.6 Binomial Theorem（C(n, r)、nCr、(n r)）"
      },
      {
        "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（(n k) を n choose k と読む）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-24",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "n choose *": 31
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    }
  },
  {
    "id": "common-log-notation",
    "latex": "\\log x,\\ \\log_{10} x",
    "spoken_en": [
      {
        "text": "log of x",
        "register": "standard"
      },
      {
        "text": "log base ten of x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ログ x ／ 常用対数 log₁₀ x",
    "name_en": "common logarithm",
    "name_ja": "常用対数",
    "category": "logarithms",
    "term_ref": "common-logarithm",
    "related": [
      "log-and-ln"
    ],
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "log x は log of x（log x）と読む。底を言うときは log base ten of x。",
      "底を省いた log x の底は、慣習差 log-and-ln を参照。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "4.3 Logarithmic Functions（log(x) is read log base 10 of x）"
      },
      {
        "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 4.9 What is a Logarithm?（read the log of 250、the log, base 10, of 250）"
      },
      {
        "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II 指数関数・対数関数: 常用対数"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "!natural log of *": 232,
        "log base ten": 67
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "complement-notation",
    "latex": "\\bar{A},\\ A^c,\\ A'",
    "spoken_en": [
      {
        "text": "the complement of A",
        "register": "standard"
      }
    ],
    "spoken_ja": "A バー ／ A の補集合",
    "name_en": "complement",
    "name_ja": "補集合",
    "category": "sets",
    "term_ref": "complement",
    "related": [
      "complement-bar-vs-c-prime"
    ],
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Discrete Math"
      ]
    },
    "notes": [
      "補集合・余事象は the complement of A と読む。A′ は OpenStax Introductory Statistics（3.1 Terminology）が A prime と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（The complement of event A is denoted A′、read A prime）"
      },
      {
        "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 2.4（complement。E′・Ē・E^C の書き方）"
      },
      {
        "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（Ā（A の補集合））・数学A 場合の数と確率（余事象）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets（the complement of A）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "the complement of *": 12
      },
      "sources": [
        "mit-6.042",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "complex-a-plus-bi",
    "latex": "a + bi",
    "spoken_en": [
      {
        "text": "a plus b i",
        "register": "standard"
      }
    ],
    "spoken_ja": "a たす b i",
    "name_en": "complex number a + bi",
    "name_ja": "複素数（a + bi）",
    "category": "complex-numbers",
    "term_ref": "complex-number",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "a + bi は a plus b i と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "2.4 Complex Numbers（a + bi）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "3.1 Complex Numbers（a + bi）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "plus b i | plus bi | minus b i | minus bi": 40
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "complex-conjugate-bar",
    "latex": "\\bar{z}",
    "spoken_en": [
      {
        "text": "z bar",
        "register": "standard"
      },
      {
        "text": "the complex conjugate of z",
        "register": "standard"
      }
    ],
    "spoken_ja": "z バー ／ z の共役複素数",
    "name_en": "complex conjugate",
    "name_ja": "共役複素数（z̄）",
    "category": "complex-numbers",
    "term_ref": "complex-conjugate",
    "level": {
      "jp": [
        "数II",
        "数C"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "notes": [
      "z̄ は z bar と読む（文字の上の横線が共役を表す）。the complex conjugate of z とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "2.4 Complex Numbers（the complex conjugate of a + bi）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "3.1 Complex Numbers（the complex conjugate of a + bi）"
      },
      {
        "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（z̄ を the conjugate of z と定義）・8.7 Complex Matrices"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "8.8 Use the Complex Number System（complex conjugate。level.us の Algebra 2）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "z bar | z-bar | w bar | lambda bar | c1 bar | c2 bar | z1 bar | z2 bar | zn bar": 17,
        "the complex conjugate of *": 9
      },
      "sources": [
        "mit-18.03",
        "mit-18.06"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "complex-modulus-bars",
    "latex": "|z|",
    "spoken_en": [
      {
        "text": "the absolute value of z",
        "register": "standard"
      },
      {
        "text": "the modulus of z",
        "register": "spoken"
      }
    ],
    "spoken_ja": "z の絶対値",
    "name_en": "modulus of a complex number",
    "name_ja": "複素数の絶対値（|z|）",
    "category": "complex-numbers",
    "term_ref": "modulus-of-a-complex-number",
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "notes": [
      "複素数の |z| は the absolute value of z と読む。the modulus of z とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "8.5 Polar Form of Complex Numbers（|z| を the absolute value of z と定義。modulus とも呼ぶ。読みを決めた参照）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "10.5 Polar Form of Complex Numbers（the absolute value of z、modulus）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the modulus of *": 4,
        "the absolute value of z | the absolute value of a complex": 1
      },
      "sources": [
        "mit-18.03",
        "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: 対象外）。読みは OpenStax の呼び方 the absolute value of z | the absolute value of a complex（本文 12 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "complex-numbers-symbol",
    "latex": "\\mathbb{C}",
    "spoken_en": [
      {
        "text": "the complex numbers",
        "register": "standard"
      }
    ],
    "spoken_ja": "複素数全体の集合 C",
    "name_en": "complex numbers",
    "name_ja": "複素数全体の集合（ℂ）",
    "category": "sets",
    "term_ref": "complex-number",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "notes": [
      "ℂ は the complex numbers と読む。"
    ],
    "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": "8.7 Complex Matrices（The set of complex numbers is denoted ℂ）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the complex numbers | the set of complex numbers": 18
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "composition-notation",
    "latex": "(f \\circ g)(x),\\ f(g(x))",
    "spoken_en": [
      {
        "text": "f of g of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "f マル g (x)",
    "name_en": "composition of functions",
    "name_ja": "合成関数の記号",
    "category": "functions",
    "term_ref": "composite-function",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB"
      ]
    },
    "notes": [
      "f(g(x)) も (f ∘ g)(x) も f of g of x と読む。OpenStax Algebra and Trigonometry（3.4 Composition of Functions）は (f ∘ g)(x) を f composed with g at x、f(g(x)) を f of g of x と読む。",
      "内側の g(x) を先に計算する（the inside function first）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "3.4 Composition of Functions（f composed with g at x、f of g of x）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "f of g of *": 61
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "compound-inequality",
    "latex": "a < x < b",
    "spoken_en": [
      {
        "text": "x is between a and b",
        "register": "standard"
      },
      {
        "text": "a is less than x is less than b",
        "register": "standard"
      },
      {
        "text": "x is greater than a and less than b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a 小なり x 小なり b ／ x は a より大きく b より小さい",
    "name_en": "compound inequality",
    "name_ja": "連なった不等式",
    "category": "arithmetic",
    "term_ref": "inequality",
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Algebra 1"
      ]
    },
    "notes": [
      "a < x < b は左から a is less than x is less than b と読むか、x を主語にして x is between a and b と言う。",
      "端を含むか（≤）どうかが大事なときは、between だけでは伝わらないので記号どおりに読む（between a and b, inclusive とも言う）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Solve Compound Inequalities（double inequality、x is between 3 and 8）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "is between * and *": 119,
        "is less than * is less than *": 88,
        "is greater than * and less than *": 10
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "conditional-probability-bar",
    "latex": "P(A \\mid B)",
    "spoken_en": [
      {
        "text": "the probability of A given B",
        "register": "standard"
      }
    ],
    "spoken_ja": "B が起こったときの A の条件付き確率",
    "name_en": "conditional probability",
    "name_ja": "条件付き確率（P(A | B)）",
    "category": "probability-statistics",
    "term_ref": "conditional-probability",
    "related": [
      "conditional-probability-notation"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics",
        "Discrete Math"
      ]
    },
    "notes": [
      "P(A | B) は the probability of A given B と読む。縦線を given と読み、条件は縦線の右。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（The conditional probability of A given B is written P(A|B)）・付録 Mathematical Phrases, Symbols, and Formulas（P(A | B)、probability of A given 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)）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "3.7 Probability（Definition 3.7.14: the conditional probability of A given B, P(A|B)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the probability of * given *": 68
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "conditional-probability-subscript-jp",
    "latex": "P_A(B)",
    "spoken_en": [
      {
        "text": "the probability of B given A",
        "register": "standard"
      }
    ],
    "spoken_ja": "A が起こったときの B が起こる条件付き確率 ／ P A B",
    "name_en": "conditional probability (Japanese notation)",
    "name_ja": "条件付き確率（日本の P_A(B)）",
    "category": "probability-statistics",
    "term_ref": "conditional-probability",
    "related": [
      "conditional-probability-notation"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "notes": [
      "P_A(B) は P(B | A) と同じ意味で、英語では the probability of B given A と読む（条件の A を後に言う）。書き方の違いは conditional-probability-notation。"
    ],
    "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 場合の数と確率 条件付き確率: P(A∩B) = P(A)P_A(B) = P(B)P_B(A)（p.94）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the probability of * given *": 68
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "congruence-mod",
    "latex": "a \\equiv b \\pmod{n}",
    "spoken_en": [
      {
        "text": "a is congruent to b modulo n",
        "register": "standard"
      }
    ],
    "spoken_ja": "a と b は n を法として合同 ／ a 合同 b モッド n",
    "name_en": "congruence modulo n",
    "name_ja": "合同式",
    "category": "number-theory",
    "term_ref": "congruence-modulo-n",
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "a ≡ b (mod n) は a is congruent to b modulo n と読む。書くときは mod と略す。",
      "≡ は中学校学習指導要領〔用語・記号〕（中2 B 図形）では図形の合同の記号（congruent-sign）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction (4th edition)",
        "url": "https://discrete.openmathbooks.org/pdfs/dmoi4.pdf",
        "note": "6.2 Introduction to Number Theory（a ≡ b (mod n) を a is congruent to b modulo n と読む。mod five とだけ読むこともある）"
      },
      {
        "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: ≡"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "congruent to * modulo *": 56
      },
      "sources": [
        "mit-6.042"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "congruent-sign",
    "latex": "\\triangle ABC \\equiv \\triangle DEF,\\ \\triangle ABC \\cong \\triangle DEF",
    "spoken_en": [
      {
        "text": "triangle ABC is congruent to triangle DEF",
        "register": "standard"
      }
    ],
    "spoken_ja": "三角形 ABC 合同 三角形 DEF",
    "name_en": "congruence sign",
    "name_ja": "合同の記号（日本は ≡、米国は ≅）",
    "category": "geometry",
    "term_ref": "congruent",
    "related": [
      "congruence-symbol"
    ],
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "notes": [
      "△ABC ≅ △DEF は triangle ABC is congruent to triangle DEF と読む。日本の記号 ≡ で書いた合同も、英語では同じく is congruent to と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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": "≅（414 件）。5.25 Corresponding Parts of Congruent Figures・4.10 Congruence Statements・4.12 Congruent Triangles ほかで … is congruent to …（35 件。読みを裏づける参照）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 1〜2 ほか。… is congruent to triangle …（69 件。読みを決めた参照）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-26",
    "evidence": {
      "spoken": {
        "is congruent to triangle *": 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: 対象外）。読みは IM の呼び方 is congruent to triangle *（Geometry 2.2 Congruent Parts, Part 2・Geometry 2.3 Congruent Triangles, Part 1・Geometry 2.6 Side-Angle-Side Triangle Congruence・計 69 件）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-26",
      "human": null
    }
  },
  {
    "id": "cosecant-of-theta",
    "latex": "\\csc\\theta",
    "spoken_en": [
      {
        "text": "cosecant of theta",
        "register": "standard"
      },
      {
        "text": "cosecant theta",
        "register": "standard"
      }
    ],
    "spoken_ja": "コセカント シータ",
    "name_en": "cosecant",
    "name_ja": "コセカント",
    "category": "trigonometry",
    "term_ref": "cosecant",
    "related": [
      "reciprocal-trig-functions"
    ],
    "level": {
      "jp": [],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "csc θ は cosecant of theta か cosecant theta と読む。書くときは csc（英語版 Wikipedia「Trigonometric functions」は cosec という書き方も挙げる）。csc θ = 1/sin θ。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "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/Trigonometric_functions",
        "note": "記事「Trigonometric functions」: cosecant の略記は csc か cosec"
      },
      {
        "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": "読み（sin: サイン、cos: コサイン、tan: タンジェント、sec: セカント、cosec: コセカント、cot: コタンジェント）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "cosecant of *": 36,
        "cosecant theta | cosecant x | cosecant squared": 34
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "cosine-of-theta",
    "latex": "\\cos\\theta",
    "spoken_en": [
      {
        "text": "cosine of theta",
        "register": "standard"
      },
      {
        "text": "cosine theta",
        "register": "standard"
      },
      {
        "text": "the cosine of theta",
        "register": "standard"
      }
    ],
    "spoken_ja": "コサイン シータ",
    "name_en": "cosine",
    "name_ja": "コサイン",
    "category": "trigonometry",
    "term_ref": "cosine",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "cos θ は cosine of theta か cosine theta と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 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）〔用語・記号〕",
        "note": "数I 図形と計量: cos"
      },
      {
        "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": "読み（sin: サイン、cos: コサイン、tan: タンジェント、sec: セカント、cosec: コセカント、cot: コタンジェント）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "cosine theta | cosine x | cosine alpha | cosine two x": 594,
        "!the !inverse !arc cosine of *": 993,
        "the cosine of *": 144
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-10-09"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "cotangent-of-theta",
    "latex": "\\cot\\theta",
    "spoken_en": [
      {
        "text": "cotangent theta",
        "register": "standard"
      },
      {
        "text": "cotangent of theta",
        "register": "standard"
      }
    ],
    "spoken_ja": "コタンジェント シータ",
    "name_en": "cotangent",
    "name_ja": "コタンジェント",
    "category": "trigonometry",
    "term_ref": "cotangent",
    "related": [
      "reciprocal-trig-functions"
    ],
    "level": {
      "jp": [],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "cot θ は cotangent theta（cotangent of theta）と読む。cot θ = 1/tan θ = cos θ / sin θ。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "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": "読み（sin: サイン、cos: コサイン、tan: タンジェント、sec: セカント、cosec: コセカント、cot: コタンジェント）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "cotangent theta | cotangent x | cotangent squared": 29,
        "cotangent of *": 15
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "cross-product-cross",
    "latex": "\\vec{u} \\times \\vec{v}",
    "spoken_en": [
      {
        "text": "u cross v",
        "register": "standard"
      },
      {
        "text": "the cross product of u and v",
        "register": "spoken"
      }
    ],
    "spoken_ja": "u クロス v ／ u と v の外積",
    "name_en": "cross product",
    "name_ja": "外積（u⃗ × v⃗）",
    "category": "vectors",
    "term_ref": "cross-product",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "notes": [
      "u⃗ × v⃗ は u cross v か the cross product of u and v と読む。× を cross と読む。",
      "× は数の掛け算（times）と同じ形の記号。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.4 The Cross Product（the cross product of r and F: τ = r × F ほか。演習の解答の図の説明は u × v を u cross v と書く）"
      },
      {
        "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（Definition 4.8 Cross Product）・4.3 More on the Cross Product"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "the cross product of *": 9,
        "u cross v | a cross b | v cross w | v cross u | b cross a | u cross w | a cross c | b cross c | r cross v | v cross v | w cross v": 23
      },
      "sources": [
        "mit-18.02",
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "cube-root",
    "latex": "\\sqrt[3]{x}",
    "spoken_en": [
      {
        "text": "the cube root of x",
        "register": "standard"
      },
      {
        "text": "cube root of x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "3 乗根 x ／ x の 3 乗根",
    "name_en": "cube root",
    "name_ja": "3 乗根（立方根）",
    "category": "roots",
    "term_ref": "nth-root",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "notes": [
      "∛x は the cube root of x と読む。∛8 = 2 は the cube root of eight is two。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "8.1 Simplify Expressions with Roots（cube root）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "9.7 Higher Roots・9.8 Rational Exponents（cube root。level.us Algebra 1）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 8.12 Edge Lengths and Volumes（the cube root of 64）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「立方根」",
        "url": "https://ja.wikipedia.org/wiki/%E7%AB%8B%E6%96%B9%E6%A0%B9",
        "note": "記事名 立方根（三乗根ともいう）。name_ja"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "the cube root of *": 109,
        "!the cube root of *": 28
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "curl-del-cross",
    "latex": "\\nabla \\times \\mathbf{F},\\ \\operatorname{curl} \\mathbf{F}",
    "spoken_en": [
      {
        "text": "the curl of F",
        "register": "standard"
      },
      {
        "text": "curl F",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ローテーション F ／ rot F",
    "name_en": "curl",
    "name_ja": "回転（curl）",
    "category": "vectors",
    "term_ref": "curl-vector",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "∇ × F と curl F は the curl of F（curl F）と読む。",
      "日本語版 Wikipedia「回転 (ベクトル解析)」は rot F・curl F・∇ × F を挙げ、OpenStax Calculus Volume 3（6.5 Divergence and Curl）は curl F と ∇ × F を使う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "6.5 Divergence and Curl（curl F = ∇ × F）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「回転 (ベクトル解析)」",
        "url": "https://ja.wikipedia.org/wiki/%E5%9B%9E%E8%BB%A2_%28%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB%E8%A7%A3%E6%9E%90%29",
        "note": "記法（rot F・curl F・∇×F）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "curl of *": 50,
        "!the curl f | !the curl g | !the curl v | !the curl e | !the curl b": 10
      },
      "sources": [
        "mit-18.02"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "curly-braces",
    "latex": "\\{\\ \\}",
    "spoken_en": [
      {
        "text": "braces",
        "register": "standard"
      },
      {
        "text": "curly brackets",
        "register": "spoken"
      }
    ],
    "spoken_ja": "中かっこ",
    "name_en": "braces",
    "name_ja": "波かっこ（{ }）",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "中1",
        "数I"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "{ } は braces と呼び、curly brackets とも言う。集合を読むときは波かっこの名前を言わず、{1, 2, 3} は the set containing one, two, and three（roster-braces）、{x | x > 0} は the set of all x such that …（set-builder-braces）と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "1.1 Use the Language of Algebra（Grouping Symbols: braces { }）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.2 Use the Language of Algebra（Grouping Symbols: braces { }）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "1.1・2.7（braces are used to indicate a set）・3.2 Domain and Range（Use braces）。グループ化の記号と集合の { } を braces と呼ぶ（OpenStax 4 冊で 9 件。読みを決めた参照）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "2.1 Use the Language of Algebra（Common Grouping Symbols: braces { }）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「括弧」",
        "url": "https://ja.wikipedia.org/wiki/%E6%8B%AC%E5%BC%A7",
        "note": "波括弧（なみかっこ）。中括弧とも"
      },
      {
        "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 集合と命題（A={x|x>2}）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "curly braces | curly brace | curly brackets | curly bracket": 5
      },
      "sources": [
        "khan-algebra",
        "mit-18.03",
        "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 5 件（10 未満） ／ 書: 対象外）。読みは OpenStax の呼び方 !curly braces（本文 9 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "decimal-point",
    "latex": "3.14",
    "spoken_en": [
      {
        "text": "three point one four",
        "register": "standard"
      }
    ],
    "spoken_ja": "3 てん 1 4",
    "name_en": "decimal point",
    "name_ja": "小数点",
    "category": "arithmetic",
    "term_ref": "decimal",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "小数点は point と読み、小数点より下は 1 桁ずつ読む。3.14 は three point one four、0.5 は zero point five か point five。",
      "3.14 を three point fourteen とは読まない。",
      "OpenStax Prealgebra（Decimals）は 5.03 を five and three hundredths と位の名前でも読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Decimals（five and three hundredths）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 3.15・Grade 7 8.17 ほかの図の説明（zero point four nine）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "zero point * | one point * | two point * | three point * | four point * | five point * | six point * | seven point * | eight point * | nine point *": 331
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "degree-sign",
    "latex": "30^{\\circ}",
    "spoken_en": [
      {
        "text": "thirty degrees",
        "register": "standard"
      },
      {
        "text": "a thirty-degree angle",
        "register": "spoken"
      }
    ],
    "spoken_ja": "30 度",
    "name_en": "degree sign",
    "name_ja": "度（°）",
    "category": "geometry",
    "term_ref": "degree-measure",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "notes": [
      "30° は thirty degrees と読む。名詞の前では a thirty-degree angle のように degree と単数で言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.9 Measuring Angles・1.10 Classifying Angles（° の記号は degrees）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles（90 degrees = 90°）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* degrees !of !fahrenheit !celsius !f !c": 1185,
        "* degree angle": 95
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "delta-lowercase",
    "latex": "\\delta",
    "spoken_en": [
      {
        "text": "delta",
        "register": "standard"
      }
    ],
    "spoken_ja": "デルタ",
    "name_en": "delta (lowercase)",
    "name_ja": "ギリシャ文字 δ",
    "category": "notation-other",
    "term_ref": "epsilon-delta-definition",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "notes": [
      "δ は delta と読む。",
      "ε-δ 論法の δ。大文字の Δ（変化量 Δx）も delta と読む（Δx は delta x）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "2.5 The Precise Definition of a Limit（δ > 0）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "1.3 Rates of Change and Behavior of Graphs（The Greek letter Δ (delta) signifies the change in a quantity）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Δ」",
        "url": "https://ja.wikipedia.org/wiki/%CE%94",
        "note": "デルタ（英: delta）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "there exists a delta | find a delta | find you a delta | come up with a delta | epsilon delta | delta greater than zero | delta is greater than zero | delta neighborhood | within delta | pick a delta | choose a delta": 26
      },
      "sources": [
        "khan-ap-calc",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "delta-x",
    "latex": "\\Delta x,\\ \\dfrac{\\Delta y}{\\Delta x}",
    "spoken_en": [
      {
        "text": "delta x",
        "register": "standard"
      },
      {
        "text": "change in x",
        "register": "standard"
      }
    ],
    "spoken_ja": "x の増分",
    "name_en": "delta x (change in x)",
    "name_ja": "Δx（x の増分）",
    "category": "derivatives",
    "term_ref": "average-rate-of-change",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "Δx は delta x と読むか、意味を言って the change in x と言う。",
      "Δy/Δx は delta y over delta x、または the change in y over the change in x（平均変化率・傾き）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "Basic Classes of Functions（Δx・Δy denote the changes in x and y）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "Rates of Change and Behavior of Graphs（read delta-y over delta-x、the change in y divided by the change in x）"
      },
      {
        "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（x の増分 Δx、Δy ≒ f′(x)Δx）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "delta x | delta y | delta t": 834,
        "change in x | change in y | change in t": 744
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "derivative-evaluated-at",
    "latex": "\\left.\\dfrac{dy}{dx}\\right|_{x = a}",
    "spoken_en": [
      {
        "text": "d y d x evaluated at x equals a",
        "register": "standard"
      },
      {
        "text": "d y d x at x equals a",
        "register": "spoken"
      }
    ],
    "spoken_ja": "x = a における dy/dx の値",
    "name_en": "derivative evaluated at a point",
    "name_ja": "x = a での微分係数",
    "category": "derivatives",
    "term_ref": "derivative-at-a-point",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "縦線と下の x = a は evaluated at x equals a と読む。短く d y d x at x equals a とも言う。",
      "f′(a) と書けば f prime of a で同じ値。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Derivative as a Function（dy/dx |_{x=a} in place of f′(a)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "evaluated at *": 305,
        "dy dx at *": 3
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "derivative-leibniz",
    "latex": "\\frac{dy}{dx}",
    "spoken_en": [
      {
        "text": "d y d x",
        "register": "standard"
      },
      {
        "text": "d y over d x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ディーワイ ディーエックス",
    "name_en": "derivative (Leibniz notation)",
    "name_ja": "導関数（ライプニッツの記法）",
    "category": "derivatives",
    "term_ref": "leibniz-notation",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "dy/dx は over を入れずに d y d x と読む（d y over d x とも言う）。",
      "d/dx だけのときは d d x と読み、d/dx (x²) は the derivative of x squared（with respect to x）とも言う。",
      "日本語版 Wikipedia「微分の記法」も dy/dx を「d y d x」と読むのが一般的とする。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「微分の記法」",
        "url": "https://ja.wikipedia.org/wiki/%E5%BE%AE%E5%88%86%E3%81%AE%E8%A8%98%E6%B3%95",
        "note": "dy/dx は d y d x と読む"
      },
      {
        "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 2.2（dy/dx）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Derivative as a Function（Leibniz notation dy/dx）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "d y d x": 335,
        "d y over d x": 20
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "derivative-prime",
    "latex": "f'(x)",
    "spoken_en": [
      {
        "text": "f prime of x",
        "register": "standard"
      },
      {
        "text": "f prime",
        "register": "spoken"
      }
    ],
    "spoken_ja": "エフ ダッシュ エックス",
    "name_en": "derivative (prime notation)",
    "name_ja": "導関数（プライム記法）",
    "category": "derivatives",
    "term_ref": "prime-notation",
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "f′(x) は f prime of x と読む（′ は prime）。",
      "二階微分 f''(x) は 'f double prime of x'。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「プライム」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%97%E3%83%A9%E3%82%A4%E3%83%A0",
        "note": "ダッシュと呼ぶ"
      },
      {
        "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 2.2（f′(x)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Derivative as a Function（f′(x)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "f prime of x": 742
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "determinant-notation",
    "latex": "\\det A,\\ |A|",
    "spoken_en": [
      {
        "text": "the determinant of A",
        "register": "standard"
      }
    ],
    "spoken_ja": "行列式 A ／ デターミナント A",
    "name_en": "determinant",
    "name_ja": "行列式（det A、|A|）",
    "category": "matrices",
    "term_ref": "determinant",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "notes": [
      "det A と |A| はどちらも the determinant of A と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "9.8 Solving Systems with Cramer's Rule（det(A)、|A|）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.8 Solving Systems with Cramer's Rule（det(A)、|A|）"
      },
      {
        "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 ほか"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the determinant of *": 201
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "df-degrees-of-freedom",
    "latex": "\\text{df}",
    "spoken_en": [
      {
        "text": "degrees of freedom",
        "register": "standard"
      }
    ],
    "spoken_ja": "自由度",
    "name_en": "degrees of freedom (df)",
    "name_ja": "自由度（df）",
    "category": "probability-statistics",
    "term_ref": "degrees-of-freedom",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "df は degrees of freedom の略で、名前のまま読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "8.2 A Single Population Mean using the Student t Distribution（df = n – 1）・付録 Mathematical Phrases, Symbols, and Formulas（df、degrees of freedom）"
      },
      {
        "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（degrees of freedom、df）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "degrees of freedom": 41
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.01",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "difference-quotient-limit",
    "latex": "\\lim_{h \\to 0} \\dfrac{f(x + h) - f(x)}{h}",
    "spoken_en": [
      {
        "text": "the limit as h approaches zero of f of x plus h minus f of x, all over h",
        "register": "standard"
      },
      {
        "text": "the limit as h goes to zero of the quantity f of x plus h minus f of x over h",
        "register": "spoken"
      }
    ],
    "spoken_ja": "リミット h → 0 h 分の f(x + h) − f(x)",
    "name_en": "limit of the difference quotient",
    "name_ja": "導関数の定義の極限",
    "category": "derivatives",
    "term_ref": "limit-definition-of-the-derivative",
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "all over h で、分子の f(x + h) − f(x) 全体を h で割ることを示す。",
      "分数の部分だけを difference quotient と呼ぶ。"
    ],
    "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 2.2（f′(x) = lim_{h→0} (f(x + h) − f(x))/h）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the limit as h approaches zero of": 18,
        "the limit as h goes to zero of": 2
      },
      "sources": [
        "khan-ap-calc",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "directional-derivative-notation",
    "latex": "D_{\\mathbf{u}} f",
    "spoken_en": [
      {
        "text": "the directional derivative of f in the direction of u",
        "register": "standard"
      }
    ],
    "spoken_ja": "u 方向の方向微分",
    "name_en": "directional derivative",
    "name_ja": "方向微分",
    "category": "derivatives",
    "term_ref": "directional-derivative",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "D_u f は the directional derivative of f in the direction of u と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Directional Derivatives and the Gradient（the directional derivative of f in the direction of u）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「方向微分」",
        "url": "https://ja.wikipedia.org/wiki/%E6%96%B9%E5%90%91%E5%BE%AE%E5%88%86",
        "note": "方向微分"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "directional derivative": 24
      },
      "sources": [
        "mit-18.02",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "distributed-as-tilde",
    "latex": "X \\sim N(\\mu, \\sigma)",
    "spoken_en": [
      {
        "text": "X is normally distributed with mean mu and standard deviation sigma",
        "register": "standard"
      },
      {
        "text": "X follows a normal distribution",
        "register": "spoken"
      }
    ],
    "spoken_ja": "X は N(μ, σ) に従う",
    "name_en": "distributed as (tilde)",
    "name_ja": "確率変数が分布に従う（X ~ N(μ, σ)）",
    "category": "probability-statistics",
    "term_ref": "distribution",
    "related": [
      "normal-distribution-parameter"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "X ~ N(μ, σ) は X is normally distributed with mean mu and standard deviation sigma と読む。~ は分布に従うことを表す。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "6.1 The Standard Normal Distribution（X ~ N(μ, σ): X is a normally distributed random variable with mean μ and standard deviation σ）ほか"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "is normally distributed": 3,
        "follows a normal distribution | follows a * distribution": 3
      },
      "sources": [
        "khan-ap-stats",
        "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 6 件（10 未満） ／ 書: 対象外）。読みは OpenStax の呼び方 is normally distributed（本文 29 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "divergence-del-dot",
    "latex": "\\nabla \\cdot \\mathbf{F},\\ \\operatorname{div} \\mathbf{F}",
    "spoken_en": [
      {
        "text": "the divergence of F",
        "register": "standard"
      },
      {
        "text": "div F",
        "register": "spoken"
      },
      {
        "text": "del dot F",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ダイバージェンス F ／ ナブラ ドット F",
    "name_en": "divergence",
    "name_ja": "発散（div）",
    "category": "vectors",
    "term_ref": "divergence-vector",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "∇ · F と div F は the divergence of F と読む。div F はそのまま div F、∇ · F は del dot F とも読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "6.5 Divergence and Curl（div F = ∇ · F）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the divergence of *": 22,
        "div f | div g | div v | div e | div b": 8,
        "del dot *": 3
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "divides-bar",
    "latex": "a \\mid b",
    "spoken_en": [
      {
        "text": "a divides b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a は b を割り切る ／ a は b の約数",
    "name_en": "divides",
    "name_ja": "割り切る（|）",
    "category": "number-theory",
    "term_ref": "divisibility",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "a | b は a divides b と読む。a/b（a over b、分数）と取り違えない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction (4th edition)",
        "url": "https://discrete.openmathbooks.org/pdfs/dmoi4.pdf",
        "note": "6.2 Introduction to Number Theory（m | n を m divides n と読む）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* divides *": 138
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "division-sign",
    "latex": "a \\div b",
    "spoken_en": [
      {
        "text": "a divided by b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a わる b",
    "name_en": "division sign",
    "name_ja": "わる（÷）",
    "category": "arithmetic",
    "term_ref": "division",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "a ÷ b は a divided by b と読む。分数の形 a/b も divided by と読める（fraction-a-over-b）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Divide Whole Numbers（12 ÷ 4 is read twelve divided by four）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* divided by *": 2635
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "dne-abbreviation",
    "latex": "\\text{DNE}",
    "spoken_en": [
      {
        "text": "does not exist",
        "register": "standard"
      },
      {
        "text": "D N E",
        "register": "spoken"
      }
    ],
    "spoken_ja": "（極限が）存在しない",
    "name_en": "does not exist (DNE)",
    "name_ja": "存在しない（DNE）",
    "category": "limits",
    "term_ref": "the-limit-does-not-exist",
    "related": [
      "dne-in-answers"
    ],
    "level": {
      "jp": [],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "DNE は does not exist の略で、読むときもふつう does not exist と言う。文字どおり D N E と読むこともある。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Limit of a Function（abbreviate does not exist as DNE）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "does not exist | doesn't exist": 139,
        "d n e | dne": 6
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "dot-product-dot",
    "latex": "\\vec{a} \\cdot \\vec{b}",
    "spoken_en": [
      {
        "text": "a dot b",
        "register": "standard"
      },
      {
        "text": "a dotted with b",
        "register": "standard"
      },
      {
        "text": "the dot product of a and b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a ドット b ／ a と b の内積",
    "name_en": "dot product",
    "name_ja": "内積（a⃗ · b⃗）",
    "category": "vectors",
    "term_ref": "dot-product",
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "notes": [
      "a⃗ · b⃗ は a dot b、a dotted with b、the dot product of a and b と読む。点（·）を dot と読む。",
      "· は数の掛け算にも使う（times-sign）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.3 The Dot Product（u · v）"
      },
      {
        "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（Definition 4.4 Dot Product in R³: v · w）・2.2 Matrix-Vector Multiplication（Definition 2.6 Dot Product in Rⁿ）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "8.8 Vectors（dot product）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "u dot v | a dot b | v dot w | u dot w | v dot u | b dot a | v dot v | u dot u | a dot a | f dot dr | f dot n | r dot v | r dot r | v dot r | g dot dr | g dot n | f dot t | f dot ds | pq dot n | qp dot qr": 143,
        "!del dotted with": 55,
        "the dot product of *": 21
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "double-integral-sign",
    "latex": "\\iint_R f(x, y)\\,dA",
    "spoken_en": [
      {
        "text": "the double integral over R of f of x y d A",
        "register": "standard"
      }
    ],
    "spoken_ja": "二重積分 ∬ f(x, y) dA",
    "name_en": "double integral",
    "name_ja": "二重積分",
    "category": "integrals",
    "term_ref": "double-integral",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "∬_R f dA は the double integral over R of f d A と読む。dA は d A と 2 音で読み、面積要素を表す。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Double Integrals over Rectangular Regions（the double integral over a region R）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the double integral over *": 15
      },
      "sources": [
        "mit-18.02",
        "mit-18.03"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "e-constant",
    "latex": "e",
    "spoken_en": [
      {
        "text": "the number e",
        "register": "standard"
      }
    ],
    "spoken_ja": "e ／ ネイピア数",
    "name_en": "the number e",
    "name_ja": "自然対数の底 e",
    "category": "exponents",
    "term_ref": "base-of-the-natural-logarithm",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "定数 e は文字のまま e と読み、数であることを言うときは the number e。",
      "Euler's number とも呼ぶ（OpenStax Algebra and Trigonometry、IM Algebra 2 4.12）が、講義では the number e と言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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III 微分法: e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Exponential Functions（the natural base or Euler's number）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 4.12 The Number e（the number e、Euler's number）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「ネイピア数」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%8D%E3%82%A4%E3%83%94%E3%82%A2%E6%95%B0",
        "note": "オイラー数 (Euler's number) とも呼ぶ"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the number e": 39
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "e-to-the-x",
    "latex": "e^x",
    "spoken_en": [
      {
        "text": "e to the x",
        "register": "standard"
      },
      {
        "text": "e to the power of x",
        "register": "spoken"
      },
      {
        "text": "e to the x power",
        "register": "spoken"
      }
    ],
    "spoken_ja": "e の x 乗",
    "name_en": "e to the x",
    "name_ja": "e の x 乗",
    "category": "exponents",
    "term_ref": "exponential-function",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "eˣ は e to the x と読む。e^{2x} は e to the two x、e^{−x} は e to the negative x。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「指数関数」",
        "url": "https://ja.wikipedia.org/wiki/%E6%8C%87%E6%95%B0%E9%96%A2%E6%95%B0",
        "note": "ネイピア数 e を底とする関数 x ↦ eˣ"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 4.12 The Number e"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "e to the x !power": 579,
        "e to the power of *": 20,
        "e to the x power": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "element-of-sign",
    "latex": "a \\in A",
    "spoken_en": [
      {
        "text": "a is a member of A",
        "register": "standard"
      },
      {
        "text": "a is an element of A",
        "register": "spoken"
      },
      {
        "text": "a belongs to A",
        "register": "spoken"
      }
    ],
    "spoken_ja": "a は A に属する ／ a は A の要素",
    "name_en": "element of",
    "name_ja": "属する（∈）",
    "category": "sets",
    "term_ref": "element",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Calculus I",
        "Discrete Math",
        "Linear Algebra"
      ]
    },
    "notes": [
      "a ∈ A は a is a member of A、a is an element of A、a belongs to A と読む。短く a is in A とも言う。",
      "x ∈ ℝ は x is a real number と言いかえることもある。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.1 Review of Functions（x is an element of A）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "0.2（The ∈ symbol means is an element of）・5.1 Sets（2 is an element of the set {1, 2, 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": "6.4（a ∈ X、a is an element of the set X）"
      },
      {
        "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（a∈A）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "belongs to *": 7,
        "is a member of *": 12,
        "is an element of *": 4
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "ellipsis",
    "latex": "1, 2, 3, \\dots",
    "spoken_en": [
      {
        "text": "and so on",
        "register": "standard"
      },
      {
        "text": "all the way up to",
        "register": "spoken"
      },
      {
        "text": "dot dot dot",
        "register": "spoken"
      }
    ],
    "spoken_ja": "…（てんてんてん）",
    "name_en": "ellipsis",
    "name_ja": "…（省略記号）",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "中1",
        "数B"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "notes": [
      "1, 2, 3, … は one, two, three, and so on と読む。a₁, a₂, …, aₙ のように終わりがあるときは a sub one, a sub two, all the way up to a sub n。",
      "記号どおりに dot dot dot と読むこともある。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.1 Sequences（a₁, a₂, a₃, … の書き方）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.1 Sequences（three dots (ellipsis) は列が終わらないことを示す）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "1.1 Introduction to Whole Numbers（… is called an ellipsis、and so on）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.1 Introduction to Whole Numbers（ellipsis、and so on）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「省略記号」",
        "url": "https://ja.wikipedia.org/wiki/%E7%9C%81%E7%95%A5%E8%A8%98%E5%8F%B7",
        "note": "数学における省略記号（数式・数列・行列）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "and so on": 271,
        "all the way up to": 112,
        "dot dot dot": 15
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "empty-set-symbol",
    "latex": "\\emptyset",
    "spoken_en": [
      {
        "text": "the empty set",
        "register": "standard"
      }
    ],
    "spoken_ja": "空集合",
    "name_en": "empty set",
    "name_ja": "空集合（∅）",
    "category": "sets",
    "term_ref": "empty-set",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Calculus III",
        "Discrete Math"
      ]
    },
    "notes": [
      "∅ は the empty set と読む。ギリシャ文字の φ（phi）とは別の記号。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "5.1 Double Integrals over Rectangular Regions・5.2 Double Integrals over General Regions（S ∩ T = ∅）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets ほか。∅ を the empty set と書く（12 件。読みを決めた参照）"
      },
      {
        "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",
        "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1問 集合と命題（空集合を ∅ と表す）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the empty set | an empty set": 9
      },
      "sources": [
        "mit-18.06",
        "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 9 件（10 未満） ／ 書: 対象外）。読みは Levin の呼び方 the empty set | an empty set（5.1 Sets・1.5 Proofs about Discrete Structures・3.1 Pascal’s Arithmetical Triangle）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "epsilon",
    "latex": "\\varepsilon,\\ \\epsilon",
    "spoken_en": [
      {
        "text": "epsilon",
        "register": "standard"
      }
    ],
    "spoken_ja": "イプシロン",
    "name_en": "epsilon",
    "name_ja": "ギリシャ文字 ε",
    "category": "notation-other",
    "term_ref": "epsilon-delta-definition",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I",
        "Intro Statistics"
      ]
    },
    "notes": [
      "ε は epsilon と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "2.5 The Precise Definition of a Limit（ε > 0）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "12.3 The Regression Equation（ε = the Greek letter epsilon）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Ε」",
        "url": "https://ja.wikipedia.org/wiki/%CE%95",
        "note": "イプシロン（英: epsilon）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "epsilon": 209
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "equals-question-mark",
    "latex": "3(2) + 1 \\stackrel{?}{=} 7",
    "spoken_en": [
      {
        "text": "does three times two plus one equal seven",
        "register": "standard"
      }
    ],
    "spoken_ja": "（検算で使う等号）",
    "name_en": "equals sign with a question mark",
    "name_ja": "検算の等号（≟）",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Geometry"
      ]
    },
    "notes": [
      "OpenStax Prealgebra 2e（2.3）は ≟ を左辺と右辺が等しいかを問う記号とし、OpenStax の Prealgebra・Elementary Algebra・Intermediate Algebra は解の確かめ（Check）で使う。CK-12 Geometry は三平方の定理の確かめに使う。does … equal …? と疑問文で読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "2.1 ほか（解の検算（Check）で ≟ を使う）"
      },
      {
        "type": "reference",
        "title": "CK-12 Foundation, CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "4.27 The Pythagorean Theorem・4.30・4.33（a² + b² ≟ c² で確かめる）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "2.3 Solving Equations Using the Subtraction and Addition Properties of Equality（≟ asks whether the left side is equal to the right side）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "does * equal *": 189
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "equals-sign",
    "latex": "a = b",
    "spoken_en": [
      {
        "text": "a is equal to b",
        "register": "standard"
      },
      {
        "text": "a equals b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a イコール b ／ a は b に等しい",
    "name_en": "equal sign",
    "name_ja": "等号（＝）",
    "category": "arithmetic",
    "term_ref": "equal-sign",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "= は equals か is equal to と読む。3 + 4 = 7 は three plus four is seven とも言う。",
      "等号の名前は equal sign ／ equals sign。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Use the Language of Algebra（a = b is read a is equal to b、the equal sign）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「等号」",
        "url": "https://ja.wikipedia.org/wiki/%E7%AD%89%E5%8F%B7",
        "note": "読み（AイコールB、AはBに等しい）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* is equal to *": 9250,
        "* equals *": 5525
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "evaluation-bar",
    "latex": "F(x)\\Big|_a^b",
    "spoken_en": [
      {
        "text": "F of x evaluated from a to b",
        "register": "standard"
      },
      {
        "text": "F of x evaluated at b minus F of x evaluated at a",
        "register": "spoken"
      }
    ],
    "spoken_ja": "F(x) の a から b ／ [F(x)] a から b",
    "name_en": "evaluation bar",
    "name_ja": "定積分の計算の縦線",
    "category": "integrals",
    "term_ref": null,
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "F(x)|ₐᵇ は F of x evaluated from a to b と読み、F(b) − F(a) を計算する印。",
      "角かっこの [F(x)]ₐᵇ も読みは同じ。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "The Fundamental Theorem of Calculus（F(x)|ₐᵇ denotes F(b) − F(a)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "evaluated from * to *": 14,
        "evaluated at * minus * evaluated at": 5
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "exp-function",
    "latex": "\\exp x,\\ \\exp(x)",
    "spoken_en": [
      {
        "text": "exp of x",
        "register": "standard"
      },
      {
        "text": "the exponential of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "エクスポネンシャル x",
    "name_en": "exponential function (exp)",
    "name_ja": "exp",
    "category": "exponents",
    "term_ref": "exponential-function",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I",
        "Calculus II"
      ]
    },
    "notes": [
      "exp x は exp of x か the exponential of x と読み、中身は eˣ と同じ。指数が長い式のときに exp(…) と書く。",
      "exp of の読みはほとんどが 3Blue1Brown で、MIT OCW の講義は the exponential of と言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "Integrals, Exponential Functions, and Logarithms（exp x）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "Integrals, Exponential Functions, and Logarithms（exp x）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「指数関数」",
        "url": "https://ja.wikipedia.org/wiki/%E6%8C%87%E6%95%B0%E9%96%A2%E6%95%B0",
        "note": "exp x のようにも書く"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the exponential of *": 17,
        "exp of *": 48
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "expected-value-e-of-x",
    "latex": "E(X),\\ \\mu_X",
    "spoken_en": [
      {
        "text": "the expected value of X",
        "register": "standard"
      },
      {
        "text": "the mean of X",
        "register": "spoken"
      }
    ],
    "spoken_ja": "X の期待値 ／ E(X)",
    "name_en": "expected value",
    "name_ja": "期待値（E(X)、μ_X）",
    "category": "probability-statistics",
    "term_ref": "expected-value",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "E(X) は the expected value of X と読む。AP Statistics の CED は確率変数の平均を μ_X とも書く。"
    ],
    "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 2.9（expected value、E(X)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "5.2 Double Integrals over General Regions（expected values E(X) and E(Y)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "付録 Mathematical Phrases, Symbols, and Formulas（μX、mean of X）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the expected value of *": 211,
        "the mean of x | the mean of the random variable": 13
      },
      "sources": [
        "khan-ap-stats",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "expression-exponent",
    "latex": "2^{x+1}",
    "spoken_en": [
      {
        "text": "two to the x plus one",
        "register": "standard"
      },
      {
        "text": "two to the x plus one power",
        "register": "spoken"
      },
      {
        "text": "two raised to the quantity x plus one",
        "register": "spoken"
      }
    ],
    "spoken_ja": "2 の x+1 乗",
    "name_en": "exponent that is an expression",
    "name_ja": "指数が式のとき",
    "category": "exponents",
    "term_ref": "power",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "2^{x+1} はふつう two to the x plus one と読む。2ˣ + 1 と同じ音になるので、指数の終わりをはっきりさせたいときは power を付けて two to the x plus one power と言う。",
      "the quantity で指数をくくる読み（two raised to the quantity x plus one）はまれ。"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "to the n plus one !power | to the x plus one !power | to the k plus one !power | to the n minus one !power | to the x minus one !power": 203,
        "to the quantity *": 2,
        "to the n plus one power | to the x plus one power | to the k plus one power | to the n minus one power": 18
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "factorial-n",
    "latex": "n!",
    "spoken_en": [
      {
        "text": "n factorial",
        "register": "standard"
      }
    ],
    "spoken_ja": "n の階乗",
    "name_en": "factorial",
    "name_ja": "階乗（!）",
    "category": "probability-statistics",
    "term_ref": "factorial",
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus BC",
        "Intro Statistics",
        "Discrete Math"
      ]
    },
    "notes": [
      "n! は n factorial と読む。5! は five factorial。",
      "(n + 1)! は n plus one factorial と読み、n + 1! と区別したいときは the quantity n plus one, factorial と言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "11.1 Sequences and Their Notations（n factorial, written as n!）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "12.1 Sequences（5!, read five factorial）"
      },
      {
        "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A 場合の数と確率: n!"
      },
      {
        "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（n! を n factorial と読む）"
      },
      {
        "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（n!）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* factorial": 542
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "floor-brackets",
    "latex": "\\lfloor x \\rfloor",
    "spoken_en": [
      {
        "text": "the floor of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "x の床関数",
    "name_en": "floor function",
    "name_ja": "床関数（⌊x⌋）",
    "category": "functions",
    "term_ref": "floor-function",
    "related": [
      "floor-function-notation",
      "gauss-bracket-jp"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I",
        "Calculus II"
      ]
    },
    "notes": [
      "⌊x⌋ は the floor of x と読む。意味を言うと the greatest integer less than or equal to x（x を超えない最大の整数）。",
      "天井関数 ⌈x⌉ は the ceiling of x（ceiling-brackets）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "4.4 The Mean Value Theorem の練習問題（⌊x⌋、This is called the 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": "床関数"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "floor of x | floor of n | floor of one | floor of two | floor of something": 4
      },
      "sources": [
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-27。docs/DECISIONS.md「Phase 5 監査（セッション 10）の前の決定」の 3）。the floor of x。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "for-all-quantifier",
    "latex": "\\forall x",
    "spoken_en": [
      {
        "text": "for all x",
        "register": "standard"
      },
      {
        "text": "for every x",
        "register": "spoken"
      },
      {
        "text": "for each x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "すべての x について",
    "name_en": "for all",
    "name_ja": "すべての（∀）",
    "category": "logic",
    "term_ref": "universal-quantifier",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "∀x は for all x と読む。for every x、for each x とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1 Mathematical Statements（The universal quantifier is written ∀ and is read, for all）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "for each x | for each n | for each t | for each k | for each epsilon": 12,
        "for every x | for every n | for every t | for every k | for every epsilon": 12,
        "for all x | for all n | for all t | for all k | for all epsilon": 110
      },
      "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",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "logic-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "fraction-a-over-b",
    "latex": "\\dfrac{a}{b}",
    "spoken_en": [
      {
        "text": "a over b",
        "register": "standard"
      },
      {
        "text": "a divided by b",
        "register": "spoken"
      }
    ],
    "spoken_ja": "b 分の a",
    "name_en": "fraction",
    "name_ja": "分数",
    "category": "fractions",
    "term_ref": "fraction",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "a/b は a over b と読む。日本語は分母から読むが、英語は分子から読む。",
      "分子・分母が式のときは all over や the quantity でまとまりを示す（rational-expression-fraction）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.2 Use the Language of Algebra の表（a ÷ b, a/b — Say: a divided by b）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「分数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%88%86%E6%95%B0",
        "note": "読み（d 分の n、英語は分子から）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "* over *": 20456,
        "* divided by *": 2635
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "fractional-exponent",
    "latex": "a^{1/2},\\ a^{1/n},\\ a^{m/n}",
    "spoken_en": [
      {
        "text": "a to the one half",
        "register": "standard"
      },
      {
        "text": "a to the m over n",
        "register": "standard"
      }
    ],
    "spoken_ja": "a の n 分の 1 乗 ／ a の n 分の m 乗",
    "name_en": "rational exponent",
    "name_ja": "分数の指数",
    "category": "exponents",
    "term_ref": "rational-exponent",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "a^{1/2} は a to the one half、a^{1/n} は a to the one over n、a^{m/n} は a to the m over n と読む。",
      "a^{1/n} は ⁿ√a と同じ値（OpenStax Algebra and Trigonometry 1.3「Radicals and Rational Exponents」）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "1.3 Radicals and Rational Exponents（a^{1/n} = ⁿ√a）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "to the one half | to the one third | to the two thirds | to the three halves": 67,
        "to the one over * | to the two over * | to the three over * | to the m over n | to the p over q | to the r over s | to the a over b": 54
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "function-f-of-x",
    "latex": "f(x)",
    "spoken_en": [
      {
        "text": "f of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "エフ エックス",
    "name_en": "function notation",
    "name_ja": "関数記号 f(x)",
    "category": "functions",
    "term_ref": "function-notation",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "notes": [
      "f(x) は f of x と読む。括弧は掛け算ではない。f(2) は f of 2。",
      "英語では of を入れて f of x と読む。"
    ],
    "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I 二次関数（記号 f(x) を使用することも考えられる）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "f of x": 1948
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    }
  },
  {
    "id": "function-maps-arrow",
    "latex": "f\\colon A \\to B,\\ x \\mapsto f(x)",
    "spoken_en": [
      {
        "text": "f maps A to B",
        "register": "standard"
      }
    ],
    "spoken_ja": "f は A から B への写像 ／ x を f(x) にうつす",
    "name_en": "function arrow",
    "name_ja": "写像の矢印",
    "category": "functions",
    "term_ref": "function",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math",
        "Linear Algebra"
      ]
    },
    "notes": [
      "f: A → B は f maps A to B、または f is a function from A to B と読む。",
      "x ↦ f(x) は x maps to f of xと読み、矢印の形が → と違う（根元に縦棒）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "0.2（a function maps one set to another set）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.1（f maps S to S′）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "maps * to *": 20
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.02",
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "gamma",
    "latex": "\\gamma",
    "spoken_en": [
      {
        "text": "gamma",
        "register": "standard"
      }
    ],
    "spoken_ja": "ガンマ",
    "name_en": "gamma",
    "name_ja": "ギリシャ文字 γ",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "notes": [
      "γ は gamma と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "5.1 Angles（ギリシャ文字の表: γ gamma）・8.1 Non-right Triangles: Law of Sines（angle γ (gamma)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "7.1 Angles・10.1 Non-right Triangles: Law of Sines（angle γ (gamma)）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Γ」",
        "url": "https://ja.wikipedia.org/wiki/%CE%93",
        "note": "ガンマ（英: gamma）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "!alloy gamma !ray !rays !function !functions !strain": 3
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.06"
      ],
      "counted": "2026-09-28"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた。gamma（2026-09-28。docs/DECISIONS.md「Phase 5 監査（セッション 11）の前の決定」の 2）。register は主張しない。",
        "raised": "2026-09-28"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "gauss-bracket-jp",
    "latex": "[x]",
    "spoken_en": [
      {
        "text": "the greatest integer less than or equal to x",
        "register": "standard"
      },
      {
        "text": "the floor of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "ガウス x ／ x を超えない最大の整数",
    "name_en": "Gauss bracket (Japanese notation)",
    "name_ja": "ガウス記号",
    "category": "functions",
    "term_ref": "floor-function",
    "related": [
      "floor-function-notation",
      "floor-brackets"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Calculus II"
      ]
    },
    "notes": [
      "[x] は the greatest integer less than or equal to x（x を超えない最大の整数）か the floor of x と読む。",
      "OpenStax Calculus Volume 1 は ⌊x⌋ と書いて floor function と呼ぶ（Volume 2 は [x] を使うときに断る）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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": "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（ガウス記号）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.6 Ratio and Root Tests の練習問題（[x] is the greatest integer less than or equal to x。1 つ目の読み）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "4.4 The Mean Value Theorem の練習問題（f(x) = ⌊x⌋、This is called the floor function。notes の ⌊x⌋）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "floor of x | floor of n | floor of one | floor of two | floor of something": 4
      },
      "sources": [
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-27。docs/DECISIONS.md「Phase 5 監査（セッション 10）の前の決定」の 3）。the greatest integer less than or equal to x ／ the floor of x。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "gcd-notation",
    "latex": "\\gcd(a, b),\\ \\text{GCF}(a, b)",
    "spoken_en": [
      {
        "text": "the greatest common divisor of a and b",
        "register": "standard"
      },
      {
        "text": "the greatest common factor of a and b",
        "register": "standard"
      },
      {
        "text": "the GCF of a and b",
        "register": "standard"
      },
      {
        "text": "the gcd of a and b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a と b の最大公約数",
    "name_en": "greatest common divisor",
    "name_ja": "最大公約数",
    "category": "number-theory",
    "term_ref": "greatest-common-divisor",
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Discrete Math"
      ]
    },
    "notes": [
      "gcd(a, b) は the greatest common divisor of a and b か the gcd of a and b（G C D）と読む。GCF(a, b) は the greatest common factor of a and b か the GCF of a and b と読む。",
      "IM Grade 6（7.16 Common Factors）と OpenStax Prealgebra 2e（10.6）は greatest common factor（GCF）と呼ぶ。"
    ],
    "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 Introduction to Number Theory（gcd(d, n), for greatest common divisor）"
      },
      {
        "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（The greatest common factor (sometimes written as GCF)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "10.6 Introduction to Factoring Polynomials（greatest common factor、GCF）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "7.1 Greatest Common Factor and Factor by Grouping（The greatest common factor (GCF) of two or more expressions。level.us の Algebra 1）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "greatest common divisor": 91,
        "gcd | g c d": 29,
        "greatest common factor": 85,
        "gcf | g c f": 39
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-6.042",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "geq-sign",
    "latex": "a \\geq b",
    "spoken_en": [
      {
        "text": "a is greater than or equal to b",
        "register": "standard"
      },
      {
        "text": "a is at least b",
        "register": "spoken"
      }
    ],
    "spoken_ja": "a 大なりイコール b ／ a は b 以上",
    "name_en": "greater-than-or-equal-to sign",
    "name_ja": "以上（≧）",
    "category": "arithmetic",
    "term_ref": "greater-than-or-equal-to",
    "related": [
      "inequality-symbols"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "x ≥ 0 は x is greater than or equal to zero と読む。is at least zero（少なくとも 0）とも言う。",
      "日本の記号（≧）との違いは慣習差 inequality-symbols。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.2 Use the Language of Algebra の Inequality Symbols の表（a ≥ b is read a is greater than or equal to b）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「不等号」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%8D%E7%AD%89%E5%8F%B7",
        "note": "読み（大なり・小なり・大なりイコール・小なりイコール、以上・以下）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "greater than or equal to *": 355,
        "is at least *": 67
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "greater-than-sign",
    "latex": "a > b",
    "spoken_en": [
      {
        "text": "a is greater than b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a 大なり b ／ a は b より大きい",
    "name_en": "greater-than sign",
    "name_ja": "大なり（>）",
    "category": "arithmetic",
    "term_ref": "greater-than",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "a > b は a is greater than b と読む（OpenStax Elementary Algebra 1.1）。bigger than は授業の口語で言うこともあるが、記号の読みは greater than。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.2 Use the Language of Algebra の Inequality Symbols の表（a > b is read a is greater than b）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「不等号」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%8D%E7%AD%89%E5%8F%B7",
        "note": "読み（大なり・小なり・大なりイコール・小なりイコール、以上・以下）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "is greater than !or": 439
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "h-naught-null",
    "latex": "H_0",
    "spoken_en": [
      {
        "text": "the null hypothesis",
        "register": "standard"
      },
      {
        "text": "H sub zero",
        "register": "spoken"
      }
    ],
    "spoken_ja": "帰無仮説 H₀ ／ エイチ ゼロ",
    "name_en": "null hypothesis (H-naught)",
    "name_ja": "帰無仮説（H₀）",
    "category": "probability-statistics",
    "term_ref": "null-hypothesis",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "H₀ は the null hypothesis と読むか、H sub zero と読む。OpenStax Introductory Statistics 2e の付録は H-naught とも読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "9.1 Null and Alternative Hypotheses・付録 Mathematical Phrases, Symbols, and Formulas（H₀: H-naught, H-sub 0）"
      },
      {
        "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（有意性検定）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the null hypothesis": 201,
        "h zero | h sub zero": 22
      },
      "sources": [
        "khan-ap-stats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "h-sub-a-alternative",
    "latex": "H_a,\\ H_1",
    "spoken_en": [
      {
        "text": "the alternative hypothesis",
        "register": "standard"
      },
      {
        "text": "H sub one",
        "register": "spoken"
      },
      {
        "text": "H sub a",
        "register": "spoken"
      }
    ],
    "spoken_ja": "対立仮説 H₁ ／ エイチ ワン",
    "name_en": "alternative hypothesis",
    "name_ja": "対立仮説（Hₐ、H₁）",
    "category": "probability-statistics",
    "term_ref": "alternative-hypothesis",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "Hₐ は H sub a、H₁ は H sub one と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "9.1 Null and Alternative Hypotheses・付録 Mathematical Phrases, Symbols, and Formulas（Hₐ: H-a, H-sub a、H₁: H-1, H-sub 1）"
      },
      {
        "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（有意性検定）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "H sub a": 1,
        "the alternative hypothesis | the alternate hypothesis": 25,
        "h sub one !h": 9
      },
      "sources": [
        "khan-ap-stats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "half-open-interval",
    "latex": "[a, b)",
    "spoken_en": [
      {
        "text": "the half-open interval from a to b",
        "register": "standard"
      },
      {
        "text": "from a to b, including a but not b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a 以上 b 未満の区間",
    "name_en": "half-open interval",
    "name_ja": "半開区間",
    "category": "sets",
    "term_ref": "interval-notation",
    "related": [
      "interval-notation-vs-inequalities"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Algebra 1",
        "Precalculus"
      ]
    },
    "notes": [
      "[a, b) は the half-open interval from a to b、または from a to b, including a but not b と読む。",
      "OpenStax Precalculus は三角方程式の解の範囲を [0, 2π) のような半開区間の記号で書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "7.5 Solving Trigonometric Equations ほか（区間の記号 [0, 2π)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "4.3 Maxima and Minima（the half-open interval [0, 2)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "including * but not *": 3
      },
      "sources": [
        "yt:nancypi",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 3 フレーズの前の修正」の 2）。the half-open interval from a to b ／ from a to b, including a but not b。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "hyperbolic-functions-notation",
    "latex": "\\sinh x,\\ \\cosh x,\\ \\tanh x",
    "spoken_en": [
      {
        "text": "cosh x",
        "register": "standard"
      },
      {
        "text": "hyperbolic sine of x",
        "register": "standard"
      },
      {
        "text": "cinch x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ハイパボリックサイン x ／ ハイパボリックコサイン x",
    "name_en": "hyperbolic functions",
    "name_ja": "双曲線関数",
    "category": "functions",
    "term_ref": "hyperbolic-functions",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I",
        "Calculus II"
      ]
    },
    "notes": [
      "cosh x は cosh と読み（OpenStax Calculus Volume 1 1.5 は gosh と韻を踏むと書く）、hyperbolic cosine of x とも言う。sinh は cinch と読むか hyperbolic sine と言う。",
      "tanh は tanch と読む（OpenStax Calculus Volume 1 1.5）。hyperbolic tangent とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.5 Exponential and Logarithmic Functions（cosh は gosh と韻、sinh は cinch、tanh は tanch と発音）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "6.9 Calculus of the Hyperbolic Functions（level.us Calculus II）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「双曲線関数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8F%8C%E6%9B%B2%E7%B7%9A%E9%96%A2%E6%95%B0",
        "note": "ハイパボリックサイン・ハイパボリックコサイン"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "cosh": 69,
        "hyperbolic sine | hyperbolic cosine | hyperbolic tangent": 26,
        "!a cinch | sinch": 2
      },
      "sources": [
        "mit-18.01",
        "mit-18.03",
        "yt:blackpenredpen",
        "yt:patrickjmt"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "identity-matrix-i",
    "latex": "I,\\ I_n,\\ E",
    "spoken_en": [
      {
        "text": "the identity",
        "register": "standard"
      },
      {
        "text": "the identity matrix",
        "register": "standard"
      }
    ],
    "spoken_ja": "単位行列 E",
    "name_en": "identity matrix",
    "name_ja": "単位行列（I、Iₙ、E）",
    "category": "matrices",
    "term_ref": "identity-matrix",
    "level": {
      "jp": [
        "数C",
        "大学"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "notes": [
      "行列の式の中では I を the identity、または the identity matrix と読む（four times the identity のように言う）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "9.7 Solving Systems with Inverses（I_n）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.7 Solving Systems with Inverses（I_n）"
      },
      {
        "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": "2.2 Matrix-Vector Multiplication（Definition 2.7）ほか"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "the identity !matrix !function !property !map !transformation !operator !element !permutation": 159,
        "the identity matrix | an identity matrix": 66
      },
      "sources": [
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "iff-arrow",
    "latex": "p \\Leftrightarrow q,\\ p \\leftrightarrow q",
    "spoken_en": [
      {
        "text": "p if and only if q",
        "register": "standard"
      },
      {
        "text": "p is equivalent to q",
        "register": "spoken"
      }
    ],
    "spoken_ja": "p と q は同値 ／ p ⇔ q",
    "name_en": "if and only if",
    "name_ja": "同値（⇔）",
    "category": "logic",
    "term_ref": "biconditional",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "notes": [
      "p ⇔ q は p if and only if q か p is equivalent to q と読む。iff は書くときの略で、読むときは if and only if と言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "4.3 Logarithmic Functions（⇔ の使用）"
      },
      {
        "type": "reference",
        "title": "CK-12 Foundation, CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.12 Converse, Inverse, and Contrapositive Statements（p ↔ q, said p if and only if 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（P ↔ Q is read P if and only if Q）"
      },
      {
        "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 の脚注（written p ⇔ q, spoken p if and only if q）"
      },
      {
        "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,十分条件や対偶などの指導。「⇔」「同値」は出てこない。level.jp の数I はこの必要条件・十分条件による）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "if and only if": 76,
        "is equivalent to *": 109
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "imaginary-unit-i",
    "latex": "i",
    "spoken_en": [
      {
        "text": "i",
        "register": "standard"
      },
      {
        "text": "the imaginary unit",
        "register": "spoken"
      }
    ],
    "spoken_ja": "アイ ／ 虚数単位 i",
    "name_en": "imaginary unit",
    "name_ja": "虚数単位（i）",
    "category": "complex-numbers",
    "term_ref": "imaginary-unit",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "i は文字どおり i と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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II いろいろな式: i"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "2.4 Complex Numbers（The imaginary number i is defined as the square root of −1）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "3.1 Complex Numbers（i is defined as the square root of −1）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "8.8 Use the Complex Number System（the imaginary unit i）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "i squared equals negative one | i squared is negative one | i squared equals minus one | i squared is minus one | i squared is equal to negative one": 13,
        "the imaginary unit": 2
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.03",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "implies-arrow",
    "latex": "p \\Rightarrow q,\\ p \\to q",
    "spoken_en": [
      {
        "text": "p implies q",
        "register": "standard"
      }
    ],
    "spoken_ja": "p ならば q",
    "name_en": "implies",
    "name_ja": "ならば（⇒）",
    "category": "logic",
    "term_ref": "conditional-statement",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "notes": [
      "p ⇒ q は p implies q と読むか、if p, then q と文で言う。",
      "CK-12 Geometry（2.8・2.12）と Levin（1.1）は条件文を p → q と書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "8.8 Vectors（⇒ の使用）"
      },
      {
        "type": "reference",
        "title": "CK-12 Foundation, CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.8 Truth Tables・2.12（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（P → Q is read if P then Q）"
      },
      {
        "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 の脚注（p implies q (written p ⇒ q)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* implies *": 142
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "logic-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "indeterminate-forms",
    "latex": "\\tfrac{0}{0},\\ \\tfrac{\\infty}{\\infty}",
    "spoken_en": [
      {
        "text": "zero over zero",
        "register": "standard"
      },
      {
        "text": "infinity over infinity",
        "register": "standard"
      }
    ],
    "spoken_ja": "0 分の 0 ／ ∞ 分の ∞",
    "name_en": "indeterminate forms",
    "name_ja": "不定形",
    "category": "limits",
    "term_ref": "indeterminate-form",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "AP Calculus BC",
        "Calculus I"
      ]
    },
    "notes": [
      "0/0 は zero over zero、∞/∞ は infinity over infinity と読む。まとめて indeterminate forms と呼ぶ。"
    ],
    "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 4.7（indeterminate forms）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「ロピタルの定理」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%AD%E3%83%94%E3%82%BF%E3%83%AB%E3%81%AE%E5%AE%9A%E7%90%86",
        "note": "不定形"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "zero over zero": 45,
        "infinity over infinity": 12
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "infinity-symbol",
    "latex": "\\infty,\\ -\\infty",
    "spoken_en": [
      {
        "text": "infinity",
        "register": "standard"
      },
      {
        "text": "negative infinity",
        "register": "standard"
      },
      {
        "text": "minus infinity",
        "register": "spoken"
      }
    ],
    "spoken_ja": "無限大 ／ マイナス無限大",
    "name_en": "infinity",
    "name_ja": "無限大（∞）",
    "category": "limits",
    "term_ref": "infinity",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "∞ は infinity、−∞ は negative infinity と読む。minus infinity と読む講義もある。",
      "x → ∞ は x approaches infinity（x goes to infinity）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Limit of a Function（is positive infinity、is negative infinity）"
      },
      {
        "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"
      },
      {
        "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III 極限: ∞"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!negative !minus infinity": 1847,
        "minus infinity": 85,
        "negative infinity": 371
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "integers-symbol",
    "latex": "\\mathbb{Z}",
    "spoken_en": [
      {
        "text": "the integers",
        "register": "standard"
      }
    ],
    "spoken_ja": "整数全体の集合 Z",
    "name_en": "integers",
    "name_ja": "整数全体の集合（ℤ）",
    "category": "sets",
    "term_ref": "integer",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "ℤ は the integers と読む。文字どおり Z（米国の発音は zee。Merriam-Webster: ˈzē、カナダ・英国・オーストラリアは ˈzed）とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「z」",
        "url": "https://www.merriam-webster.com/dictionary/z",
        "note": "発音 ˈzē（カナダ・英国・オーストラリアは ˈzed）（2026-09-26 に確かめた）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets（ℤ The set of integers）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the integers | the set of integers": 33
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "integral-definite",
    "latex": "\\int_a^b f(x)\\,dx",
    "spoken_en": [
      {
        "text": "the integral from a to b of f of x d x",
        "register": "standard"
      },
      {
        "text": "the integral of f of x from a to b",
        "register": "spoken"
      }
    ],
    "spoken_ja": "インテグラル a から b、f(x) dx ／ f(x) の a から b までの定積分",
    "name_en": "definite integral",
    "name_ja": "定積分",
    "category": "integrals",
    "term_ref": "definite-integral",
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "dx は d x と 2 音で読む。",
      "英語は the integral from a to b of f of x dx と、of f of x で被積分関数を読み、最後に dx を読む。",
      "∫₀¹ は the integral from zero to one と読む。上端・下端が文字でも数でも同じ形で読む。"
    ],
    "tts_text": "the integral from a to b of f of x, d x",
    "sources": [
      {
        "type": "reference",
        "title": "Nekovar, Mathematical English (a brief summary)"
      },
      {
        "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（definite integral notation）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Definite Integral"
      },
      {
        "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": "インテグラル"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the integral from * to * of": 246,
        "the integral of * from * to *": 11
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "integral-indefinite",
    "latex": "\\int f(x)\\,dx",
    "spoken_en": [
      {
        "text": "the integral of f of x d x",
        "register": "standard"
      },
      {
        "text": "the indefinite integral of f of x",
        "register": "standard"
      },
      {
        "text": "the integral of f of x with respect to x",
        "register": "standard"
      }
    ],
    "spoken_ja": "インテグラル f(x) dx ／ f(x) の不定積分",
    "name_en": "indefinite integral",
    "name_ja": "不定積分",
    "category": "integrals",
    "term_ref": "indefinite-integral",
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "∫ f(x) dx は the integral of f of x d x と読む。the indefinite integral of f of x とも言う。",
      "dx は d x と 2 音で読む。with respect to x と言いかえることもある。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "Antiderivatives（∫f(x)dx is called the indefinite integral of f）"
      },
      {
        "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"
      },
      {
        "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": "インテグラル … ディーエックス"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "integral of * dx": 81,
        "the indefinite integral of *": 33,
        "integral of * with respect to": 12
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "intersection-sign",
    "latex": "A \\cap B",
    "spoken_en": [
      {
        "text": "A intersect B",
        "register": "standard"
      },
      {
        "text": "the intersection of A and B",
        "register": "spoken"
      }
    ],
    "spoken_ja": "A キャップ B ／ A と B の共通部分",
    "name_en": "intersection",
    "name_ja": "共通部分（∩）",
    "category": "sets",
    "term_ref": "intersection",
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics",
        "Discrete Math"
      ]
    },
    "notes": [
      "A ∩ B は A intersect B か the intersection of A and B と読む。∩ を cap と読む例は、用例コーパスと参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）には出てこない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "11.7 Probability（the intersection of E and F, written E ∩ F）"
      },
      {
        "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（intersect）・Topic 2.7（intersect）"
      },
      {
        "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（A∩B）・数学A 場合の数と確率（n(A∪B)=n(A)+n(B)−n(A∩B)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（A AND B can also be written as A Intersection B、A ∩ B。level.us の Intro Statistics）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.5 Proofs about Discrete Structures・5.1 Sets（the intersection of A and B, written A ∩ B。2 つ目の読みと level.us の Discrete Math）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.7 Probability（the intersection of E and F and is written as E ∩ F。level.us の Algebra 2）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「キャップ」",
        "url": "https://ja.wikipedia.org/wiki/%E3%82%AD%E3%83%A3%E3%83%83%E3%83%97",
        "note": "キャップ（∩）: 数学の記号。共通部分を表す（spoken_ja の A キャップ B）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "a intersect b | b intersect a | a intersect c | b intersect c | s intersect u | s intersect t | p intersect l": 16,
        "the intersection of a and b | the intersection of b and a | the intersection of a and c | the intersection of b and c | the intersection of set a and set b | the intersection of sets a and b": 1
      },
      "sources": [
        "mit-18.06",
        "mit-6.042",
        "yt:organicchem"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    },
    "related": [
      "probability-and-or-notation"
    ]
  },
  {
    "id": "inverse-cosine-notation",
    "latex": "\\cos^{-1} x,\\ \\arccos x",
    "spoken_en": [
      {
        "text": "arccosine of x",
        "register": "standard"
      },
      {
        "text": "inverse cosine of x",
        "register": "standard"
      },
      {
        "text": "cosine inverse of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "アークコサイン x",
    "name_en": "inverse cosine",
    "name_ja": "逆余弦（アークコサイン）",
    "category": "trigonometry",
    "term_ref": "arccosine",
    "related": [
      "inverse-trig-functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "cos⁻¹ x も arccos x も、arccosine of x、inverse cosine of x、cosine inverse of x と読む。1/cos x（secant）ではない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "3.7 Derivatives of Inverse Functions"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "6.3 Inverse Trigonometric Functions（the inverse cosine function is sometimes called the arccosine function）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Geometry 4.9 Using Trigonometric Ratios to Find Angles（arccosine）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "arccosine | arc cosine": 36,
        "inverse cosine": 17,
        "cosine inverse": 17
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "inverse-function-notation",
    "latex": "f^{-1}(x)",
    "spoken_en": [
      {
        "text": "f inverse",
        "register": "standard"
      },
      {
        "text": "f inverse of x",
        "register": "standard"
      },
      {
        "text": "the inverse of f",
        "register": "standard"
      }
    ],
    "spoken_ja": "f インバース x",
    "name_en": "inverse function notation",
    "name_ja": "逆関数の記号",
    "category": "functions",
    "term_ref": "inverse-function",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Calculus I"
      ]
    },
    "notes": [
      "f⁻¹(x) は f inverse of x と読み、関数だけなら f inverse。the inverse of f とも言う。",
      "−1 は指数ではないので f to the negative one とは読まない。1/f(x) は one over f of x で、f⁻¹(x) とは別のもの。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.4 Inverse Functions（f⁻¹ is read as f inverse）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "Inverse Functions（f⁻¹ is read f inverse）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "1.7 Inverse Functions（f⁻¹(x) is read f inverse of x、the −1 is not an exponent、not 1/f(x)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "f inverse !of": 29,
        "the inverse of f | the inverse function of f": 15,
        "f inverse of *": 24
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "inverse-sine-notation",
    "latex": "\\sin^{-1} x,\\ \\arcsin x",
    "spoken_en": [
      {
        "text": "arcsine of x",
        "register": "standard"
      },
      {
        "text": "sine inverse of x",
        "register": "standard"
      },
      {
        "text": "inverse sine of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "アークサイン x",
    "name_en": "inverse sine",
    "name_ja": "逆正弦（アークサイン）",
    "category": "trigonometry",
    "term_ref": "arcsine",
    "related": [
      "inverse-trig-functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "sin⁻¹ x も arcsin x も、arcsine of x、inverse sine of x、sine inverse of x と読む。",
      "sin⁻¹ x の −1 は指数ではなく、1/sin x（cosecant）ではない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "Inverse Trigonometric Functions（the inverse sine function is sometimes called the arcsine function）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Geometry 4.9 Using Trigonometric Ratios to Find Angles（arcsine）"
      },
      {
        "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": "逆正弦関数（inverse sine、sin⁻¹x）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "arcsine | arc sine | arcsin": 26,
        "sine inverse": 20,
        "inverse sine": 17
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "inverse-tangent-notation",
    "latex": "\\tan^{-1} x,\\ \\arctan x",
    "spoken_en": [
      {
        "text": "arctangent of x",
        "register": "standard"
      },
      {
        "text": "inverse tangent of x",
        "register": "standard"
      },
      {
        "text": "tangent inverse of x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "アークタンジェント x",
    "name_en": "inverse tangent",
    "name_ja": "逆正接（アークタンジェント）",
    "category": "trigonometry",
    "term_ref": "arctangent",
    "related": [
      "inverse-trig-functions"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "tan⁻¹ x も arctan x も、arctangent of x か inverse tangent of x と読む。短く arctan x とも言う。",
      "∫ 1/(1 + x²) dx = arctan x + C は the integral of one over the quantity one plus x squared is arctangent of x plus C。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "Inverse Trigonometric Functions（the inverse tangent function is sometimes called the arctangent function）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Geometry 4.9 Using Trigonometric Ratios to Find Angles（arctangent）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "inverse tangent": 37,
        "arctangent | arc tangent | arctan": 95,
        "!hyperbolic tangent inverse": 4
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "iqr-abbreviation",
    "latex": "\\text{IQR}",
    "spoken_en": [
      {
        "text": "the interquartile range",
        "register": "standard"
      },
      {
        "text": "I Q R",
        "register": "standard"
      }
    ],
    "spoken_ja": "四分位範囲",
    "name_en": "interquartile range (IQR)",
    "name_ja": "四分位範囲（IQR）",
    "category": "probability-statistics",
    "term_ref": "interquartile-range",
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "IQR は the interquartile range と読むか、文字どおり I Q R と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "2.3 Measures of the Location of the Data（IQR = Q3 – Q1）・付録 Mathematical Phrases, Symbols, and Formulas（IQR、interquartile range）"
      },
      {
        "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（interquartile range）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "interquartile range": 37,
        "i q r | iqr": 24
      },
      "sources": [
        "khan-ap-stats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "jacobian-notation",
    "latex": "\\dfrac{\\partial(x, y)}{\\partial(u, v)}",
    "spoken_en": [
      {
        "text": "the Jacobian of x and y with respect to u and v",
        "register": "standard"
      }
    ],
    "spoken_ja": "ヤコビアン",
    "name_en": "Jacobian",
    "name_ja": "ヤコビアン",
    "category": "derivatives",
    "term_ref": "jacobian",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "∂(x, y)/∂(u, v) は the Jacobian of x and y with respect to u and v と読み、行列式の値を言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Change of Variables in Multiple Integrals（J(u, v) = ∂(x, y)/∂(u, v)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the jacobian": 35
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "lambda",
    "latex": "\\lambda",
    "spoken_en": [
      {
        "text": "lambda",
        "register": "standard"
      }
    ],
    "spoken_ja": "ラムダ",
    "name_en": "lambda",
    "name_ja": "ギリシャ文字 λ",
    "category": "notation-other",
    "term_ref": "eigenvalue",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra",
        "Calculus III",
        "Intro Statistics"
      ]
    },
    "notes": [
      "λ は lambda と読む。",
      "固有値やラグランジュ乗数に使う。"
    ],
    "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 ほか"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "4.8 Lagrange Multipliers（λ）・7.1 Second-Order Linear Equations（λ (the lowercase Greek letter lambda)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "4.6 Poisson Distribution（λ (lambda)）・付録 Mathematical Phrases, Symbols, and Formulas（λ、Lambda）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Λ」",
        "url": "https://ja.wikipedia.org/wiki/%CE%9B",
        "note": "ラムダ（英: lambda）。ラグランジュ乗数を表す"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "lambda": 739
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "lcm-notation",
    "latex": "\\operatorname{lcm}(a, b)",
    "spoken_en": [
      {
        "text": "the least common multiple of a and b",
        "register": "standard"
      },
      {
        "text": "the lcm of a and b",
        "register": "spoken"
      }
    ],
    "spoken_ja": "a と b の最小公倍数",
    "name_en": "least common multiple",
    "name_ja": "最小公倍数",
    "category": "number-theory",
    "term_ref": "least-common-multiple",
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "lcm(a, b) は the least common multiple of a and b と読む。略して the L C M of a and b とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "2.5 Prime Factorization and the Least Common Multiple（least common multiple（LCM））"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.1 Introduction to Whole Numbers（find the least common multiple）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 7.17 Common Multiples（The least common multiple (sometimes written as LCM)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "least common multiple": 109,
        "lcm | l c m": 7
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "yt:nancypi",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "length-ab",
    "latex": "AB",
    "spoken_en": [
      {
        "text": "the length of segment AB",
        "register": "standard"
      },
      {
        "text": "the length of AB",
        "register": "standard"
      }
    ],
    "spoken_ja": "線分 AB の長さ ／ AB",
    "name_en": "length of a segment",
    "name_ja": "線分の長さ（AB）",
    "category": "geometry",
    "term_ref": null,
    "related": [
      "figure-and-measure-notation"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry"
      ]
    },
    "notes": [
      "上線のない AB は長さを表し、the length of segment AB（the length of AB とも）と読む。"
    ],
    "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（線分は上線、長さは上線なしの AB）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Geometry 1.1・1.12・3.2 ほか（the length of segment AB）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "the length of segment *": 14,
        "the length of a b | the length of ab | the length of b c | the length of bc | the length of a c | the length of ac | the length of c d | the length of cd | the length of a d | the length of ad | the length of b d | the length of bd | the length of d c | the length of dc | the length of e a | the length of ea | the length of o p | the length of op | the length of j l | the length of jl | the length of k l | the length of kl": 11
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "leq-sign",
    "latex": "a \\leq b",
    "spoken_en": [
      {
        "text": "a is less than or equal to b",
        "register": "standard"
      },
      {
        "text": "a is at most b",
        "register": "spoken"
      }
    ],
    "spoken_ja": "a 小なりイコール b ／ a は b 以下",
    "name_en": "less-than-or-equal-to sign",
    "name_ja": "以下（≦）",
    "category": "arithmetic",
    "term_ref": "less-than-or-equal-to",
    "related": [
      "inequality-symbols"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "x ≤ 5 は x is less than or equal to five と読む。is at most five（多くても 5）とも言い、離散数学の講義（MIT 6.042）に多い。",
      "日本の記号（≦）との違いは慣習差 inequality-symbols。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 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"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.2 Use the Language of Algebra の Inequality Symbols の表（a ≤ b is read a is less than or equal to b）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「不等号」",
        "url": "https://ja.wikipedia.org/wiki/%E4%B8%8D%E7%AD%89%E5%8F%B7",
        "note": "読み（大なり・小なり・大なりイコール・小なりイコール、以上・以下）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "less than or equal to *": 439,
        "is at most *": 60
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "less-than-sign",
    "latex": "a < b",
    "spoken_en": [
      {
        "text": "a is less than b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a 小なり b ／ a は b より小さい ／ a は b 未満",
    "name_en": "less-than sign",
    "name_ja": "小なり（<）",
    "category": "arithmetic",
    "term_ref": "less-than",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "a < b は a is less than b と読む。左から右へ読むので、x < 3 は x is less than three。",
      "3 > x を x を主語にして読みかえて x is less than three と言うこともある。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.2 Use the Language of Algebra の Inequality Symbols の表（a < b is read a is less than b、左から右へ読む）"
      },
      {
        "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": "読み（小なり、未満）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "is less than !or": 688
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "limit-from-left",
    "latex": "\\lim_{x \\to a^-} f(x)",
    "spoken_en": [
      {
        "text": "the limit as x approaches a from the left of f of x",
        "register": "standard"
      },
      {
        "text": "the left-hand limit of f at a",
        "register": "spoken"
      }
    ],
    "spoken_ja": "x が a に左から近づくときの極限 ／ リミット x → a マイナス 0",
    "name_en": "limit from the left",
    "name_ja": "左側極限",
    "category": "limits",
    "term_ref": "left-hand-limit",
    "related": [
      "one-sided-limit-notation"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "x → a⁻ は x approaches a from the left と読む。AP の CED（Unit 1）と OpenStax Calculus Volume 1 は the limit of f(x) as x approaches a from the left と関数を先にも読む。",
      "a⁻ の右上の − は「左から」の印で、a から数を引くのではない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Limit of a Function（the limit of f(x) as x approaches a 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": "x → a − 0 ／ a + 0 の書き方"
      },
      {
        "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 の概要（The limit of x cubed as x approaches 0 from the left）・Topic 1.3"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "from the left": 264,
        "left-hand limit | left hand limit": 6
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "limit-from-right",
    "latex": "\\lim_{x \\to a^+} f(x)",
    "spoken_en": [
      {
        "text": "the limit as x approaches a from the right of f of x",
        "register": "standard"
      },
      {
        "text": "the right-hand limit of f at a",
        "register": "spoken"
      }
    ],
    "spoken_ja": "x が a に右から近づくときの極限 ／ リミット x → a プラス 0",
    "name_en": "limit from the right",
    "name_ja": "右側極限",
    "category": "limits",
    "term_ref": "right-hand-limit",
    "related": [
      "one-sided-limit-notation"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "x → a⁺ は x approaches a from the right と読む。OpenStax Calculus Volume 1 は the limit of f(x) as x approaches a from the right と関数を先にも読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Limit of a Function（the limit of f(x) as x approaches a from the right）"
      },
      {
        "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": "x → a − 0 ／ a + 0 の書き方"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "from the right": 220,
        "right-hand limit | right hand limit": 11
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "limit-x-to-a",
    "latex": "\\lim_{x \\to a} f(x)",
    "spoken_en": [
      {
        "text": "the limit as x approaches a of f of x",
        "register": "standard"
      },
      {
        "text": "the limit of f of x as x approaches a",
        "register": "standard"
      },
      {
        "text": "the limit as x goes to a of f of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "リミット x → a f(x) ／ x が a に限りなく近づくときの f(x) の極限値",
    "name_en": "limit",
    "name_ja": "極限（lim）",
    "category": "limits",
    "term_ref": "limit",
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "lim は limit と読み、the limit as x approaches a of f of x と、条件を先、関数を後に読む。the limit of f of x as x approaches a と関数を先にも言える。",
      "大学の講義（MIT OCW）は approaches より goes to を多く使う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Limit of a Function（the limit of f(x) as x approaches a is L）"
      },
      {
        "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"
      },
      {
        "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II 微分・積分の考え: lim"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「極限」",
        "url": "https://ja.wikipedia.org/wiki/%E6%A5%B5%E9%99%90",
        "note": "lim（リミット）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the limit as * goes to * of": 90,
        "the limit as * approaches * of": 314,
        "the limit of * as * approaches": 161
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "line-ab-arrow",
    "latex": "\\overleftrightarrow{AB}",
    "spoken_en": [
      {
        "text": "line AB",
        "register": "standard"
      }
    ],
    "spoken_ja": "直線 AB",
    "name_en": "line notation",
    "name_ja": "直線の記号（両矢印の AB）",
    "category": "geometry",
    "term_ref": "line",
    "related": [
      "figure-and-measure-notation"
    ],
    "level": {
      "jp": [],
      "us": [
        "Geometry"
      ]
    },
    "notes": [
      "両矢印の AB は line AB と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.1 Geometry Terms（Say It: line AB、could be read line AB or line BA）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "両矢印の AB（20 件）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "line a b | line ab | line c d | line cd | line b c | line bc | line a c | line ac | line p q | line pq | line x y | line xy": 29
      },
      "sources": [
        "khan-middle",
        "mit-18.01",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "line-integral-sign",
    "latex": "\\int_C \\mathbf{F} \\cdot d\\mathbf{r},\\ \\oint_C",
    "spoken_en": [
      {
        "text": "the integral over C of F dot d r",
        "register": "standard"
      },
      {
        "text": "the integral around C",
        "register": "spoken"
      },
      {
        "text": "the line integral over C of F dot d r",
        "register": "spoken"
      },
      {
        "text": "the line integral of F along C",
        "register": "spoken"
      }
    ],
    "spoken_ja": "線積分",
    "name_en": "line integral",
    "name_ja": "線積分",
    "category": "integrals",
    "term_ref": "line-integral",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "∫_C F · dr は the integral over C of F dot d r（the line integral over C …）と読む。",
      "∮_C は閉じた曲線に沿った積分で、the integral around C と言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Line Integrals（∮ の丸は、C が端点の無い circular な曲線であることを表すと書く）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "the integral over c | the integral along c": 18,
        "the integral around *": 8,
        "the line integral over * | the line integral along *": 6,
        "the line integral of *": 6
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "log-base-a",
    "latex": "\\log_a x",
    "spoken_en": [
      {
        "text": "log base a of x",
        "register": "standard"
      },
      {
        "text": "the logarithm of x to the base a",
        "register": "spoken"
      }
    ],
    "spoken_ja": "a を底とする x の対数",
    "name_en": "logarithm with base a",
    "name_ja": "a を底とする対数",
    "category": "logarithms",
    "term_ref": "logarithm",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "logₐ x は log base a of x と読む。log₂ 8 = 3 は log base two of eight is three。",
      "OpenStax Precalculus（4.3 Logarithmic Functions）は log_b(x) を the logarithm with base b of x とも読むと書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "4.3 Logarithmic Functions（log_b(x) is read the log base b of x）"
      },
      {
        "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II 指数関数・対数関数: log_a x"
      },
      {
        "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?（read the log, base 10, of 250）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「対数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%AF%BE%E6%95%B0",
        "note": "底を b とする x の対数"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "log base *": 413,
        "to the base *": 7
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "log-e-jp",
    "latex": "\\log x,\\ \\log_e x",
    "spoken_en": [
      {
        "text": "the natural log of x",
        "register": "standard"
      },
      {
        "text": "log base e of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "ログ x（自然対数）",
    "name_en": "natural log written log x (Japanese 数III)",
    "name_ja": "数III の log x（底 e）",
    "category": "logarithms",
    "term_ref": "natural-logarithm",
    "related": [
      "log-and-ln"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "notes": [
      "log x（底 e）は the natural log of x と読む。底を言うなら log base e of x。英語で ln x と書いたときの読みは natural-log-ln。",
      "OpenStax Calculus Volume 1（Antiderivatives）は ∫ 1/x dx = ln|x| + C と書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "Antiderivatives（∫ 1/x dx = ln|x| + C）"
      },
      {
        "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III 微分法: 自然対数・e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「自然対数」",
        "url": "https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%AF%BE%E6%95%B0",
        "note": "底を省略して log x と書く"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "natural log of * | natural log *": 874,
        "log base e": 33
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "logical-and-wedge",
    "latex": "p \\land q",
    "spoken_en": [
      {
        "text": "p and q",
        "register": "standard"
      }
    ],
    "spoken_ja": "p かつ q",
    "name_en": "logical and",
    "name_ja": "かつ（∧）",
    "category": "logic",
    "term_ref": "logical-connective",
    "level": {
      "jp": [
        "理数",
        "大学"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "notes": [
      "p ∧ q は p and q と読む。LaTeX では \\land（\\wedge）と書くが、読むときは and。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Foundation, CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.8 Truth Tables（We would write p and q as 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（P ∧ Q is read P and Q）"
      },
      {
        "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": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "p and q": 16
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "logic-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "logical-or-vee",
    "latex": "p \\lor q",
    "spoken_en": [
      {
        "text": "p or q",
        "register": "standard"
      }
    ],
    "spoken_ja": "p または q",
    "name_en": "logical or",
    "name_ja": "または（∨）",
    "category": "logic",
    "term_ref": "logical-connective",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "notes": [
      "p ∨ q は p or q と読む。数学の or は両方成り立つ場合も含む（inclusive or）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Foundation, CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.8 Truth Tables（p or q を p∨q と書く）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1 Mathematical Statements（P ∨ Q is read P or Q、4 件。読みを決めた参照）"
      }
    ],
    "confidence": "verified",
    "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: 対象外）。読みは Levin の呼び方 p or q（1.1 Mathematical Statements・1.6 Chapter Summary）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "related": [
      "logic-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "long-division-bracket",
    "latex": "5\\overline{)20}",
    "spoken_en": [
      {
        "text": "twenty divided by five",
        "register": "standard"
      },
      {
        "text": "five goes into twenty",
        "register": "spoken"
      }
    ],
    "spoken_ja": "20 ÷ 5 の筆算",
    "name_en": "long division bracket",
    "name_ja": "割り算の筆算",
    "category": "arithmetic",
    "term_ref": "division",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "筆算の記号の読みも twenty divided by five。OpenStax Prealgebra（Divide Whole Numbers）は筆算の 4)12 も twelve divided by four と読み、筆算の途中では goes into を使う（The divisor 3 goes into 18 six times）。",
      "割られる数を dividend、割る数を divisor、商を quotient と呼ぶ。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Divide Whole Numbers（4)12 is read twelve divided by four、goes into）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* divided by *": 2635,
        "* goes into *": 454
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "matrix-brackets",
    "latex": "\\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix}",
    "spoken_en": [
      {
        "text": "the matrix a, b, c, d",
        "register": "standard"
      },
      {
        "text": "the two-by-two matrix with entries a, b, c, d",
        "register": "spoken"
      }
    ],
    "spoken_ja": "行列 a b c d",
    "name_en": "matrix (brackets)",
    "name_ja": "行列（[ ] と ( )）",
    "category": "matrices",
    "term_ref": "matrix",
    "level": {
      "jp": [
        "数C",
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "notes": [
      "行列は成分を左上から行ごとに読む。",
      "[ ] で囲んでも ( ) で囲んでも読み方は変わらない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "9.5 Matrices and Matrix Operations（[ ] か ( ) で囲む）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.5 Matrices and Matrix Operations（[ ] か ( ) で囲む）"
      },
      {
        "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（行列を [ ] で書く）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the matrix zero | the matrix one | the matrix two | the matrix three | the matrix four | the matrix five | the matrix six | the matrix seven | the matrix eight | the matrix nine | the matrix minus | the matrix negative | the matrix a b": 10,
        "matrix with entries *": 4
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "matrix-dimensions-by",
    "latex": "2 \\times 3,\\ m \\times n",
    "spoken_en": [
      {
        "text": "a two-by-three matrix",
        "register": "standard"
      },
      {
        "text": "a matrix with two rows and three columns",
        "register": "spoken"
      }
    ],
    "spoken_ja": "2 行 3 列の行列 ／ m 行 n 列",
    "name_en": "matrix dimensions",
    "name_ja": "行列の型（m × n）",
    "category": "matrices",
    "term_ref": "matrix",
    "level": {
      "jp": [
        "数C",
        "大学"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "notes": [
      "2 × 3 行列は a two-by-three matrix と読む。× を by と読み、行の数が先。m × n は m by n。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "9.5 Matrices and Matrix Operations（m × n、a 2 by 3 matrix）"
      },
      {
        "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": "2.1 Matrix Addition, Scalar Multiplication, and Transposition ほか"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "4.5 Solve Systems of Equations Using Matrices（a 2 by 3 matrix）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "by two matrix | by three matrix | by n matrix | by-two matrix | by-three matrix | by-n matrix | by one matrix | by four matrix | by m matrix | by-one matrix": 80,
        "rows and * columns": 8
      },
      "sources": [
        "mit-18.02",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "matrix-entry-aij",
    "latex": "a_{ij}",
    "spoken_en": [
      {
        "text": "the entry in row i, column j",
        "register": "standard"
      },
      {
        "text": "the i j entry of A",
        "register": "spoken"
      }
    ],
    "spoken_ja": "a i j ／ (i, j) 成分",
    "name_en": "matrix entry",
    "name_ja": "行列の (i, j) 成分（aᵢⱼ）",
    "category": "matrices",
    "term_ref": "entry",
    "level": {
      "jp": [
        "数C",
        "大学"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "notes": [
      "aᵢⱼ は the entry in row i, column j（the i j entry of A）と読む。i 行 j 列の成分で、添字は行が先。",
      "OpenStax Precalculus 2e（9.5）は aᵢⱼ を the entry in row i, column j と説明する。Nicholson は the (i, j)-entry と書く（24 件）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "9.5 Matrices and Matrix Operations（aᵢⱼ、the entry in row i, column j。読みを決めた参照）"
      },
      {
        "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・2.3 ほか。the (i, j)-entry（24 件）。2.1 は aᵢⱼ を the entry in row i and column j of A とも説明する"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "the entry in row *": 2,
        "the i j entry | the i j-th entry | the ij entry | i comma j entry | the i j th entry | i-j entry | i jth entry | i-jth entry | (i, j)-entry | (i, j) entry | i, j entry | i, j-entry | ij-entry": 3
      },
      "sources": [
        "mit-18.06",
        "mit-6.042"
      ],
      "counted": "2026-09-28"
    },
    "flags": [
      {
        "code": "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: 対象外）。読みは OpenStax の呼び方 the entry in row *（本文 17 件）。register は主張しない。",
        "raised": "2026-09-28"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "matrix-inverse-notation",
    "latex": "A^{-1}",
    "spoken_en": [
      {
        "text": "A inverse",
        "register": "standard"
      },
      {
        "text": "the inverse of A",
        "register": "spoken"
      }
    ],
    "spoken_ja": "A インバース ／ A の逆行列",
    "name_en": "inverse matrix",
    "name_ja": "逆行列（A⁻¹）",
    "category": "matrices",
    "term_ref": "inverse-matrix",
    "level": {
      "jp": [
        "理数",
        "大学"
      ],
      "us": [
        "Precalculus",
        "Linear Algebra"
      ]
    },
    "notes": [
      "A⁻¹ は A inverse と読む。数の a⁻¹ = 1/a に当たる記号だが、行列の割り算は無いので one over A とは読まない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "9.7 Solving Systems with Inverses（A⁻¹）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "11.7 Solving Systems with Inverses（A⁻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": "2.4 Matrix Inverses"
      },
      {
        "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"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "a inverse | b inverse | m inverse | p inverse | s inverse | c inverse | ab inverse | a b inverse": 175,
        "the inverse of a | the inverse of the matrix | the inverse of b | the inverse of m | the inverse matrix": 56
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "max-min-notation",
    "latex": "\\max\\{a, b\\},\\ \\min\\{a, b\\}",
    "spoken_en": [
      {
        "text": "the maximum of a and b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a と b の大きい方",
    "name_en": "max and min notation",
    "name_ja": "max・min の記号",
    "category": "functions",
    "term_ref": "maximum",
    "level": {
      "jp": [],
      "us": [
        "Calculus II",
        "Discrete Math",
        "AP Statistics"
      ]
    },
    "notes": [
      "max{a, b} は the maximum of a and b と読む（min{a, b} は the minimum of a and b）。言葉で the larger of a and b（min は the smaller of a and b）と言い換えることもある。",
      "AP Statistics の CED は、2 標本の t の自由度の範囲を the smaller of … and … と言葉で書く（topic 4.7・4.10）。"
    ],
    "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 4.7・4.10。min を the smaller of … and … と書く（2 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Calculus of Parametric Curves で分割の幅の最大を max{ … } と書く（2 つの値の max{a, b} の形ではない）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "6.2（Pick the smaller of a and b）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "Sequences（Let N be the larger of N1 and N2）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the maximum of * and * | the minimum of * and *": 5
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-26。docs/DECISIONS.md「Phase 3 フレーズ 2 の前の修正」の 4）。the maximum of a and b。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "measure-of-angle",
    "latex": "m\\angle ABC",
    "spoken_en": [
      {
        "text": "the measure of angle ABC",
        "register": "standard"
      }
    ],
    "spoken_ja": "角 ABC の大きさ",
    "name_en": "measure of an angle",
    "name_ja": "角の大きさ（m∠ABC）",
    "category": "geometry",
    "term_ref": null,
    "related": [
      "figure-and-measure-notation"
    ],
    "level": {
      "jp": [],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Geometry"
      ]
    },
    "notes": [
      "m∠ABC は the measure of angle ABC と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.7 Introduction to Angles（the measure of the angle ABC を m∠ABC と書く。m∠ 399 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem（we read m∠A as the measure of angle A）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the measure of angle *": 63
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "yt:blackpenredpen",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "minus-plus-sign",
    "latex": "\\mp",
    "spoken_en": [
      {
        "text": "minus or plus",
        "register": "standard"
      }
    ],
    "spoken_ja": "マイナスプラス（複号同順）",
    "name_en": "minus-plus sign",
    "name_ja": "複号（∓）",
    "category": "arithmetic",
    "term_ref": null,
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Calculus I"
      ]
    },
    "notes": [
      "∓ は minus or plus と読む。cos(α ± β) = cos α cos β ∓ sin α sin β のように ± と組で使い、上の符号どうし・下の符号どうしを組にする（日本語の「複号同順」）。OpenStax Calculus Volume 1（1.3 Trigonometric Functions）の公式がこの形で ∓ を使う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.3 Trigonometric Functions の公式 cos(α ± β) = cos α cos β ∓ sin α sin β で ∓ を使う"
      },
      {
        "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": "加法定理（複号同順、cos(x ± y) = cos x cos y ∓ sin x sin y）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Plus%E2%80%93minus_sign",
        "note": "リード（minus or plus を太字）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "minus or plus": 3
      },
      "sources": [
        "khan-algebra"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 3 フレーズの前の修正」の 2）。minus or plus。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "minus-sign",
    "latex": "a - b",
    "spoken_en": [
      {
        "text": "a minus b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a ひく b ／ a マイナス b",
    "name_en": "minus sign",
    "name_ja": "ひく（−）",
    "category": "arithmetic",
    "term_ref": "subtraction",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "2 つの数の引き算の − は minus と読む。7 − 3 は seven minus three。",
      "負の数の符号 −3 は negative three と読む（negative-sign）。同じ記号でも読み方が違う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Subtract Whole Numbers・Introduction to Integers（10 − 4 is read 10 minus 4、−8 is read negative eight）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* minus *": 13318
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "negative-vs-minus"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "mixed-number",
    "latex": "2\\tfrac{3}{4}",
    "spoken_en": [
      {
        "text": "two and three fourths",
        "register": "standard"
      },
      {
        "text": "two and three quarters",
        "register": "spoken"
      }
    ],
    "spoken_ja": "2 と 4 分の 3",
    "name_en": "mixed number",
    "name_ja": "帯分数",
    "category": "fractions",
    "term_ref": null,
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "帯分数 2¾ は two and three fourths と読み、and で整数部分と分数部分をつなぐ。",
      "OpenStax Prealgebra（Visualize Fractions）と IM Grade 6 は帯分数（mixed number）を使う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「分数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%88%86%E6%95%B0",
        "note": "帯分数は「k と d 分の n」と読む、英語は and で読む"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "4.1 Visualize Fractions（mixed numbers）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6（three and one half cups ほか）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "and a quarter | and three quarters | and one quarter": 4,
        "and one fourth | and three fourths | and one third | and two thirds | and one fifth | and two fifths | and three fifths | and four fifths | and one half": 16
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "mu-population-mean",
    "latex": "\\mu,\\ m",
    "spoken_en": [
      {
        "text": "mu",
        "register": "standard"
      },
      {
        "text": "the population mean",
        "register": "standard"
      }
    ],
    "spoken_ja": "ミュー ／ 母平均 m",
    "name_en": "population mean (mu)",
    "name_ja": "母平均（μ、m）",
    "category": "probability-statistics",
    "term_ref": "population-mean",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "μ は mu と読む（Merriam-Webster: ˈmyü, ˈmü。OpenStax Introductory Statistics 2.5 は pronounced mew と書く）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "2.5 Measures of the Center of the Data（μ (pronounced mew)）"
      },
      {
        "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（量的データの推測）"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「mu」",
        "url": "https://www.merriam-webster.com/dictionary/mu",
        "note": "発音 ˈmyü, ˈmü（2026-09-27 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": "数学B 統計的な推測（母平均m、母標準偏差σの母集団）"
      },
      {
        "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（μ is the mean）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the population mean": 44,
        "mu": 97
      },
      "sources": [
        "khan-ap-stats",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "nabla",
    "latex": "\\nabla f",
    "spoken_en": [
      {
        "text": "del f",
        "register": "standard"
      },
      {
        "text": "the gradient of f",
        "register": "standard"
      }
    ],
    "spoken_ja": "ナブラ エフ",
    "name_en": "del (nabla)",
    "name_ja": "ナブラ",
    "category": "derivatives",
    "term_ref": "gradient",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "∇f は del f か the gradient of f と読む。OpenStax Calculus Volume 3 は ∇ を nabla と呼び、∇f を del f と読むと書く（grad f とも書く）。",
      "∂ も del と読むことがあるので、del と聞いたら ∇ か ∂ かを式で確かめる。",
      "∇ · F（発散）と ∇ × F（回転）の読みは divergence-del-dot・curl-del-cross を見る。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Directional Derivatives and the Gradient（∇ is called nabla、∇f is read del f）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the gradient of f": 12,
        "del f": 16
      },
      "sources": [
        "mit-18.02",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "natural-log-ln",
    "latex": "\\ln x",
    "spoken_en": [
      {
        "text": "the natural log of x",
        "register": "standard"
      },
      {
        "text": "l n x",
        "register": "standard"
      }
    ],
    "spoken_ja": "x の自然対数",
    "name_en": "natural logarithm",
    "name_ja": "自然対数（ln）",
    "category": "logarithms",
    "term_ref": "natural-logarithm",
    "related": [
      "log-and-ln"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "ln x は the natural log of x と読むか、文字どおり L N of x（l n x）と読む。",
      "ln |x| は the natural log of the absolute value of x。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 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"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「自然対数」",
        "url": "https://ja.wikipedia.org/wiki/%E8%87%AA%E7%84%B6%E5%AF%BE%E6%95%B0",
        "note": "ln x"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 2 4.14（the natural logarithm of x, written ln(x)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "natural log of * | natural log *": 874,
        "l n | ln": 744
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "natural-numbers-symbol",
    "latex": "\\mathbb{N}",
    "spoken_en": [
      {
        "text": "the natural numbers",
        "register": "standard"
      }
    ],
    "spoken_ja": "自然数全体の集合 N",
    "name_en": "natural numbers",
    "name_ja": "自然数全体の集合（ℕ）",
    "category": "sets",
    "term_ref": "natural-number",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "ℕ は the natural numbers と読む。文字どおり N（エヌ）とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "0.2（the ℕ symbol represents the natural numbers）・5.1 Sets（ℕ The set of natural numbers）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "the natural numbers | the set of natural numbers": 35
      },
      "sources": [
        "khan-middle",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "negation-sign",
    "latex": "\\neg p,\\ {\\sim}p,\\ \\bar{p}",
    "spoken_en": [
      {
        "text": "not p",
        "register": "standard"
      },
      {
        "text": "the negation of p",
        "register": "spoken"
      }
    ],
    "spoken_ja": "p でない ／ p バー ／ p の否定",
    "name_en": "negation",
    "name_ja": "否定",
    "category": "logic",
    "term_ref": "negation",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "notes": [
      "¬p（~p）は not p と読み、名前は the negation of p。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Foundation, CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.12 Converse, Inverse, and Contrapositive Statements（not p を ~p と書く）・2.10 Negative Statements"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1 Mathematical Statements（¬P is read not P）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "not p | not q": 11,
        "the negation of *": 1
      },
      "sources": [
        "mit-18.03",
        "mit-6.042",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "logic-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "negative-exponent-power",
    "latex": "x^{-1},\\ x^{-n}",
    "spoken_en": [
      {
        "text": "x to the negative one",
        "register": "standard"
      },
      {
        "text": "x to the negative first power",
        "register": "standard"
      }
    ],
    "spoken_ja": "x のマイナス 1 乗 ／ x のマイナス n 乗",
    "name_en": "negative exponent",
    "name_ja": "負の指数",
    "category": "exponents",
    "term_ref": "negative-exponent",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "notes": [
      "x⁻¹ は x to the negative one と読む（negative first power とも）。指数の − も negative と読む。",
      "x inverse と読むこともあるが、f⁻¹（逆関数、f inverse）や A⁻¹（逆行列）と同じ読みになる。数の逆数なら one over x とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 Unit 7（10 to the negative 4 power）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Properties of Exponents and Scientific Notation（a to the negative n equals one divided by a to the n）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "to the negative one !power !half | to the negative n !power": 91,
        "to the negative first power | to the negative one power": 35
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "negative-sign",
    "latex": "-3,\\ -x",
    "spoken_en": [
      {
        "text": "negative three",
        "register": "standard"
      },
      {
        "text": "minus three",
        "register": "spoken"
      },
      {
        "text": "the opposite of x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "マイナス 3 ／ 負の 3 ／ マイナス x",
    "name_en": "negative sign",
    "name_ja": "負の符号（−）",
    "category": "arithmetic",
    "term_ref": "negative-number",
    "related": [
      "negative-vs-minus"
    ],
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "x = −3 は x equals negative three と読む。",
      "−x は negative x とも the opposite of x とも読む。the opposite of は中学の教材（Khan Academy の中学）に多い。",
      "minus three と読むのは大学の講義にもある。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Introduction to Integers（−8 is read negative eight、−x is read the opposite of x）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "equals minus * | is equal to minus *": 145,
        "the opposite of *": 99,
        "equals negative * | is equal to negative *": 1126
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "normal-distribution-n",
    "latex": "N(\\mu, \\sigma),\\ N(m, \\sigma^2)",
    "spoken_en": [
      {
        "text": "normal with mean mu and standard deviation sigma",
        "register": "standard"
      }
    ],
    "spoken_ja": "正規分布 N(m, σ²)",
    "name_en": "normal distribution notation",
    "name_ja": "正規分布の記号（N(μ, σ)）",
    "category": "probability-statistics",
    "term_ref": "normal-distribution",
    "related": [
      "normal-distribution-parameter"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "N(μ, σ) は normal with mean mu and standard deviation sigma と読む。2 つ目の値の違いは normal-distribution-parameter。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "6.1 The Standard Normal Distribution ほか（X ~ N(μ, σ)）"
      },
      {
        "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": "試験の例題。normal with mean … and standard deviation …（2 件）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "normal with mean *": 2
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-26。docs/DECISIONS.md「Phase 3 フレーズ 2 の前の修正」の 4）。normal with mean mu and standard deviation sigma。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "not-element-of-sign",
    "latex": "a \\notin A",
    "spoken_en": [
      {
        "text": "a is not an element of A",
        "register": "standard"
      },
      {
        "text": "a is not a member of A",
        "register": "spoken"
      }
    ],
    "spoken_ja": "a は A に属さない",
    "name_en": "not an element of",
    "name_ja": "属さない（∉）",
    "category": "sets",
    "term_ref": "element",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "a ∉ A は a is not an element of A と読む。",
      "∉ は米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない。Levin は a ∉ A を is not an element of と言葉で書く（6 件）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets ほか。is not an element of（6 件。読みを決めた参照）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "is not a member of * | isn't a member of *": 2
      },
      "sources": [
        "khan-algebra",
        "khan-middle"
      ],
      "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: 対象外）。読みは Levin の呼び方 is not an element of * | isn't an element of *（5.1 Sets・1.5 Proofs about Discrete Structures・2.1 Problems and Definitions）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "not-equal-sign",
    "latex": "a \\neq b",
    "spoken_en": [
      {
        "text": "a does not equal b",
        "register": "standard"
      },
      {
        "text": "a is not equal to b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a は b に等しくない",
    "name_en": "not-equal sign",
    "name_ja": "等しくない（≠）",
    "category": "arithmetic",
    "term_ref": null,
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "≠ は does not equal（doesn't equal）か is not equal to と読む。x ≠ 0 は x is not equal to zero。",
      "OpenStax Elementary Algebra（1.1）は a ≠ b を a is not equal to b と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 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(3) 一次方程式（Ax = B（A ≠ 0））"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.2 Use the Language of Algebra の Inequality Symbols の表（a ≠ b is read a is not equal to b）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "is not equal to *": 72,
        "does not equal * | doesn't equal *": 117
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "nth-derivative-notation",
    "latex": "f^{(n)}(x),\\ f'''(x)",
    "spoken_en": [
      {
        "text": "the nth derivative of f",
        "register": "standard"
      },
      {
        "text": "f triple prime of x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "f の n 次導関数",
    "name_en": "nth derivative (higher-order derivatives)",
    "name_ja": "第 n 次導関数",
    "category": "derivatives",
    "term_ref": "nth-derivative",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "AP Calculus BC",
        "Calculus I",
        "Calculus II"
      ]
    },
    "notes": [
      "f⁽ⁿ⁾(x) は the nth derivative of f と読む。かっこ付きの (n) は指数ではない。",
      "f‴(x) は f triple prime of x。OpenStax Calculus Volume 1（The Derivative as a Function）は third derivative, fourth derivative, and so on と呼び、4 次以上を f⁽⁴⁾(x), …, f⁽ⁿ⁾(x) と書く。f⁽⁴⁾(x) は the fourth derivative of f と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Derivative as a Function（f″, f‴, f⁽⁴⁾, …, f⁽ⁿ⁾）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "the nth derivative": 32,
        "triple prime": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "nth-root-radical",
    "latex": "\\sqrt[n]{x}",
    "spoken_en": [
      {
        "text": "the nth root of x",
        "register": "standard"
      },
      {
        "text": "nth root of x",
        "register": "standard"
      }
    ],
    "spoken_ja": "n 乗根 x ／ x の n 乗根",
    "name_en": "nth root",
    "name_ja": "n 乗根",
    "category": "roots",
    "term_ref": "nth-root",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "ⁿ√x は the nth root of x と読み、4 乗根は the fourth root of x のように序数で言う。",
      "根号の左上の小さな数を index と呼ぶ。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the nth root of * | the fourth root of * | the fifth root of *": 34,
        "!the nth root of * | !the fourth root of * | !the fifth root of *": 12
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "numeric-fraction",
    "latex": "\\tfrac{3}{4},\\ \\tfrac{1}{2}",
    "spoken_en": [
      {
        "text": "one half",
        "register": "standard"
      },
      {
        "text": "three over four",
        "register": "standard"
      },
      {
        "text": "three fourths",
        "register": "standard"
      },
      {
        "text": "three quarters",
        "register": "standard"
      }
    ],
    "spoken_ja": "4 分の 3 ／ 2 分の 1",
    "name_en": "numerical fraction",
    "name_ja": "数の分数",
    "category": "fractions",
    "term_ref": "fraction",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "数の分数は分母を序数で読む: 3/4 は three fourths、2/3 は two thirds、分子が 2 以上なら序数に s を付ける。分母が 2 のときは half（1/2 は one half か a half、3/2 は three halves）。",
      "three over four とも読む。quarter は 4 分の 1 の別の言い方で、fourth と同じ意味。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「分数」",
        "url": "https://ja.wikipedia.org/wiki/%E5%88%86%E6%95%B0",
        "note": "読み（d 分の n）と英語の読み（one half、three halves）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6（three fourths、one half ほか）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "4.1 Visualize Fractions"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "one over two | one over four | three over four | one over three | two over three | one over five | two over five | three over five": 496,
        "one half": 608,
        "one quarter | three quarters": 25,
        "one fourth | three fourths | one third | two thirds | one fifth | two fifths | three fifths": 371
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "omega",
    "latex": "\\omega",
    "spoken_en": [
      {
        "text": "omega",
        "register": "standard"
      }
    ],
    "spoken_ja": "オメガ",
    "name_en": "omega",
    "name_ja": "ギリシャ文字 ω",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "notes": [
      "ω は omega と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "3.4 Motion in Space（angular speed ω）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "5.1 Angles（ω (read as omega) is angular speed）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Ω」",
        "url": "https://ja.wikipedia.org/wiki/%CE%A9",
        "note": "オメガ（英: omega）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "omega": 608
      },
      "sources": [
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "open-interval-parentheses",
    "latex": "(a, b)",
    "spoken_en": [
      {
        "text": "the open interval from a to b",
        "register": "standard"
      }
    ],
    "spoken_ja": "開区間 a b ／ a より大きく b より小さい",
    "name_en": "open interval",
    "name_ja": "開区間",
    "category": "sets",
    "term_ref": "open-interval",
    "related": [
      "interval-notation-vs-inequalities"
    ],
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "(a, b) は the open interval from a to b と読む。点 (a, b) と同じ形なので、区間なら interval を付けて言う。"
    ],
    "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 5.1（open interval）・Topic 5.6（open interval）・Topic 10.13（open interval）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "7.6 Solve Rational Inequalities（区間 (−3, 1)。level.us の Algebra 2）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "1.3 Rates of Change and Behavior of Graphs（increasing on the interval (1, 3)。level.us の Precalculus）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "4.4 The Mean Value Theorem（differentiable over the open interval (a, b)。level.us の Calculus I）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "open interval from * | open interval, from *": 16
      },
      "sources": [
        "khan-ap-calc"
      ],
      "counted": "2026-09-28"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "ordered-pair",
    "latex": "(x, y),\\ P(3, 4)",
    "spoken_en": [
      {
        "text": "the point three comma four",
        "register": "standard"
      },
      {
        "text": "the ordered pair x, y",
        "register": "spoken"
      }
    ],
    "spoken_ja": "点 (x, y) ／ 点 P(3, 4)",
    "name_en": "ordered pair",
    "name_ja": "座標（(x, y)）",
    "category": "geometry",
    "term_ref": "coordinates",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "(3, 4) は the point three comma four と読む。カンマを comma と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "11.1 Use the Rectangular Coordinate System（ordered pair (x, y)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "4.1 Use the Rectangular Coordinate System（ordered pair (x, y)）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 4.12・4.13 の図の説明（the point two comma eight）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the ordered pair *": 7,
        "the point * comma *": 184
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "p-hat-sample-proportion",
    "latex": "\\hat{p},\\ p'",
    "spoken_en": [
      {
        "text": "p hat",
        "register": "standard"
      },
      {
        "text": "the sample proportion",
        "register": "standard"
      }
    ],
    "spoken_ja": "p ハット ／ 標本比率",
    "name_en": "sample proportion (p-hat)",
    "name_ja": "標本比率（p̂）",
    "category": "probability-statistics",
    "term_ref": "sample-proportion",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "p̂ は p hat と読む。帽子の形の印を hat と読む。",
      "OpenStax Introductory Statistics は標本比率を p′ と書き、P prime と読む（8.3）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "8.3 A Population Proportion（p′ read P prime、P̂ read P hat）"
      },
      {
        "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（カテゴリデータの推測）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the sample proportion": 44,
        "p hat | p-hat": 63
      },
      "sources": [
        "khan-ap-stats",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "parallel-sign",
    "latex": "AB \\parallel CD,\\ AB /\\!/ CD",
    "spoken_en": [
      {
        "text": "AB is parallel to CD",
        "register": "standard"
      }
    ],
    "spoken_ja": "AB 平行 CD ／ AB と CD は平行",
    "name_en": "parallel sign",
    "name_ja": "平行の記号（∥ ／ //）",
    "category": "geometry",
    "term_ref": "parallel",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry"
      ]
    },
    "notes": [
      "AB ∥ CD は AB is parallel to CD と読む。直線 AB と直線 CD なら line AB is parallel to line CD。",
      "中学校学習指導要領〔用語・記号〕（中1）は // と書き、CK-12 Geometry は ∥ と書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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: //"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "3.2 Parallel and Skew Lines（Say It: Line AB is parallel to line MN）・∥（52 件）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "∥（26 件）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "is parallel to *": 35
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "parallelogram-abcd",
    "latex": "\\text{▱}ABCD",
    "spoken_en": [
      {
        "text": "parallelogram ABCD",
        "register": "standard"
      }
    ],
    "spoken_ja": "平行四辺形 ABCD",
    "name_en": "parallelogram notation",
    "name_ja": "平行四辺形の記号（▱ABCD）",
    "category": "geometry",
    "term_ref": "parallelogram",
    "level": {
      "jp": [
        "中2"
      ],
      "us": [
        "Geometry"
      ]
    },
    "notes": [
      "▱ABCD は parallelogram ABCD と読む。"
    ],
    "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 2.13 Proofs about Parallelograms ほか。parallelogram ABCD の形（15 件。決めた読みの言い方）"
      },
      {
        "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": "第2学年 B 図形 (2) イ(ア)（三角形の合同条件などを基にして三角形や平行四辺形の基本的な性質を論理的に確かめる。level.jp の中2: 平行四辺形を扱う学年）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "sources": [],
      "counted": "2026-09-28"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "字形と読みは人間が決めた。▱ABCD、parallelogram ABCD（IM）（2026-09-28。docs/DECISIONS.md「Phase 5 監査（セッション 11）の前の決定」の 2）。register は主張しない。",
        "raised": "2026-09-28"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "parenthesized-quantity",
    "latex": "(a + b)^2",
    "spoken_en": [
      {
        "text": "the quantity a plus b, squared",
        "register": "standard"
      },
      {
        "text": "a plus b, all squared",
        "register": "spoken"
      }
    ],
    "spoken_ja": "かっこ a たす b の 2 乗",
    "name_en": "the quantity (parentheses read aloud)",
    "name_ja": "かっこでくくった式",
    "category": "arithmetic",
    "term_ref": "parentheses",
    "level": {
      "jp": [
        "中1",
        "中3"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "かっこは読まず、the quantity でまとまりを示す。(a + b)² は the quantity a plus b, squared、2(x + 3) は two times the quantity x plus three。",
      "a plus b squared と続けて読むと a + b² と区別できない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 1 6.8 Equivalent Quadratic Expressions（x times the quantity x plus three）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the quantity *": 138,
        "all squared": 6
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "partial-derivative-leibniz",
    "latex": "\\dfrac{\\partial f}{\\partial x}",
    "spoken_en": [
      {
        "text": "the partial derivative of f with respect to x",
        "register": "standard"
      },
      {
        "text": "the partial of f with respect to x",
        "register": "standard"
      },
      {
        "text": "partial f partial x",
        "register": "standard"
      },
      {
        "text": "del f del x",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ラウンド f ラウンド x ／ デル f デル x",
    "name_en": "partial derivative",
    "name_ja": "偏微分（∂）",
    "category": "derivatives",
    "term_ref": "partial-derivative",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "∂f/∂x は the partial derivative of f with respect to x（短く the partial of f with respect to x）と言うか、記号どおりに partial f partial x と読む。",
      "∂ を del と読む講義もあるが、del は ∇ の読みでもある。",
      "OpenStax Calculus Volume 3（4.3）は ∂ を partial と呼び、∂f/∂x を the partial of f with respect to x と読むと書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "4.3 Partial Derivatives（the partial derivative of f with respect to x, written ∂f/∂x or f_x。∂f/∂x is spoken as the partial of f with respect to x）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the partial derivative of * with respect to": 40,
        "del * del *": 7,
        "the partial of * with respect to": 32,
        "partial f partial * | partial g partial * | partial h partial * | partial u partial * | partial v partial * | partial w partial * | partial x partial * | partial y partial * | partial z partial * | partial t partial * | partial r partial * | partial s partial * | partial p partial * | partial q partial *": 27
      },
      "sources": [
        "mit-18.02",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "partial-derivative-subscript",
    "latex": "f_x,\\ f_{xy}",
    "spoken_en": [
      {
        "text": "f sub x",
        "register": "standard"
      }
    ],
    "spoken_ja": "f x（偏導関数）",
    "name_en": "partial derivative (subscript notation)",
    "name_ja": "偏導関数の添字記法",
    "category": "derivatives",
    "term_ref": "partial-derivative",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "fₓ は f sub x（短く f x）と読み、x での偏導関数。fₓᵧ は f sub x y で、x で偏微分してから y で偏微分する。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Partial Derivatives（f_x）・Higher-Order Partial Derivatives（f_xy）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "f sub x | f sub y | f sub t": 19
      },
      "sources": [
        "mit-18.02"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "percent-sign",
    "latex": "25\\%",
    "spoken_en": [
      {
        "text": "twenty-five percent",
        "register": "standard"
      }
    ],
    "spoken_ja": "25 パーセント",
    "name_en": "percent sign",
    "name_ja": "パーセント（%）",
    "category": "arithmetic",
    "term_ref": "percent",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "25% は twenty-five percent。percent は数と組で使い、割合の大きさを言う名詞は percentage。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Understand Percent（percent）"
      },
      {
        "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"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* percent": 330
      },
      "sources": [
        "khan-algebra",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:nancypi",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "permutation-npr",
    "latex": "{}_n\\mathrm{P}_r = P(n, r)",
    "spoken_en": [
      {
        "text": "the number of permutations of n things taken r at a time",
        "register": "standard"
      },
      {
        "text": "n P r",
        "register": "standard"
      }
    ],
    "spoken_ja": "エヌ ピー アール",
    "name_en": "number of permutations (nPr)",
    "name_ja": "順列の数（ₙPᵣ）",
    "category": "probability-statistics",
    "term_ref": "permutation",
    "related": [
      "combination-permutation-notation"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "Discrete Math"
      ]
    },
    "notes": [
      "〔用語・記号〕（数A）は ₙPᵣ と書き、OpenStax Algebra and Trigonometry 2e ／ Precalculus 2e（13.5 ／ 11.5 Counting Principles）は P(n, r) と書く（各 11 件、数を入れた P(12, 9) の形が各 12 件）。nPr は同じ節で計算機の関数の名前として出てくる。日米の違いは combination-permutation-notation。",
      "ₙPᵣ・P(n, r) は the number of permutations of n things taken r at a time と読み、短く n P r とも言う（n P r は計算機の関数の名前）。OpenStax は公式の見出しでも taken r at a time の言い方を使う（Key Equations）。",
      "組合せの ₙCᵣ は C(n, r) と書き、n choose r と読む（combination-ncr）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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A 場合の数と確率: nPr"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.5 Counting Principles（P(n, r)、nPr（計算機の関数））"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "11.5 Counting Principles（P(n, r)、nPr（計算機の関数））"
      },
      {
        "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（We write this number P(n, k)）"
      }
    ],
    "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-human-settled",
        "note": "読みと記法は人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 2 幾何・離散の単元 2 の前の修正」の 5）。主の読みは the number of permutations of n things taken r at a time、n P r は別の読み。OpenStax の P(n, r) との記法の違いを notes に書いた（symbols に pitfalls の欄はない）。register は主張しない。",
        "raised": "2026-09-25"
      }
    ]
  },
  {
    "id": "perpendicular-sign",
    "latex": "AB \\perp CD",
    "spoken_en": [
      {
        "text": "AB is perpendicular to CD",
        "register": "standard"
      }
    ],
    "spoken_ja": "AB 垂直 CD ／ AB と CD は垂直",
    "name_en": "perpendicular sign",
    "name_ja": "垂直の記号（⊥）",
    "category": "geometry",
    "term_ref": "perpendicular",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Geometry"
      ]
    },
    "notes": [
      "AB ⊥ CD は AB is perpendicular to CD と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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: ⊥"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "⊥（52 件）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "⊥（32 件）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "is perpendicular to *": 63
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "phi",
    "latex": "\\varphi,\\ \\phi",
    "spoken_en": [
      {
        "text": "phi",
        "register": "standard"
      }
    ],
    "spoken_ja": "ファイ",
    "name_en": "phi",
    "name_ja": "ギリシャ文字 φ",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "φ は phi と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.7 Cylindrical and Spherical Coordinates（φ (the Greek letter phi)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "5.1 Angles（φ or ϕ、phi）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Φ」",
        "url": "https://ja.wikipedia.org/wiki/%CE%A6",
        "note": "ファイ（英: phi）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "phi": 226
      },
      "sources": [
        "khan-middle",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "pi-symbol",
    "latex": "\\pi",
    "spoken_en": [
      {
        "text": "pi",
        "register": "standard"
      }
    ],
    "spoken_ja": "パイ ／ 円周率",
    "name_en": "pi",
    "name_ja": "円周率（π）",
    "category": "geometry",
    "term_ref": "pi",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "notes": [
      "π は pi（パイ）と読む（Merriam-Webster: ˈpī。OpenStax Prealgebra 2e 5.3 は pronounced pie と書く）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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: π"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "5.3 Decimals and Fractions（π (pronounced pie)）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "円・扇形の節ほか"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「pi」",
        "url": "https://www.merriam-webster.com/dictionary/pi",
        "note": "発音 ˈpī（2026-09-27 に確かめた）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "pi": 4447
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "piecewise-brace",
    "latex": "f(x) = \\begin{cases} x^2 & (x \\geq 0) \\\\ -x & (x < 0) \\end{cases}",
    "spoken_en": [
      {
        "text": "f of x equals x squared when x is greater than or equal to zero, and negative x when x is less than zero",
        "register": "standard"
      },
      {
        "text": "x squared if x is greater than or equal to zero",
        "register": "spoken"
      },
      {
        "text": "x squared for x greater than or equal to zero",
        "register": "spoken"
      }
    ],
    "spoken_ja": "f(x) は x ≥ 0 のとき x²、x < 0 のとき −x",
    "name_en": "piecewise-defined function",
    "name_ja": "場合分けで定義した関数",
    "category": "functions",
    "term_ref": "piecewise-function",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB"
      ]
    },
    "notes": [
      "波かっこは読まず、式ごとに when x is …（if x is …、for x …）で条件を添える。",
      "OpenStax Precalculus（1.2 Domain and Range）は条件を x² if x ≥ 0 のように式の右に if を付けて書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "1.2 Domain and Range（piecewise notation、x if x ≥ 0）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Algebra 1 4.14 Absolute Value Functions（when x is less than zero and when it is greater than zero）"
      },
      {
        "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（g(x) = 0 (x < 0)、x² (x ≧ 0)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "when x is greater than | when x is less than": 91,
        "if x is greater than | if x is less than": 23,
        "for x greater than | for x less than": 18
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "plus-c-constant",
    "latex": "+\\, C",
    "spoken_en": [
      {
        "text": "plus C",
        "register": "standard"
      },
      {
        "text": "plus a constant",
        "register": "spoken"
      }
    ],
    "spoken_ja": "プラス C（C は積分定数）",
    "name_en": "constant of integration",
    "name_ja": "積分定数（+ C）",
    "category": "integrals",
    "term_ref": "constant-of-integration",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "不定積分の + C は plus C と読む。plus a constant とも言う。",
      "C の意味は the constant of integration。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      },
      {
        "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"
      },
      {
        "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（F(x) + C）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "plus a constant": 60,
        "!b !bx !a plus c !sub !times !prime !to !one !two": 243
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "constant-of-integration-remark"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "plus-minus-sign",
    "latex": "\\pm",
    "spoken_en": [
      {
        "text": "plus or minus",
        "register": "standard"
      }
    ],
    "spoken_ja": "プラスマイナス",
    "name_en": "plus-minus sign",
    "name_ja": "複号（±）",
    "category": "arithmetic",
    "term_ref": null,
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "notes": [
      "x = ±3 は x equals plus or minus three と読む。解の公式の ± も plus or minus。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「プラス記号とマイナス記号」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%97%E3%83%A9%E3%82%B9%E8%A8%98%E5%8F%B7%E3%81%A8%E3%83%9E%E3%82%A4%E3%83%8A%E3%82%B9%E8%A8%98%E5%8F%B7",
        "note": "± はプラスマイナス記号と呼ぶ"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "8 章 Confidence Intervals（the point estimate plus or minus the margin of error）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "plus or minus": 359
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "plus-sign",
    "latex": "a + b",
    "spoken_en": [
      {
        "text": "a plus b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a たす b ／ a プラス b",
    "name_en": "plus sign",
    "name_ja": "たす（＋）",
    "category": "arithmetic",
    "term_ref": "addition",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "+ は plus と読む。3 + 4 = 7 は three plus four equals seven（is seven とも）。",
      "式の中の + は項をつなぐ記号で、正の数の符号の +3 は positive three と読む（ふつうは書かない）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Use the Language of Algebra の表（a + b — Say: a plus b）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「プラス記号とマイナス記号」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%97%E3%83%A9%E3%82%B9%E8%A8%98%E5%8F%B7%E3%81%A8%E3%83%9E%E3%82%A4%E3%83%8A%E3%82%B9%E8%A8%98%E5%8F%B7",
        "note": "加算記号としては「足す（たす）」と読む"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* plus *": 14340
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "polar-form-cis",
    "latex": "r(\\cos\\theta + i\\sin\\theta),\\ r\\operatorname{cis}\\theta",
    "spoken_en": [
      {
        "text": "r cis theta",
        "register": "standard"
      },
      {
        "text": "r times cosine theta plus i sine theta",
        "register": "standard"
      }
    ],
    "spoken_ja": "極形式 r(cos θ + i sin θ)",
    "name_en": "polar form of a complex number",
    "name_ja": "複素数の極形式（r(cos θ + i sin θ)、r cis θ）",
    "category": "complex-numbers",
    "term_ref": "polar-form",
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus"
      ]
    },
    "notes": [
      "r cis θ は r(cos θ + i sin θ) の略記で、r cis theta と読む。略さずに書いたときは r times cosine theta plus i sine theta と読む。",
      "cis は OpenStax Precalculus（と Algebra and Trigonometry）の略記（We often use the abbreviation … の形で導入する）。日本の高等学校学習指導要領解説には cis が出てこない（日本語版 Wikipedia「複素数」は r cis(φ) と書くこともあると説明する）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "8.5 Polar Form of Complex Numbers（We often use the abbreviation r cis θ）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "10.5 Polar Form of Complex Numbers（r cis θ）"
      },
      {
        "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": "極形式（r cis(φ) と書くこともある）"
      },
      {
        "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C 複素数平面（極形式は出てくるが cis は出てこない）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "cosine theta plus i sine theta | cos theta plus i sin theta | cosine theta plus i sin theta": 5
      },
      "sources": [
        "mit-18.03",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 3 フレーズの前の修正」の 2）。r cis theta ／ r times cosine theta plus i sine theta。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "power-cubed",
    "latex": "x^3",
    "spoken_en": [
      {
        "text": "x cubed",
        "register": "standard"
      },
      {
        "text": "x to the third",
        "register": "standard"
      },
      {
        "text": "x to the third power",
        "register": "standard"
      }
    ],
    "spoken_ja": "x の 3 乗",
    "name_en": "cube (third power)",
    "name_ja": "3 乗",
    "category": "exponents",
    "term_ref": "cube",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "x³ は x cubed か x to the third（power）と読む。2 乗の squared と同じく、3 乗だけに cubed という特別な読みがある。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "指数の読み方の表（five to the third power, or five cubed）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 1.17 Squares and Cubes（four cubed、to the third power）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* cubed": 809,
        "to the third !power": 577,
        "to the third power": 477
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "power-fourth",
    "latex": "x^4",
    "spoken_en": [
      {
        "text": "x to the fourth",
        "register": "standard"
      },
      {
        "text": "x to the fourth power",
        "register": "spoken"
      },
      {
        "text": "x to the power of four",
        "register": "spoken"
      }
    ],
    "spoken_ja": "x の 4 乗",
    "name_en": "fourth power",
    "name_ja": "4 乗",
    "category": "exponents",
    "term_ref": "power",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "notes": [
      "4 乗以上は序数で to the fourth、to the fifth と読み、power を付けてもよい。x⁵ は x to the fifth。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 6.12 Meaning of Exponents（raised to the power of five、to the fifth power、to the fifth）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "指数の読み方の表（nine to the fourth power）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "to the fourth !power": 529,
        "to the fourth power": 169,
        "to the power of four": 1
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "power-n",
    "latex": "x^n",
    "spoken_en": [
      {
        "text": "x to the n",
        "register": "standard"
      },
      {
        "text": "x to the nth power",
        "register": "spoken"
      },
      {
        "text": "x to the nth",
        "register": "spoken"
      },
      {
        "text": "x to the power of n",
        "register": "spoken"
      }
    ],
    "spoken_ja": "x の n 乗",
    "name_en": "nth power",
    "name_ja": "n 乗",
    "category": "exponents",
    "term_ref": "power",
    "level": {
      "jp": [
        "中1",
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "notes": [
      "xⁿ は x to the n と読む。x to the nth（power）とも言う。",
      "x^{n+1} は x to the n plus one と読み、xⁿ + 1 と聞き分けにくい（expression-exponent）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "指数の読み方（aⁿ is read a to the nth power）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "to the n !power !plus !minus": 408,
        "to the nth power": 32,
        "to the nth !power !derivative !point !component !term": 13,
        "to the power of n": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "power-set-notation",
    "latex": "\\mathcal{P}(A)",
    "spoken_en": [
      {
        "text": "the power set of A",
        "register": "standard"
      }
    ],
    "spoken_ja": "A のべき集合",
    "name_en": "power set",
    "name_ja": "べき集合",
    "category": "sets",
    "term_ref": "power-set",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "𝒫(A) は the power set of A と読む。",
      "べき集合は米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない。Levin は 𝒫(A) を the power set of A と書く（5.1 Sets、4 件）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets。𝒫(A) を the power set of A と書く（4 件。読みを決めた参照）"
      }
    ],
    "confidence": "verified",
    "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: 対象外）。読みは Levin の呼び方 the power set of * | power set（5.1 Sets）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "power-squared",
    "latex": "x^2",
    "spoken_en": [
      {
        "text": "x squared",
        "register": "standard"
      },
      {
        "text": "x to the second power",
        "register": "spoken"
      }
    ],
    "spoken_ja": "x の 2 乗",
    "name_en": "square (second power)",
    "name_ja": "2 乗",
    "category": "exponents",
    "term_ref": "square",
    "level": {
      "jp": [
        "中1",
        "中2"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "2 乗は squared、3 乗は cubed、4 乗以上は 'to the fourth (power)'。"
    ],
    "sources": [
      {
        "type": "editorial"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "x squared": 2719,
        "x to the second power": 11
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "prime-label-image",
    "latex": "A',\\ A''",
    "spoken_en": [
      {
        "text": "A prime",
        "register": "standard"
      }
    ],
    "spoken_ja": "A ダッシュ ／ A ツーダッシュ",
    "name_en": "prime (image of a point)",
    "name_ja": "移動後の点（A′、A″）",
    "category": "geometry",
    "term_ref": "image",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "notes": [
      "A′ は A prime、A″ は A double prime と読む。日本語版 Wikipedia「プライム」は ′ を日本ではダッシュと呼ぶとする（″ はツーダッシュ）。英語では prime。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.5 Geometric Translation（A′ said a prime、A′′ said a double prime）・A′（89 件）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「プライム」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%97%E3%83%A9%E3%82%A4%E3%83%A0",
        "note": "呼称（日本ではダッシュ、″ はツーダッシュ）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 1.4 Making the Moves（A′ (pronounced A prime)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!is !not !have !of !all !like !it's !what a prime !number !numbers !factor !factors !factorization !suspect !or !source !course | b prime | c prime | d prime": 57
      },
      "sources": [
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "probability-of-a",
    "latex": "P(A)",
    "spoken_en": [
      {
        "text": "the probability of A",
        "register": "standard"
      },
      {
        "text": "P of A",
        "register": "spoken"
      }
    ],
    "spoken_ja": "A の確率 ／ P(A)",
    "name_en": "probability of an event",
    "name_ja": "事象 A の確率（P(A)）",
    "category": "probability-statistics",
    "term_ref": "probability",
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "P(A) は the probability of A か P of A と読む。P of A は関数の値 p(a) と同じ読みになる。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（The probability of an event A is written P(A)）・付録 Mathematical Phrases, Symbols, and Formulas（P(A)、probability of A）"
      },
      {
        "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 2.4（P(E)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the probability of event a | the probability of event b | the probability of event e | the probability of a occurring | the probability of b occurring | the probability of a happening | the probability of b happening": 13,
        "p of a | p of b | p of e | p of not a | p of a complement": 12
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-6.042",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "probability-of-intersection",
    "latex": "P(A \\cap B)",
    "spoken_en": [
      {
        "text": "the probability of A and B",
        "register": "standard"
      },
      {
        "text": "the probability of A intersect B",
        "register": "spoken"
      }
    ],
    "spoken_ja": "A かつ B の確率 ／ P(A キャップ B)",
    "name_en": "probability of A and B",
    "name_ja": "積事象の確率（P(A ∩ B)）",
    "category": "probability-statistics",
    "term_ref": "intersection-of-events",
    "related": [
      "probability-and-or-notation"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "P(A ∩ B) は the probability of A and B と読む。OpenStax Introductory Statistics は P(A AND B) と書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（P(A AND B)、A Intersection 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 2.5（P(A intersect B)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the probability of a and b | the probability of both *": 18,
        "the probability of a intersect b | probability of a intersection b": 1
      },
      "sources": [
        "mit-6.042",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "probability-of-union",
    "latex": "P(A \\cup B)",
    "spoken_en": [
      {
        "text": "the probability of A or B",
        "register": "standard"
      },
      {
        "text": "the probability of A union B",
        "register": "standard"
      }
    ],
    "spoken_ja": "A または B の確率 ／ P(A カップ B)",
    "name_en": "probability of A or B",
    "name_ja": "和事象の確率（P(A ∪ B)）",
    "category": "probability-statistics",
    "term_ref": "union-of-events",
    "related": [
      "probability-and-or-notation"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": [
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "P(A ∪ B) は the probability of A or B、または the probability of A union B と読む。OpenStax Introductory Statistics は P(A OR B) と書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（P(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。P(A ∪ B) の記号を使う"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the probability of a union b": 1,
        "the probability of a or b | the probability of either *": 4
      },
      "sources": [
        "mit-6.042",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-26。docs/DECISIONS.md「Phase 3 フレーズ 2 の前の修正」の 4）。the probability of A or B ／ the probability of A union B。register は主張しない。",
        "raised": "2026-09-26"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "probability-x-equals-k",
    "latex": "P(X = k)",
    "spoken_en": [
      {
        "text": "the probability that X equals k",
        "register": "standard"
      },
      {
        "text": "P of X equals k",
        "register": "spoken"
      }
    ],
    "spoken_ja": "X = k となる確率",
    "name_en": "probability that X equals k",
    "name_ja": "確率変数が値をとる確率（P(X = k)）",
    "category": "probability-statistics",
    "term_ref": "probability-distribution",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "P(X = k) は the probability that X equals k と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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 2（確率変数）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "p of x equals * | p of x is equal to": 8,
        "the probability that x equals * | the probability that x is equal to * | the probability that x is *": 40
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "product-pi-notation",
    "latex": "\\prod_{i=1}^{n} a_i",
    "spoken_en": [
      {
        "text": "the product from i equals one to n of a sub i",
        "register": "standard"
      }
    ],
    "spoken_ja": "パイ（総乗）、i = 1 から n まで a i",
    "name_en": "product notation",
    "name_ja": "総乗（∏）",
    "category": "sums",
    "term_ref": "product",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "∏ は the product from i equals one to n of a sub i と読む（Σ の the sum from k equals one to n of a sub k と同じ形）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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": "総乗記号 ∏"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "4.1 Describing Sequences の練習問題（Use summation (Σ) or product (Π) notation。name_en と level.us の Discrete Math）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-27。docs/DECISIONS.md「Phase 5 監査（セッション 10）の前の決定」の 3）。the product from i equals one to n of a sub i。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "projection-notation",
    "latex": "\\operatorname{proj}_{\\vec{v}} \\vec{u}",
    "spoken_en": [
      {
        "text": "the projection of u onto v",
        "register": "standard"
      }
    ],
    "spoken_ja": "u の v への正射影ベクトル",
    "name_en": "vector projection",
    "name_ja": "正射影ベクトル（proj_v u）",
    "category": "vectors",
    "term_ref": "vector-projection",
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "notes": [
      "proj_v u は the projection of u onto v と読む。onto の後ろが射影する先のベクトル。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.3 The Dot Product（proj_u v、the projection of v onto u。読みを決めた参照）"
      },
      {
        "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"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the projection of * onto *": 8
      },
      "sources": [
        "mit-18.02",
        "mit-18.06",
        "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 8 件（10 未満） ／ 書: 対象外）。読みは OpenStax の呼び方 the projection of * onto *（本文 20 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "proper-subset-sign",
    "latex": "A \\subsetneq B",
    "spoken_en": [
      {
        "text": "A is a proper subset of B",
        "register": "standard"
      }
    ],
    "spoken_ja": "A は B の真部分集合",
    "name_en": "proper subset",
    "name_ja": "真部分集合（⊊）",
    "category": "sets",
    "term_ref": "subset",
    "related": [
      "subset-symbol"
    ],
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "A ⊊ B は A is a proper subset of B と読む。Levin は真部分集合を A ⊂ B と書き、A ⊆ B（部分集合）と分ける（5.1 Sets）。",
      "真部分集合は米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない。Levin は is a proper subset of と書く（3 件）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.5・5.1 Sets。is a proper subset of（3 件。読みを決めた参照）"
      }
    ],
    "confidence": "verified",
    "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: 対象外）。読みは Levin の呼び方 proper subset（1.5 Proofs about Discrete Structures・5.1 Sets）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "proportion-colon",
    "latex": "a : b = c : d",
    "spoken_en": [
      {
        "text": "a is to b as c is to d",
        "register": "standard"
      }
    ],
    "spoken_ja": "a 対 b イコール c 対 d",
    "name_en": "proportion",
    "name_ja": "比例式",
    "category": "arithmetic",
    "term_ref": "proportion",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "a : b = c : d は a is to b as c is to d と読む。OpenStax Prealgebra（Solve Proportions and their Applications）は比例式を a/b = c/d と書き、a is to b as c is to d と読むと示す。分数の形は a over b equals c over d とも読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Solve Proportions and their Applications（a is to b, as c is to d。規則 2 で読みを決めた参照。OpenStax 全体で 18 件）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「比」",
        "url": "https://ja.wikipedia.org/wiki/%E6%AF%94",
        "note": "a:b は「a対b」と読む、比例式"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "is to * as * is to": 3
      },
      "sources": [
        "khan-middle",
        "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 3 件（10 未満） ／ 書: 対象外）。読みは OpenStax の呼び方 is to * as * is to（本文 18 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "qed-end-of-proof",
    "latex": "\\blacksquare,\\ \\square,\\ \\text{Q.E.D.}",
    "spoken_en": [
      {
        "text": "end of proof",
        "register": "standard"
      },
      {
        "text": "Q.E.D.",
        "register": "standard"
      }
    ],
    "spoken_ja": "証明終わり",
    "name_en": "end-of-proof mark",
    "name_ja": "証明終わりの印",
    "category": "logic",
    "term_ref": null,
    "related": [
      "written-solution-as-desired"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Calculus I",
        "Linear Algebra",
        "Discrete Math"
      ]
    },
    "notes": [
      "記号そのものは読まないことが多い。書き言葉では This completes the proof. のような文で締める（Nicholson・Levin）。",
      "Q.E.D. は文字どおり Q E D と読む（Merriam-Webster: ˌkyü-ˌē-ˈdē）。",
      "OpenStax Calculus は証明の終わりに □ を置く。"
    ],
    "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": "2.5・2.7・3.2・5.7 と巻末の Selected Exercise Answers（2.7 の練習問題の解答）（this completes the proof、5 件。notes[0] の書き言葉の締め）"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「QED」",
        "url": "https://www.merriam-webster.com/dictionary/QED",
        "note": "発音 ˌkyü-ˌē-ˈdē（2026-09-27 に確かめた）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.5 Exponential and Logarithmic Functions・2.4 Continuity・2.5 The Precise Definition of a Limit ほか（証明の終わりに □。notes[2] と level.us の Calculus I）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.4 Proofs（This completes the proof. ■。notes[0] の書き言葉の締め）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "end of proof | end of the proof": 5,
        "q e d | qed | q.e.d": 2
      },
      "sources": [
        "mit-18.01",
        "mit-18.06",
        "mit-6.042"
      ],
      "counted": "2026-09-28"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた。end of proof ／ Q.E.D.（2026-09-28。docs/DECISIONS.md「Phase 5 監査（セッション 11）の前の決定」の 2）。register は主張しない。",
        "raised": "2026-09-28"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "quartile-q1-q3",
    "latex": "Q_1,\\ Q_3",
    "spoken_en": [
      {
        "text": "Q one",
        "register": "standard"
      },
      {
        "text": "the first quartile",
        "register": "spoken"
      }
    ],
    "spoken_ja": "第 1 四分位数 Q₁",
    "name_en": "quartiles (Q1, Q3)",
    "name_ja": "四分位数（Q₁、Q₃）",
    "category": "probability-statistics",
    "term_ref": "quartile",
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "Q₁ は Q one、Q₃ は Q three と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "2.3 Measures of the Location of the Data・付録 Mathematical Phrases, Symbols, and Formulas（Q₁、Quartile one）"
      },
      {
        "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（量的データの記述）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "q one | q three | q1 | q3": 126,
        "the first quartile | the third quartile": 20
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.03",
        "mit-18.06",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "r-squared",
    "latex": "r^2",
    "spoken_en": [
      {
        "text": "r squared",
        "register": "standard"
      },
      {
        "text": "the coefficient of determination",
        "register": "spoken"
      }
    ],
    "spoken_ja": "r の 2 乗 ／ 決定係数",
    "name_en": "coefficient of determination (r-squared)",
    "name_ja": "決定係数（r²）",
    "category": "probability-statistics",
    "term_ref": "coefficient-of-determination",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "r² は r squared と読む。記号の読みなので、円の面積 πr² の r squared と同じ言い方。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "12.3 The Regression Equation（The variable r² is called the coefficient of determination）"
      },
      {
        "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（回帰）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "r squared | r-squared": 259,
        "coefficient of determination": 4
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "radian-unit",
    "latex": "\\tfrac{\\pi}{2}\\ \\text{rad}",
    "spoken_en": [
      {
        "text": "pi over two radians",
        "register": "standard"
      }
    ],
    "spoken_ja": "2 分の π ラジアン",
    "name_en": "radian",
    "name_ja": "ラジアン",
    "category": "trigonometry",
    "term_ref": "radian",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "π/2 rad は pi over two radians と読む。OpenStax Precalculus（5.1 Angles）は radian を単位のない量とし、書かなくてよいとする。単位を書かない π/2 も radians で測った角。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "5.1 Angles（radians are a unitless measure）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Geometry 7.12 Radian Sense（π/3 radians）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「ラジアン」",
        "url": "https://ja.wikipedia.org/wiki/%E3%83%A9%E3%82%B8%E3%82%A2%E3%83%B3",
        "note": "ラジアン"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* radians": 252
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "ratio-colon",
    "latex": "a : b",
    "spoken_en": [
      {
        "text": "the ratio of a to b",
        "register": "standard"
      }
    ],
    "spoken_ja": "a 対 b",
    "name_en": "ratio",
    "name_ja": "比（:）",
    "category": "arithmetic",
    "term_ref": "ratio",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "a : b は a to b と読み、比であることを言うときは the ratio of a to b。3 : 4 は three to four。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "Ratios and Rate（The ratio of a to b is written a to b, a/b, or a : b）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 2.1 Introducing Ratios and Ratio Language（the ratio of … to …）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「比」",
        "url": "https://ja.wikipedia.org/wiki/%E6%AF%94",
        "note": "a:b は「a対b」と読む"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the ratio of * to *": 122
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "rational-expression-fraction",
    "latex": "\\dfrac{x + 1}{x - 1}",
    "spoken_en": [
      {
        "text": "x plus one, all over x minus one",
        "register": "standard"
      },
      {
        "text": "the quantity x plus one over the quantity x minus one",
        "register": "spoken"
      }
    ],
    "spoken_ja": "x − 1 分の x + 1",
    "name_en": "fraction with expressions",
    "name_ja": "式の分数",
    "category": "fractions",
    "term_ref": "rational-expression",
    "level": {
      "jp": [
        "数II"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2"
      ]
    },
    "notes": [
      "分子・分母が式の分数は、all over で分子全体を分母で割ることを示す: x plus one, all over x minus one。",
      "the quantity x plus one over the quantity x minus one と、かっこの代わりに the quantity を付けても言える。"
    ],
    "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 (1) いろいろな式（分数式）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "7 章 Rational Expressions and Functions（rational expression）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "all over *": 130,
        "over the quantity *": 6
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "rationals-symbol",
    "latex": "\\mathbb{Q}",
    "spoken_en": [
      {
        "text": "the rational numbers",
        "register": "standard"
      }
    ],
    "spoken_ja": "有理数全体の集合 Q",
    "name_en": "rational numbers",
    "name_ja": "有理数全体の集合（ℚ）",
    "category": "sets",
    "term_ref": "rational-number",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "ℚ は the rational numbers と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "1.1 Real Numbers: Algebra Essentials（the rational numbers ／ the set of rational numbers。OpenStax 4 冊で 11 件。読みを決めた参照）"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "the rational numbers（5 件）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets（ℚ The set of rational numbers）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the rational numbers | the set of rational numbers": 4
      },
      "sources": [
        "khan-algebra",
        "khan-middle"
      ],
      "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: 対象外）。読みは OpenStax の呼び方 the rational numbers | the set of rational numbers（本文 11 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "ray-ab-arrow",
    "latex": "\\overrightarrow{AB}",
    "spoken_en": [
      {
        "text": "ray AB",
        "register": "standard"
      }
    ],
    "spoken_ja": "半直線 AB",
    "name_en": "ray notation",
    "name_ja": "半直線の記号（矢印の AB）",
    "category": "geometry",
    "term_ref": "ray",
    "level": {
      "jp": [],
      "us": [
        "Geometry",
        "Precalculus"
      ]
    },
    "notes": [
      "矢印の AB は ray AB と読む（CK-12 Geometry 1.1 Geometry Terms）。始点の A を先に言う。同じ形の矢印はベクトル AB にも使う（vector-ab-arrow）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.1 Geometry Terms（Say It: Ray CD）・矢印の AB（39 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "5.1 Angles（ray EF, or in symbol form EF→）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "ray a b | ray ab | ray b a | ray ba | ray b c | ray bc | ray a c | ray ac | ray c d | ray cd | ray a d | ray ad | ray b d | ray bd": 17
      },
      "sources": [
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "figure-and-measure-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "reals-symbol",
    "latex": "\\mathbb{R},\\ \\mathbb{R}^2,\\ \\mathbb{R}^n",
    "spoken_en": [
      {
        "text": "the real numbers",
        "register": "standard"
      },
      {
        "text": "the reals",
        "register": "spoken"
      }
    ],
    "spoken_ja": "実数全体の集合 R ／ R 2",
    "name_en": "real numbers",
    "name_ja": "実数全体の集合（ℝ）",
    "category": "sets",
    "term_ref": "real-number",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus I",
        "Calculus III",
        "Linear Algebra",
        "Discrete Math"
      ]
    },
    "notes": [
      "ℝ は the real numbers（the reals）と読む。ℝ² は R two（平面）、ℝⁿ は R n と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.2 Vectors in Three Dimensions（ℝ・ℝ³）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.1 Review of Functions（a subset of all real numbers, denoted ℝ）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets（ℝ The set of real numbers）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the real numbers | the set of real numbers": 34,
        "the reals": 4
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "repeated-combination-h-jp",
    "latex": "{}_{n}\\mathrm{H}_{r}",
    "spoken_en": [
      {
        "text": "the number of combinations with repetition of n things taken r at a time",
        "register": "standard"
      }
    ],
    "spoken_ja": "エヌ エイチ アール ／ 重複組合せの数",
    "name_en": "combinations with repetition (Japanese notation)",
    "name_ja": "重複組合せの記号（ₙHᵣ）",
    "category": "probability-statistics",
    "term_ref": "combination-with-repetition",
    "related": [
      "repeated-combination-notation"
    ],
    "level": {
      "jp": [
        "数A"
      ],
      "us": []
    },
    "notes": [
      "ₙHᵣ は日本語版 Wikipedia「重複組合せ」が使う記号。読みは the number of combinations with repetition of n things taken r at a time。",
      "英語で書くときは二項係数 C(n + r − 1, r) に直す（Levin 3.5。repeated-combination-notation）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction (4th edition)",
        "url": "https://discrete.openmathbooks.org/pdfs/dmoi4.pdf",
        "note": "3.5 Counting Multisets（sticks and stones。stars and bars・balls and bins とも）"
      },
      {
        "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": "H の記号"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "sources": [],
      "counted": "2026-09-26"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 3 フレーズの前の修正」の 2）。the number of combinations with repetition of n things taken r at a time。米国にこの記号はない。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "repeating-decimal-bar",
    "latex": "0.\\overline{3}",
    "spoken_en": [
      {
        "text": "zero point three repeating",
        "register": "standard"
      },
      {
        "text": "zero point three with a bar over the three",
        "register": "spoken"
      }
    ],
    "spoken_ja": "れい てん さん さん …",
    "name_en": "repeating decimal bar",
    "name_ja": "循環小数の上線",
    "category": "arithmetic",
    "term_ref": "repeating-decimal",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "0.3̄ は zero point three repeating と読むか、横線を指して with a bar over the three と言う。",
      "OpenStax Prealgebra・Elementary Algebra は循環する部分全体の上に横線（bar）を引く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "循環節の上に横線（bar）を引く"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "循環節全体の上に横線を引く書き方"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 8.14 Decimal Representations of Rational Numbers（the bar over part of the expansion）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Rational and irrational numbers の説明（the bar over the 3）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "point * repeating | three repeating | six repeating | nine repeating | one repeating | two repeating | repeating forever": 24,
        "bar over the * | line over the * | bar on top": 5
      },
      "sources": [
        "khan-ap-stats",
        "khan-middle",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "rho",
    "latex": "\\rho",
    "spoken_en": [
      {
        "text": "rho",
        "register": "standard"
      }
    ],
    "spoken_ja": "ロー",
    "name_en": "rho",
    "name_ja": "ギリシャ文字 ρ",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus II",
        "Calculus III",
        "Intro Statistics"
      ]
    },
    "notes": [
      "ρ は rho と読む。",
      "球面座標の ρ、母相関係数の ρ に使う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.7 Cylindrical and Spherical Coordinates（ρ (the Greek letter rho)）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2",
        "note": "5.6 Ratio and Root Tests（ρ）・2.5 Physical Applications（density ρ）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "12.4 Testing the Significance of the Correlation Coefficient（ρ, the Greek letter rho）・付録 Mathematical Phrases, Symbols, and Formulas（rho）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Ρ」",
        "url": "https://ja.wikipedia.org/wiki/%CE%A1",
        "note": "ロー（英: rho）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "rho": 157
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "roster-braces",
    "latex": "\\{1, 2, 3\\}",
    "spoken_en": [
      {
        "text": "the set containing one, two, and three",
        "register": "standard"
      },
      {
        "text": "the set one, two, three",
        "register": "spoken"
      }
    ],
    "spoken_ja": "1, 2, 3 を要素とする集合",
    "name_en": "roster notation",
    "name_ja": "要素を列挙して表す集合",
    "category": "sets",
    "term_ref": "set",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Intro Statistics",
        "Discrete Math"
      ]
    },
    "notes": [
      "{1, 2, 3} は the set one, two, three か the set containing one, two, and three と読む。波かっこは読まない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets（{1, 2, 3} を the set containing 1, 2, and 3 と読む。読みを決めた参照）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "11.7 Probability（roster notation）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「集合」",
        "url": "https://ja.wikipedia.org/wiki/%E9%9B%86%E5%90%88",
        "note": "記法（要素をすべて列挙する方法、条件を明示する方法）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the set one | the set zero | the set a comma": 7
      },
      "sources": [
        "khan-algebra",
        "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 7 件（10 未満） ／ 書: 対象外）。読みは Levin の呼び方 the set containing *（5.1 Sets・1.5 Proofs about Discrete Structures）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "s-sample-sd",
    "latex": "s,\\ s_x",
    "spoken_en": [
      {
        "text": "the sample standard deviation",
        "register": "standard"
      }
    ],
    "spoken_ja": "s ／ 標本標準偏差",
    "name_en": "sample standard deviation",
    "name_ja": "標本標準偏差（s、sₓ）",
    "category": "probability-statistics",
    "term_ref": "standard-deviation",
    "level": {
      "jp": [
        "数I",
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "s は the sample standard deviation と読む。記号は文字どおり s。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "2.7 Measures of the Spread of the Data・付録 Mathematical Phrases, Symbols, and Formulas（s、sample standard deviation）"
      },
      {
        "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（量的データの記述）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the sample standard deviation": 36
      },
      "sources": [
        "khan-ap-stats",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "sample-variance-n-minus-1"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "scientific-notation-form",
    "latex": "3.2 \\times 10^5",
    "spoken_en": [
      {
        "text": "three point two times ten to the fifth",
        "register": "standard"
      }
    ],
    "spoken_ja": "3.2 かける 10 の 5 乗",
    "name_en": "scientific notation",
    "name_ja": "a × 10ⁿ の形",
    "category": "arithmetic",
    "term_ref": "scientific-notation",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "3.2 × 10⁵ は three point two times ten to the fifth（power）と読む。10⁻³ は ten to the negative three（power）、ten to the negative third とも。",
      "× はここでは times と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "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 D(1) 近似値（a × 10ⁿ の形で表す）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 Unit 7（10 to the negative 4 power）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "times ten to the *": 276
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "se-standard-error",
    "latex": "\\mathrm{SE},\\ \\mathrm{SE}(\\bar{x})",
    "spoken_en": [
      {
        "text": "the standard error",
        "register": "standard"
      }
    ],
    "spoken_ja": "標準誤差",
    "name_en": "standard error (SE)",
    "name_ja": "標準誤差（SE）",
    "category": "probability-statistics",
    "term_ref": "standard-error",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "SE は the standard error と読む。"
    ],
    "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": "Unit 3・4"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the standard error": 18
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "secant-of-theta",
    "latex": "\\sec\\theta",
    "spoken_en": [
      {
        "text": "secant theta",
        "register": "standard"
      },
      {
        "text": "secant of theta",
        "register": "spoken"
      },
      {
        "text": "sec theta",
        "register": "spoken"
      }
    ],
    "spoken_ja": "セカント シータ",
    "name_en": "secant",
    "name_ja": "セカント",
    "category": "trigonometry",
    "term_ref": "secant",
    "related": [
      "reciprocal-trig-functions"
    ],
    "level": {
      "jp": [],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "sec θ は secant theta（secant of theta）と読む。sec² x は secant squared x。",
      "sec θ = 1/cos θ。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e"
      },
      {
        "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": "読み（sin: サイン、cos: コサイン、tan: タンジェント、sec: セカント、cosec: コセカント、cot: コタンジェント）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "secant theta | secant x | secant squared": 255,
        "sec theta | sec x": 35,
        "secant of *": 61
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "second-derivative-leibniz",
    "latex": "\\dfrac{d^2 y}{dx^2}",
    "spoken_en": [
      {
        "text": "the second derivative of y with respect to x",
        "register": "standard"
      },
      {
        "text": "d squared y d x squared",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ディー 2 ワイ ディーエックス 2 乗",
    "name_en": "second derivative (Leibniz notation)",
    "name_ja": "第 2 次導関数（ライプニッツの記法）",
    "category": "derivatives",
    "term_ref": "second-derivative",
    "level": {
      "jp": [
        "数III"
      ],
      "us": [
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "d²y/dx² は記号どおりに d squared y d x squared とも読み、the second derivative of y with respect to x とも言う。",
      "2 の位置に注意: 分子は d の右上（d²y）、分母は x の右上（dx²）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "The Derivative as a Function（d²y/dx²）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the second derivative of y": 34,
        "d squared y dx squared | d squared y over dx squared": 2
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "segment-ab-overline",
    "latex": "\\overline{AB}",
    "spoken_en": [
      {
        "text": "segment AB",
        "register": "standard"
      },
      {
        "text": "line segment AB",
        "register": "spoken"
      }
    ],
    "spoken_ja": "線分 AB",
    "name_en": "line segment notation",
    "name_ja": "線分の記号（上線の AB）",
    "category": "geometry",
    "term_ref": "segment",
    "related": [
      "figure-and-measure-notation"
    ],
    "level": {
      "jp": [],
      "us": [
        "Pre-Algebra",
        "Geometry"
      ]
    },
    "notes": [
      "上線の AB は segment AB と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.1 Geometry Terms（Say It: Segment AB）・上線の AB（710 件）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "上線の AB（129 件）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 1.10・2.2・2.12（line segment に上線）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!line segment a b | !line segment ab | !line segment b c | !line segment bc | !line segment a c | !line segment ac | !line segment c d | !line segment cd | !line segment d e | !line segment de | !line segment x y | !line segment xy | !line segment p q | !line segment pq": 27,
        "line segment a b | line segment ab | line segment b c | line segment bc | line segment a c | line segment ac | line segment c d | line segment cd | line segment p q | line segment pq": 1
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "mit-18.02",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "sequence-braces",
    "latex": "\\{a_n\\}",
    "spoken_en": [
      {
        "text": "the sequence a sub n",
        "register": "standard"
      },
      {
        "text": "the sequence a n",
        "register": "standard"
      }
    ],
    "spoken_ja": "数列 {aₙ} ／ 数列 a n",
    "name_en": "sequence notation",
    "name_ja": "数列の記号",
    "category": "subscripts",
    "term_ref": "sequence",
    "related": [
      "sequence-starting-index"
    ],
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus BC",
        "Calculus II"
      ]
    },
    "notes": [
      "{aₙ} は the sequence a sub n と読み、波かっこは読まない。",
      "OpenStax Calculus Volume 2 は数列を {aₙ} と書く。添字が 0 から始まることもある（sequence-starting-index）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 2"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the sequence a sub * | the sequence of a sub *": 2,
        "the sequence a n | the sequence of a n": 1
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "yt:blackpenredpen"
      ],
      "counted": "2026-09-27"
    },
    "flags": [
      {
        "code": "corpus-human-settled",
        "note": "読みは人間が決めた（2026-09-25。docs/DECISIONS.md「Phase 3 フレーズの前の修正」の 2）。the sequence a sub n ／ the sequence a n。register は主張しない。",
        "raised": "2026-09-25"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "set-builder-braces",
    "latex": "\\{x \\mid x > 0\\}",
    "spoken_en": [
      {
        "text": "the set of all x such that x is greater than zero",
        "register": "standard"
      }
    ],
    "spoken_ja": "x > 0 を満たす x 全体の集合",
    "name_en": "set-builder notation",
    "name_ja": "条件で表す集合（内包的記法）",
    "category": "sets",
    "term_ref": "set-builder-notation",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 1",
        "Algebra 2",
        "Precalculus",
        "Discrete Math"
      ]
    },
    "notes": [
      "{x | x > 0} は the set of all x such that x is greater than zero と読む。縦線は such that。",
      "Levin（5.1）は {x : x > 2} のようにコロンで書き、コロンも such that と読む（縦棒 | を使うこともあると断る）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "1.1 Review of Functions（We read this as the set of real numbers x such that x has some property）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "3.2 Domain and Range（the braces { } are read as the set of、the vertical bar | as such that）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets（{x : x > 2} is the set of all x such that x is greater than 2。1 つ目の読みの文。コロンは such that と読み、| も使われると注記。notes[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 集合と命題（A={x|x>2}）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "the set of all * such that": 4
      },
      "sources": [
        "khan-algebra",
        "mit-6.042",
        "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: 対象外）。読みは OpenStax の呼び方 the set of all * such that（本文 5 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "set-difference",
    "latex": "A \\setminus B,\\ A - B",
    "spoken_en": [
      {
        "text": "the set difference of A and B",
        "register": "standard"
      }
    ],
    "spoken_ja": "A から B を除いた集合",
    "name_en": "set difference",
    "name_ja": "差集合",
    "category": "sets",
    "term_ref": null,
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "A − B とも書く（英語版 Wikipedia「Complement (set theory)」の Relative complement）。A ∖ B は A ∩ B̄ と同じ集合（Levin 5.1）。",
      "差集合は米国の高校課程（CED・OpenStax・IM・CK-12）では扱わない。Levin は A ∖ B を set difference ／ difference of sets と書く（3 件）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.5 Proofs about Discrete Structures・5.1 Sets。set difference ／ difference of sets（3 件。読みを決めた参照）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「差集合」",
        "url": "https://ja.wikipedia.org/wiki/%E5%B7%AE%E9%9B%86%E5%90%88",
        "note": "差集合（B ∖ A、B − A）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Complement_(set_theory)",
        "note": "節 Relative complement（B − A とも書く）"
      }
    ],
    "confidence": "verified",
    "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: 対象外）。読みは Levin の呼び方 set difference | difference of sets（5.1 Sets・1.5 Proofs about Discrete Structures）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "similar-sign",
    "latex": "\\triangle ABC \\backsim \\triangle DEF,\\ \\triangle ABC \\sim \\triangle DEF",
    "spoken_en": [
      {
        "text": "triangle ABC is similar to triangle DEF",
        "register": "standard"
      }
    ],
    "spoken_ja": "三角形 ABC 相似 三角形 DEF",
    "name_en": "similarity sign",
    "name_ja": "相似の記号（日本は ∽、米国は ∼）",
    "category": "geometry",
    "term_ref": "similar",
    "related": [
      "similarity-symbol"
    ],
    "level": {
      "jp": [
        "中3"
      ],
      "us": [
        "Geometry"
      ]
    },
    "notes": [
      "△ABC ∼ △DEF は triangle ABC is similar to triangle DEF と読む。中学校学習指導要領〔用語・記号〕（中3）の ∽ で書いた相似も、英語では同じく is similar to と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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: ∽"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "7.3 Similar Polygons and Scale Factors（∼）・∼（165 件）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math / IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/",
        "note": "Grade 8 2・Geometry 3。Geometry 3.6 Connecting Similarity and Transformations が ∼ を is similar to と読む。… is similar to triangle …（27 件。読みを決めた参照）"
      }
    ],
    "confidence": "verified",
    "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: 対象外）。読みは IM の呼び方 is similar to triangle *（Geometry 3.13 Using the Pythagorean Theorem and Similarity・Geometry 3.6 Connecting Similarity and Transformations・Grade 8 2.6 Similarity・計 27 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "sine-of-theta",
    "latex": "\\sin\\theta",
    "spoken_en": [
      {
        "text": "sine of theta",
        "register": "standard"
      },
      {
        "text": "sine theta",
        "register": "standard"
      },
      {
        "text": "the sine of theta",
        "register": "standard"
      }
    ],
    "spoken_ja": "サイン シータ",
    "name_en": "sine",
    "name_ja": "サイン",
    "category": "trigonometry",
    "term_ref": "sine",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "sin θ は sine of theta か sine theta と読む。sine は sign と同じ音（Merriam-Webster: ˈsīn）で、つづりどおりに「シン」とは読まない。",
      "sin 2x は sine of two x。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 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）〔用語・記号〕",
        "note": "数I 図形と計量: sin"
      },
      {
        "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": "読み（sin: サイン、cos: コサイン、tan: タンジェント、sec: セカント、cosec: コセカント、cot: コタンジェント）"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「sine」",
        "url": "https://www.merriam-webster.com/dictionary/sine",
        "note": "発音 ˈsīn（2026-09-27 に確かめた）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "sine theta | sine x | sine alpha | sine two x": 418,
        "!the !inverse !arc sine of *": 931,
        "the sine of *": 126
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-10-09"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "sine-squared",
    "latex": "\\sin^2\\theta",
    "spoken_en": [
      {
        "text": "sine squared theta",
        "register": "standard"
      },
      {
        "text": "sine squared of theta",
        "register": "standard"
      },
      {
        "text": "sine of theta, squared",
        "register": "spoken"
      }
    ],
    "spoken_ja": "サイン 2 乗 シータ",
    "name_en": "sine squared",
    "name_ja": "sin² θ",
    "category": "trigonometry",
    "term_ref": "sine",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "sin²θ は sine squared theta と読む。(sin θ)² の意味で、sin(θ²) ではない。",
      "sine of theta, squared と区切って読むこともあるが、sine of theta squared と続けると sin(θ²) にも聞こえる。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Trigonometric_functions",
        "note": "節 Notation（sin² x は (sin x)²）"
      },
      {
        "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": "三角関数の累乗は sin²x のように書く"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "sine squared of *": 31,
        "sine squared theta | sine squared x | sine squared alpha": 55,
        "sine of theta squared | sine of x squared": 32
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "square-brackets",
    "latex": "[\\ ]",
    "spoken_en": [
      {
        "text": "brackets",
        "register": "standard"
      },
      {
        "text": "square brackets",
        "register": "spoken"
      }
    ],
    "spoken_ja": "大かっこ ／ ブラケット",
    "name_en": "brackets",
    "name_ja": "角かっこ（[ ]）",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra"
      ]
    },
    "notes": [
      "[ ] は brackets と呼ぶ。( ) の parentheses（単数 parenthesis）と区別する。",
      "閉区間 [a, b] の [ ] も brackets（closed-interval-brackets）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "1.2 Domain and Range（Brackets, [ or ], are used to indicate that an endpoint is included）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "2.1 Use the Language of Algebra（Common Grouping Symbols: brackets [ ]）"
      },
      {
        "type": "reference",
        "title": "CK-12 Algebra (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Algebra",
        "note": "1.4.1 Order of Operations（Brackets [ ], second level of grouping symbols）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「括弧」",
        "url": "https://ja.wikipedia.org/wiki/%E6%8B%AC%E5%BC%A7",
        "note": "角括弧（ブラケット）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!square !curly !angle !round brackets | !square !curly !angle !round bracket": 173,
        "square brackets | square bracket": 2
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "square-root",
    "latex": "\\sqrt{x^2+1}",
    "spoken_en": [
      {
        "text": "the square root of x squared plus one",
        "register": "standard"
      },
      {
        "text": "root x squared plus one",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ルート x の 2 乗 プラス 1",
    "name_en": "square root",
    "name_ja": "平方根（ルート）",
    "category": "roots",
    "term_ref": "square-root-of-a-number",
    "level": {
      "jp": [
        "中3",
        "数I"
      ],
      "us": [
        "Algebra 1"
      ]
    },
    "notes": [
      "根号が全体にかかっていることをはっきりさせたいときは the square root of the quantity x squared plus one と言うことがある。the quantity が括弧の役割をする。",
      "3 乗根は 'the cube root of', n 乗根は 'the nth root of'。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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 A 数と式: √"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 8 8.2 Side Lengths and Areas（the square root of 73）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the square root of *": 1677,
        "!square !cube !nth !the !a root two | !square !cube !nth !the !a root three | !square !cube !nth !the !a root x": 151
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "standard-deviation-sigma",
    "latex": "\\sigma,\\ \\sigma(X),\\ \\sigma_X",
    "spoken_en": [
      {
        "text": "the standard deviation of X",
        "register": "standard"
      },
      {
        "text": "sigma",
        "register": "standard"
      }
    ],
    "spoken_ja": "シグマ ／ X の標準偏差",
    "name_en": "standard deviation (sigma)",
    "name_ja": "標準偏差（σ、σ(X)、σ_X）",
    "category": "probability-statistics",
    "term_ref": "standard-deviation",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "σ は sigma と読む。σ(X)・σ_X は the standard deviation of X と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "付録 Mathematical Phrases, Symbols, and Formulas（σ: sigma、σ_X: standard deviation of 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": "Unit 1・2"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!capital !u !matrix !plus sigma !notation !transpose !inverse !plus !one !two !r !n !v !squared": 75,
        "the standard deviation of *": 93
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "subset-sign",
    "latex": "A \\subseteq B",
    "spoken_en": [
      {
        "text": "A is a subset of B",
        "register": "standard"
      },
      {
        "text": "A is contained in B",
        "register": "spoken"
      }
    ],
    "spoken_ja": "A は B の部分集合 ／ A は B に含まれる",
    "name_en": "subset",
    "name_ja": "部分集合（⊆）",
    "category": "sets",
    "term_ref": "subset",
    "related": [
      "subset-symbol"
    ],
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Calculus III",
        "Discrete Math",
        "Linear Algebra"
      ]
    },
    "notes": [
      "A ⊆ B は A is a subset of B と読む。A is contained in B とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "5.2 Double Integrals over General Regions（D is a subset of R (D ⊆ R)）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.5 Proofs about Discrete Structures・5.1 Sets（A ⊆ B asserts that A is a subset of B）"
      },
      {
        "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（X ⊆ Y、X is a subset of Y）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "is contained in *": 10,
        "is a subset of *": 20
      },
      "sources": [
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "summation-sigma",
    "latex": "\\sum_{k=1}^{n} a_k",
    "spoken_en": [
      {
        "text": "the sum from k equals one to n of a sub k",
        "register": "standard"
      },
      {
        "text": "the sum of a k, k from one to n",
        "register": "spoken"
      }
    ],
    "spoken_ja": "シグマ、k イコール 1 から n まで、a k",
    "name_en": "summation",
    "name_ja": "総和（シグマ）",
    "category": "sums",
    "term_ref": "summation-notation",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "Algebra 2",
        "Precalculus",
        "AP Calculus AB",
        "AP Calculus BC",
        "Calculus I",
        "Calculus II"
      ]
    },
    "notes": [
      "添字は a sub k と読む。",
      "Σ そのものは sigma と読む。OpenStax Calculus Volume 1 は sigma notation（summation notation）と呼ぶ。"
    ],
    "tts_text": "the sum from k equals one to n of a sub k",
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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B 数列: Σ"
      },
      {
        "type": "textbook",
        "title": "OpenStax Intermediate Algebra 2e",
        "note": "Sequences（We read this as the sum of a sub i from i equals one to five）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 1",
        "note": "Approximating Areas（sigma notation）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "the sum from * to * of": 52,
        "the sum of * from * to *": 9
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "tangent-of-theta",
    "latex": "\\tan\\theta",
    "spoken_en": [
      {
        "text": "tangent theta",
        "register": "standard"
      },
      {
        "text": "tangent of theta",
        "register": "standard"
      },
      {
        "text": "tan theta",
        "register": "standard"
      },
      {
        "text": "the tangent of theta",
        "register": "standard"
      }
    ],
    "spoken_ja": "タンジェント シータ",
    "name_en": "tangent",
    "name_ja": "タンジェント",
    "category": "trigonometry",
    "term_ref": "tangent",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "tan θ は tangent theta か tangent of theta と読み、短く tan theta とも言う。",
      "接線は tangent line と言う（AP の CED topic 4.6、IM Geometry 7.3。IM は円の接線を tangent to a circle とも書く）。OpenStax Precalculus（12.4 Derivatives）は the tangent line to the curve と書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "12.4 Derivatives: the tangent line to the curve"
      },
      {
        "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）〔用語・記号〕",
        "note": "数I 図形と計量: tan"
      },
      {
        "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": "読み（sin: サイン、cos: コサイン、tan: タンジェント、sec: セカント、cosec: コセカント、cot: コタンジェント）"
      },
      {
        "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（tangent line）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math",
        "url": "https://im.kendallhunt.com/HS/students/index.html",
        "note": "Geometry 7.3 Tangent Lines"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "tan theta | !arc tan x | tan alpha": 76,
        "the tangent of *": 49,
        "!the !inverse !arc tangent of *": 168,
        "tangent theta | tangent x | tangent alpha": 185
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-10-09"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "there-exists-quantifier",
    "latex": "\\exists x",
    "spoken_en": [
      {
        "text": "there exists an x",
        "register": "standard"
      },
      {
        "text": "there is an x such that",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ある x が存在して",
    "name_en": "there exists",
    "name_ja": "存在する（∃）",
    "category": "logic",
    "term_ref": "existential-quantifier",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Discrete Math"
      ]
    },
    "notes": [
      "∃x は there exists an x（such that …）と読む。there is an x such that … とも言う。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1 Mathematical Statements（∃ is read there exists or for some）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "there exists *": 86,
        "there is * such that": 12
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.01",
        "mit-18.02",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "logic-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "therefore-sign",
    "latex": "\\therefore",
    "spoken_en": [
      {
        "text": "therefore",
        "register": "standard"
      }
    ],
    "spoken_ja": "ゆえに",
    "name_en": "therefore sign",
    "name_ja": "ゆえに（∴）",
    "category": "logic",
    "term_ref": null,
    "related": [
      "therefore-because-symbols"
    ],
    "level": {
      "jp": [
        "中2",
        "数I"
      ],
      "us": [
        "Geometry",
        "Discrete Math"
      ]
    },
    "notes": [
      "∴ は therefore と読む（Levin 1.1、CK-12 Geometry 2.8）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "CK-12 Foundation, CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "2.8 Truth Tables（∴ therefore）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "1.1 Mathematical Statements（The symbol ∴ means therefore）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Therefore_sign",
        "note": "冒頭（read therefore）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「∴」",
        "url": "https://ja.wikipedia.org/wiki/%E2%88%B4",
        "note": "数学などで「ゆえに（故に）」「したがって」と読む記号（spoken_ja）。level.jp は慣習差 therefore-because-symbols の jp（ユーザーの決定）に合わせる（DECISIONS）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "therefore": 581
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "theta",
    "latex": "\\theta",
    "spoken_en": [
      {
        "text": "theta",
        "register": "standard"
      }
    ],
    "spoken_ja": "シータ",
    "name_en": "theta (a letter for an angle)",
    "name_ja": "θ（角を表す文字）",
    "category": "trigonometry",
    "term_ref": "angle",
    "level": {
      "jp": [
        "数I"
      ],
      "us": [
        "Geometry",
        "Algebra 2",
        "Precalculus"
      ]
    },
    "notes": [
      "英語の読みは theta。",
      "sin θ は sine theta、θ = π/3 は theta equals pi over three と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「Θ」",
        "url": "https://ja.wikipedia.org/wiki/%CE%98",
        "note": "θ（シータ）"
      }
    ],
    "confidence": "verified",
    "reviewed": {
      "machine": "2026-09-25",
      "audit": "2026-09-27",
      "human": null
    },
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "theta": 3109
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    }
  },
  {
    "id": "times-sign",
    "latex": "a \\times b,\\ a \\cdot b,\\ ab",
    "spoken_en": [
      {
        "text": "a times b",
        "register": "standard"
      },
      {
        "text": "a multiplied by b",
        "register": "spoken"
      }
    ],
    "spoken_ja": "a かける b",
    "name_en": "multiplication sign",
    "name_ja": "かける（× ・）",
    "category": "arithmetic",
    "term_ref": "multiplication",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1"
      ]
    },
    "notes": [
      "× も · も並べて書いた ab も、掛け算の読みは a times b（OpenStax Elementary Algebra 1.1 の表）。並べて書いた 3x はふつう three x と文字を続けて読む。",
      "× は外積（u × v、cross）や行列の型（2 × 3、two by three）にも使い、そのときは times と読まない（cross-product-cross、matrix-dimensions-by）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "1.2 Use the Language of Algebra の表（a · b, ab, (a)(b) — Say: a times b）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-28",
    "evidence": {
      "spoken": {
        "* times *": 15702,
        "* multiplied by *": 279
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-28",
      "human": null
    }
  },
  {
    "id": "transpose-notation",
    "latex": "A^{T},\\ {}^{t}\\!A",
    "spoken_en": [
      {
        "text": "A transpose",
        "register": "standard"
      },
      {
        "text": "the transpose of A",
        "register": "spoken"
      }
    ],
    "spoken_ja": "A の転置 ／ A トランスポーズ ／ t A",
    "name_en": "transpose",
    "name_ja": "転置行列（Aᵀ、ᵗA）",
    "category": "matrices",
    "term_ref": "transpose",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Linear Algebra"
      ]
    },
    "notes": [
      "Aᵀ は A transpose と読む。左肩に t を付ける ᵗA も読みは同じ。"
    ],
    "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": "2.1 Matrix Addition, Scalar Multiplication, and Transposition"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Transpose",
        "note": "節 Definition（Aᵀ と ᵗA）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the transpose of *": 12,
        "a transpose | b transpose | x transpose | v transpose | q transpose | u transpose | m transpose | c transpose": 498
      },
      "sources": [
        "mit-18.02",
        "mit-18.06"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "triangle-abc",
    "latex": "\\triangle ABC",
    "spoken_en": [
      {
        "text": "triangle ABC",
        "register": "standard"
      }
    ],
    "spoken_ja": "三角形 ABC",
    "name_en": "triangle sign",
    "name_ja": "三角形の記号（△ABC）",
    "category": "geometry",
    "term_ref": "triangle",
    "level": {
      "jp": [
        "中1"
      ],
      "us": [
        "Pre-Algebra",
        "Algebra 1",
        "Geometry"
      ]
    },
    "notes": [
      "△ABC は triangle ABC と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "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: △"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 9–12 Math (Geometry)",
        "url": "https://im.kendallhunt.com/HS/students/2/index.html",
        "note": "Geometry 2.1・2.2・3.6 ほか（△ 20 件）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem（△ABC）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Prealgebra 2e",
        "note": "9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem（△ABC is read triangle ABC）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "triangle a b c | triangle abc | triangle d e f | triangle def | triangle x y z | triangle xyz | triangle p q r | triangle pqr | triangle a b d | triangle abd | triangle a c d | triangle acd | triangle b c d | triangle bcd": 21
      },
      "sources": [
        "khan-algebra",
        "khan-middle",
        "yt:organicchem"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "triple-integral-sign",
    "latex": "\\iiint_E f\\,dV",
    "spoken_en": [
      {
        "text": "the triple integral over E of f d V",
        "register": "standard"
      },
      {
        "text": "the triple integral of f over E",
        "register": "spoken"
      }
    ],
    "spoken_ja": "三重積分",
    "name_en": "triple integral",
    "name_ja": "三重積分",
    "category": "integrals",
    "term_ref": "triple-integral",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Calculus III"
      ]
    },
    "notes": [
      "∭_E f dV は the triple integral over E of f d V と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "Triple Integrals（the triple integral over a solid region）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the triple integral over *": 12,
        "the triple integral of *": 5
      },
      "sources": [
        "mit-18.02"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "union-sign",
    "latex": "A \\cup B",
    "spoken_en": [
      {
        "text": "A union B",
        "register": "standard"
      },
      {
        "text": "the union of A and B",
        "register": "standard"
      }
    ],
    "spoken_ja": "A カップ B ／ A と B の和集合",
    "name_en": "union",
    "name_ja": "和集合（∪）",
    "category": "sets",
    "term_ref": "union",
    "level": {
      "jp": [
        "数I",
        "数A"
      ],
      "us": [
        "Algebra 2",
        "AP Statistics",
        "Intro Statistics",
        "Discrete Math"
      ]
    },
    "notes": [
      "A ∪ B は A union B、the union of A and B と読む。∪ を cup と読む例は、用例コーパスと参照（CED・OpenStax・IM・CK-12・Nicholson・Levin）には出てこない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "1.2 Domain and Range（a symbol known as the union, ∪）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Algebra and Trigonometry 2e",
        "note": "13.7 Probability（The union of two events E and F, written E ∪ F）"
      },
      {
        "type": "reference",
        "title": "Oscar Levin, Discrete Mathematics: An Open Introduction, 4th edition",
        "url": "https://discrete.openmathbooks.org/dmoi4.html",
        "note": "5.1 Sets（C = A ∪ B, read, C is the union of A and B）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "3.1 Terminology（A OR B can also be written as A Union B、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 2.7（union）"
      },
      {
        "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（A∪B）・数学A 場合の数と確率（n(A∪B)）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the union of *": 14,
        "!the !their !labor !soviet !this !for !use union !of !sign !bound": 42
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.06",
        "mit-6.042",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "related": [
      "probability-and-or-notation"
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "unit-vector-hat",
    "latex": "\\hat{\\imath},\\ \\hat{\\jmath},\\ \\hat{k},\\ \\hat{u}",
    "spoken_en": [
      {
        "text": "i hat",
        "register": "standard"
      }
    ],
    "spoken_ja": "i ハット ／ 単位ベクトル",
    "name_en": "unit vector hat",
    "name_ja": "単位ベクトルのハット",
    "category": "vectors",
    "term_ref": "standard-unit-vectors",
    "level": {
      "jp": [
        "大学"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "notes": [
      "î は i hat、û は u hat と読む。帽子の形の印を hat と呼ぶ。",
      "OpenStax Calculus Volume 3 は i, j, k を太字で書き、hat を付けない。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.1 Vectors in the Plane（i, j を太字で書く）"
      },
      {
        "type": "reference",
        "title": "English Wikipedia",
        "url": "https://en.wikipedia.org/wiki/Unit_vector",
        "note": "冒頭（circumflex, or hat。v̂ is read v-hat）"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「単位ベクトル」",
        "url": "https://ja.wikipedia.org/wiki/%E5%8D%98%E4%BD%8D%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB",
        "note": "r̂ のように表す書き方"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "i hat | j hat | k hat | u hat": 79
      },
      "sources": [
        "mit-18.02",
        "yt:3blue1brown"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "variance-of-x",
    "latex": "V(X),\\ \\operatorname{Var}(X),\\ \\sigma^2",
    "spoken_en": [
      {
        "text": "the variance of X",
        "register": "standard"
      },
      {
        "text": "sigma squared",
        "register": "spoken"
      }
    ],
    "spoken_ja": "X の分散 ／ V(X)",
    "name_en": "variance",
    "name_ja": "分散（V(X)、Var(X)、σ²）",
    "category": "probability-statistics",
    "term_ref": "variance",
    "level": {
      "jp": [
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "V(X)・Var(X) は the variance of X と読む。σ² は sigma squared。",
      "AP Statistics の CED（topic 2.9）は確率変数の分散を V(X) か σ²_X と書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "付録 Mathematical Phrases, Symbols, and Formulas（σ²: sigma squared）"
      },
      {
        "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（variance、V(X) か σ²_X。notes[1]）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-10-09",
    "evidence": {
      "spoken": {
        "sigma squared": 10,
        "the variance of *": 106
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.06",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-10-09",
      "human": null
    }
  },
  {
    "id": "vector-ab-arrow",
    "latex": "\\overrightarrow{AB}",
    "spoken_en": [
      {
        "text": "vector AB",
        "register": "standard"
      },
      {
        "text": "the vector from A to B",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ベクトル AB",
    "name_en": "vector from A to B",
    "name_ja": "ベクトル AB",
    "category": "vectors",
    "term_ref": "vector",
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III"
      ]
    },
    "notes": [
      "AB⃗ は vector A B か the vector from A to B と読む。",
      "CK-12 Geometry（1.1 Geometry Terms）は同じ形の矢印の AB を ray AB と読む（ray-ab-arrow）。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.1 Vectors in the Plane（PQ→）・2.2・2.3（vector PQ→ ／ vector AB→。OpenStax Algebra and Trigonometry と合わせて 8 件。読みを決めた参照）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "1.1 Geometry Terms（矢印の CD を Ray CD と読む）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "8.8 Vectors（PQ→）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the vector from * to *": 3,
        "vector a b | vector ab | vector p q | vector pq": 4
      },
      "sources": [
        "mit-18.02",
        "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 7 件（10 未満） ／ 書: 対象外）。読みは OpenStax の呼び方 vector a b | vector ab | vector p q | vector pq（本文 8 件）。register は主張しない。",
        "raised": "2026-09-27"
      }
    ],
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "vector-arrow-notation",
    "latex": "\\vec{a},\\ \\mathbf{v}",
    "spoken_en": [
      {
        "text": "vector v",
        "register": "standard"
      },
      {
        "text": "the vector v",
        "register": "spoken"
      },
      {
        "text": "v vector",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ベクトル a",
    "name_en": "vector notation",
    "name_ja": "ベクトルの記号",
    "category": "vectors",
    "term_ref": "vector",
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Geometry",
        "Precalculus",
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "notes": [
      "a⃗ も太字の v も、vector v（the vector v）と読む。v vector とも言う。",
      "OpenStax Calculus Volume 3（2.1）は太字で書き、手で書くときは文字の上に矢印を付けるとする。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.1 Vectors in the Plane（太字。手書きでは矢印）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "8.8 Vectors（太字と v→）"
      },
      {
        "type": "reference",
        "title": "CK-12 Geometry (K12 LibreTexts)",
        "url": "https://k12.libretexts.org/Bookshelves/Mathematics/Geometry",
        "note": "8.1 Identify Transformation Types・8.7（along vector v→）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!the vector v | !the vector a | !the vector u | !the vector b": 69,
        "the vector v | the vector a | the vector u | the vector b": 16,
        "v vector | u vector": 3
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.03",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "vector-magnitude-bars",
    "latex": "\\lVert \\vec{v} \\rVert,\\ |\\vec{a}|",
    "spoken_en": [
      {
        "text": "the magnitude of v",
        "register": "standard"
      },
      {
        "text": "the length of v",
        "register": "standard"
      },
      {
        "text": "the norm of v",
        "register": "spoken"
      }
    ],
    "spoken_ja": "ベクトル a の大きさ",
    "name_en": "magnitude of a vector",
    "name_ja": "ベクトルの大きさ（‖v‖、|a⃗|）",
    "category": "vectors",
    "term_ref": "magnitude",
    "level": {
      "jp": [
        "数C"
      ],
      "us": [
        "Precalculus",
        "Calculus III",
        "Linear Algebra"
      ]
    },
    "notes": [
      "‖v‖ は the magnitude of v と読む。",
      "OpenStax Calculus Volume 3 はベクトルの大きさを 2 本の縦線 ‖v‖ で書く。1 本の縦線 |a⃗| も読みは同じ。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Calculus Volume 3",
        "note": "2.1 Vectors in the Plane（‖v‖）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Precalculus 2e",
        "note": "8.8 Vectors（|v|）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the length of the vector | the length of vector * | the length of v | the length of w | the length of f": 37,
        "the magnitude of * vector | the magnitude of v | the magnitude of u | the magnitude of w | the magnitude of f | the magnitude of t | the magnitude of * gradient | the magnitude of * displacement | the magnitude of the cross product": 59,
        "the norm of *": 1
      },
      "sources": [
        "khan-ap-calc",
        "mit-18.02",
        "mit-18.06",
        "yt:3blue1brown",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "wave-dash-range-jp",
    "latex": "3\\text{〜}5",
    "spoken_en": [
      {
        "text": "from three to five",
        "register": "standard"
      },
      {
        "text": "between three and five",
        "register": "spoken"
      }
    ],
    "spoken_ja": "3 から 5",
    "name_en": "range written with a wave dash (Japanese)",
    "name_ja": "範囲の〜",
    "category": "notation-other",
    "term_ref": null,
    "level": {
      "jp": [
        "中1"
      ],
      "us": []
    },
    "notes": [
      "「3〜5」は from three to five か between three and five と言う（範囲の「〜」は日本語の書き方で、「から」と読む: 日本語版 Wikipedia「波ダッシュ」）。",
      "「〜」は前後の数を含む（日本語版 Wikipedia「波ダッシュ」）。度数分布表の階級「10〜20」は 10 以上 20 未満（中学校学習指導要領解説 中1）で、IM Grade 6 8.7 もヒストグラムの階級は左端を含み右端を含まないとする。英語で端を含むことをはっきりさせるときは inclusive を添える（OpenStax Introductory Statistics 1.3・3.1）か、不等号で書く。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "日本語版 Wikipedia「波ダッシュ」",
        "url": "https://ja.wikipedia.org/wiki/%E6%B3%A2%E3%83%80%E3%83%83%E3%82%B7%E3%83%A5",
        "note": "範囲を表す波ダッシュ（「から」と読む。前後の語を含む）"
      },
      {
        "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 D データの活用 表1（階級 0〜10、10〜20 …、以上・未満）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "1.3 Frequency, Frequency Tables, and Levels of Measurement（from five to 20 years, inclusive）・3.1 Terminology（between zero and one, inclusive）"
      },
      {
        "type": "reference",
        "title": "Illustrative Mathematics, IM 6–8 Math",
        "url": "https://im.kendallhunt.com/MS/students/index.html",
        "note": "Grade 6 8.7（ヒストグラムの階級は左端を含み右端を含まない）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!went !get from one to * | !went !get from two to * | !went !get from three to * | !went !get from four to * | !went !get from five to * | !went !get from ten to *": 459,
        "!difference !distance !halfway !average !way between one and * | !difference !distance !halfway !average !way between two and * | !difference !distance !halfway !average !way between three and * | !difference !distance !halfway !average !way between four and * | !difference !distance !halfway !average !way between five and * | !difference !distance !halfway !average !way between ten and *": 122
      },
      "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",
        "yt:3blue1brown",
        "yt:blackpenredpen",
        "yt:nancypi",
        "yt:organicchem",
        "yt:patrickjmt",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "x-bar-sample-mean",
    "latex": "\\bar{x},\\ \\bar{X}",
    "spoken_en": [
      {
        "text": "the sample mean",
        "register": "standard"
      },
      {
        "text": "x bar",
        "register": "standard"
      }
    ],
    "spoken_ja": "x バー ／ 標本平均 X̄",
    "name_en": "sample mean (x-bar)",
    "name_ja": "標本平均（x̄、X̄）",
    "category": "probability-statistics",
    "term_ref": "sample-mean",
    "level": {
      "jp": [
        "数I",
        "数B"
      ],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "x̄ は x bar と読む。文字の上の横線を bar と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "2.5 Measures of the Center of the Data（x̄ (pronounced x bar)）"
      },
      {
        "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（x-bar）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "!lambda !ax x bar !transpose | x-bar !transpose": 39,
        "the sample mean": 105
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.02",
        "mit-18.06",
        "yt:blackpenredpen",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "x-naught",
    "latex": "x_0,\\ x_1",
    "spoken_en": [
      {
        "text": "x sub zero",
        "register": "standard"
      },
      {
        "text": "x naught",
        "register": "standard"
      },
      {
        "text": "x zero",
        "register": "standard"
      }
    ],
    "spoken_ja": "x ゼロ ／ x 1",
    "name_en": "x naught (subscript zero)",
    "name_ja": "添字 0",
    "category": "subscripts",
    "term_ref": "subscript",
    "level": {
      "jp": [
        "数II",
        "数III"
      ],
      "us": [
        "Algebra 1",
        "AP Calculus AB",
        "Calculus I"
      ]
    },
    "notes": [
      "x₀ は x sub zero、x naught、x zero と読む。naught は 0（zero）を指す語（Merriam-Webster）で、添字の 0 と初期値（t₀ は t naught）に使う。",
      "x₁ は x sub one か x one。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "reference",
        "title": "Merriam-Webster「naught」",
        "url": "https://www.merriam-webster.com/dictionary/naught",
        "note": "名詞 2: the arithmetical symbol 0 : ZERO。発音 ˈnȯt（2026-09-27 に確かめた）"
      },
      {
        "type": "textbook",
        "title": "OpenStax Elementary Algebra 2e",
        "note": "傾きの公式の (x₁, y₁)（read x sub one, y sub one）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "* naught": 89,
        "* sub zero": 119,
        "x zero": 25
      },
      "sources": [
        "khan-algebra",
        "khan-ap-calc",
        "khan-ap-stats",
        "khan-middle",
        "mit-18.01",
        "mit-18.02",
        "mit-18.03",
        "mit-6.042",
        "yt:3blue1brown",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "y-hat-predicted-value",
    "latex": "\\hat{y}",
    "spoken_en": [
      {
        "text": "y hat",
        "register": "standard"
      }
    ],
    "spoken_ja": "y ハット ／ 予測値",
    "name_en": "predicted value (y-hat)",
    "name_ja": "回帰直線による予測値（ŷ）",
    "category": "probability-statistics",
    "term_ref": "regression-line",
    "level": {
      "jp": [],
      "us": [
        "Algebra 1",
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "ŷ は y hat と読む。"
    ],
    "sources": [
      {
        "type": "editorial"
      },
      {
        "type": "textbook",
        "title": "OpenStax Introductory Statistics 2e",
        "note": "12.3 The Regression Equation（ŷ is read y hat）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "y hat | y-hat": 11
      },
      "sources": [
        "khan-ap-stats"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  },
  {
    "id": "z-star-critical-value",
    "latex": "z^{*},\\ t^{*}",
    "spoken_en": [
      {
        "text": "z star",
        "register": "standard"
      },
      {
        "text": "the critical value",
        "register": "standard"
      }
    ],
    "spoken_ja": "臨界値 z",
    "name_en": "critical value (z-star)",
    "name_ja": "臨界値（z*、t*）",
    "category": "probability-statistics",
    "term_ref": "critical-value",
    "level": {
      "jp": [],
      "us": [
        "AP Statistics",
        "Intro Statistics"
      ]
    },
    "notes": [
      "z* は z star、t* は t star と読む。星印を star と読む。"
    ],
    "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 3.3（critical value、z*）・Topic 4.2（critical value、t*）"
      }
    ],
    "confidence": "verified",
    "updated": "2026-09-27",
    "evidence": {
      "spoken": {
        "the critical value | a critical value": 21,
        "z star | z-star | t star | t-star": 42
      },
      "sources": [
        "khan-ap-stats",
        "mit-18.01",
        "mit-18.03",
        "mit-6.042",
        "yt:organicchem",
        "yt:profleonard"
      ],
      "counted": "2026-09-27"
    },
    "reviewed": {
      "machine": null,
      "audit": "2026-09-27",
      "human": null
    }
  }
]