{
  "$schema":"../../../schemas/mko-v0.4.schema.json","schema_version":"mko-v0.4","id":"mko-rational-number","semantic_family_id":"sf-rational-number","type":"concept","version":"0.10.0","language":"zh-Hant",
  "titles":{"zh-Hant":"有理數","en":"Rational number"},
  "summary":{"zh-Hant":"有理數可表示為兩個整數之比，且分母不可為零。"},
  "statement":{"zh-Hant":"數 x 是有理數，若存在整數 p、q 且 q≠0，使 x=p/q。不同整數對可以表示同一個有理數。"},
  "formula":{"tex":"\\frac{p}{q}","mathml":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mi>p</mi><mi>q</mi></mfrac></math>","semantic_ast":{"type":"fraction","numerator":{"type":"symbol","name":"p","glyph":"p"},"denominator":{"type":"symbol","name":"q","glyph":"q"}},"compiler":{"id":"ocme-formula-core","version":"0.4.0","source_sha256":"8c0cd9aee7ba0b87686d7f95c8f5629ea9e5c24579191af1cf0001aa1fd992c1"}},
  "assumptions":[{"id":"a1","zh-Hant":"p 與 q 是整數。"},{"id":"a2","zh-Hant":"分母 q 不等於 0。"}],
  "symbols":[{"token":"p","role":"numerator","role_zh":"整數分子","scope":"definition"},{"token":"q","role":"denominator","role_zh":"非零整數分母","scope":"definition"},{"token":"/","role":"quotient_representation","role_zh":"比值表示","scope":"definition"}],
  "dependencies":[{"id":"mko-integer","title_zh":"整數","reason_zh":"有理數表示使用整數分子與非零整數分母。"}],
  "explanation":{"paragraphs_zh":["每個整數 n 都是有理數，因為可寫成 n/1。分母為零的字串不表示有理數。","不同分數可表示同一值；在分母皆非零時，可用交叉乘積比較 p/q 與 r/s。","約分、把負號移到分子並令分母為正，是 canonical display convention；它改變表示但不改變有理數值。"],"common_misconception_zh":"常見誤解是允許零分母、把不同分數外觀當成不同數值，或把約分視為近似。"},
  "proofs":[],"computational_companions":[],
  "verification":{"computational_status":"not_evaluated_no_computational_claim","warning_zh":"有限分數重算只驗證具體表示，不構成有理數構造、次序、稠密性或域公理的證明。","evidence_refs":[],"producers":[{"id":"lean-mathlib","adapter":"formal_system","config_path":"formal/lean/lakefile.toml","status":"configured"}]},
  "formalization":{"status":"not_formalized","target_systems":["Lean 4","Mathlib"],"next_obligation_zh":"綁定 Rat.divInt、非零分母、表示等價與 normalized numerator-denominator 的精確聲明及版本證據。"},
  "provenance":{"origin":"ocme_local_ai_materialization_candidate","note_zh":"依 research/atlas-rational-number.md 整理；待 AI 治理與形式化審查。","content_license":"CC-BY-SA-4.0","code_license":"MIT","created_at":"2026-09-20"}
}