{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"paths","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"from":"/en/paths/"},"path":{"id":"path-pulse-foundations-20260930","title":{"zh-Hant":"七輪基礎擴充：數值、函數、計數與座標","en":"Seven-Round Foundation Expansion: Numbers, Functions, Counting, and Coordinates"},"summary":{"zh-Hant":"在既有數系與函數基礎上依相依順序完成七個可追溯知識物件。","en":"Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations."},"audience":["general_learner","upper_secondary"],"review":"candidate","nodes":[{"id":"mko-rational-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-arithmetic-operations","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-function-mapping","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-set","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-natural-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-real-number","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-counting-principle","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-factorial","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-point-line-plane","position":1,"requirement":"required","rationale":{"zh-Hant":"本脈衝新物件的既有 Canonical 前置。","en":"Existing canonical prerequisite for the new objects in this pulse."}},{"id":"mko-fraction","position":2,"requirement":"optional","rationale":{"zh-Hant":"分數 p/q 是有理數的一種表示；分子與分母為整數，分母必須非零。","en":"Treat p/q as a rational-number representation with integer numerator and denominator and a nonzero denominator."}},{"id":"mko-ratio-proportion","position":3,"requirement":"optional","rationale":{"zh-Hant":"本條目把數值比 a:b 解讀為 a/b，把比例定義為兩個合法比值相等。","en":"Interpret the numerical ratio a:b as a/b and a proportion as equality between two valid ratios."}},{"id":"mko-domain-codomain-range","position":4,"requirement":"optional","rationale":{"zh-Hant":"總函數 f:A→B 的定義域是 A，陪域是 B，值域是實際取得的輸出集合 f[A]。","en":"Distinguish the domain A, declared codomain B, and actually attained range f[A] of a total function f:A→B."}},{"id":"mko-composition","position":5,"requirement":"optional","rationale":{"zh-Hant":"合成 g∘f 先求 f 的輸出，再把它送入 g；合成前須確認中間值可作為 g 的輸入。","en":"Evaluate g∘f by applying f first and verify that the intermediate value is an allowed input of g."}},{"id":"mko-sequence","position":6,"requirement":"optional","rationale":{"zh-Hant":"本條目採從 0 起始的實數數列 a:N→R；數列項依索引排列，不能任意重排成集合。","en":"Use zero-based real sequences a:N→R whose terms retain their index order rather than becoming an unordered set."}},{"id":"mko-permutation-combination","position":7,"requirement":"optional","rationale":{"zh-Hant":"由 n 個相異物件不重複取 k 個，有序選取是排列，無序選取是組合。","en":"Distinguish ordered permutations from unordered combinations when selecting distinct objects without replacement."}},{"id":"mko-coordinate-plane","position":8,"requirement":"optional","rationale":{"zh-Hant":"固定原點、互相垂直的有向實數軸與单位後，平面點用有序實數對 (x,y) 表示。","en":"Represent plane points by ordered real pairs after fixing an origin, perpendicular oriented axes, and units."}}]},"title":"Seven-Round Foundation Expansion: Numbers, Functions, Counting, and Coordinates"}