{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","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":{"q":"mko-euclidean-length"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-euclidean-length-01","title":{"zh-Hant":"歐幾里得長度：由座標精確計算兩點距離","en":"Euclidean length: computing the distance between two points exactly from coordinates"},"summary":{"zh-Hant":"在以 ℝ² 表示的歐幾里得平面中，設 P=(1,−2)、Q=(4,2)、R=(2,3)。精確算出 d(P,Q)=5；算出 d(P,R)=√26，並以平方比較證明 5.09<√26<5.1，說明 5.1 只是近似值。另對一般的實數坐標證明：距離有定義且非負、d(Q,P)=d(P,Q)、d(P,P)=0，以及 d(P,Q)=0 ⇔ P=Q。","en":"In the Euclidean plane represented by ℝ², let P=(1,−2), Q=(4,2), R=(2,3). Compute d(P,Q)=5 exactly; compute d(P,R)=√26 and prove by comparing squares that 5.09<√26<5.1, showing that 5.1 is only an approximation. For general real coordinates, also prove that the distance is defined and nonnegative, that d(Q,P)=d(P,Q), that d(P,P)=0, and that d(P,Q)=0 ⇔ P=Q."},"mko_ids":["mko-euclidean-length"]}],"title":"概念推導導讀"}