{"schema_version":"ocme-site-results-v1","locale":"en","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-complex-number"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-complex-number-01","title":{"zh-Hant":"複數：以有序對計算 (2+3i)(4−5i) 並判定相等","en":"Complex numbers: computing (2+3i)(4−5i) as an ordered pair and deciding equality"},"summary":{"zh-Hant":"設 z=2+3i、w=4−5i。以有序對（實部, 虛部係數）求 z+w、z−w 與 z·w。乘法先由分配律與 i²=−1 推出一般公式 (a+bi)(c+di)=(ac−bd)+(ad+bc)i，再代入數值，並以直接展開交叉核對；最後判定 z·w 是否等於實數 23。","en":"Let z=2+3i and w=4−5i. Find z+w, z−w and z·w as ordered pairs (real part, imaginary coefficient). For the product, first derive the general formula (a+bi)(c+di)=(ac−bd)+(ad+bc)i from distributivity and i²=−1, then substitute the numbers and cross-check by direct expansion; finally decide whether z·w equals the real number 23."},"mko_ids":["mko-complex-number"]}],"title":"Concept derivation guides"}