第 10345 題：複數模與共軛

原題與已知條件

複數以整數有序對表示。操作=complexmod-conjugation-product；資料={"left":[7,2],"right":[2,29]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。

• G001 — 複數以整數有序對表示。操作=complexmod-conjugation-product；資料={"left":[7,2],"right":[2,29]}。請依 [left,right,product,conjugate_product,product_of_conjugates,equal] 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-complex-modulus-conjugate

逐步推導

declared complex ordered pair=[7, 2]

exact integer coefficients in the algebra with i^2=-1

original product real-real := mul(7, 2)

original product real-real = 14

original product imaginary-imaginary := mul(2, 29)

original product imaginary-imaginary = 58

original product real coefficient := sub(14, 58)

original product real coefficient = -44

original product real-imaginary := mul(7, 29)

original product real-imaginary = 203

original product imaginary-real := mul(2, 2)

original product imaginary-real = 4

original product imaginary coefficient := sum(203, 4)

original product imaginary coefficient = 207

original product:left=[7, 2];right=[2, 29];i^2=-1

complex product=[-44, 207]

original product negate imaginary coefficient := sub(0, 207)

original product negate imaginary coefficient = -207

original product:original ordered pair=[-44, 207]

conjugate pair=[-44, -207]

left value negate imaginary coefficient := sub(0, 2)

left value negate imaginary coefficient = -2

left value:original ordered pair=[7, 2]

conjugate pair=[7, -2]

right value negate imaginary coefficient := sub(0, 29)

right value negate imaginary coefficient = -29

right value:original ordered pair=[2, 29]

conjugate pair=[2, -29]

product of conjugates real-real := mul(7, 2)

product of conjugates real-real = 14

product of conjugates imaginary-imaginary := mul(-2, -29)

product of conjugates imaginary-imaginary = 58

product of conjugates real coefficient := sub(14, 58)

product of conjugates real coefficient = -44

product of conjugates real-imaginary := mul(7, -29)

product of conjugates real-imaginary = -203

product of conjugates imaginary-real := mul(-2, 2)

product of conjugates imaginary-real = -4

product of conjugates imaginary coefficient := sum(-203, -4)

product of conjugates imaginary coefficient = -207

product of conjugates:left=[7, -2];right=[2, -29];i^2=-1

complex product=[-44, -207]

conjugate of computed product=[-44, -207];computed product of conjugates=[-44, -207]

equal=是

逐步反向核對

• S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 以精確有理數重算 product of conjugates imaginary coefficient，不以小數近似替代。
• S019 · premise_and_result_audit — 以精確有理數重算 product of conjugates imaginary-real，不以小數近似替代。
• S018 · premise_and_result_audit — 以精確有理數重算 product of conjugates real-imaginary，不以小數近似替代。
• S017 · premise_and_result_audit — 以精確有理數重算 product of conjugates real coefficient，不以小數近似替代。
• S016 · premise_and_result_audit — 以精確有理數重算 product of conjugates imaginary-imaginary，不以小數近似替代。
• S015 · premise_and_result_audit — 以精確有理數重算 product of conjugates real-real，不以小數近似替代。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 以精確有理數重算 right value negate imaginary coefficient，不以小數近似替代。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 以精確有理數重算 left value negate imaginary coefficient，不以小數近似替代。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 以精確有理數重算 original product negate imaginary coefficient，不以小數近似替代。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 以精確有理數重算 original product imaginary coefficient，不以小數近似替代。
• S006 · premise_and_result_audit — 以精確有理數重算 original product imaginary-real，不以小數近似替代。
• S005 · premise_and_result_audit — 以精確有理數重算 original product real-imaginary，不以小數近似替代。
• S004 · premise_and_result_audit — 以精確有理數重算 original product real coefficient，不以小數近似替代。
• S003 · premise_and_result_audit — 以精確有理數重算 original product imaginary-imaginary，不以小數近似替代。
• S002 · premise_and_result_audit — 以精確有理數重算 original product real-real，不以小數近似替代。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([7, 2], [2, 29], [-44, 207], [-44, -207], [-44, -207], 是)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。