第 33983 題：數值近似誤差

原題與已知條件

exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={"mode":"signed_vs_absolute","model":{"approximation":8,"reference":-8}}；請依 exact_ordered_tuple_declared_by_interface 回答。

• G001 — exact residual、forward error、conditioning與tolerance convention。操作=numerr-boundary；明示資料={"mode":"signed_vs_absolute","model":{"approximation":8,"reference":-8}}；請依 exact_ordered_tuple_declared_by_interface 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-numerical-approximation-error

逐步推導

source roles = [qf-mod50-numerr-boundary, [domain, mode, model, operation]]

derived = [numerr-boundary, Exact_numerical_error_density]

source roles = [signed ORIGINAL approximation, 100000000, 8, 1]

derived = [[8, 1], 8]

signed ORIGINAL approximation ORIGINAL numerator over denominator := div(8, 1)

signed ORIGINAL approximation ORIGINAL numerator over denominator = 8

source roles = [signed ORIGINAL reference, 100000000, -8, 1]

derived = [[-8, 1], -8]

signed ORIGINAL reference ORIGINAL numerator over denominator := div(-8, 1)

signed ORIGINAL reference ORIGINAL numerator over denominator = -8

source roles = [Signed original approximation minus reference SOURCE naive sub, 100000000, 16, 1]

derived = [[16, 1], 16]

Signed original approximation minus reference SOURCE naive sub ORIGINAL numerator over denominator := div(16, 1)

Signed original approximation minus reference SOURCE naive sub ORIGINAL numerator over denominator = 16

Signed original approximation minus reference ORIGINAL rational sub := sub(8, -8)

Signed original approximation minus reference ORIGINAL rational sub = 16

source roles = [8, -8]

derived = [16, 16, 否]

逐步反向核對

• S009 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S008 · premise_and_result_audit — 以精確有理數重算 Signed original approximation minus reference ORIGINAL rational sub，不以小數近似替代。
• S007 · premise_and_result_audit — 以精確有理數重算 Signed original approximation minus reference SOURCE naive sub ORIGINAL numerator over denominator，不以小數近似替代。
• S006 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S005 · premise_and_result_audit — 以精確有理數重算 signed ORIGINAL reference ORIGINAL numerator over denominator，不以小數近似替代。
• S004 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S003 · premise_and_result_audit — 以精確有理數重算 signed ORIGINAL approximation ORIGINAL numerator over denominator，不以小數近似替代。
• S002 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。
• S001 · premise_and_result_audit — 反向核對本步原始角色、完整前提與欄序；只接受這些條件下的結論，不把單向結論還原為唯一前提，也不把有限核對稱一般形式證明。

結論

(signed_vs_absolute, 8, -8, 16, 16, 否, signed_error_and_absolute_error_preserve_different_information)

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。