# Question 29032: The Negative Real Contribution from the Product of Imaginary Parts [Return to catalogue page 146](https://ocme.evemisslab.com/en/questions/page/146/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261022-r03-complex-number-foundations/#q-20261013-advance-20261022-r03-complex-number-0057) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0057-29032 · q-20261013-advance-20261022-r03-complex-number-0057 ## Original question and givens Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-6, 3], right=[c,d]=[-1, 3]. Give (left,right,[a*c−b*d,a*d+b*c]). - G001 — Represent a complex number a+b*i by the ordered integer pair [a,b], with i²=−1. Keep the real component first and the imaginary coefficient second; graph appearance supplies no missing values. No complex division or polar form is used here. left=[a,b]=[-6, 3], right=[c,d]=[-1, 3]. Give (left,right,[a*c−b*d,a*d+b*c]). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-complex-number](https://ocme.evemisslab.com/en/mko/mko-complex-number/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-str22-complex-multiplication, complex-multiplication] After derived = [[left, right, result]] Conditions: none Use this bank exact ordered tuple and hyphenated operation.Rational outputs reduce with positive denominators and OMIT/1 for integers,unlike the preceding legacy bank.Do not change original bilingual snapshots or meaning. Step identifiers and dependenciesRule: STR22_exact_interface_and_source_ORDERED_tuple_roles · Uses: G001 S002 · calculationBefore Complex ORIGINAL real-real product := mul(-6, -1) After Complex ORIGINAL real-real product = 6 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Complex ORIGINAL imaginary-imaginary product := mul(3, 3) After Complex ORIGINAL imaginary-imaginary product = 9 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Complex ORIGINAL real-imaginary product := mul(-6, 3) After Complex ORIGINAL real-imaginary product = -18 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Complex ORIGINAL imaginary-real product := mul(3, -1) After Complex ORIGINAL imaginary-real product = -3 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Complex ORIGINAL real part ac minus bd := sub(6, 9) After Complex ORIGINAL real part ac minus bd = -3 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Complex ORIGINAL imaginary coefficient ad plus bc := sum(-18, -3) After Complex ORIGINAL imaginary coefficient ad plus bc = -21 Conditions: none Use the ORIGINAL operands exactly,without floating point.Only the declared SOURCE rational operations use their profile;additional integer teaching algebra adds no new input-domain restriction. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore source roles = [[-6, 3], [-1, 3]] After derived = [6, 9, -18, -3, [-3, -21]] Conditions: none Distribute to ac+(ad+bc)i+bd*i²,and use i²=-1 to obtain(ac-bd,ad+bc).Derived components may exceed raw1000/default1e6 bounds;do not impose a new cap or use a Hadamard product. Step identifiers and dependenciesRule: Complex_distribute_i_squared_MINUSone_not_Hadamard · Uses: G001, S007 ## Reverse checks - S008 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. - S007 · premise_and_result_audit — Recompute Complex ORIGINAL imaginary coefficient ad plus bc with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Complex ORIGINAL real part ac minus bd with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Complex ORIGINAL imaginary-real product with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Complex ORIGINAL real-imaginary product with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Complex ORIGINAL imaginary-imaginary product with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Complex ORIGINAL real-real product with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Reverse-check every ORIGINAL coordinate/table entry/alignment and strict premise,not just the final answer.Keep finite candidates,general reasoning,formal proof and publication responsibilities distinct. ## Result ([-6, 3], [-1, 3], [-3, -21]) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: dd9ef007c34b225632fd939cbd8f02c9fe1d2e44b7a03aace584707276f95d98 - candidate record SHA-256: fe8a992a5f844b9e3c4acfc1a961b8b0afbb1eaf854907d4237ebd1620f9bba6 - semantic review receipt SHA-256: db445bb0caa8a249fcf78eadd2550b49b820e279cc80b2327f0d757d3ae920b6 - question-first witness SHA-256: 58d12a50ea9943c50b65927607347101b1ddaf692d62e091f2cae8a9f0362eb5 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261022-r03-complex-number-0057.json)