# Question 10339: Conjugating a Product versus Multiplying the Conjugates [Return to catalogue page 52](https://ocme.evemisslab.com/en/questions/page/52/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261023-r01-complex-modulus-conjugate-foundations/#q-20261013-advance-20261023-r01-complex-modulus-conjugate-0089) 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. R0019-10339 · q-20261013-advance-20261023-r01-complex-modulus-conjugate-0089 ## Original question and givens Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[1, 3], right=[1, 17]. Give (left,right,product,conjugate of product,product of conjugates,equal?). - G001 — Use integer ordered pairs [a,b] for a+b*i, with i²=−1. Conjugation reverses the imaginary coefficient, not the real part. Modulus squared is a²+b²; do not replace it with a square root or a decimal approximation. No complex-analysis conclusion is claimed. left=[1, 3], right=[1, 17]. Give (left,right,product,conjugate of product,product of conjugates,equal?). ## 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-modulus-conjugate](https://ocme.evemisslab.com/en/mko/mko-complex-modulus-conjugate/) ## Step-by-step derivation S001 · impliesBefore declared complex ordered pair=[1, 3] After exact integer coefficients in the algebra with i^2=-1 Conditions: none Use the declared complex algebra,not componentwise real-vector multiplication.The zero complex number is allowed. Step identifiers and dependenciesRule: exact_complex_pair_scope · Uses: G001 S002 · calculationBefore original product real-real := mul(1, 1) After original product real-real = 1 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore original product imaginary-imaginary := mul(3, 17) After original product imaginary-imaginary = 51 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore original product real coefficient := sub(1, 51) After original product real coefficient = -50 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore original product real-imaginary := mul(1, 17) After original product real-imaginary = 17 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore original product imaginary-real := mul(3, 1) After original product imaginary-real = 3 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore original product imaginary coefficient := sum(17, 3) After original product imaginary coefficient = 20 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore original product:left=[1, 3];right=[1, 17];i^2=-1 After complex product=[-50, 20] Conditions: none Expand(a+bi)(c+di)=(ac−bd)+(ad+bc)i,retaining each of the four products' source positions and signs. Step identifiers and dependenciesRule: complex_product_ordered_component_expansion · Uses: G001, S007 S009 · calculationBefore original product negate imaginary coefficient := sub(0, 20) After original product negate imaginary coefficient = -20 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore original product:original ordered pair=[-50, 20] After conjugate pair=[-50, -20] Conditions: none The pair(a,b) denotes a+bi.Conjugation fixes the real part and negates only b;it does not exchange coordinates. Step identifiers and dependenciesRule: complex_conjugate_real_fixed_imaginary_negated · Uses: G001, S009 S011 · calculationBefore left value negate imaginary coefficient := sub(0, 3) After left value negate imaginary coefficient = -3 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore left value:original ordered pair=[1, 3] After conjugate pair=[1, -3] Conditions: none The pair(a,b) denotes a+bi.Conjugation fixes the real part and negates only b;it does not exchange coordinates. Step identifiers and dependenciesRule: complex_conjugate_real_fixed_imaginary_negated · Uses: G001, S011 S013 · calculationBefore right value negate imaginary coefficient := sub(0, 17) After right value negate imaginary coefficient = -17 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · impliesBefore right value:original ordered pair=[1, 17] After conjugate pair=[1, -17] Conditions: none The pair(a,b) denotes a+bi.Conjugation fixes the real part and negates only b;it does not exchange coordinates. Step identifiers and dependenciesRule: complex_conjugate_real_fixed_imaginary_negated · Uses: G001, S013 S015 · calculationBefore product of conjugates real-real := mul(1, 1) After product of conjugates real-real = 1 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore product of conjugates imaginary-imaginary := mul(-3, -17) After product of conjugates imaginary-imaginary = 51 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore product of conjugates real coefficient := sub(1, 51) After product of conjugates real coefficient = -50 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore product of conjugates real-imaginary := mul(1, -17) After product of conjugates real-imaginary = -17 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore product of conjugates imaginary-real := mul(-3, 1) After product of conjugates imaginary-real = -3 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore product of conjugates imaginary coefficient := sum(-17, -3) After product of conjugates imaginary coefficient = -20 Conditions: none Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore product of conjugates:left=[1, -3];right=[1, -17];i^2=-1 After complex product=[-50, -20] Conditions: none Expand(a+bi)(c+di)=(ac−bd)+(ad+bc)i,retaining each of the four products' source positions and signs. Step identifiers and dependenciesRule: complex_product_ordered_component_expansion · Uses: G001, S020 S022 · impliesBefore conjugate of computed product=[-50, -20];computed product of conjugates=[-50, -20] After equal=Yes Conditions: none Expand both routes completely,retaining original left/right roles and every intermediate pair.These exact instances are not new general formal Evidence. Step identifiers and dependenciesRule: conjugation_product_two_routes · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S020 · premise_and_result_audit — Recompute product of conjugates imaginary coefficient with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute product of conjugates imaginary-real with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute product of conjugates real-imaginary with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute product of conjugates real coefficient with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute product of conjugates imaginary-imaginary with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute product of conjugates real-real with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Recompute right value negate imaginary coefficient with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute left value negate imaginary coefficient with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute original product negate imaginary coefficient with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute original product imaginary coefficient with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute original product imaginary-real with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute original product real-imaginary with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute original product real coefficient with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute original product imaginary-imaginary with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute original product real-real with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([1, 3], [1, 17], [-50, 20], [-50, -20], [-50, -20], Yes) 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: 6375e7e9d1bbde00485c0ad55d365a594b6ce5f7ed5806374f28ca48da9ad7df - candidate record SHA-256: 23bec019d924742c75d8618dd4952f9f58ad2b0bc7cdebc7219c3e06e4753135 - semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5 - question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261023-r01-complex-modulus-conjugate-0089.json)