# Question 10386: The Product with the Conjugate Is a Real Embedding [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-0061) 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. R0019-10386 · q-20261013-advance-20261023-r01-complex-modulus-conjugate-0061 ## 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. value=[-2, 0]. Give (value,conjugate,product as a complex pair,modulus squared). - 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. value=[-2, 0]. Give (value,conjugate,product as a complex pair,modulus squared). ## 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=[-2, 0] 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 given value negate imaginary coefficient := sub(0, 0) After given value negate imaginary coefficient = 0 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 · impliesBefore given value:original ordered pair=[-2, 0] After conjugate pair=[-2, 0] 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, S002 S004 · calculationBefore value times conjugate real-real := mul(-2, -2) After value times conjugate real-real = 4 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 value times conjugate imaginary-imaginary := mul(0, 0) After value times conjugate imaginary-imaginary = 0 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 value times conjugate real coefficient := sub(4, 0) After value times conjugate real coefficient = 4 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 value times conjugate real-imaginary := mul(-2, 0) After value times conjugate real-imaginary = 0 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 · calculationBefore value times conjugate imaginary-real := mul(0, -2) After value times conjugate imaginary-real = 0 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, S007 S009 · calculationBefore value times conjugate imaginary coefficient := sum(0, 0) After value times conjugate imaginary coefficient = 0 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 value times conjugate:left=[-2, 0];right=[-2, 0];i^2=-1 After complex product=[4, 0] 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, S009 S011 · calculationBefore real coefficient squared := mul(-2, -2) After real coefficient squared = 4 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 · calculationBefore imaginary coefficient squared := mul(0, 0) After imaginary coefficient squared = 0 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, S011 S013 · calculationBefore exact squared modulus := sum(4, 0) After exact squared modulus = 4 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 value=[-2, 0];real square=4;imaginary square=0 After modulus squared=4 Conditions: none The requested value is|z|²=a²+b²,not|z|.Do not substitute a radical,approximation or negative square root. Step identifiers and dependenciesRule: modulus_squared_not_unrequested_square_root · Uses: G001, S013 S015 · impliesBefore computed product=[4, 0];squared modulus=4 After product is real embedding [modulus squared,0] Conditions: none The computed imaginary coefficient cancels to0,and the real coefficient equals the sum of squares.The product was not merely filled from an expected answer. Step identifiers and dependenciesRule: complex_times_conjugate_imaginary_cancellation · Uses: G001, S014 ## Reverse checks - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 exact squared modulus with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute imaginary coefficient squared with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute real coefficient squared 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 value times conjugate imaginary coefficient with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute value times conjugate imaginary-real with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute value times conjugate real-imaginary with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute value times conjugate real coefficient with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute value times conjugate imaginary-imaginary with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute value times conjugate real-real with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Recompute given value negate imaginary coefficient 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 ([-2, 0], [-2, 0], [4, 0], 4) 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: 2cec670ba82cc1bcef59a5b85182e74a0e7a5d07792bc9afe63ca34a9f48f227 - candidate record SHA-256: add0f8bf734f61204d3e54503e3b1bc5906e466a3935f781b3e67860823208f3 - 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-0061.json)