Question 10346: Conjugating a Product versus Multiplying the Conjugates

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=[8, 3], right=[-2, 31]. 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=[8, 3], right=[-2, 31]. 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

Step-by-step derivation

declared complex ordered pair=[8, 3]

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

original product real-real := mul(8, -2)

original product real-real = -16

original product imaginary-imaginary := mul(3, 31)

original product imaginary-imaginary = 93

original product real coefficient := sub(-16, 93)

original product real coefficient = -109

original product real-imaginary := mul(8, 31)

original product real-imaginary = 248

original product imaginary-real := mul(3, -2)

original product imaginary-real = -6

original product imaginary coefficient := sum(248, -6)

original product imaginary coefficient = 242

original product:left=[8, 3];right=[-2, 31];i^2=-1

complex product=[-109, 242]

original product negate imaginary coefficient := sub(0, 242)

original product negate imaginary coefficient = -242

original product:original ordered pair=[-109, 242]

conjugate pair=[-109, -242]

left value negate imaginary coefficient := sub(0, 3)

left value negate imaginary coefficient = -3

left value:original ordered pair=[8, 3]

conjugate pair=[8, -3]

right value negate imaginary coefficient := sub(0, 31)

right value negate imaginary coefficient = -31

right value:original ordered pair=[-2, 31]

conjugate pair=[-2, -31]

product of conjugates real-real := mul(8, -2)

product of conjugates real-real = -16

product of conjugates imaginary-imaginary := mul(-3, -31)

product of conjugates imaginary-imaginary = 93

product of conjugates real coefficient := sub(-16, 93)

product of conjugates real coefficient = -109

product of conjugates real-imaginary := mul(8, -31)

product of conjugates real-imaginary = -248

product of conjugates imaginary-real := mul(-3, -2)

product of conjugates imaginary-real = 6

product of conjugates imaginary coefficient := sum(-248, 6)

product of conjugates imaginary coefficient = -242

product of conjugates:left=[8, -3];right=[-2, -31];i^2=-1

complex product=[-109, -242]

conjugate of computed product=[-109, -242];computed product of conjugates=[-109, -242]

equal=Yes

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

([8, 3], [-2, 31], [-109, 242], [-109, -242], [-109, -242], 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.