Question 10344: 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=[6, 1], right=[1, 27]. 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=[6, 1], right=[1, 27]. 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=[6, 1]

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

original product real-real := mul(6, 1)

original product real-real = 6

original product imaginary-imaginary := mul(1, 27)

original product imaginary-imaginary = 27

original product real coefficient := sub(6, 27)

original product real coefficient = -21

original product real-imaginary := mul(6, 27)

original product real-imaginary = 162

original product imaginary-real := mul(1, 1)

original product imaginary-real = 1

original product imaginary coefficient := sum(162, 1)

original product imaginary coefficient = 163

original product:left=[6, 1];right=[1, 27];i^2=-1

complex product=[-21, 163]

original product negate imaginary coefficient := sub(0, 163)

original product negate imaginary coefficient = -163

original product:original ordered pair=[-21, 163]

conjugate pair=[-21, -163]

left value negate imaginary coefficient := sub(0, 1)

left value negate imaginary coefficient = -1

left value:original ordered pair=[6, 1]

conjugate pair=[6, -1]

right value negate imaginary coefficient := sub(0, 27)

right value negate imaginary coefficient = -27

right value:original ordered pair=[1, 27]

conjugate pair=[1, -27]

product of conjugates real-real := mul(6, 1)

product of conjugates real-real = 6

product of conjugates imaginary-imaginary := mul(-1, -27)

product of conjugates imaginary-imaginary = 27

product of conjugates real coefficient := sub(6, 27)

product of conjugates real coefficient = -21

product of conjugates real-imaginary := mul(6, -27)

product of conjugates real-imaginary = -162

product of conjugates imaginary-real := mul(-1, 1)

product of conjugates imaginary-real = -1

product of conjugates imaginary coefficient := sum(-162, -1)

product of conjugates imaginary coefficient = -163

product of conjugates:left=[6, -1];right=[1, -27];i^2=-1

complex product=[-21, -163]

conjugate of computed product=[-21, -163];computed product of conjugates=[-21, -163]

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

([6, 1], [1, 27], [-21, 163], [-21, -163], [-21, -163], 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.