Question 10347: 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=[9, -3], right=[-1, 33]. 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=[9, -3], right=[-1, 33]. 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=[9, -3]

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

original product real-real := mul(9, -1)

original product real-real = -9

original product imaginary-imaginary := mul(-3, 33)

original product imaginary-imaginary = -99

original product real coefficient := sub(-9, -99)

original product real coefficient = 90

original product real-imaginary := mul(9, 33)

original product real-imaginary = 297

original product imaginary-real := mul(-3, -1)

original product imaginary-real = 3

original product imaginary coefficient := sum(297, 3)

original product imaginary coefficient = 300

original product:left=[9, -3];right=[-1, 33];i^2=-1

complex product=[90, 300]

original product negate imaginary coefficient := sub(0, 300)

original product negate imaginary coefficient = -300

original product:original ordered pair=[90, 300]

conjugate pair=[90, -300]

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

left value negate imaginary coefficient = 3

left value:original ordered pair=[9, -3]

conjugate pair=[9, 3]

right value negate imaginary coefficient := sub(0, 33)

right value negate imaginary coefficient = -33

right value:original ordered pair=[-1, 33]

conjugate pair=[-1, -33]

product of conjugates real-real := mul(9, -1)

product of conjugates real-real = -9

product of conjugates imaginary-imaginary := mul(3, -33)

product of conjugates imaginary-imaginary = -99

product of conjugates real coefficient := sub(-9, -99)

product of conjugates real coefficient = 90

product of conjugates real-imaginary := mul(9, -33)

product of conjugates real-imaginary = -297

product of conjugates imaginary-real := mul(3, -1)

product of conjugates imaginary-real = -3

product of conjugates imaginary coefficient := sum(-297, -3)

product of conjugates imaginary coefficient = -300

product of conjugates:left=[9, 3];right=[-1, -33];i^2=-1

complex product=[90, -300]

conjugate of computed product=[90, -300];computed product of conjugates=[90, -300]

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

([9, -3], [-1, 33], [90, 300], [90, -300], [90, -300], 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.