Question 10334: 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=[-4, -2], right=[1, 7]. 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=[-4, -2], right=[1, 7]. 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=[-4, -2]

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

original product real-real := mul(-4, 1)

original product real-real = -4

original product imaginary-imaginary := mul(-2, 7)

original product imaginary-imaginary = -14

original product real coefficient := sub(-4, -14)

original product real coefficient = 10

original product real-imaginary := mul(-4, 7)

original product real-imaginary = -28

original product imaginary-real := mul(-2, 1)

original product imaginary-real = -2

original product imaginary coefficient := sum(-28, -2)

original product imaginary coefficient = -30

original product:left=[-4, -2];right=[1, 7];i^2=-1

complex product=[10, -30]

original product negate imaginary coefficient := sub(0, -30)

original product negate imaginary coefficient = 30

original product:original ordered pair=[10, -30]

conjugate pair=[10, 30]

left value negate imaginary coefficient := sub(0, -2)

left value negate imaginary coefficient = 2

left value:original ordered pair=[-4, -2]

conjugate pair=[-4, 2]

right value negate imaginary coefficient := sub(0, 7)

right value negate imaginary coefficient = -7

right value:original ordered pair=[1, 7]

conjugate pair=[1, -7]

product of conjugates real-real := mul(-4, 1)

product of conjugates real-real = -4

product of conjugates imaginary-imaginary := mul(2, -7)

product of conjugates imaginary-imaginary = -14

product of conjugates real coefficient := sub(-4, -14)

product of conjugates real coefficient = 10

product of conjugates real-imaginary := mul(-4, -7)

product of conjugates real-imaginary = 28

product of conjugates imaginary-real := mul(2, 1)

product of conjugates imaginary-real = 2

product of conjugates imaginary coefficient := sum(28, 2)

product of conjugates imaginary coefficient = 30

product of conjugates:left=[-4, 2];right=[1, -7];i^2=-1

complex product=[10, 30]

conjugate of computed product=[10, 30];computed product of conjugates=[10, 30]

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

([-4, -2], [1, 7], [10, -30], [10, 30], [10, 30], 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.