Question 10380: The Product with the Conjugate Is a Real Embedding

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=[-8, 1]. 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=[-8, 1]. 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

Step-by-step derivation

declared complex ordered pair=[-8, 1]

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

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

given value negate imaginary coefficient = -1

given value:original ordered pair=[-8, 1]

conjugate pair=[-8, -1]

value times conjugate real-real := mul(-8, -8)

value times conjugate real-real = 64

value times conjugate imaginary-imaginary := mul(1, -1)

value times conjugate imaginary-imaginary = -1

value times conjugate real coefficient := sub(64, -1)

value times conjugate real coefficient = 65

value times conjugate real-imaginary := mul(-8, -1)

value times conjugate real-imaginary = 8

value times conjugate imaginary-real := mul(1, -8)

value times conjugate imaginary-real = -8

value times conjugate imaginary coefficient := sum(8, -8)

value times conjugate imaginary coefficient = 0

value times conjugate:left=[-8, 1];right=[-8, -1];i^2=-1

complex product=[65, 0]

real coefficient squared := mul(-8, -8)

real coefficient squared = 64

imaginary coefficient squared := mul(1, 1)

imaginary coefficient squared = 1

exact squared modulus := sum(64, 1)

exact squared modulus = 65

value=[-8, 1];real square=64;imaginary square=1

modulus squared=65

computed product=[65, 0];squared modulus=65

product is real embedding [modulus squared,0]

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

([-8, 1], [-8, -1], [65, 0], 65)

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.