Question 10352: Exact Modulus Squared without an Unrequested Root

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=[-11, -2]. Give (value,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=[-11, -2]. Give (value,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=[-11, -2]

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

real coefficient squared := mul(-11, -11)

real coefficient squared = 121

imaginary coefficient squared := mul(-2, -2)

imaginary coefficient squared = 4

exact squared modulus := sum(121, 4)

exact squared modulus = 125

value=[-11, -2];real square=121;imaginary square=4

modulus squared=125

Reverse checks

• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute exact squared modulus with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute imaginary coefficient squared with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute real coefficient squared 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

([-11, -2], 125)

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.