# Question 10371: Exact Modulus Squared without an Unrequested Root

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0019-10371 · q-20261013-advance-20261023-r01-complex-modulus-conjugate-0046



## 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, 3]. 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=[8, 3]. 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](https://ocme.evemisslab.com/en/mko/mko-complex-modulus-conjugate/)





## Step-by-step derivation

S001 · impliesBefore

declared complex ordered pair=[8, 3]

After

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

Conditions: none

Use the declared complex algebra,not componentwise real-vector multiplication.The zero complex number is allowed.

Step identifiers and dependenciesRule: exact_complex_pair_scope · Uses: G001





S002 · calculationBefore

real coefficient squared := mul(8, 8)

After

real coefficient squared = 64

Conditions: none

Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001





S003 · calculationBefore

imaginary coefficient squared := mul(3, 3)

After

imaginary coefficient squared = 9

Conditions: none

Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · calculationBefore

exact squared modulus := sum(64, 9)

After

exact squared modulus = 73

Conditions: none

Compute exactly from this step's original operands in the specified order,retaining signs and denominator conditions.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

value=[8, 3];real square=64;imaginary square=9

After

modulus squared=73

Conditions: none

The requested value is|z|²=a²+b²,not|z|.Do not substitute a radical,approximation or negative square root.

Step identifiers and dependenciesRule: modulus_squared_not_unrequested_square_root · Uses: G001, S004









## 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

([8, 3], 73)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: ed4740ae39ee5de7accf75bd3bef813f2a964c83ce4700f41dc5988c25c7f481
- candidate record SHA-256: d8fa48ab2a266404e1c258f8c13746988bbbacbe17f99e713dbc56f365075087
- semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
- question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
- Lean: NOT_RUN · human review: NOT_PERFORMED

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