# Complex modulus and conjugation

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Complex conjugation preserves the real part and reverses the imaginary coefficient; modulus squared is the sum of their squares.

mko-complex-modulus-conjugate · concept · v0.10.0



## Mathematical statement

For z=a+b*i, the conjugate is a−b*i and modulus squared is a²+b². Multiplying z by its conjugate gives the real embedding of that squared modulus. Conjugation preserves addition and multiplication.





## Representative formula (exact TeX)

M(z)





## Assumptions


- Exercises represent complex numbers by integer ordered pairs and calculate only conjugates and modulus squares, not decimal square-root approximations presented as exact moduli.
- Arguments, polar form and complex differentiability are not handled in this slice.





## Symbol scope


- M — complex-modulus/conjugation record
- z — complex number





## Intuition, definition, and misconceptions

The conjugate of (a,b) is (a,−b).

The product of z and its conjugate has zero imaginary coefficient and real part a²+b².

Modulus squared remains an exact integer, avoiding decimal approximations for non-perfect squares.

Do not negate only the real part or confuse modulus squared with the modulus itself.





## Adopted derivation guide

[Complex modulus and conjugate: proving z·z̄=|z|² and that conjugation preserves the operations](https://ocme.evemisslab.com/en/derivations/derivation-mko-complex-modulus-conjugate-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Complex number](https://ocme.evemisslab.com/en/mko/mko-complex-number/) — Conjugation and modulus build on ordered-pair complex arithmetic.
- [Absolute value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/) — Complex modulus extends the magnitude semantics of real absolute value.
- [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) — Modulus squared uses a sum of squares.





## Verification boundary

Only integer ordered-pair algebra is checked; no complex-analysis result is asserted.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Add complex division, polar form and complex functions.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/complex-modulus-conjugate.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-complex-modulus-conjugate/index.md)