Complex modulus and conjugation

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Complex number — Conjugation and modulus build on ordered-pair complex arithmetic.
• Absolute value — Complex modulus extends the magnitude semantics of real absolute value.
• Natural-number powers and principal square roots — 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.