# Modulus in Terms of Conjugate

## Theorem

Let $z = a + i b$ be a complex number.

Let $\cmod z$ be the modulus of $z$.

Let $\overline z$ be the conjugate of $z$.

Then:

$\cmod z^2 = z \overline z$

## Proof

Let $z = a + i b$.

Then:

 $\ds z \overline z$ $=$ $\ds a^2 + b^2$ Product of Complex Number with Conjugate $\ds$ $=$ $\ds \cmod z^2$ Definition of Complex Modulus

$\blacksquare$