# Product of Complex Number with Conjugate

## Theorem

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

Let $\overline z$ denote the complex conjugate of $z$.

Then:

$z \overline z = a^2 + b^2$

and thus is wholly real.

## Proof

By the definition of a complex number, let $z = a + i b$ where $a$ and $b$ are real numbers.

Then:

 $\displaystyle z \overline z$ $=$ $\displaystyle \left({a + i b}\right) \left({a - i b}\right)$ Definition of Complex Conjugate $\displaystyle$ $=$ $\displaystyle a^2 + a \cdot i b + a \cdot \left({-i b}\right) + i \cdot \left({-i}\right) \cdot b^2$ Definition of Complex Multiplication $\displaystyle$ $=$ $\displaystyle a^2 + i a b - i a b + b^2$ $\displaystyle$ $=$ $\displaystyle a^2 + b^2$

As $a^2 + b^2$ is wholly real, the result follows.

$\blacksquare$