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:

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