# Complex Modulus is Non-Negative

## Theorem

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

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

Then:

$\cmod z \ge 0$

## Proof

 $\ds \cmod z$ $=$ $\ds \cmod {a + b i}$ Definition of $z$ $\ds$ $=$ $\ds +\sqrt {a^2 + b^2}$ Definition of Complex Modulus $\ds$ $\ge$ $\ds 0$ Definition of Positive Square Root

$\blacksquare$