Complex Modulus is Norm

Theorem
The complex modulus is a norm on the set of complex numbers $\C$.

Proof
We prove the norm axioms.

Positive definiteness
As $0 = 0 + 0i$, it follows that:


 * $\left\vert{ 0 }\right\vert = \sqrt{ 0^2 + 0^2 } = 0$

Suppose instead that $\left\vert{a + bi}\right\vert = 0$ for some $z = a+bi \in \C$ with $a, b \in \R$.

As $\sqrt{a^2 + b^2} = 0$, it follows from squaring both sides that:


 * $a^2 + b^2 = 0$

From Even Powers are Positive, it follows that $a = 0$ and $b = 0$.

Hence, $z = 0 + 0i = 0$.

Multiplicativity
Follows from Modulus of Product.

Triangle Inequality
Follows from Triangle Inequality/Complex Numbers