Complex Modulus is Non-Negative

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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

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

$\blacksquare$


Sources