Modulus Larger than Real Part and Imaginary Part

Theorem

Let $z \in \C$ be a complex number.

Let $\operatorname{Re} \left({z}\right)$ denote the real part of $z$, and $\operatorname{Im} \left({z}\right) $ the imaginary part of $z$.

Then:

Modulus Larger than Real Part

$\cmod z \ge \size {\map \Re z}$

Modulus Larger than Imaginary Part

$\cmod z \ge \size {\map \Im z}$