Modulus Larger than Real Part and Imaginary Part

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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

$\quad \left\vert{z}\right\vert \ge \left\vert{\operatorname{Re} \left({z}\right) }\right\vert$


Modulus Larger than Imaginary Part

$\quad \left\vert{z}\right\vert \ge \left\vert{\operatorname{Im} \left({z}\right) }\right\vert$