Modulus Larger than Real Part and Imaginary Part

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

Then the modulus of $z$ is larger than the real part of $z$ and the the imaginary part of $z$:


 * $(1): \quad \left\vert{z}\right\vert \ge \left\vert{\operatorname{Re} \left({z}\right) }\right\vert$


 * $(2): \quad \left\vert{z}\right\vert \ge \left\vert{\operatorname{Im} \left({z}\right) }\right\vert$

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

Proof
By the definition of a complex number, we have $z = \operatorname{Re} \left({z}\right) + i \operatorname{Im} \left({z}\right)$.

Then:

This proves $(1)$.

The proof of $(2)$ follows similarly:

This proves $(2)$.