# Modulus Larger than Real Part

## Theorem

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

Then the modulus of $z$ is larger than the real part $\map \Re z$ of $z$:

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

## Proof

By the definition of a complex number, we have:

$z = \map \Re z + i \map \Im z$

Then:

 $\displaystyle \cmod z$ $=$ $\displaystyle \sqrt {\paren {\map \Re z}^2 + \paren {\map \Im z}^2}$ Definition of Complex Modulus $\displaystyle$ $\ge$ $\displaystyle \sqrt {\paren {\map \Re z}^2 }$ Square of Real Number is Non-Negative, as $\map \Im z$ is real $\displaystyle$ $=$ $\displaystyle \cmod {\map \Re z}$ Square of Real Number is Non-Negative, as $\map \Re z$ is real

$\blacksquare$