# Complex Modulus of Real Number equals Absolute Value

## Theorem

Let $x \in \R$ be a real number.

Then the complex modulus of $x$ equals the absolute value of $x$.

## Proof

Let $x = x + 0 i \in \R$.

Then:

 $\ds \cmod {x + 0 i}$ $=$ $\ds \sqrt {x^2 + 0^2}$ Definition of Complex Modulus $\ds$ $=$ $\ds \sqrt {x^2}$ $\ds$ $=$ $\ds \size x$ Definition 2 of Absolute Value

$\blacksquare$