# Absolute Value Function is Completely Multiplicative/Proof 3

## Theorem

The absolute value function on the real numbers $\R$ is completely multiplicative:

$\forall x, y \in \R: \left\vert{x y}\right\vert = \left\vert{x}\right\vert \, \left\vert{y}\right\vert$

where $\left \vert{a}\right \vert$ denotes the absolute value of $a$.

## Proof

 $\displaystyle \size {x y}$ $=$ $\displaystyle \sqrt {\paren {x y}^2}$ Definition 2 of Absolute Value $\displaystyle$ $=$ $\displaystyle \sqrt {x^2 y^2}$ $\displaystyle$ $=$ $\displaystyle \sqrt {x^2} \sqrt{y^2}$ $\displaystyle$ $=$ $\displaystyle \size x \cdot \size y$

$\blacksquare$