# Definition:Absolute Value/Definition 1

## Definition

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

The absolute value of $x$ is denoted $\size x$, and is defined using the usual ordering on the real numbers as follows:

$\size x = \begin{cases} x & : x > 0 \\ 0 & : x = 0 \\ -x & : x < 0 \end{cases}$

## Also presented as

Note that since $0 = -0$, the value of $\size x$ at $x = 0$ is often included in one of the other two cases, most commonly:

$\size x = \begin{cases} x & : x \ge 0 \\ -x & : x < 0 \end{cases}$

but this can be argued as being less symmetrically aesthetic.

## Also known as

The absolute value can also be found referred to as the numerical value.

## Also see

• Results about the absolute value function can be found here.