Definition:Absolute Value/Definition 1

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

The absolute value of $x$ is denoted $\left\vert{x}\right\vert$, and is defined using the ordering on the real numbers as follows:
 * $\left\vert{x}\right\vert = \begin{cases}

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

Also presented as
Note that since $0 = -0$, the value of $\left\vert{x}\right\vert$ at $x = 0$ is often included in one of the other two cases, most commonly:
 * $\left\vert{x}\right\vert = \begin{cases}

x & : x \ge 0 \\ -x & : x < 0 \end{cases}$ but this can be argued as being less symmetrically aesthetic.

Also see

 * Absolute Value Equals Square Root of Square

Generalizations

 * Definition:Complex Modulus, as shown at Complex Modulus of Real Number equals Absolute Value