Absolute value

The standard absolute value function on the real numbers.

Absolute value

An absolute value is the function $|\cdot|:\mathbb R\to[0,\infty)$ defined by

|x|=\begin{cases} x,& x\ge 0,\\ -x,& x<0. \end{cases}

It is defined using the order axioms on $\mathbb R$ and measures distance via $d(x,y)=|x-y|$ , a basic example of a metric . Absolute value is built into $\varepsilon$ – $\delta$ definitions such as the limit at a point .

Examples:

$|-3|=3$ and $|3|=3$ .
The distance between $2$ and $5$ in $\mathbb R$ is $|2-5|=3$ .