Let \(x,y\in \mathbb R\). Based on the ordering relation for real numbers, we define a function \(|~|:\mathbb R\times \mathbb R\mapsto \mathbb R\) by

\[|x-y| :=

\begin{cases}

x-y & \text{ if } x\ge y \\

y-x & \text{ if } x < y

\end{cases}\]

and call it the **distance** of \(x\) and \(y\).

The distance of any real number \(x\) from \(0\)

\[|x| := |x-0|=

\begin{cases}

x & \text{ if } x\ge 0 \\

-x & \text{ if } x < 0

\end{cases}\]

is called the **absolute value** of \(x\).

| | | | | created: 2014-04-26 22:17:29 | modified: 2020-07-04 15:53:01 | by: *bookofproofs*

## 1.**Corollary**: Properties of the Absolute Value