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 \(x\).
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| created: 2014-04-26 22:17:29 | modified: 2020-07-04 15:53:01 | by: bookofproofs
