absolute value
Let $R$ be an ordered ring and let $a\in R$. The absolute value^{} of $a$ is defined to be the function $:R\to R$ given by
$$a:=\{\begin{array}{cc}a\hfill & \text{if}a\ge 0,\hfill \\ a\hfill & \text{otherwise.}\hfill \end{array}$$ 
In particular, the usual absolute value $$ on the field $\mathbb{R}$ of real numbers is defined in this manner. An equivalent^{} definition over the real numbers is $a:=\mathrm{max}\{a,a\}$.
Absolute value has a different meaning in the case of complex numbers^{}: for a complex number $z\in \u2102$, the absolute value $z$ of $z$ is defined to be $\sqrt{{x}^{2}+{y}^{2}}$, where $z=x+yi$ and $x,y\in \mathbb{R}$ are real.
All absolute value functions satisfy the defining properties of a valuation, including:

•
$a\ge 0$ for all $a\in R$, with equality if and only if $a=0$

•
$ab=a\cdot b$ for all $a,b\in R$

•
$a+b\le a+b$ for all $a,b\in R$ (triangle inequality^{})
However, in general they are not literally valuations, because valuations are required to be real valued. In the case of $\mathbb{R}$ and $\u2102$, the absolute value is a valuation, and it induces a metric in the usual way, with distance function defined by $d(x,y):=xy$.
Title  absolute value 

Canonical name  AbsoluteValue 
Date of creation  20130322 11:52:09 
Last modified on  20130322 11:52:09 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  10 
Author  djao (24) 
Entry type  Definition 
Classification  msc 1300 
Classification  msc 11A15 