# absolute value

Let $R$ be an ordered ring and let $a\in R$. The absolute value of $a$ is defined to be the function $|\ |\colon R\to R$ given by

 $|a|:=\begin{cases}a&\text{\ \ if }a\geq 0,\\ -a&\text{\ \ otherwise.}\end{cases}$

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 $|a|:=\max\{a,-a\}$.

Absolute value has a different meaning in the case of complex numbers: for a complex number $z\in\mathbb{C}$, 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:

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

• $|ab|=|a|\cdot|b|$ for all $a,b\in R$

• $|a+b|\leq|a|+|b|$ 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 $\mathbb{C}$, the absolute value is a valuation, and it induces a metric in the usual way, with distance function defined by $d(x,y):=|x-y|$.

Title absolute value AbsoluteValue 2013-03-22 11:52:09 2013-03-22 11:52:09 djao (24) djao (24) 10 djao (24) Definition msc 13-00 msc 11A15