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

In particular, the usual absolute value $\left|\right|$ on the field $ℝ$ of real numbers is defined in this manner. An equivalent definition over the real numbers is $\left|a\right|:=\max \{a,-a\}$.

Absolute value has a different meaning in the case of complex numbers: for a complex number $z\in ℂ$, the absolute value $\left|z\right|$ of $z$ is defined to be $\sqrt{x^2+y^2}$, where $z=x+yi$ and $x,y\in ℝ$ are real.

All absolute value functions satisfy the defining properties of a valuation, including:

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

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

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  $\left|a+b\right|\le \left|a\right|+\left|b\right|$ 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 $ℝ$ and $ℂ$, the absolute value is a valuation, and it induces a metric in the usual way, with distance function defined by $d(x,y):=\left|x-y\right|$.