Let$R$ be anordered ring and let$a\in R$. Theabsolute value of$a$ is defined to be thefunction$||:R\to R$ given by

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

Absolute value has a different meaning in the case ofcomplex numbers: for a complex number$z\in ℂ$, the absolute value$|z|$ 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 avaluation, including:

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

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

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