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

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