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 $$ |a| := \begin{cases} a & \text{\ \ if } a \geq 0, \\ -a & \text{\ \ otherwise.} \end{cases} $$ In particular, the usual absolute value $|\ |$ on the field $\mathbb{R}$ of real numbers is defined in this manner.

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:

• $|a| \ge 0$ 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 $\mathbb{C}$ the absolute value is a valuation, and it induces a metric in the usual way, with distance function defined by $d(x,y) := |x-y|$