AbsoluteValue 2001-10-21 02:21:17 2003-01-19 08:09:08 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} Let $R$ be an ordered ring and let $a \in R$. The {\em 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: \begin{itemize} \item $|a| \ge 0$ for all $a \in R$, with equality if and only if $a = 0$ \item $|ab| = |a| \cdot |b|$ for all $a,b \in R$ \item $|a+b| \le |a| + |b|$ for all $a, b \in R$ (triangle inequality) \end{itemize} 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 $d(x,y) := |x-y|$.