PlanetMath: absolute value

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absolute value

(Definition)

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

$\begin{displaymath} \vert a\vert := \begin{cases} a & \text{\ \ if } a \geq 0, \ -a & \text{\ \ otherwise.} \end{cases}\end{displaymath}$

In particular, the usual absolute value $\vert\ \vert$ 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 $\vert z\vert$ of is defined to be $\sqrt{x^2+y^2}$ , where and $x,y \in \mathbb{R}$ are real.

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

$\vert a\vert \ge 0$ for all $a \in R$ , with equality if and only if
$\vert ab\vert = \vert a\vert \cdot \vert b\vert$ for all $a,b \in R$
$\vert a+b\vert \le \vert a\vert + \vert b\vert$ 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 $d(x,y) := \vert x-y\vert$ .

"absolute value" is owned by djao. [ full author list (2) | owner history (1) ]

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Cross-references: distance, metric, induces, triangle inequality, equality, valuation, defining properties, complex numbers, real numbers, field, function, ordered ring
There are 30 references to this entry.

This is version 4 of absolute value, born on 2001-10-21, modified 2003-01-19.
Object id is 448, canonical name is AbsoluteValue.
Accessed 19719 times total.

Classification:

AMS MSC:

13-00 (Commutative rings and algebras :: General reference works )

Pending Errata and Addenda

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