Absolute value

Introduction

The absolute value of a real number $x$ , denoted $|x|$ , is its distance from 0. Therefore, if $x\ge 0$ , then $|x|=x$ , and if $x<0$ , then |x|=-x.

Similarly, the absolute value of a complex number $z=x+iy$ , where $x,y\in\mathbb{R}$ , is $|z|=\sqrt{x^2+y^2}$ .

Generalized absolute values

The absolute value functions listed above have three very important properties:

$|x|\ge 0$ for all x, and $|x|=0$ iff x=0.
$|x\times y|=|x|\times |y|$ .
$|x+y| \le |x|+|y|$ . (Triangle inequality)

We call any function satisfying these three properties an absolute value.

Another example of an absolute value is the p-adic absolute value on $\mathbb{Q}$ , the rational numbers. Let $x=\prod_{i=1}^n p_i^{e_i}$ , where the $p_{i}$ 's are distinct prime numbers, and the $e_i$ 's are (positive, negative, or zero) integers. Define $|x|_{p_i}=p_i^{-e_i}$ . This defines an absolute value on $\mathbb{Q}$ . This absolute value satisfies a stronger triangle inequality:

$|x+y|\le\max(|x|,|y|)$ .

An absolute value satisfying this strong triangle inequality is called nonarchimedian. If an absolute value does not satisfy the strong triangle inequality, then it is called archimedian. The ordinary absolute value on $\mathbb{R}$ or $\mathbb{C}$ is archimedian.

The theory of absolute values is important in algebraic number theory. Let $K/\mathbb{Q}$ be a finite Galois extension with $[K:\mathbb{Q}]=n$ , and let $\sigma_1,\ldots,\sigma_n$ be the field automorphisms of $K$ over $\mathbb{Q}$ . Then the only absolute values are the archimedian ones given by $|x|_i=|\sigma_i(x)|$ (the ordinary real or complex absolute values) and the nonarchimedian ones given by $|x|_{\mathfrak{p}}$ for some prime ${\mathfrak{p}}$ of $K$ .

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Absolute value

Introduction

Generalized absolute values

See also