Absolute value
Introduction
The absolute value of a real number , denoted
, is its distance from 0. Therefore, if
, then
, and if
, then |x|=-x.
Similarly, the absolute value of a complex number , where
, is
.
Generalized absolute values
The absolute value functions listed above have three very important properties:
for all x, and
iff x=0.
.
. (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 , the rational numbers. Let
, where the
's are distinct prime numbers, and the
's are (positive, negative, or zero) integers. Define
. This defines an absolute value on
. This absolute value satisfies a stronger triangle inequality:
.
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 or
is archimedian.
The theory of absolute values is important in algebraic number theory. Let be a finite Galois extension with
, and let
be the field automorphisms of
over
. Then the only absolute values are the archimedian ones given by
(the ordinary real or complex absolute values) and the nonarchimedian ones given by
for some prime
of
.