absolute value

Let R be an ordered ring and let aR. The absolute valueMathworldPlanetmathPlanetmathPlanetmath of a is defined to be the function ||:RR given by

|a|:={a if a0,-a otherwise.

In particular, the usual absolute value || on the field of real numbers is defined in this manner. An equivalentPlanetmathPlanetmath definition over the real numbers is |a|:=max{a,-a}.

Absolute value has a different meaning in the case of complex numbersMathworldPlanetmathPlanetmath: for a complex number z, the absolute value |z| of z is defined to be x2+y2, where z=x+yi and x,y are real.

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

  • |a|0 for all aR, with equality if and only if a=0

  • |ab|=|a||b| for all a,bR

  • |a+b||a|+|b| for all a,bR (triangle inequalityPlanetmathPlanetmath)

However, in general they are not literally valuations, because valuations are required to be real valued. In the case of and , 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|.

Title absolute value
Canonical name AbsoluteValue
Date of creation 2013-03-22 11:52:09
Last modified on 2013-03-22 11:52:09
Owner djao (24)
Last modified by djao (24)
Numerical id 10
Author djao (24)
Entry type Definition
Classification msc 13-00
Classification msc 11A15