Absolute value (algebra)

This article is about the generalization of the basic concept. For the basic concept, see Absolute value. For other uses, see Absolute value (disambiguation).

In mathematics, an absolute value is a function which measures the "size" of elements in a field or integral domain. More precisely, if D is an integral domain, then an absolute value is any mapping |x| from D to the real numbers R satisfying:

It follows from these axioms that |1| = 1 and |−1| = 1. Furthermore, for every positive integer n,