What is Absolute value (algebra)?

In algebra, 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,

The classical "absolute value" is one in which, for example, |2|=2, but many other functions fulfill the requirements stated above, for instance the square root of the classical absolute value.

An absolute value induces a metric by d = | f − g | . {\displaystyle d=|f-g|.}