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: * | x | ≥ 0, * | x | = 0 if and only if x = 0, * | xy | = | x || y |, * | x + y | ≤ | x | + | y |. It follows from these axioms that | 1 | = 1 and | −1 | = 1. Furthermore, for every positive integer n, | n | = | 1 + 1 + ...(n times) | = | −1 − 1 − ...(n times) | ≤ n.