Absolute Value (algebra)

Absolute Value (algebra)

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 |.

By setting x = −1 and y = 1, it follows from the second and third of these that | 1 | = 1 and | −1 | = 1. Furthermore, for any positive integer n,

| n | = | 1+1+...(n times) | = | −1−1...(n times) | ≤ n.

Note that some authors use the terms valuation, norm, or magnitude instead of "absolute value".