Absolute value (algebra)

In algebra, an absolute value (also called a valuation, magnitude, or norm, although "norm" usually refers to a specific kind of absolute value on a field) 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:

	•	${\displaystyle \left|x\right|\geq 0}$		(non-negativity)
	•	${\displaystyle \left|x\right|=0}$ if and only if ${\displaystyle x=0}$		(positive definiteness)
	•	${\displaystyle \left|xy\right|=\left|x\right|\left|y\right|}$		(multiplicativity)
	•	${\displaystyle \left|x+y\right|\leq \left|x\right|+\left|y\right|}$		(triangle inequality)

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

|n| = |1 + 1 + ... + 1 (n times)| = |−1 − 1 − ... − 1 (n times)| ≤ 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 (but not the square thereof).

An absolute value induces a metric (and thus a topology) by ${\displaystyle d(f,g)=|f-g|.}$