Absolute value (algebra)
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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 algebra, an absolute value (also called a valuation, magnitude, or norm,[1] 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:
• | (non-negativity) | |||
• | if and only if | (positive definiteness) | ||
• | (multiplicativity) | |||
• | (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).