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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 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''. Note that some authors use the terms valuation, norm, or magnitude instead of "absolute value". However, the word "norm" usually refers to a specific kind of absolute value on a field (and which is also applied to other vector spaces). 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). == Types of absolute value == The trivial absolute value is the absolute value with | ''x'' | = 0 when ''x'' = 0 and | ''x'' | = 1 otherwise. Every integral domain can carry at least the trivial absolute value. The trivial value is the only possible absolute value on a finite field because any element can be raised to some power to yield 1. If | ''x'' + ''y'' | satisfies the stronger property | ''x'' + ''y'' | ≤ max(|''x''|, |''y''|), then | ''x'' | is called an ultrametric or non-Archimedean absolute value, and otherwise an Archimedean absolute value. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Absolute value (algebra)」の詳細全文を読む スポンサード リンク
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