Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example an absolute value is also defined for the Complex Number s, the Quaternion s, Ordered Ring s, Field s and Vector Space s.
The absolute value is closely related to the notions of Magnitude , Distance , and Norm in various mathematical and physical contexts.
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\begin{cases} a, & \mbox{if } a \ge 0 \ -a, & \mbox{if } a < 0 \end{cases} </math>
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\sqrt{a^2}</math>
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10
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10
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\sqrt{a^2}</math>
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\sqrt{x^2 + y^2}</math>
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\sqrt{x^2 + 0^2} = \sqrt{x^2} = x</math>
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r\,</math>
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\bar{z}</math>
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"http://wwwinformationdelightinfo/encyclopedia/entry/ordered_ring" class="copylinks">Ordered Ring That is, if <math>a</math> is an element of an ordered ring <math>R</math>, then the '''absolute value''' of <math>a</math>, denoted by <math>a </math>, is defined to be:
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\begin{cases} a, & \mbox{if } a \ge 0 \ -a, & \mbox{if } a < 0, \end{cases} </math>
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\sqrt{(a - b)^2}</math>
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10
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10
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10
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"http://wwwinformationdelightinfo/encyclopedia/entry/vector_space" class="copylinks">Vector Space <math>V</math> a over a field <math>F</math>, is called an '''absolute value''' (or more usually a ''' Norm ''') if it satisfies the following axioms:
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10
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\sqrt{\sum_{i=1}^{n}(x_i)^2}</math>
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"http://wwwinformationdelightinfo/encyclopedia/entry/Euclidean_norm" class="copylinks">Euclidean Norm When the real numbers '''R''' are considered as the one-dimensional Vector Space '''R'''<sup>1</sup> , the absolute value is a Norm , and is the ''p''-norm for any ''p'' In fact the absolute value is the "only" norm in '''R'''<sup>1</sup>, in the sense that, for every norm · in '''R'''<sup>1</sup>, ''x''=1·''x'' The complex absolute value is a special case of the Norm in an Inner Product Space It is identical to the Euclidean norm, if the Complex Plane is identified with the Euclidean Plane '''R'''<sup>2</sup>
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"http://functionswolframcom/ComplexComponents/Abs/35/" class="copylinks" target="_blank">functionsWolframcom credits Karl Weierstrass with introducing the notation ''x'' in 1841 <br>
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