absolute value
Let be an ordered ring and let . The absolute value of is defined to be the function given by
In particular, the usual absolute value on the field of real numbers is defined in this manner. An equivalent definition over the real numbers is .
Absolute value has a different meaning in the case of complex numbers: for a complex number , the absolute value of is defined to be , where and are real.
All absolute value functions satisfy the defining properties of a valuation, including:
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for all , with equality if and only if
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for all
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for all (triangle inequality)
However, in general they are not literally valuations, because valuations are required to be real valued. In the case of and , the absolute value is a valuation, and it induces a metric in the usual way, with distance function defined by .
Title | absolute value |
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Canonical name | AbsoluteValue |
Date of creation | 2013-03-22 11:52:09 |
Last modified on | 2013-03-22 11:52:09 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 10 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 13-00 |
Classification | msc 11A15 |