absolute value
Let R be an ordered ring and let a∈R. The absolute value of a is defined to be the function ||:R→R given by
|a|:= |
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:
-
•
for all , with equality if and only if
-
•
for all
-
•
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 |