Mathematics-Online lexicon: absolute value of real numbers

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	Mathematics-Online lexicon:
	absolute value of real numbers

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

The absolute value of a real number

is defined by

$\displaystyle \vert a\vert = \max(a,-a) = \left\{ \begin{array}{lll} a & \mbox{for $a \geq 0$} \\ -a & \mbox{for $a \leq 0$} \end{array}\right.$

and has the following properties.

positivity:

$\displaystyle \vert a\vert\geq 0,\quad \vert a\vert = 0 \Longleftrightarrow a = 0$
multiplicativity:

$\displaystyle \vert a b\vert = \vert a\vert\,\vert b\vert,\quad \vert a/c\vert = \vert a\vert/\vert c\vert,\, c\ne0$
triangle inequality:

$\displaystyle \big\vert\vert a\vert-\vert b\vert\big\vert \leq \vert a+b\vert \leq \vert a\vert+\vert b\vert$

(Authors: Höllig/Kimmerle/Abele)

Annotation:

automatically generated 6/11/2007