Mathematics-Online lexicon: Absolute Value of Complex Numbers

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	Mathematics-Online lexicon:
	Absolute Value of Complex 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 complex number $z=x+\mathrm{i}y$ is defined as

$\displaystyle \vert z\vert = \sqrt{x^2 + y^2} = \sqrt{z\bar z} .$

For $z\in\mathbb{R}$ this definition is consistent with the definition of the absolute value of real numbers and has corresponding properties.

Positivity:

$\displaystyle \vert z\vert\geq 0,\quad \vert z\vert = 0 \Longleftrightarrow z = 0$
Multiplicativity:

$\displaystyle \vert z_1 z_2\vert = \vert z_1\vert\,\vert z_2\vert,\quad \vert z_1/z_2\vert = \vert z_1\vert/\vert z_2\vert,\, z_2\ne0$
Triangle Inequality:

$\displaystyle \big\vert\vert z_1\vert-\vert z_2\vert\big\vert \leq \vert z_1+z_2\vert \leq \vert z_1\vert+\vert z_2\vert$

(Authors: Höllig/Kimmerle/Abele)

Annotation:

automatically generated 6/11/2007