Mathematics-Online lexicon: Complex Numbers

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	Mathematics-Online lexicon:
	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

In order to determine the square roots of negative numbers, the imaginary unit $\mathrm{i}$ has been introduced as one of the solutions of

$\displaystyle \mathrm{i}^2 = -1 .$

Then

$\displaystyle \mathbb{C} = \{ z = x + \mathrm{i}y,\quad x,y\in\mathbb{R}\} \,,$

is referred to as the set of complex numbers. The real numbers

and

are called ,,real part`` and ,,imaginary part`` respectively:

$\displaystyle x = \operatorname{Re}z,\quad y = \operatorname{Im}z .$

In particular, $\mathbb{R} = \{z\in\mathbb{C}:\ \operatorname{Im}(z)=0\}$ .

Complex numbers form a field. Defining addition and multiplication by

$\displaystyle z_1+z_2$	$\displaystyle =$	$\displaystyle x_1+x_2 + \mathrm{i} (y_1+y_2)$
$\displaystyle z_1\cdot z_2$	$\displaystyle =$	$\displaystyle x_1x_2-y_1y_2 + \mathrm{i} (x_1y_2+x_2y_1)\, ,$

the usual rules of arithmetics apply.

(Authors: Höllig/Kopf/Abele)

automatically generated 6/11/2007