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Mathematics-Online course: Basic Mathematics - Complex Numbers

Complex Numbers

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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 $ x$ and $ y$ 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)
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  automatically generated 10/31/2008