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Mathematics-Online course: Basic Mathematics - Complex Numbers

Gaussian Plane

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Complex numbers $ z=x+\mathrm{i}y$ can be identified with the points of the real plane. Their absolute value corresponds to their distance from the origin, while their real and imaginary parts are their projections onto the real and imaginary axes respectively. The complex conjugate of a complex number is obtained by reflection with respect to the real axis.

\includegraphics[width=0.4\moimagesize]{a_gausssche_bild1} \includegraphics[width=0.4\moimagesize]{a_gausssche_bild2}

In polar coordinates, the Euler-Moivre-formula yields the representation

$\displaystyle z = r(\cos\varphi + \mathrm{i}\sin\varphi) =
r \exp(\mathrm{i}\varphi)

with $ r = \vert z\vert$. The angle $ \varphi$ is determined only up to multiples of $ 2\pi$; it is called the argument of $ z$:

$\displaystyle \varphi = \operatorname{arg}(z)

It is common practice to use $ (-\pi,\pi]$ as the standard interval (principal value). Moreover,

$\displaystyle \tan\varphi = \frac{\operatorname{Im}(z)}{\operatorname{Re}(z)}\,,

i.e., the argument $ \operatorname{arg}(z)$ can be determined from the quotient $ y/x$. However, one has to select the correct branch. If $ \operatorname{Re}(z)<0$, $ \pi$ or $ -\pi$ must be added to the inverse function's value.

In the table below some complex numbers are given in polar coordinates.

$ z$ $ 1$ $ -1$ $ \pm\mathrm{i}$ $ 1\pm\mathrm{i}$ $ \sqrt{3}\pm\mathrm{i}$ $ 1\pm\sqrt{3}\mathrm{i}$
$ r$ $ 1$ $ 1$ $ 1$ $ \sqrt{2}$ $ 2$ $ 2$
$ \varphi$ 0 $ \pi$ $ \pm\pi/2$ $ \pm\pi/4$ $ \pm\pi/6$ $ \pm\pi/3$

(Authors: Höllig/Kopf/Abele)

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  automatically generated 10/31/2008