Editing Polar coordinate system (section)

===Complex numbers===
[[Image:Imaginarynumber2.svg|thumb|An illustration of a complex number ''z'' plotted on the complex plane]]
[[Image:Euler's formula.svg|thumb|An illustration of a complex number plotted on the complex plane using [[Euler's formula]]]]
Every [[complex number]] can be represented as a point in the [[complex plane]], and can therefore be expressed by specifying either the point's Cartesian coordinates (called rectangular or Cartesian form) or the point's polar coordinates (called polar form).

In polar form, the distance and angle coordinates are often referred to as the number's '''magnitude''' and '''argument''' respectively. Two complex numbers can be multiplied by adding their arguments and multiplying their magnitudes.

The complex number ''z'' can be represented in rectangular form as
<math display="block">z = x + iy</math>
where ''i'' is the [[imaginary unit]], or can alternatively be written in polar form as
<math display="block">z = r(\cos\varphi + i\sin\varphi)</math>
and from there, by [[Euler's formula]],<ref>{{Cite book |last=Smith |first=Julius O. |title=Mathematics of the Discrete Fourier Transform (DFT) |publisher=W3K Publishing |year=2003 |isbn=0-9745607-0-7 |chapter=Euler's Identity |access-date=2006-09-22 |chapter-url=http://ccrma-www.stanford.edu/~jos/mdft/Euler_s_Identity.html |archive-url=https://web.archive.org/web/20060915004724/http://ccrma-www.stanford.edu/~jos/mdft/Euler_s_Identity.html |archive-date=2006-09-15 |url-status=dead}}</ref> as
<math display="block">z = re^{i\varphi} = r \exp i \varphi. </math>
where ''e'' is [[e (mathematical constant)|Euler's number]], and ''φ'', expressed in radians, is the [[principal value]] of the complex number function [[argument (complex analysis)|arg]] applied to ''x'' + ''iy''. To convert between the rectangular and polar forms of a complex number, the conversion formulae given above can be used. Equivalent are the {{math|[[Cis (mathematics)|cis]]}} and [[angle notation]]s:
<math display="block"> z = r \operatorname\mathrm{cis} \varphi = r \angle \varphi .</math>

For the operations of [[multiplication]], [[division (mathematics)|division]], [[exponentiation]], and [[root extraction]] of complex numbers, it is generally much simpler to work with complex numbers expressed in polar form rather than rectangular form. From the laws of exponentiation:

; Multiplication: <math>r_0 e^{i\varphi_0}\, r_1 e^{i\varphi_1} = r_0 r_1 e^{i\left(\varphi_0 + \varphi_1\right)} </math>
; Division: <math>\frac{r_0 e^{i\varphi_0}}{r_1 e^{i\varphi_1}} = \frac{r_0}{r_1}e^{i(\varphi_0 - \varphi_1)} </math>
; Exponentiation ([[De Moivre's formula]]): <math>\left(re^{i\varphi}\right)^n = r^n e^{in\varphi} </math>
; Root Extraction (Principal root): <math>\sqrt[n]{re^{i\varphi}} = \sqrt[n]{r} e^{i\varphi \over n} </math>