Editing Complex number (section)

===Polar form{{anchor|Polar form}}===
{{Main|Polar coordinate system}}
{{Redirect|Polar form|the higher-dimensional analogue|Polar decomposition}}

[[File:Complex multi.svg|right|thumb|Multiplication of {{math|2 + ''i''}} (blue triangle) and {{math|3 + ''i''}} (red triangle). The red triangle is rotated to match the vertex of the blue one (the adding of both angles in the terms ''φ''<sub>1</sub>+''φ''<sub>2</sub> in the equation) and stretched by the length of the [[hypotenuse]] of the blue triangle (the multiplication of both radiuses, as per term ''r''<sub>1</sub>''r''<sub>2</sub> in the equation).]]

For any complex number ''z'', with absolute value <math>r = |z|</math> and argument <math>\varphi</math>, the equation 
:<math>z=r(\cos\varphi +i\sin\varphi) </math>
holds. This identity is referred to as the polar form of ''z''. It is sometimes abbreviated as <math display="inline"> z = r \operatorname\mathrm{cis} \varphi </math>. 
In electronics, one represents a [[Phasor (sine waves)|phasor]] with amplitude {{mvar|r}} and phase {{mvar|φ}} in [[angle notation]]:<ref>
{{cite book |last1=Nilsson |first1=James William |title=Electric circuits |last2=Riedel |first2=Susan A. |publisher=Prentice Hall |year=2008 |isbn=978-0-13-198925-2 |edition=8th |page=338 |chapter=Chapter 9 |chapter-url=https://books.google.com/books?id=sxmM8RFL99wC&pg=PA338}}
</ref><math display="block">z = r \angle \varphi . </math>

If two complex numbers are given in polar form, i.e., {{math|1=''z''<sub>1</sub> = ''r''<sub>1</sub>(cos ''φ''<sub>1</sub> + ''i'' sin ''φ''<sub>1</sub>)}} and {{math|1=''z''<sub>2</sub> = ''r''<sub>2</sub>(cos ''φ''<sub>2</sub> + ''i'' sin ''φ''<sub>2</sub>)}}, the product and division can be computed as 
<math display=block>z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).</math>
<math display=block>\frac{z_1}{z_2} = \frac{r_1}{r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right), \text{if }z_2 \ne 0.</math>
(These are a consequence of the [[trigonometric identities]] for the sine and cosine function.)
In other words, the absolute values are ''multiplied'' and the arguments are ''added'' to yield the polar form of the product. The picture at the right illustrates the multiplication of
<math display=block>(2+i)(3+i)=5+5i. </math>
Because the real and imaginary part of {{math|5 + 5''i''}} are equal, the argument of that number is 45 degrees, or {{math|''π''/4}} (in [[radian]]). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are [[arctan]](1/3) and arctan(1/2), respectively. Thus, the formula
<math display=block>\frac{\pi}{4} = \arctan\left(\frac{1}{2}\right) + \arctan\left(\frac{1}{3}\right) </math>
holds. As the [[arctan]] function can be approximated highly efficiently, formulas like this – known as [[Machin-like formula]]s – are used for high-precision approximations of [[pi|{{pi}}]]:<ref>{{cite book |title=Modular Forms: A Classical And Computational Introduction |author1=Lloyd James Peter Kilford |edition= 2nd|publisher=World Scientific Publishing Company |year=2015 |isbn=978-1-78326-547-3 |page=112 |url=https://books.google.com/books?id=qDk8DQAAQBAJ}} [https://books.google.com/books?id=qDk8DQAAQBAJ&pg=PA112 Extract of page 112]</ref>
<math display=block>\frac{\pi}{4} = 4 \arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right) </math>