section_path: Definition and basic operations/Polar form
section_name: Polar form
paragraph_text: Polar form
  
  For any complex number z, with absolute value r = | z | {\displaystyle r=|z|} and argument φ {\displaystyle \varphi }, the equation
  
  holds. This identity is referred to as the polar form of z. It is sometimes abbreviated as z = r c i s φ {\textstyle z=r\operatorname {\mathrm {cis} } \varphi }. In electronics, one represents a phasor with amplitude r and phase φ in angle notation: z = r ∠ φ. {\displaystyle z=r\angle \varphi.}
  
  If two complex numbers are given in polar form, i.e., z = r (cos φ + i sin φ) and z = r (cos φ + i sin φ), the product and division can be computed as z 1 z 2 = r 1 r 2 (cos (φ 1 + φ 2) + i sin (φ 1 + φ 2)). {\displaystyle z_{1}z_{2}=r_{1}r_{2}(\cos(\varphi _{1}+\varphi _{2})+i\sin(\varphi _{1}+\varphi _{2})).} z 1 z 2 = r 1 r 2 (cos (φ 1 − φ 2) + i sin (φ 1 − φ 2)), if z 2 ≠ 0. {\displaystyle {\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}\neq 0.} (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 (2 + i) (3 + i) = 5 + 5 i. {\displaystyle (2+i)(3+i)=5+5i.} Because the real and imaginary part of 5 + 5 i are equal, the argument of that number is 45 degrees, or 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 π 4 = arctan (1 2) + arctan (1 3) {\displaystyle {\frac {\pi }{4}}=\arctan \left({\frac {1}{2}}\right)+\arctan \left({\frac {1}{3}}\right)} holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of π.