Complex numbers/Polar coordinates/Angle naive/Example

Every complex number ${}z\in \mathbb {C}$, ${}z\neq 0$, can be written uniquely as

${}z=r(\cos \alpha ,\sin \alpha )=(r\cos \alpha ,r\sin \alpha )=r\cos \alpha +(r\sin \alpha ){\mathrm {i} }\,$

with a positive real number ${}r$, which is the distance between ${}z$ and the zero point (thus, ${}r=\vert {z}\vert$) and an angle ${}\alpha$ between ${}0$ and below ${}360$ degree, measured counterclockwise starting with the positive real axis. The pair ${}(r,\alpha )$ constitutes the polar coordinates of the complex number.

Polar coordinates in the real plane and for complex numbers are the same. However, the polar coordinates allow a new interpretation of the multiplication of complex numbers: Because of

${}{\begin{aligned}(r\cos \alpha +{\mathrm {i} }r\sin \alpha )\cdot (s\cos \beta +{\mathrm {i} }s\sin \beta )&=rs(\cos \alpha \cos \beta -\sin \alpha \sin \beta )+{\mathrm {i} }rs(\cos \alpha \sin \beta +\sin \alpha \cos \beta )\\&=rs(\cos(\alpha +\beta )+{\mathrm {i} }\sin(\alpha +\beta ))\end{aligned}}$

(where we have used the addition theorems for sine and cosine), one can multiply two complex numbers by multiplying their modulus and adding their angles.