Complex numbers/Polar coordinates/Angle naive/Example

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Every complex number , , can be written uniquely as

with a positive real number , which is the distance between and the zero point (thus, ) and an angle between and below degree, measured counterclockwise starting with the positive real axis. The pair 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

(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.