The argument of a complex number and its absolute value allow for another unique representation of a complex number, which is different from its vector representation $z=(a,b)=a+bi$ but particularly useful in some applications.

Proposition: Polar Coordinates of a Complex Number

Every complex number $z\in\mathbb C$ can be written as $z=r\exp(i\phi),$ for some real numbers $\phi\in\mathbb R$ and $r=|z|\in\mathbb R_+,$ where $\exp$ denotes the complex exponential function and $|z|$ denotes the absolute value of $z$. For $z\neq 0,$ the number $\phi$ is uniquely determined apart of a multiple of $2\pi,$ i.e. for all $k\in Z$:

$$z=r\exp(i\phi)=r\exp(i\phi+2\pi k).$$

The number $r$ can be interpreted as the radius of the circle drawn at the origin of the complex plane which equals the distance of $z$ to the origin. The number $\phi$ can be interpreted as the angle between the positive $x-$axis and the ray from the origin to the point $z.$

Proofs: 1

Proofs: 1 2 3
Theorems: 4

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  1. Modler, F.; Kreh, M.: "Tutorium Analysis 1 und Lineare Algebra 1", Springer Spektrum, 2018, 4th Edition