The argument of a complex number is the angle of its vector with the real axis of the complex plane.

Definition: Argument of a Complex Number

Let $z=(a,b)=a+bi$ be a complex number. The argument of $z$ is defined by $$\phi=\arg(z):=\begin{cases} \arctan\left(\frac ba\right)&\text{ for }a > 0\\ \arctan\left(\frac ba\right)+\pi&\text{ for }a < 0,\;b\ge 0\\ \arctan\left(\frac ba\right)-\pi&\text{ for }a < 0,\;b < 0\\ \pi/2&\text{ for }a = 0,\;b > 0\\ -\pi/2&\text{ for }a = 0,\;b < 0\\ \text{undefined}&\text{ for }a = 0,\;b = 0 \end{cases}$$

where $\pi$ denotes the number pi and $\arctan$ denotes the inverse tangent function.

Geometrically, the argument of $z$ measures the angle between the vector $z$ and the $x$ axis.

Propositions: 1

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