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Revision as of 19:58, 31 July 2020

Given a complex number $z$, the argument $\arg z$ is the measure of the signed angle the ray $\overrightarrow{0z}$ makes with the positive real axis. (Note that this means the argument of the complex number 0 is undefined.)

Unfortunately, this means that $\arg$ is not a proper function but is instead a "multi-valued function": for example, any positive real number has argument 0, but also has argument $2 \pi, -2\pi, 4\pi, \ldots$. This means that the argument may be best considered as an equivalence class $\mathbf r = \{r + 2\pi n, n \in \mathbb{Z}\}$. The advantages of this are several: most importantly, they make $\arg$ into a continuous function. They also make some properties of the argument "look nicer." For example, under this interpretation, we can write $\arg(w \cdot z) = \arg(w) + \arg(z)$. The other common solution is to restrict the range of $\arg$ to some interval, usually $[0, 2\pi)$ or $(-\pi, \pi]$. This forces us to state this equality modulo $2\pi$.

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