Argument of a complex number

Definition

Let $z$ be a complex number with $z \neq 0$, and let $|z|$ be its modulus. An argument of $z$ is a real number $\theta$ with $$z = |z| (\cos \theta + i \sin \theta).$$ The notation $\arg z$ means the set of all arguments of $z$, or one chosen argument.

Remarks

If $\theta$ is an argument of $z$, then the arguments of $z$ are exactly the numbers $\theta + 2 \pi k$ with $k \in \mathbb{Z}$. The number $0$ has no argument in this sense. Some conventions set $\arg 0 = 0$. For example, the arguments of $i$ are $\pi/2 + 2 \pi k$, and those of $-1$ are $\pi + 2 \pi k$.

Explanations

L1 human

The argument says in which direction a complex number points from $0$. The number $1 + i$ points at $45$ degrees, so $\pi/4$ is an argument of it. A full turn brings you back to the same direction, so adding $2\pi$ gives another argument.