---
tag: 0ZZRSX91QR
kind: definition
name: Argument of a complex number
msc: [30A99]
defines: 070PFSYJJR
uses: [09FTAET5Z4, 0KW2X0TFBX]
sources:
  - url: https://en.wikipedia.org/wiki/Argument_(complex_analysis)
    locator: Definition
---

## Definition

Let $z$ be a [[complex number|09FTAET5Z4]] with $z \neq 0$, and let $|z|$ be its [[modulus|0KW2X0TFBX]].
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.