Arguments add under multiplication

Statement

Let $z$ and $w$ be complex numbers with $z \neq 0$ and $w \neq 0$. If $\theta$ is an argument of $z$ and $\varphi$ is an argument of $w$, then $\theta + \varphi$ is an argument of $z w$. Thus $\arg(z w) \equiv \arg z + \arg w \pmod{2 \pi}$.

Remarks

Proof: write $z = |z| e^{i \theta}$ and $w = |w| e^{i \varphi}$. Then $z w = |z| |w| e^{i (\theta + \varphi)}$ by the product of exponentials. For the principal values, the equality can fail. For example, $\operatorname{Arg}(-1) + \operatorname{Arg}(-1) = 2\pi$, but $\operatorname{Arg}((-1)(-1)) = 0$.

Explanations

L1 human

When you multiply two complex numbers, the angles add. For example, $i$ points at $90$ degrees, and $i \cdot i = -1$ points at $180$ degrees.