Argument Identity

Question

When does $Arg(\bar{z})=Arg(1/z)$? To me it seems to hold for any nonzero $z$, since $1/z=\bar{z}/|z|^2$ and $Arg$ is unaffected by scalar multiplication by real numbers

Answer 1

You are correct, but there are a couple notes I'd like to add:

1) If $z=0$, neither quantity is properly defined.

2) $\arg(z)$ is not affected by positive scalar multiplication. The fact that $|z|^2>0$ for $z\neq 0$ is important as, for example $\arg(i)=\frac{\pi}{2}\neq-\frac{\pi}{2}=\arg(-i)$ (the $-\frac{\pi}{2}$ may be replaced with $\frac{3\pi}{2}$ depending on how you define $\arg$).