Arg(x/y) = Arg(x) - Arg(y)

 

It is easy to see that $\arg \left(\frac{a}{b}\right)=\arg \left(a\right)-\arg \left(b\right)$ for any complex number $a$ and $b$. But, this is not the case for the principle value that is $\mathrm{Arg}$ does not always equal to $\mathrm{Arg}\left(a\right)-\mathrm{Arg}\left(b\right)$.

For example, the principle value argument of $\frac{-i}{i}$ is $\mathrm{Arg}(-1)=\pi$ but $\mathrm{Arg}(-i)-\mathrm{Arg}\left(i\right)=-\frac{\pi }{2}-\frac{\pi }{2}=-\pi \ne \pi =\mathrm{Arg}\left(\frac{-i}{i}\right)$.

In order $\mathrm{Arg}=\mathrm{Arg}\left(a\right)-\mathrm{Arg}\left(b\right)$ to be true, the necessary condition is $-\pi <\mathrm{Arg}\left(a\right)-\mathrm{Arg}\left(b\right)\le \pi$. In other words, $\mathrm{Arg}\left(a\right)-\mathrm{Arg}\left(b\right)$ must be in the range of the function $\mathrm{Arg}$ that is $(-\pi ,\pi ]$.