A question from my textbook asks, given $\operatorname{Arg}(z) = 3\pi/4$ and $\operatorname{Arg}(w) = \pi/2$, to find $\operatorname{Arg}(zw)$. So, I add $\operatorname{Arg}(z)$ and $\operatorname{Arg}(w)$ and get $5\pi/4$.
But when I check the solution they mention it in the following form: $5\pi/4 - 2\pi = -3\pi/4$.
Why is it mentioned in this form?
If $\theta$ is an argument for $z$, then so are the numbers $\theta + 2\pi k$ for all integers $k$ (since $re^{i\theta}=re^{i\theta +2ki\pi}$).
But $\operatorname{Arg}(z)$, with capital "A", is usually defined to be the specific argument that lies in $(-\pi,\pi]$. That is, you always must have $-\pi< \operatorname{Arg}(z)\leq \pi$. (There are other definitions of the "principle" range, but this is pretty common.)
So you need to add or subtract integral multiples of $2\pi$ to your answer until the result is in the required range.