Propositions from Polar form

Principal arguments of products and quotients

Adding or subtracting principal arguments gives an argument of the result, but it may lie outside the principal interval $(-\pi,\pi]$. The integers $k$ and $\ell$ record the reduction back into that interval. For instance, $\operatorname{Arg}((-1)(-1))=0$, whereas $\operatorname{Arg}(-1)+\operatorname{Arg}(-1)=2\pi$; the latter is an argument of the product, but not its principal argument.

Two polar forms describe the same non-zero number exactly when their moduli agree and their arguments differ by an integer multiple of $2\pi$.