Why isn't this equation holding: arg (z1/z2) =arg z1 - arg z2?

Question

$\arg (z_1/z_2) =\arg z_1 - \arg z_2$? When $z_1=-1–2i$, and $z_2=-1+2i$, I get $\arg (z_1/z_2)=2.21…$, but $\arg z_1 - \arg z_2=-4.069…$ Why is that?

Answer 1

As pointed out in the comment there is a factor $2\pi$ of difference. To avoid that we need to use properly the derivation for the $\arg$ on a same range, we obtain

• $\frac{z_1}{z_2}=\frac{-1-2i}{-1+2i}=\frac{-1-2i}{-1+2i}\frac{1-2i}{-1-2i}=-\frac35+\frac45 i\implies \arg \frac{z_1}{z_2}=\pi+\arctan (-4/3)=2.2142...$

• $\arg{z_1}=\arg(-1-2i)=\pi+\arctan (2)=4.2487...$

• $\arg{z_2}=\arg(-1+2i)=\pi+\arctan (-2)=2.0344$