Find the argument of a complex number

Question

I know about the conditions for computing the Argument, and by that

$\mbox{if } x < 0 \mbox{ and } y \ge 0, \\$ it should be $ \arctan(\frac{y}{x}) + \pi$.

but why is it then the argument of $z = -1+i$ equal to $\frac{3\pi}{4}$?

Shouldn't it be $\frac{3\pi}{4}$ + $\pi$ which is $\frac{7\pi}{4}$?

Answer 1

It seems that you suggest that summing two complex numbers one sums their arguments. This is not true. Instead one sums their real and imaginary parts. Therefore the principal argument of $z= -1+i$ is $3\pi/4$, since $\frac{\Im(z)}{\Re(z)}=-1$ and $z$ lies in the II quadrant.

Answer 2

$$\arctan\left(\frac1{-1}\right)+\pi=-\frac\pi4+\pi=\frac{3\pi}4.$$

Answer 3

What actually was meant to be expressed is that:

$\mbox{if } x < 0 \mbox{ and } y \ge 0, \mbox{it should be } \arctan(\dfrac{y}{|x|}) + \pi$.

In the usual manner for $ \tan^{-1}\dfrac{1}{-1}$ add $\pi$ for positive argument arctan.