The point is that we are thinking of the complex number "a+ bi" as a point in the xy-plane. a+ bi corresponds to the point (a, b). In polar coordinates, since \(\displaystyle x= r cos(\theta)\) and \(\displaystyle y= r sin(\theta)\), \(\displaystyle \frac{y}{x}= \frac{r sin(\theta)}{r cos(\theta)}= \frac{sin(\theta)}{cos(\theta)}= tan(\theta)\) so that \(\displaystyle \theta= arctan(\frac{y}{x})\) as long as x is not 0. Even for x= 0, since tan(\theta) goes to infinity as \(\displaystyle \theta\) goes to \(\displaystyle \pi/2\) and to negative infinity as \(\displaystyle \theta\) goes to \(\displaystyle -\pi/2\), we have \(\displaystyle \theta= \pi/2\) or \(\displaystyle \theta= -\pi/2\).

Since \(\displaystyle \theta\) is the angle the line from (0, 0) to (a, b) makes with the positive real axis, you should be able to see, without any calculation that (1) for any positive real number, \(\displaystyle \theta= 0\), (2) for any positive real number times i (like 2i or \(\displaystyle i\sqrt{3}\)) \(\displaystyle \theta= \pi/2\), (3) for any negative real number \(\displaystyle \theta= \pi\) (or \(\displaystyle -\pi\) since we can add or subtract any multiple of \(\displaystyle 2\pi\) and have the same position), (4) for any negative real number times i (like -2i or \(\displaystyle -i\sqrt{3}\)) \(\displaystyle \theta= 2\pi/2\) (or \(\displaystyle -\pi/2\)).