# Determining the principal argument from a complex exponential

How do I determine the principal argument of a complex number in exponential form?

For example, if my numbers are $6e^{-i\frac{3\pi}{2}}$ and $13e^{i13\pi}$, how do I get them in the interval of $(-\pi,\pi]$? I know that the arguments are in the exponents, but I don't know how to get them in the above mentioned interval.

$e^{2i\pi}=1$, so multiplying through by it has no effect on the absolute value of any $z$, it only changes the angle by $2\pi$ which has no effect on the orientation of the complex $z$.
So you can multiply by any of $e^{2ki\pi}$ with $k\in\mathbb{Z}$ until your angle is in the range $(-\pi,\pi]$.
• I see. So for the first number, the principal argument would be $\frac{\pi}{2}$. The second number, however, would be $13\pi-2*6\pi=\pi$. Is that right? – Steve Sep 25 '16 at 16:48