\begin{definition}[Definition:Argument of Complex Number/Principal Range]
It is understood that the argument of a complex number $z$ is unique only up to multiples of $2 k \pi$.
With this understanding, we can limit the choice of what $\theta$ can be for any given $z$ by requiring that $\theta$ lie in some half open interval of length $2 \pi$.
The most usual of these are:
:$\hointr 0 {2 \pi}$
:$\hointl {-\pi} \pi$
but in theory any such interval may be used.
This interval is known as the '''principal range'''.
\end{definition}