Definition:Argument of Complex Number/Principal Argument

Definition
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:
 * $\left[{0 \,.\,.\, 2 \pi}\right)$
 * $\left({-\pi \,.\,.\, \pi}\right]$

but in theory any such interval may be used.

The unique value of $\theta$ in the interval $\left({-\pi \,.\,.\, \pi}\right]$ is known as the principal value of the argument, or just principal argument, of $z$.

This is denoted $\operatorname{Arg} \left({z}\right)$.

Note the capital $A$.

This standard practice ensures that the principal argument is continuous on the real axis.