Definition:Argument of Complex Number/Principal Range

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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.

Linguistic Note

The word principal is an adjective which means main.

Do not confuse with the word principle, which is a noun.