Definition:Argument of Complex Number/Principal Range

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:

$\hointr 0 {2 \pi}$

$\hointl {-\pi} \pi$

but in theory any such interval may be used.

This interval is known as the principal range.

Linguistic Note

The word principal is (except in the context of economics) an adjective which means main.

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

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: $(2.9)$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Polar Form of Complex Numbers
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): argument
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): argument (of a complex number)