# Definition:Argument of Complex Number/Principal Range

< Definition:Argument of Complex Number(Redirected from Definition:Principal Range)

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## 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 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