# Definition:Argument of Complex Number/Principal Argument

## Definition

Let $R$ be the principal range of the complex numbers $\C$.

The unique value of $\theta$ in $R$ 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 $A$.

The standard practice is for $R$ to be $\left({-\pi \,.\,.\, \pi}\right]$.

This ensures that the **principal argument** is continuous on the real axis for positive numbers.

## Also known as

Some sources give this as just the **principal value**.

## Linguistic Note

The word **principal** is an adjective which means **main**.

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

## Sources

- 1964: Murray R. Spiegel:
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