# 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 $\Arg z$.

Note the capital $A$.

The standard practice is for $R$ to be $\hointl {-\pi} \pi$.

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

- 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Polar Form of Complex Numbers