# 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.

Thus, if $z$ is represented in the complex plane, the principal argument $\Arg z$ is intuitively defined as the angle which $z$ yields with the real ($y = 0$) axis.

## Also known as

Some sources refer to the principal argument as just the principal value.

Some sources use the term principal phase.

## 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

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