# 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 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.

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## Also known as

Some sources refer to the **principal argument** as the **principal value of the argument**, or 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

- 1957: E.G. Phillips:
*Functions of a Complex Variable*(8th ed.) ... (previous) ... (next): Chapter $\text I$: Functions of a Complex Variable: $\S 3$. Geometric Representation of Complex Numbers - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Polar Form of Complex Numbers - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**phase** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**argument**:**1.**(**amplitude**) - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**argument**:**1.**(**amplitude**)