Argument of a Complex Number

Argument of a Complex Number

## Formula

For:

$$z=x+iy$$

the argument $$\theta$$ satisfies:

$$\tan\theta=\frac{y}{x}$$

The general argument is:

$$\arg(z)=\theta+2n\pi,\qquad n\in\mathbb Z$$

The principal argument is generally denoted by:

$$\operatorname{Arg}(z)$$

with:

$$-\pi<\operatorname{Arg}(z)\le\pi$$

## Conditions / Special Cases

For $x>0$ and $y>0$:

$$\operatorname{Arg}(z)=\tan^{-1}\left(\frac{y}{x}\right)$$

The correct quadrant must always be considered when determining the argument.

For positive real numbers:

$$\operatorname{Arg}(z)=0$$

For negative real numbers:

$$\operatorname{Arg}(z)=\pi$$

For positive imaginary numbers:

$$\operatorname{Arg}(z)=\frac{\pi}{2}$$

For negative imaginary numbers:

$$\operatorname{Arg}(z)=-\frac{\pi}{2}$$

## Usage

- Used to determine the angular position of a complex number in the Argand plane.