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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.
