Roots of a Complex Number

Roots of a Complex Number

## Definition / Concept

The $$n$$th roots of a non-zero complex number are the $$n$$ distinct complex numbers whose $$n$$th power equals the given number.

## Formula

If:

$$z=r(\cos\theta+i\sin\theta)$$

then the $$n$$th roots are:

$$z_k=r^{1/n}\left[\cos\left(\frac{\theta+2k\pi}{n}\right)+i\sin\left(\frac{\theta+2k\pi}{n}\right)\right]$$

where:

$$k=0,1,2,\ldots,n-1$$

## Conditions / Special Cases

The $$n$$ roots have equal modulus:

$$|z_k|=r^{1/n}$$

The arguments differ by:

$$\frac{2\pi}{n}$$

## Usage

- Used to solve equations of the form $$w^n=z$$.

- Used to locate roots geometrically on the Argand plane.