Powers of i

Powers of $$i$$

## Formula

$$i^0=1$$

$$i^1=i$$

$$i^2=-1$$

$$i^3=-i$$

$$i^4=1$$

The powers repeat after every four powers:

$$i^{4n}=1$$

$$i^{4n+1}=i$$

$$i^{4n+2}=-1$$

$$i^{4n+3}=-i$$

## Conditions / Special Cases

For any integer $$n$$:

$$i^{4n+r}=i^r$$

where $$r$$ is the remainder when the exponent is divided by $4$.

## Usage

- Used to simplify high powers of $$i$$ quickly.