# Powers of i Calculator

URL: https://speedycalculator.com/calculators/math/powers-of-i/
Last updated: 2026-10-30

> The powers of i calculator simplifies iⁿ for any integer exponent n using the four-step cycle 1, i, −1, −i.

## Inputs

- Exponent n

## Results

- iⁿ
- n mod 4
- The cycle

## Formula

```
iⁿ = i^(n mod 4):  0 → 1,  1 → i,  2 → −1,  3 → −i
```

## Explanation

Because i² = −1, the powers of i repeat every four steps. This makes even huge powers such as i²⁰²⁶ easy: only the remainder of the exponent divided by 4 matters.

### How to use the Powers of i Calculator

- **Exponent n**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The calculator reduces n modulo 4 (correctly for negative exponents) and reads off the value from the cycle.

## Worked example

Inputs — Exponent n: 2026.

Results — **iⁿ = −1**; n mod 4 = 2026 mod 4 = 2; The cycle = i⁰ = 1, i¹ = i, i² = −1, i³ = −i, then the cycle repeats.

## Key terms

- **Imaginary unit**: i, with i² = −1.

## FAQ

**What is i²?**
−1, by definition.

**What is i⁰?**
1.

**What is i⁻¹?**
−i, because i × (−i) = 1.

**What is i¹⁰⁰?**
1, since 100 is divisible by 4.


## Sources

- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [Wolfram MathWorld – i](https://mathworld.wolfram.com/i.html)