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.
Formula
iⁿ = i^(n mod 4): 0 → 1, 1 → i, 2 → −1, 3 → −i
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.
Frequently asked questions
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.