Powers of i Calculator — Imaginary Unit i^n
Find i^n for any integer exponent. The imaginary unit i satisfies i² = −1, and its powers repeat every 4 steps: i⁰=1, i¹=i, i²=−1, i³=−i, i⁴=1, and so on — enter n and see which cycle position you land on.
Write the exponent mod 4 — the cycle repeats every 4 steps
Look up the cycle position: i⁰=1, i¹=i, i²=−1, i³=−i
Final result
How does this calculator work?
i^n cycles with period 4: i⁰=1, i¹=i, i²=−1, i³=−i. To find i^n, compute n mod 4 (use the positive remainder for negative exponents). For example i^7 = i^3 = −i. Negative exponents follow the same rule.
Formula
How this is calculated
The imaginary unit i is defined by the equation i² = −1. Computing successive powers: i⁰ = 1, i¹ = i, i² = −1, i³ = i² × i = −i, i⁴ = (i²)² = (−1)² = 1 — and then the pattern repeats with period 4. Because the cycle has length 4, i^n depends only on n modulo 4.
For negative exponents, the same rule applies once you compute the remainder mod 4 correctly (using the mathematically positive remainder, not the C-style remainder that can be negative). For example, i^(−1) corresponds to remainder 3, so i^(−1) = i³ = −i — which makes sense because i × (−i) = −i² = 1, confirming −i is the multiplicative inverse of i.
Geometrically, multiplying by i is a 90° counter-clockwise rotation in the complex plane. So i^n performs n such rotations: n=0 → stays at (1, 0); n=1 → rotates to (0, 1); n=2 → rotates to (−1, 0); n=3 → rotates to (0, −1); and n=4 completes the circle back to (1, 0).
Frequently asked questions
Because i² = −1, which means i⁴ = (i²)² = (−1)² = 1 — the multiplicative identity. Multiplying by 1 resets the cycle. Geometrically, four 90° rotations in the complex plane return to the starting point, so the period is exactly 4.
i⁰ = 1. Any non-zero number raised to the power 0 equals 1 by the convention a⁰ = 1, which ensures the exponent rules a^m × a^n = a^(m+n) work consistently. So i⁰ = 1 and the cycle starts at 1.
1000 mod 4 = 0, so i^1000 = i^0 = 1. For any large integer n, just divide n by 4 and look at the remainder: 0 → 1, 1 → i, 2 → −1, 3 → −i.
Also known as
TG we-Calculate Editorial Team. (2026). Powers of i Calculator — Imaginary Unit i^n [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/powers-of-i-calculator
TG we-Calculate Editorial Team. "Powers of i Calculator — Imaginary Unit i^n." TG we-Calculate. 2026. https://we-calculate.com/calculator/powers-of-i-calculator.
TG we-Calculate Editorial Team, "Powers of i Calculator — Imaginary Unit i^n," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/powers-of-i-calculator
@misc{wecalculate_powers_of_i_calculator, title = {Powers of i Calculator — Imaginary Unit i^n}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/powers-of-i-calculator}}, year = {2026}, note = {TG we-Calculate} }
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