How this instrument works
The imaginary unit i is defined by i² = −1, and raising it to successive whole-number powers produces a short, endlessly repeating cycle: i¹ = i, i² = −1, i³ = −i, i⁴ = 1, and then the pattern starts over exactly from i¹ again at i⁵. Because the cycle length is exactly 4, any power of i can be identified immediately just by taking the exponent modulo 4, with no need to multiply i by itself the long way for large exponents.
This calculator reports both the real and imaginary parts of iⁿ for any whole-number exponent n: a remainder of 0 gives 1 (real part 1, imaginary part 0), a remainder of 1 gives i (real part 0, imaginary part 1), a remainder of 2 gives −1 (real part −1, imaginary part 0), and a remainder of 3 gives −i (real part 0, imaginary part −1).
The same four-value cycle applies just as cleanly to very large exponents — i¹⁰⁰, for instance, needs no multiplication chain at all, just 100 mod 4 = 0, landing directly on 1. This periodic behavior of i's powers is a small preview of a much bigger idea in complex analysis: every nonzero complex number's powers eventually trace out a repeating or spiraling pattern determined entirely by its own argument (angle), and i's is simply the cleanest possible case, with an argument of exactly 90°.
- Enter the exponent into the Exponent n field — any whole number, including 0 or negative values.
- Read Real part of iⁿ: 1, 0, or −1 depending on where the exponent falls in the four-step cycle.
- Read Imaginary part of iⁿ: 0, 1, or −1, together with the real part giving the exact value of i raised to that power.
Worked example — i⁵
i⁵ needs no direct multiplication: since 5 mod 4 = 1, it matches i¹ exactly, giving a real part of 0 and an imaginary part of 1 — i⁵ = i, the same value the cycle started with four steps earlier.
i⁴ = 1 exactly, since 4 mod 4 = 0, completing one full cycle and landing back on the real number 1 — every multiple of 4 (i⁰, i⁴, i⁸, i¹²...) shares this same value. And i¹⁰, since 10 mod 4 = 2, matches i² = −1, two full cycles into the pattern plus two extra steps.
Questions
Why do the powers of i repeat every 4 steps?
Because i² = −1 by definition, i⁴ = (i²)² = (−1)² = 1 — raising i to the 4th power returns exactly to 1, the same starting point multiplication by i began from, so the whole cycle repeats identically every 4 steps after that.
What is i to the power of 0?
Exactly 1 — like any nonzero number raised to the zeroth power, i⁰ = 1, matching a remainder of 0 when the exponent is divided by 4.
How do I find i raised to a very large exponent?
Take the exponent modulo 4 and use that small remainder (0, 1, 2, or 3) to look up the matching value in the four-step cycle — no need to multiply i by itself the long way, regardless of how large the original exponent is.
What about negative exponents of i?
The same four-value cycle still applies, since i⁻¹ = 1⁄i = −i (multiplying top and bottom by i), which already matches i³ in the forward cycle — negative exponents of i simply continue the pattern in the opposite direction, landing on the same four repeating values.
Is this pattern unique to i, or do other complex numbers cycle too?
Every complex number whose powers eventually repeat exactly does so because its own argument (angle) divides evenly into 360° — i's argument is exactly 90°, a quarter of a full turn, giving the shortest possible nontrivial repeating cycle of 4.