Powers of i Calculator
Simplify any power of the imaginary unit i. Enter a whole-number exponent (positive, negative or zero) to get i^n reduced through its four-step cycle, in both shorthand and a + bi form.
Example: with Exponent n 15 → Simplified i^n: i^15 = -i.
- Rectangular form (a + bi)0 - 1i
- Whyremainder 3 — i repeats every 4 powers
Computed by the calculator below using its default values. Change any input to see your own numbers.
Because i² = -1, the powers of i repeat every four steps: i, -1, -i, 1, then back to i. Only the exponent mod 4 matters, so any i^n reduces to one of four values.
A four-step loop
The imaginary unit is defined by i² = -1. Multiply again and i³ = i²·i = -i; once more and i⁴ = (i²)² = 1, which returns you to the start. From there the pattern i, -1, -i, 1 repeats forever. So raising i to a large power is really just asking where you land after that many quarter-turns.
That is why only the remainder of n divided by 4 matters. i^15 has remainder 3, so it equals i³ = -i. Negative exponents work the same once you wrap them around: i^-1 = 1/i = -i, matching remainder 3. Each value also has a rectangular a + bi form, which is handy when the result feeds into further complex arithmetic.
How it’s calculated
Reduce the exponent modulo 4 using r = ((n mod 4) + 4) mod 4 so negatives wrap correctly. The remainder maps to a fixed value: 0 gives 1, 1 gives i, 2 gives -1, 3 gives -i. Each is also shown in rectangular form a + bi. Non-integer exponents are rejected because they yield multivalued complex results.
Only integer exponents are handled; fractional powers of i (roots) are multivalued and outside this tool. i is treated as the principal imaginary unit with i² = -1.
The cycle of powers of i
| n mod 4 | Powers | Value |
|---|---|---|
| 1 | i¹, i⁵, i⁹, ... | i |
| 2 | i², i⁶, i¹⁰, ... | -1 |
| 3 | i³, i⁷, i¹¹, ... | -i |
| 0 | i⁴, i⁸, i¹², ... | 1 |
Derived from i² = -1; powers repeat every four steps.
Common mistakes
- Thinking i³ = i instead of -i — the third step is negative.
- Mishandling negative exponents; i^-1 is -i, not i.
- Assuming i⁰ is 0 rather than 1 — any nonzero base to the zero power is 1.
- Reducing by mod 2 instead of mod 4, which only captures half the cycle.
Frequently asked questions
What is i to the power of n?
It cycles through four values with period 4. Take n mod 4: remainder 1 gives i, 2 gives -1, 3 gives -i, and 0 gives 1. So i^n reduces to one of those four.
Why do powers of i repeat every four?
Because i² = -1 and i⁴ = 1. Once you reach i⁴ you are back to 1, and multiplying by i again restarts the pattern i, -1, -i, 1.
How do you handle a negative exponent like i^-3?
Wrap it into the cycle: ((-3 mod 4) + 4) mod 4 = 1, so i^-3 = i. Negative powers land in the same four-value loop.
What is i to the power of 0?
i⁰ = 1. Any nonzero number raised to the zero power equals 1, and i is no exception.