Powers of i Calculator

The imaginary unit i is defined as the number satisfying i² = −1, or equivalently, the equation x² + 1 = 0. It forms the foundation of complex numbers, which are written as a + bi where a is the real part and b is the imaginary part. In electrical engineering, the symbol j is often used instead to avoid confusion with electric current.

Because i follows a repeating cycle of length 4 — i⁰ = 1, i¹ = i, i² = −1, i³ = −i — you simply divide the exponent n by 4 and look at the remainder. For example, i⁷: 7 ÷ 4 = 1 remainder 3, so i⁷ = i³ = −i. This trick works for any integer exponent.

The four values in the cycle are: i⁰ = 1, i¹ = i, i² = −1, and i³ = −i. After that, i⁴ = 1 again and the pattern repeats. Any power of i can be reduced to one of these four values using the remainder of n divided by 4.

42 divided by 4 gives a remainder of 2 (since 4 × 10 = 40, and 42 − 40 = 2). Therefore i⁴² = i² = −1.

Yes! When the exponent n is even, the result is always a real number. Specifically, i⁰ = 1 and i² = −1 are both real. Any even multiple of these gives a real result: for example, i⁴ = 1, i⁶ = −1, i⁸ = 1, and so on.

For negative exponents, the same cycle applies but runs in reverse. You can convert a negative exponent to a positive equivalent using: i⁻ⁿ = 1 / iⁿ. Alternatively, compute n mod 4, handling negative remainders by adding 4 if the remainder is negative. For example, i⁻¹ = −i, i⁻² = −1, i⁻³ = i, i⁻⁴ = 1.

To simplify iⁿ in any expression: (1) Find the remainder r = n mod 4, ensuring r is between 0 and 3. (2) Replace iⁿ with i⁰=1, i¹=i, i²=−1, or i³=−i accordingly. (3) Combine real and imaginary parts of the full expression. This calculator performs that reduction automatically for any integer n.

i⁰ = 1. This follows the standard mathematical rule that any non-zero number raised to the power of 0 equals 1, and the imaginary unit is no exception.