Powers of i

The pattern for powers of i repeats every four steps. No need to multiply i by itself dozens of times. i¹ = i, i² = −1, i³ = −i, and i⁴ = 1. From there the cycle restarts: i⁵ = i, i⁶ = −1, and so on. The four values, i, −1, −i, 1, never change. The exponent alone determines which one you get.

Raising i to any integer power always produces one of four results, in a fixed order. The table below shows the full cycle from exponent 0 through exponent 4, and the pattern continues identically for every subsequent group of four.

To use the cycle, find the exponent's remainder when divided by 4. That remainder, 0, 1, 2, or 3, points directly to the result. This method works for any whole-number exponent, positive or negative.

n mod 4	0	1	2	3
Result	1	i	−1	−i
Example n = 0	i<sup>0</sup> = 1	—	—	—
Example n = 1	—	i<sup>1</sup> = i	—	—
Example n = 2	—	—	i<sup>2</sup> = −1	—
Example n = 3	—	—	—	i<sup>3</sup> = −i

To simplify powers of i for any integer exponent, divide the exponent by 4 and take the remainder. Map the remainder to the cycle table above.

• Remainder 0: result is 1.
• Remainder 1: result is i.
• Remainder 2: result is −1.
• Remainder 3: result is −i.

For example, i³⁷: 37 ÷ 4 = 9 remainder 1, so i³⁷ = i. For i⁵⁰: 50 ÷ 4 = 12 remainder 2, so i⁵⁰ = −1. This method removes all guesswork and is the fastest way to simplify a high power of i.

Negative powers of i follow the same cycle but require one extra step. For i⁻ⁿ, rewrite it as 1 / iⁿ. Then simplify iⁿ using the remainder method and divide 1 by that result.

Examples from the fact sheet:
i⁻¹ = 1 / i = −i (because 1 / i = −i after multiplying numerator and denominator by i).
i⁻² = 1 / (−1) = −1.
i⁻³ = 1 / (−i) = i.
i⁻⁴ = 1 / 1 = 1.

For a large negative exponent like i⁻¹⁷, first note that the positive exponent 17 gives remainder 1 (17 ÷ 4 = 4 remainder 1), so i¹⁷ = i. Then i⁻¹⁷ = 1 / i = −i.

The four-value cycle is not a coincidence. It comes from the geometric interpretation of complex numbers. On the complex plane, the number i sits at coordinates (0, 1), one unit above the origin. Multiplying by i rotates a point by 90° counterclockwise.

Start at 1 (point (1, 0)). Multiply by i: you get i at (0, 1), a 90° rotation. Multiply i by i: you get i² = −1 at (−1, 0), another 90°. Multiply −1 by i: you get −i at (0, −1), a third 90°. Multiply −i by i: you get 1 back at (1, 0), a full 360°, four rotations.

Each multiplication by i is a quarter turn. After four turns you return to the starting point. This geometric reason explains why i⁴ = 1 and why the cycle repeats every 4 steps. The same idea extends to De Moivre's theorem, which handles powers of any complex number, not just i.

Below are three problems showing the remainder method on different exponent types.

Positive Exponent: i⁸³

83 ÷ 4 = 20 remainder 3. Remainder 3 corresponds to −i. So i⁸³ = −i.

Negative Exponent: i⁻²²

Work with the positive part first. 22 ÷ 4 = 5 remainder 2, so i²² = −1. Then i⁻²² = 1 / (−1) = −1. The answer is −1.

Large Exponent with Zero Remainder: i¹⁰⁰

100 ÷ 4 = 25 remainder 0. Remainder 0 gives 1. So i¹⁰⁰ = 1.

For fractional exponents such as i^1/2 or i^1/3, the remainder method does not apply directly. Those require De Moivre's theorem and produce multiple roots. For i^1/2, the two values are (1 + i)/√2 and −(1 + i)/√2. For i^1/3, the three values are (√3 + i)/2, (−√3 + i)/2, and −i.