Powers of i

The powers of i repeat in a cycle of four: i¹ = i, i² = −1, i³ = i² × i = −i, i⁴ = (i²)² = 1, and then i⁵ = i again.

For every integer k: i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = −1 and i⁴ᵏ⁺³ = −i.

To find iⁿ, divide n by 4 and keep only the remainder. For n = 103 = 4 × 25 + 3, i¹⁰³ = i³ = −i; for n = 58 = 4 × 14 + 2, i⁵⁸ = −1.

Negative powers follow the same cycle: i⁻¹ = 1/i = −i, i⁻² = −1, i⁻³ = i and i⁻⁴ = 1. For i⁻²², write −22 = 4 × (−6) + 2, so i⁻²² = i² = −1; directly, 1/i²² = 1/(−1) = −1.

Any four consecutive powers add to zero: iⁿ + iⁿ⁺¹ + iⁿ⁺² + iⁿ⁺³ = iⁿ(1 + i − 1 − i) = 0.

Seen on the Argand plane, each extra factor of i is a quarter turn anticlockwise, so four factors make a full turn and bring the point back to where it started.