How this calculator works
The imaginary unit i is defined by i² = -1. Multiplying by i rotates a number a quarter turn around the complex origin, so the powers repeat every four steps: i¹ = i, i² = -1, i³ = -i and i⁴ = 1. Because of that cycle, any exponent reduces to its remainder modulo 4. Negative exponents use the reciprocal: i⁻¹ = -i, and the same reduction works once the remainder is taken in the range 0 to 3.
Formula
i^n = i^r where r is n mod 4 in the range 0 to 3. Then r = 0 gives 1, r = 1 gives i, r = 2 gives -1 and r = 3 gives -i.
Worked example
i^5 = i^4 × i = 1 × i = i. Similarly i^10 = i^2 = -1, i^102 = i^2 = -1, and i^-1 = i^3 = -i.
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Frequently asked questions
Why does the pattern repeat every four powers?
Multiplying by i twice multiplies by -1, and four times multiplies by (+1). Once i⁴ = 1, every four extra powers return the same value, so only n mod 4 matters.
What does i mean?
The letter i names a number whose square is -1. No real number has that property, so i extends the real line into the complex plane, where a number has a real part and an imaginary part.
How do negative exponents work?
A negative exponent means a reciprocal: i^-n = 1 / i^n. For example i^-2 = 1 / (-1) = -1. The four-value cycle still applies after converting the remainder to 0 to 3.
Is the sum of four consecutive powers of i zero?
Yes. The four values 1, i, -1 and -i add to 0, so any block of four consecutive powers cancels. That is a quick way to sum long runs of powers.