Evaluate i^888 — Powers of i (Imaginary Number)

The powers of i follow a cycle of 4: • i^1 = i • i^2 = −1 • i^3 = −i • i^4 = 1 • i^5 = i (cycle repeats)

Rule: To find i^n, find the remainder when n is divided by 4. • Remainder 0 → i^n = 1 • Remainder 1 → i^n = i • Remainder 2 → i^n = −1 • Remainder 3 → i^n = −i

For i^888: 888 ÷ 4 = 222, remainder = 0

Since remainder = 0: i^888 = 1

Alternative working: i^888 = (i^4)^222 = 1^222 = 1

Answer: i^888 = 1

Using the remainder method:

• i^1: 1÷4 = rem 1 → i • i^10: 10÷4 = rem 2 → i^10 = −1 • i^27: 27÷4 = rem 3 → i^27 = −i • i^100: 100÷4 = rem 0 → i^100 = 1 • i^50: 50÷4 = rem 2 → i^50 = −1 • i^75: 75÷4 = rem 3 → i^75 = −i • i^200: 200÷4 = rem 0 → i^200 = 1 • i^999: 999÷4 = rem 3 → i^999 = −i

Negative powers: i^(−1) = 1/i = i/i² = i/(−1) = −i i^(−2) = 1/i² = 1/(−1) = −1 i^(−3) = 1/i³ = 1/(−i) = i i^(−4) = 1

Definition: i = √(−1) i² = −1

The imaginary unit i is the fundamental unit of complex numbers.

A complex number has the form: z = a + bi where a = real part, b = imaginary part.

Cyclic pattern of i: i^1 = i i^2 = −1 (since i×i = √(−1)×√(−1) = −1) i^3 = i²×i = −1×i = −i i^4 = i²×i² = (−1)(−1) = 1 i^5 = i^4×i = 1×i = i (back to start)

Period = 4 (the pattern repeats every 4 powers)

Why does this matter? Powers of i appear in: • Complex number arithmetic • Fourier series and transforms • Quantum mechanics • Electrical engineering (impedance)