Powers of i Calculator/Visualizer

Powers of i

i⁴ = 1, so powers cycle every 4. Just find k mod 4.

r = 0

1

i⁰ = 1

r = 1

i

i¹ = i

r = 2

−1

i² = −1

r = 3

−i

i³ = −i

1Divide by 4:323 ÷ 4 = 80 remainder 3 (4×80+3=323 ✓)

2Rewrite:i³²³ = i^4×80+3 = i^4×80 · i³

3Apply i⁴=1:(i⁴)⁸⁰ · i³ = 1 · i³ = i³

4Lookup r=3:i³ = −i

i³²³ = −iiᵏ = i^r where r = k mod 4

Explanation

i⁰ = 1Any number to power 0 equals 1

i¹ = ii to the first power is i

i² = −1By definition: i² = −1

i³ = −ii³ = i² · i = −1 · i = −i