Find the powers of i

The powers of i ( iⁿ) are shown in the table below and these can be computed quite easily when n > 0 and n < 0.

If you did not quite understand the information in the table, keep reading to see the logic behind it!

Powers of i when n > 0

i¹ = i

i² = - 1

i³ = i² × i = -i

i⁴ = i² × i² = 1

i⁵ = i⁴ × i = i

i⁶ = i⁴ × i² = -1

i⁷ = i⁴ × i³ = -i

i⁸ = i⁴ × i⁴ = 1

Notice the pattern i, -1, -i, 1, ... repeats after the first four complex numbers. In general, if n is an integer bigger than zero, the value of iⁿ can be found by dividing n by 4 and examining the remainder.

Did you make the following observations about the powers of i?

For i⁴ and i⁸, the remainder is 0 when we divide 4 and 8 by 4.
For i³ and i⁷, the remainder is 3 when we divide 3 and 7 by 4.
For i² and i⁶, the remainder is 2 when we divide 2 and 6 by 4.
For i¹ and i⁵, the remainder is 1 when we divide 1 and 5 by 4.

Conclusion

Let n > 0 and R is the remainder when n is divided by 4

If R = 1, iⁿ = i

If R = 2, iⁿ = -1

If R = 3, iⁿ = -i

If R = 0, iⁿ = 1

Powers of i when n < 0

i^-1 = 1 / i = (1 × i) / (i × i) = i / i² = i / -1 = -i

i^-2= 1 / i² = 1 / -1 = -1

i^-3 = 1 / i³ = 1 / -i = (1 × i) / (-i × i) = i / 1 = i

i^-4 = 1 / i⁴ = 1 / 1 = 1

i^-5 = 1 / i⁵ = 1 / i = -i

i^-6 = 1 / i⁶ = 1 / -1 = -1

i^-7 = 1 / i⁷ = 1 / -i = i

i^-8 = 1 / i⁸ = 1 / 1 = 1

i^-1=-i

i^-2= -1

i^-3 = i

i^-4 = 1

i^-5=-i

i^-6= -1

i^-7 = i

i^-8 = 1

Notice the pattern -i, -1, i, 1, ... repeats after the first four complex numbers. In general, if n is an integer smaller than zero, the value of iⁿ can be found by dividing n by 4 and examining the remainder.

Did you make the following observations about the powers of i?