Find the powers of i 

The first eight powers of i ( iⁿ ) are shown below and they can be computed quite easily when n > 0 and n < 0

Powers of i when n > 0

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?

• 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. 

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

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?

• For i^-4 and i^-8, the remainder is 0 when we divide 4 and 8 by 4. 
• For i^-3 and i^-7, the remainder is -3 when we divide -3 and -7 by 4. 
• For i^-2 and i^-6, the remainder is -2 when we divide -2 and -6 by 4. 
• For i^-1 and i^-5, the remainder is -1 when we divide 1 and 5 by 4. 

A few examples showing how to find the powers of i.

Example #1:

i⁶⁷

67 divided 4 gives a remainder of 3. Since n is positive, i⁶⁷ = -i

Example #2:

i^-67

-67 divided 4 gives a remainder of -3. Since n is negative, i^-67 = i

Example #3:

i³⁶

36 divided 4 gives a remainder of 0. Since n is positive, i³⁶ = 1

Example #4:

i^-36

-36 divided 4 gives a remainder of 0. Since n is negative, i^-36 = 1