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

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. 

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 about the powers of i?

• 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

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