# Find the powers of i

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

## Powers of i when n > 0

i1 = i

i = - 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?

• 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

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?

• 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.

Conclusion

Let n < 0 and R is the remainder

If R  = -1, in = -i

If R = -2, in = -1

If R = -3, in = i

If R = 0, in = 1

## 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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