The powers of i ( i^{n }) 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!

i^{1} = i

i^{2 } = - 1

i^{3} = i^{2} × i = -i

i^{4} = i^{2} × i^{2} = 1

i^{5} = i^{4} × i = i

i^{6} = i^{4} × i^{2} = -1

i^{7} = i^{4} × i^{3} = -i

i^{8} = i^{4} × i^{4} = 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 i^{n} 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.

Conclusion

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

If R = 1, i^{n} = i

If R = 2, i^{n} = -1

If R = 3, i^{n} = -i

If R = 0, i^{n} = 1

i^{-1} = 1 / i = (1 × i) / (i × i) = i / i^{2} = i / -1 = -i

i^{-2 }= 1 / i^{2} = 1 / -1 = -1

i^{-3} = 1 / i^{3} = 1 / -i = (1 × i) / (-i × i) = i / 1 = i

i^{-4} = 1 / i^{4} = 1 / 1 = 1

i^{-5} = 1 / i^{5} = 1 / i = -i

i^{-6} = 1 / i^{6} = 1 / -1 = -1

i^{-7} = 1 / i^{7} = 1 / -i = i

i^{-8} = 1 / i^{8} = 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 i^{n} 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.

Conclusion

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

If R = -1, i^{n} = -i

If R = -2, i^{n} = -1

If R = -3, i^{n} = i

If R = 0, i^{n} = 1

**Example #1:**

i^{67}

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

**Example #2:**

i^{-67}

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

**Example #3:**

i^{36}

36 divided 4 gives a remainder of 0. Since n is positive, i^{36} = 1

**Example #4:**

i^{-36}

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