Find the powers of i 

The powers of i ( i) 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

  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 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?

  • 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, 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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