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?
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?
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
Jan 26, 23 11:44 AM
Jan 25, 23 05:54 AM