The first eight powers of i ( in ) are shown below and they can be computed quite easily when n > 0 and n < 0
i1 = i
i2 = - 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?
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
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?
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
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
Nov 18, 20 01:20 PM
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