 Cyclic Nature of the Powers of "i " MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts When the imaginary unit, i, is raised to increasingly higher powers, a cyclic (repetitive) pattern emerges. Remember that i 2 = -1. Repeating Pattern of Powers of i : i0 = 1 i4 = i3 • i = (-i) • i = -i2 = 1 i8 = i 4• i4 = 1 • 1 = 1 i1 = i i5 = i 4• i = 1 • (i) = i i9 = i 4• i 4• i = 1 • 1• i = i i2 = -1 i6 = i 4• i2 = 1 • (-1) = -1 i10 = (i 4)2 • i2 = 1 • (-1) = -1 i3 = i2 • i = (-1) • i = -i i7 = i 4• i3 = 1 • (-i) = -i i11 = (i 4)2 • i3 = 1 • (-i) = -i

The powers of i repeat in a definite pattern: (i, -1, -i, 1)

 Powers of i i1 i2 i3 i4 i5 i6 i7 i8 ... Simplified Form i -1 -i 1 i -1 -i 1 ...

Simplifying powers of i:
You will need to remember (or establish) the powers of 1 through 4 of i to obtain one cycle of the pattern. From that list of values, you can easily determine any other positive integer powers of i.

 Method 1: When the exponent is greater than or equal to 5, use the fact that i 4 = 1 and the rules for working with exponents to simplify higher powers of i. Break the power down to show the factors of four.  When raising i to any positive integer power, the answer is always i, -1, -i or 1.
 Another way to look at the simplification: Method 2: Divide the exponent by 4: • if the remainder is 0, the answer is 1 (i0). • if the remainder is 1, the answer is i (i1). • if the remainder is 2, the answer is -1 (i2). • if the remainder is 3, the answer is -i (i3).  Simplify i87

 By Method 1: Break down the power to show factors of 4. (84 is the largest multiple of 4) By Method 2: Divide the power by 4 to find the remainder. 87 ÷ 4 = 21 with remainder 3 The answer is i3 which is -i.  How to use your TI-83+/84+ calculator with complex numbers. Click here. You can raise i to any positive integer value using a TI-84+ calculator. Unfortunately, the older model calculators will only give an exact answer ( i, 1, -i, -1) up to a power of 6.
The newer TI-84+CE will give an exact answer ( i, 1, -i -1) up to a power of 100.
Beyond these powers, the calculators will give an estimate (in scientific notation) that will need to be interpreted as to whether the answer is i, 1, -i , or -1. NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use".

Contact Person: Donna Roberts