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When the imaginary unit, i, is raised to increasingly higher powers,
a cyclic (repetitive) pattern emerges.
Remember that i 2 = -1. |
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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.
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 When raising i to any positive integer power,
the answer is always
i, -1, -i or 1. |
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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). |
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 Simplify i87
By Method 1:
Break down the power to show factors of 4. (84 is the largest multiple of 4)
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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.
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How to use your
TI-83+/84+
calculator
with
complex numbers.
Click here. |
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 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. Read more.
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