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To be
cyclic means to be repetitive in nature. When the imaginary
unit,
i, is raised to increasingly larger
powers, it creates a cyclic pattern.
The powers of i
repeat in a definite pattern:
( i, -1, -i, 1 )
| Powers of
i |
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... |
| Simplified
form |
i |
-1 |
-i |
1 |
i |
-1 |
-i |
1 |
... |
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Think about what happens when i
is raised to a given power:
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LOOK OUT!!! |
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is not true when
a
and
b are both
negative. |
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False
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TRUE: |
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Whenever the exponent is greater than or equal to 5, you
can
use the fact that
 to
simplify a power of i.
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 When raising i to any integral power, the answer
is always i, -1, -i or 1. |
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Another way to think of this process of
simplifying powers of i is to 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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Let's examine two ways to simplify
 :
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Using the patterns
shown in the robot table above:
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Looking at
remainders when dividing by 4:
with a remainder of 3,
which means the answer is
i 3 = -i. |
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Check out how to use your
TI-83+/84+ graphing calculator with
complex numbers.
Click here. |
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