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Complex Numbers
Introduction
If we try to solve x
2
= 1, what happens?
We extract square roots to get x = +/
√
1.
But if we try to evaluate the square root of –1
on a scientific calculator, we get ERROR!
But still, we need a way to define solutions
like this so it is defined that
i
2
= 1 and thus i =
√
(1).
This means that the solutions of x
2
= 1 are x = i and x = i
We refer to such solutions as
Complex Solutions
.
Furthermore, we refer to a number containing the quantity “i”, where i =
√
1, as an
imaginary number
.
This choice of words “imaginary” is actually not appropriate, since
we use the number “i” in many
realworld
engineering applications!
Using Complex Numbers To Evaluate Square Roots
Given that i
2
= 1 and thus i =
√
(1), we can use this fact to evaluate any square
root.
For example,
√
(13) =
√
(1
•
13) =
√
(1)
•
√
13
and we can replace
√
(1)
with i to get
√
(13) = i
•√
13.
In general, we can say that
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 Fall '09
 Crandall
 Anthropology

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