Imaginary and Complex Numbers
Most of the Math IC test deals with real numbers. But
every so often, a question will appear that does not involve the
set of real numbers. These questions deal with imaginary and complex
numbers.
Imaginary Numbers
Imaginary numbers are used to represent the
even roots of negative numbers. They use the quantity
i,
where
i =
. For example:
Square roots of negative numbers are called imaginary
numbers because they do not lie on the real number line.
Complex Numbers
A
complex number is the sum of a real number
and an imaginary number. A complex number is written in the form
a +
bi,
where
a and
b are real numbers,
and
i =
.
There are two things you need to be able to do with complex
numbers:
-
Know the powers of i
- Know
how to do operations, like addition, subtraction, and multiplication, on
complex numbers
The Powers of i
The powers of i are easy to work with.
For example:
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Evaluate i2  i9. |
|
The trick is to remember that the powers of i work
in cycles of four:
- i1 = i
- i2 =
= (
)2 = –1
- i3 =
= 
(
)2 = –i
- i4 =
= (
)4 = 1
This way, the expression
i2 
i9 becomes
(–1)(
i) = –
i. If you know these
cycles, you can reduce any exponent of
i to a much
more manageable size. The expression
i2
i9 becomes (–1)(
i)
= –
i.
Operations on Complex Numbers
Algebraic manipulation of complex numbers is exactly like
dealing with real numbers. See for yourself:
|
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Simplify
the expression (3x + i)(x –
2i). |
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