Imaginary and Complex Numbers
Most of the Math IIC test deals with real numbers. But
every so often, a question will appear that takes place outside
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
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 of
a +
bi,
where
a and
b are real numbers,
and
i =
.
There are three 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
- Know
how to graph a complex number in the complex plane and how to find its
absolute value
The Powers of i
The powers of i are easy to work with.
For example:
|
|
|
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:
|
|
|
Simplify
the expression (3x + i)(x –
2i). |
|
The Complex Plane
The complex plane is a plane similar to the coordinate
plane, except that instead of having an x-axis
and a y-axis, it has axes that allow you to plot
complex numbers as well as real numbers.
The horizontal axis of a complex plane is called the real
axis, and the vertical axis is called the imaginary axis. The complex
number a + bi is plotted in the complex plane exactly
where the point (a, b) would be plotted
in the standard coordinate plane. Take a look at the picture of
the complex plane below, with various complex numbers plotted on it.
The magnitude, or absolute value, of a complex number
is the distance from the origin to that number in the complex plane.
You can use the Pythagorean theorem to calculate the magnitude of
the complex number.
The magnitude of
a + bi, which can also
be written as |
a +
bi|, is equal
to
Try the following example:
|
|
|
What
is the magnitude of the complex number 4 – 9i? |
|
The complex number 4 – 9
i would be graphed
on the plane as the point (
4, –9
i).
Using the Pythagorean theorem to calculate its magnitude:
=
.gif)
