UNIT 4
COMPLEX NUMBERS
INTRO: This Unit is about complex numbers. Unlike real numbers, which fill a onedimensional
line (in fact, we frequently say the
real number line
), the complex numbers fill an entire plane,
the socalled
complex plane
.
We wish to understand complex numbers geometrically, their
locations and transformations in the complex plane, their powers and their roots.
1. Complex Numbers
The most general
complex number
z
can be written as
z
=
a
+
i b
where
a
and
b
are real numbers and
i
=
√

1 is an imaginary number (since no real number
squares to give a negative number).
The product
ib
is called a
pure imaginary number
, or
often, simply an
imaginary number
, and signifies the product of
b
times
√

1.
We usually represent the complex numbers as filling the
x

y
plane, with the points on the
x
axis representing the real number line, and the points on the
y
axis representing the
pure
imaginary numbers
, ie., the complex numbers with zero real part.
Plotted on the complex
plane, the complex number
a
+
bi
would then be located at the point with
x
component
a
and
y
component
b
.
A graph of the complex plane is below.
You can see the real numbers 1, 2, 3, etc., located
along the horizontal axis, and the pure imaginary numbers
i,
2
i,
3
i,
etc., located along the
vertical axis. Can you locate where the number 3+2
i
would be located, or the number

2

2
i
?
Actually, if you look carefully, the graph has dots located at these numbers. Remember, each
point on the complex plane is a (complex) number, not a pair of coordinates.
3
2
1
1
2
3
Minus
3 i
Minus
2 i
Minus
i
i
2 i
3 i
The complex plane
Because you have been dealing with complex numbers since they were introduced in first
year high school algebra in order to solve the quadratic equation, we will not deal with the
interesting question of what exactly a complex number
is
, that is to say, how we can multiply
a real number
b
times a symbol
i
, or add a real number to it. Instead, we will go directly to a