Review of Complex Numbers
Introduction
This is a short review of the main concepts of
complex numbers
. Complex numbers are used
throughout mathematics and its applications. In particular, when we try to solve differential
equations it is often convenient and natural to use complex numbers to express the solutions.
Here we review those ideas and results from the theory of complex numbers that will be used
in Math 216.
A complex number
z
may be expressed as an ordered pair of
real
numbers:
z
= (
x, y
) =
x
+
iy
where
i
:=
√

1 (so
i
2
=

1) and
x
and
y
are real numbers. The following notations are
often used:
x
= Re(
z
) or
x
=
<
(
z
) denotes the
real part
of the complex number
z
y
= Im(
z
) or
y
=
=
(
z
) denotes the
imaginary part
of the complex number
z
Recall that two complex numbers are equal if and only if both the real and the imaginary
parts are equal. In other words,
z
1
:= (
x
1
, y
1
) equals
z
2
:= (
x
2
, y
2
) if and only if
x
1
=
x
2
and
y
1
=
y
2
.
A convenient way of thinking about complex numbers is to imagine them as points in
the (
x, y
) plane (in this case it is called the
complex plane
), as illustrated in the following
figure. In the complex plane, the line
y
= 0 is frequently called the
real axis
, and the line
y=Im(z)
x=Re(z)
z=(x,y)=x+iy
Figure 1: The complex number
z
=
x
+
iy
plotted in the complex plane.
x
= 0 is frequently called the
imaginary axis
.
Doing arithmetic with complex numbers
Addition and multiplication of two complex numbers
z
1
= (
x
1
, y
1
) and
z
2
= (
x
2
, y
2
) are
defined by the following rules:
1
•
Addition:
z
1
+
z
2
:= (
x
1
+
x
2
, y
1
+
y
2
) = (
x
1
+
x
2
) +
i
(
y
1
+
y
2
).
•
Multiplication:
z
1
z
2
:= (
x
1
x
2

y
1
y
2
, x
1
y
2
+
x
2
y
1
) = (
x
1
x
2

y
1
y
2
) +
i
(
x
1
y
2
+
x
2
y
1
).
Note that if we interpret
z
1
and
z
2
as points in the complex plane as in Figure 1, then addition
of complex numbers is the same as vector addition in the plane; we are just adding the real
and imaginary parts componentwise. On the other hand, the multiplication of two complex
numbers may perhaps seem different than what you might have expected it to be; this is
only an illusion, however, and when we introduce exponential forms for complex numbers
later, the multiplication will make perfect sense.
Although complex numbers obey different rules of arithmetic than do ordinary real num
bers, it is very important to keep in mind that the complex numbers simply generalize the
notion of the real numbers.
Indeed, we can think of the real number
x
as the complex
number (
x,
0) =
x
+
i
0. Such a complex number whose imaginary part is zero is said to be
purely real
. If we add or multiply two purely real complex numbers, then according to the
rules for complex arithmetic, we have
(
x
1
,
0) + (
x
2
,
0) = (
x
1
+
x
2
,
0)
and
(
x
1
,
0)(
x
2
,
0) = (
x
1
x
2
,
0)
so in each case the result is also a purely real complex number, and the real part in each case is
exactly what we would have found by applying the usual rules of addition and multiplication
for real numbers to the real parts.
This shows that all the new operations defined for
complex numbers when applied to purely real numbers give the usual familiar corresponding
operations.
One way to think of (0
,
1) is as the
new