Chapter 1
Complex Numbers
Die ganzen Zahlen hat der liebe Gott geschaﬀen, alles andere ist Menschenwerk.
(God created the integers, everything else is made by humans.)
Leopold Kronecker (1823–1891)
1.0 Introduction
The real numbers have nice properties. There are operations such as addition, subtraction, mul
tiplication as well as division by any real number except zero. There are useful laws that govern
these operations such as the commutative and distributive laws. You can also take limits and do
calculus. But you cannot take the square root of

1. Equivalently, you cannot ﬁnd a root of the
equation
x
2
+ 1 = 0
.
(1.1)
Most of you have heard that there is a “new” number
i
that is a root of the Equation (1.1).
That is,
i
2
+ 1 = 0 or
i
2
=

1. We will show that when the real numbers are enlarged to a new
system called the complex numbers that includes
i
, not only do we gain a number with interesting
properties, but we do not lose any of the nice properties that we had before.
Speciﬁcally, the complex numbers, like the real numbers, will have the operations of addition,
subtraction, multiplication as well as division by any complex number except zero. These operations
will follow all the laws that we are used to such as the commutative and distributive laws. We will
also be able to take limits and do calculus. And, there will be a root of Equation (1.1).
In the next section we show exactly how the complex numbers are set up and in the rest
of this chapter we will explore the properties of the complex numbers. These properties will be
both algebraic properties (such as the commutative and distributive properties mentioned already)
and also geometric properties. You will see, for example, that multiplication can be described
geometrically. In the rest of the book, the calculus of complex numbers will be built on the
propeties that we develop in this chapter.
1.1 Deﬁnition and Algebraic Properties
The
complex numbers
can be deﬁned as pairs of real numbers,
C
=
{
(
x,y
) :
x,y
∈
R
}
,
1
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2
equipped with the
addition
(
x,y
) + (
a,b
) = (
x
+
a,y
+
b
)
and the
multiplication
(
x,y
)
·
(
a,b
) = (
xa

yb,xb
+
ya
)
.
One reason to believe that the deﬁnitions of these binary operations are “good” is that
C
is an
extension of
R
, in the sense that the complex numbers of the form (
x,
0) behave just like real
numbers; that is, (
x,
0) + (
y,
0) = (
x
+
y,
0) and (
x,
0)
·
(
y,
0) = (
x
·
y,
0). So we can think of the
real numbers being embedded in
C
as those complex numbers whose second coordinate is zero.
The following basic theorem states the algebraic structure that we established with our deﬁni
tions. Its proof is straightforward but nevertheless a good exercise.
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 Spring '10
 MARCHESI
 Topology, Real Numbers, Integers, Complex Numbers, Metric space, Complex number

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