Chapter 1

The Complex Numbers

1.1. Definition and Simple Properties

The number system is a tool devised by humans to aid in the description of quan-

tities of the various things humans have to deal with. It has evolved as human

culture has evolved, beginning with something very primitive like: 1, 2, 3, many,

moving on to the natural numbers, then the integers, the rational numbers, the real

numbers and then the complex numbers.

At each stage of development, the number system was expanded in response

to the need to describe quantities that the old number system could not. For

example, the negative numbers were introduced in order to be able to describe a

loss as opposed to a gain, or moving backward rather than forward. The rational

numbers were introduced because we do not always deal with whole numbers of

things (we have 2/3 of a pie left). The real number system evolved from the

rational number system out of a need to be able to describe such things as the

length of the hypotenuse of a right triangle (this involves square roots) and the

area or circumference of a circle (this involves π).

In this course, we will assume students are familiar with the real number system

and its properties. We will define the complex number system as a needed extension

of the real number system and develop its properties. We will then go on to study

functions of a complex variable.

The Real Numbers are Insuﬃcient. The complex number system was devel-

oped in response to the need for solutions to polynomial equations. The simplest

polynomial equation that does not have a solution in the real number system is the

equation

x2

+ 1 = 0,

which has no real solution because −1 has no real square root. More generally, a

quadratic equation

(1.1.1)

ax2

+ bx + c = 0,

1