Chapter 1

From the Real Numbers

to the Complex Numbers

1. Introduction

Many problems throughout mathematics and physics illustrate an amazing princi-

ple: ideas expressed within the realm of real numbers ﬁnd their most elegant ex-

pression through the unexpected intervention of complex numbers. Many of these

delightful interventions arise in elementary, recreational mathematics. On the other

hand most college students either never see complex numbers in action or they wait

until the junior or senior year in college, at which time the sophisticated courses

have little time for the elementary applications. Hence too few students witness the

beauty and elegance of complex numbers. This book aims to present a variety of

elegant applications of complex analysis and geometry in an accessible but precise

fashion. We begin at the beginning, by recalling various number systems such as

the integers Z, the rational numbers Q, and the real numbers R, before even deﬁn-

ing the complex numbers C. We then provide three possible equivalent deﬁnitions.

Throughout we strive for as much geometric reasoning as possible.

2. Number systems

The ancients were well aware of the so-called natural numbers, written 1, 2, 3,....

Mathematicians write N for the collection of natural numbers together with the

usual operations of addition and multiplication. Partly because subtraction is not

always possible, but also because negative numbers arise in many settings such as

ﬁnancial debts, it is natural to expand the natural number system to the larger

system Z of integers. We assume that the reader has some understanding of the

integers; the set Z is equipped with two distinguished members, written 1 and 0,

and two operations, called addition (+) and multiplication (*), satisfying familiar

laws. These operations make Z into what mathematicians call a commutative ring

with unit 1. The integer 0 is special. We note that each n in Z has an additive

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