viii Preface

Using the complex exponential function to simplify trigonometry is a compelling as-

pect of elementary complex analysis and geometry. Students in my courses seemed

to appreciate this material to a great extent.

One of the most appealing combinations of the geometric series and the expo-

nential series appears in Chapter 4. We combine them to derive a formula for the

sums

Xn

j=1

jp,

in terms of Bernoulli numbers.

We briefly discuss ordinary and exponential generating functions, and we ﬁnd

the ordinary generating function for the Fibonacci numbers. We then derive Bi-

net’s formula for the n-th Fibonacci number and show that the ratio of successive

Fibonacci numbers tends to the golden ratio

1+

√

5

2

.

Fairly early in the book (Chapter 3) we discuss hyperbolas, ellipses, and parabo-

las. Most students have seen this material in calculus or even earlier. In order to

make the material more engaging, we describe these objects by way of Hermitian

symmetric quadratic polynomials. This approach epitomizes our focus on complex

numbers rather than on pairs of real numbers.

The geometry of the unit circle also allows us to determine the Pythagorean

triples. We identify the Pythagorean triple (a,b,c) with the complex number

a

c

+i

b

c

;

we then realize that a Pythagorean triple corresponds to a rational point (in the ﬁrst

quadrant) on the unit circle. After determining the usual rational parametrization

of the unit circle, one can easily ﬁnd all these triples. But one gains much more; for

example, one discovers the so-called

tan(θ

2

) substitution from calculus. During the

course several students followed up this idea and tracked down how the indeﬁnite

integral of the secant function arose in navigation.

This book is more formal than was the course itself. The list of approximately

two hundred eighty exercises in the book is also considerably longer than the list

of assigned exercises. These exercises (as well as the ﬁgures) are numbered by

chapter, whereas items such as theorems, propositions, lemmas, examples, and

deﬁnitions are numbered by section. Unless speciﬁed otherwise, a reference to

a section, theorem, proposition, lemma, example, or deﬁnition is to the current

chapter. The overall development in the book closely parallels that of the courses,

although each time I omitted many of the harder topics. I feel cautiously optimistic

that this book can be used for similar courses. Instructors will need to make their

own decisions about which subjects can be omitted. I hope however that the book

has a wider audience including anyone who has ever been curious about complex

numbers and the striking role they play in modern mathematics and science.

Chapter 1 starts by considering various number systems and continues by de-

scribing, slowly and carefully, what it means to say that the real numbers are

a complete ordered ﬁeld. We give an interesting proof that there is no rational

square root of 2, and we prove carefully (based on the completeness axiom) that

positive real numbers have square roots. The chapter ends by giving several possible

deﬁnitions of the ﬁeld of complex numbers.