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
1
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