I
The Expanding Universe of Numbers
For many people, numbers must seem to be the essence of mathematics.Number
theory, which is the subject of this book, is primarily concerned with the properties
of one particular type of number, the ‘whole numbers’ orintegers.However,there
are many other types, such as complex numbers andp-adic numbers. Somewhat sur-
prisingly, a knowledge of these other types turns out to be necessary for any deeper
understanding of the integers.
In this introductory chapter we describe several such types (but defer the study of
p-adic numbers to Chapter VI).To embark on number theory proper the reader may
proceed to Chapter II nowand refer back to the present chapter, via the Index, only as
occasion demands.
When one studies the properties of various types of number, one becomes aware
of formal similarities between different types. Instead of repeating the derivations of
properties for each individual case, it is more economical – and sometimes actually
clearer – to study their common algebraic structure. This algebraic structure may be
shared by objects which one would not even consider as numbers.
There is a pedagogic difficulty here. Usuallya property is discovered in one context
and only later is it realized that it has wider validity. It may be more digestible to
prove a result in the context of number theory and then simply point out its wider
range of validity. Since this is a book on number theory, and many properties were
first discovered in this context, we feel free to adopt this approach. However, to make
the statements of such generalizations intelligible, in the latter part of this chapter we
describe several basic algebraic structures. We do not attempt to study these structures
in depth, but restrict attention to the simplest properties which throw light on the work
of later chapters.
0 Sets,RelationsandMappings
The label ‘0’ given to this section may be interpreted to stand for ‘0ptional’. We collect
here some definitions of a logical nature which have become part of the common lan-
guage of mathematics. Those who are not already familiar with this language, and who
are repelled by its abstraction, should consult this section only when the need arises.
DOI: 10.1007/978-0-387-89486-7_1, © Springer Science + Business Media, LLC 2009
W.A. Coppel, Number Theory: An Introduction to Mathematics, Universitext, 1