Springer coppel number theory an introduction to mathematics part a (springer 2006)

NUMBER THEORY
An Introduction to Mathematics: Part A


NUMBER THEORY
An Introduction to Mathematics: Part A

BY
WILLIAM A. COPPEL

- Springer


Library of Congress Control Number: 2005934653
PARTA

ISBN-10: 0-387-29851-7

e-ISBN: 0-387-29852-5

ISBN-13: 978-0387-29851-1

PART B

ISBN-10: 0-387-29853-3

e-ISBN: 0-387-29854-1

ISBN-13: 978-0387-29853-5

PVOLUME SET

ISBN-10: 0-387-30019-8

e-ISBN:0-387-30529-7

ISBN-]3: 978-0387-30019-1

Printed on acid-free paper.

AMS Subiect Classifications: 1 1-xx. 05820. 33E05
O 2006 Springer Science+Business Media, Inc

All rights reserved. This work may not be translated or copied in whole or in part without the written
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Printed in the United States of America


For Jonathan, Nicholas, Philip and Stephen


Contents
Part A
Preface

I

The expanding universe of numbers

Sets, relations and mappings
Natural numbers
Integers and rational numbers
Real numbers
Metric spaces
Complex numbers
Quaternions and octonions
Groups
Rings and fields
Vector spaces and associative algebras
Inner product spaces
Further remarks
Selected references

I1

Divisibility
1
2
3
4
5
6
7
8

Greatest common divisors

The Bezout identity
Polynomials
Euclidean domains
Congruences
Sums of squares
Further remarks
Selected references


...

Contents

Vlll

I11

More on divisibility
1 The law of quadratic reciprocity
2
3
4
5
6

IV

Quadratic fields
Multiplicative functions
Linear Diophantine equations

Further remarks
Selected references

Continued fractions and their uses
1 The continued fraction algorithm

2
3
4
5
6
7
8
9
V

Diophantine approximation
Periodic continued fractions
Quadratic Diophantine equations
The modular group
Non-Euclidean geometry
Complements
Further remarks
Selected references

Hadamard's determinant problem
1 What is a determinant?
2 Hadamard matrices

3 The art of weighing

4 Some matrix theory
5 Application to Hadamard's determinant problem
6
7
8
9

Designs
Groups and codes
Further remarks
Selected references


Contents

VI

Hensel's p-adic numbers
1 Valued fields
2 Equivalence

3
4
5
6
7
8

Completions
Non-archimedean valued fields

Hensel's lemma
Locally compact valued fields
Further remarks
Selected references

Notations
Axioms
Index
Part B

VII

The arithmetic of quadratic forms
1 Quadratic spaces
2 The Hilbert symbol
3 The Hasse-Minkowski theorem
4 Supplements
5 Further remarks
6 Selected references

VIII The geometry of numbers
Minkowski's lattice point theorem
Lattices
Proof of the lattice-point theorem, and some generalizations
Voronoi cells
Densest packings
Mahler's compactness theorem
Further remarks
Selected references

Contents

IX

The number of prime numbers
1
2
3
4

5
6
7
8
9

X

Finding the problem
Chebyshev's functions
Proof of the prime number theorem
The Riemann hypothesis
Generalizations and analogues
Alternative formulations
Some further problems
Further remarks
Selected references

A character study

1 Primes in arithmetic progressions
2 Characters of finite abelian groups
3 Proof of the prime number theorem for arithmetic progressions
4 Representations of arbitrary finite groups

5 Characters of arbitrary finite groups
6 Induced representations and examples
7 Applications
8 Generalizations
9 Further remarks
10 Selected references

XI

Uniform distribution and ergodic theory
1 Uniform distribution
2 Discrepancy
3 Birkhoff's ergodic theorem
4 Applications
5 Recurrence
6 Further remarks
7 Selected references


Contents

XI1

Elliptic functions
1 Elliptic integrals

2 The arithmetic-geometric mean
3 Elliptic functions
4
5
6
7
8

Theta functions
Jacobian elliptic functions
The modular function
Further remarks
Selected references

XI11 Connections with number theory
Sums of squares
Partitions
Cubic curves
Mordell's theorem
Further results and conjectures
Some applications
Further remarks
Selected references
Notations
Axioms
Index


Preface to the revised edition

Undergraduate courses in mathematics are colnmonly of two types. On the one hand there are
courses in subjects, such as linear algebra or real analysis, with which it is considered that every
student of mathematics should be acquainted. On the other hand there are courses given by
lecturers in their own areas of specialization, which are intended to sellre as a prepasation for
research. There ase, I believe, several reasons why students need more than this.
Fhst, although the vast extent of mathematics today makes it impossible for any indvidual
to have a deep knowledge of more than a small part, it is important to have some understanding
and appreciation of the work of others. Indeed the sometimes su~prisingintei-relationships and
analogies between different branches of mathematics are both the basis for many of its
applications and the stimulus for further development. Secondly, different branches of
mathematics appeal in different ways and require different talents. It is unlikely that all students
at one university will have the same interests and aptitudes as their lecturers. Rather, they will
only discover what their own interests and aptitudes are by being exposed to a broader range.
Thirdly, many students of lnathematics will become, not professional mathematicians, but
scientists, engineers or schoolteachers. It is useful for them to have a clear understanding of the
nature and extent of mathematics, and it is in the interests of mathematicians that there should be
a body of people in the coinmunity who have this understanding.
The present book attempts to provide such an understanding of the nature and extent of
mathematics, The connecting theme is the theory of numbers, at first sight one of the most
abstruse and irrelevant branches of mathematics. Yet by exploiing its many connections with
other branches, we may obtain a broad picture. The topics chosen are not trivial and demand
some effort on the past of the reader. As Euclid already said, there is no royal road. In general
I have concentrated attention on those hard-won results which illuminate a wide area. If I am
accused of picking the eyes out of some subjects, I have no defence except to say "But what
bea~ltif~d
eyes!"
The book is divided into two parts. Past A, which deals with elementary number theory,
should be accessible to a first-year undergraduate. To provide a foundation for subsequent
work, Chapter I contains the definitions and basic propesties of various mathematical structures.

xiv

Preface

However, the reader may simply skim through this chapter and refer back to it later as required.
Chapter V, on Hadamard's determinant problem, shows that elementary number theory may
have unexpected applications.
Part B, which is more advanced, is intended to provide an undergraduate with some idea
of the scope of ~nathernaticstoday. The chapters in this part are largely independent, except that
Chapter X depends 011 Chapter IX and Chapter XI11 on Chapter XII.
Although much of the content of the book is common to any introductory work on number
theory, I wish to &aw attention to the discussion here of quadratic fields and elliptic curves.
These are quite special cases of algebraic number fields and algebraic curves, and it may be
asked why one should restrict attention to these special cases when the general cases are now
well understood and may even be developed in parallel. My answers are as follows. First, to
treat the general cases in full rigour requires a commitment of time which many will be unable to
afford. Secondly, these special cases are those most commonly encountered and more
constructive methods are available for them than for the general cases. There is yet another
reason. Sometimes in mathematics a generalization is so simple and far-reaching that the special
case is more fully understood as an instance of the generalization. For the topics mentioned,
however, the generalization is more complex and is, in my view, more fully understood as a
development from the special case.
At the end of each chapter of the book I have added a list of selected references, which will
enable readers to travel further in their own chosen directions. Since the literature is
voluminous, any such selection must be somewhat arbitrary, but I hope that mine may be found
interesting and useful.
The computer revolution has made possible calculations on a scale and with a speed
undreamt of a century ago. One consequence has been a considerable increase in 'experimental
mathematics' - the search for patterns. This book, on the other hand, is devoted to 'theoretical

mathematics' - the explanation of patterns. I do not wish to conceal the fact that the former
usually precedes the latter. Nor do I wish to conceal the fact that some of the results here have
been proved by the greatest minds of the past only after years of labour, and that their proofs
have later been improved and simplified by many other mathematicians. Once obtained,
however, a good proof organizes and provides understa~dingfor a mass of computational data.
Often it also suggests further developments.
The present book may indeed be viewed as a 'treasury of proofs'.

We concentrate

attention on this aspect of mathematics, not only because it is a distinctive feature of the subject,
but also because we consider its exposition is better suited to a book than to a blackboard or a
computer screen. In keeping with this approach, the proofs themselves have been chosen with


Preface

xv

some care and I hope that a few may be of interest even to those who are no longer students.
Proofs which depend on general principles have been given preference over proofs which offer
no particular insight.
Mathematics is a part of civilization and an achievement in which human beings may take
some pride. It is not the possession of any one national, political or religious group and any
attempt to make it so is ultimately destructive. At the present time there are strong pressures to
make academic studies more 'relevant'. At the same time, however, staff at some universities
are assessed by 'citation counts' and people are paid for giving lectures on chaos, for example,
that are demonstrably rubbish.
The theoiy of numbers provides ample evidence that topics pursued for their own intrinsic
interest can later find significant applications. I do not contend that curiosity has been the only

driving force. More mundane motives, such as ambition or the necessity of eaining a living,
have also played a role. It is also true that mathematics pursued for the sake of applications has
been of benefit to subjects such as number theory; there is a two-way trade. However, it shows
a dangerous ignorance of history and of human nature to promote utility at the expense of spirit.
This book has its origin in a course of lectures which I gave at the Victoria University of
Wellington, New Zealand, in 1975. The demands of my own research have hitherto prevented
me from completing it, although I have continued to collect material. If it succeeds at all in
conveying some idea of the power and beauty of mathematics, the labour of writing it will have
been well worthwhile.
As with a previous book, I have to thank Helge Tverberg, who has read most of the
manuscript and made many useful suggestions.
In this revised edition of my book, the original edition of which appeared in 2002, I have
removed an error in the statement and proof of Proposition 11.12 and filled a gap in the proof of
Proposition 111.12. The statements of the Weil conjectures in Chapter IX and of a result of
Heath-Brown in Chapter X have been modified, following comments by J.-P. Sene. I have
also corrected a few misprints, made many small expository changes and expanded the index.
Although I have made a few changes to the references, I have not attempted a systematic
update. For this I think the Internet has the advantage over a book. The reader is referred to the
American Mathematical Society's MathSciNet (www.ams.org/mathscinet) and to The Number
Theoiy Web maintained by Keith Matthews (www.maths.uq.edu.a~~/-lu-~nn.

Note added (September, 2005) I am grateful to Springer Science for undertaking the
commercial publication of my book. I hope you will be also.


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' or integers. However, there are many other types, such
as complex numbers andp-adic numbers. Somewhat surprisingly, 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 ZI now and 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. Usually a 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, relations and mappings
The label '0' given to this section may be interpreted to stand for 'Optional'. We collect
here some definitions of a logical nature which have become part of the common language of


2

I. The expanding universe of numbers

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.
We will not formally define a set, but will simply say that it is a collection of objects,

which are called its elenzerzts. We write a E A if a is an element of the set A and a

e A if it is

not.

A set may be specified by listing its elements. For example, A = {a,l?,c}is the set whose
elements are a,b,c. A set may also be specified by characterizing its elements. For example,

is the set of all real numbers x such that x2 < 2.
If two sets A,B have precisely the same elements, we say that they are equal and write
A = B. (If A and B are not equal, we write A # B.) For example,

Just as it is convenient to admit 0 as a number, so it is convenient to admit the empty set
0, which has no elements, as a set.
If eveiy element of a set A is also an element of a set B we say that A is a subset of B, or
that A is included in B, or that B contains A, and we write A c B. We say that A is a proper
subset of B, and write A c B, if A L B and A # B.
Thus 0c A for every set A and 0 c A if A # 0. Set inclusion has the following obvious
properties:

(i) A G A ;
(ii) ifA c B and B c A, then A = B;
(iii) if A c B and B c C, then A c C .

AuB

A n B
Figure 1: Uniort and Intenection

0. Sets, relations and mappings

For any sets A&, the set whose elements are the elements of A or B (or both) is called the
union or 'join' of A and B and is denoted by A u B:
Aorx~
B}.
A u B = {x:x~
The set whose elements are the common elements of A and B is called the intersection or
'meet' of A and B and is denoted by A n B:

A n B = {x:xA
~ a n d x ~B } .

If A n B = 0 ,the sets
A and B are said to be disjoint.
. It is easily seen that union and intersection have the following algebraic properties:

Set inclusion could have been defined in terms of either union or intersection, since A
is the same as A u B = B and also the same as A n B = A.

cB

For any sets A&, the set of all elements of B which are not also elements of A is called the
difference of B from A and is denoted by B \A:

B\A

= (x:x~
BandxP

A].

It is easily seen that

An important special case is where all sets under consideration are subsets of a given
universal set X. For any A c X, we have

The set X \ A is said to be the complement of A (in X) and may be denoted by ACfor fixed X.
Evidently
0c=x, xc=0,

AuAC=X, AnAC=O,
(AC)C = A.


I. The expanding universe of numbers

4

By taking C = X in the previous relations for differences, we obtain 'De Morgan's laws':

Since A n B = (ACu Bc)c, set intersection can be defined in terms of unions and
complements. Alternatively, since A u B = (ACnBC)C, set union can be defined in terms of
intersections and complements.
For any sets A,B, the set of all ordered pairs (a,b) with a E A and b
(Cartesian)product of A by B and is denoted by A x B.

E

B is called the

Similarly one can define the product of more than two sets. We mention only one special
case. For any positive integer n, we write An instead of A x ... x A for the set of all (ordered)

n-tuples ( a l ,...,a,) with aj E A ( 1 I j I n). We call aj the j-th coordinate of the n-tuple.
A binary relation on a set A is just a subset R of the product set A x A. For any a,b E A,
we write aRb if (a,b) E R. A binary relation R on a set A is said to be
reflexive if aRa for every a E A;
symmetric if bRu whenever aRb;
transitive if aRc whenever aRb and bRc.
It is said to be an equivalence relation if it is reflexive, symmetric and transitive.
If R is an equivalence relation on a set A and a E A, the equivalence class R, of a is the
set of all x E A such that xRa. Since R is reflexive, a E R,. Since R is symmetric, b E R ,

Rb. Since R is transitive, b E R, implies Rb G R,. It follows that, for all
a,b E A, either R, = Rb or R, n Rb = 0.
A partition % of a set A is a collection of nonempty subsets of A such that each element of
A is an element of exactly one of the subsets in %.
implies a

E

Thus the distinct equivalence classes corresponding to a given equivalence relation on a set
A form a partition of A. It is not difficult to see that, conversely, if % is a partition of A, then an
equivalence relation R is defined on A by taking R to be the set of all (a,b) E A x A for which a
and b are elements of the same subset in the collection %.
Let A and B be nonempty sets. A mapping f of A into B is a subset of A x B with the
property that, for each a E A, there is a unique b E B such that (a,b) e f. We write f(a) = b if
(a,b) E f, and say that b is the image of a under f or that b is the value off at a. We express

thatf is a mapping of A into B by writingf: A + B and we put
f(A) = (f(a):a E A } .


0. Sets, relations and mappings

5

The term function is often used instead of 'mapping', especially when A and B are sets of
real or complex numbers, and 'mapping' itself is often abbreviated to map.
Iff is a mapping of A into B, and if A' is a nonempty subset of A, then the restriction off
to A' is the set of all (a,b)E f with a E A'.
The identity map iA of a nonempty set A into itself is the set of all ordered pairs (a,a) with

a € A.
I f f is a mapping of A into B, and g a mapping of B into C, then the composite mapping
g 0f of A into C is the set of all ordered pairs (a,c),where c = g(b) and b =f(a). Composition
of mappings is associative, i.e. if h is a mapping of C into D, then

The identity map has the obvious propertiesf 0 iA =f and iB 0 f =f.
Let A,B be nonempty sets and$ A + B a mapping of A into B. The mapping f is said to
be 'one-to-one' or injective if, for each b E B, there exists at most one a E A such that

(a,b) E f. The mapping f is said to be 'onto' or surjective if, for each b E B, there exists at
least one a E A such that (a,b) E f. Iff is both injective and surjective, then it is said to be
bijective or a 'one-to-one correspondence'. The nouns injection, surjection and bijection are
also used instead of the corresponding adjectives.
It is not difficult to see that f is injective if and only if there exists a mapping g: B -+A
such that g of = iA, and surjective if and only if there exists a mapping h: B -+ A such that
f o h = iB. Furthermore, iff is bijective, then g and h are unique and equal. Thus, for any

bijective map f: A -+ B, there is a unique inverse map f l: B
fofl

-+A such that f l

=iB.
Iff: A +B and g: B -+ C are both bijective maps, then g of: A

0

f

= iA and

-+ C is also bijective and

1 Natural numbers
The natural numbers are the numbers usually denoted by l,2,3,4,5,
... . However, other
notations are also used, e.g. for the chapters of this book. Although one notation may have
considerable practical advantages over another, it is the properties of the natural numbers which
are basic.


6

I. The expanding universe of numbers

The following system of axioms for the natural numbers was essentially given by
Dedekind (1888), although it is usually attributed to Peano (1889):

The natural numbers are the elenzents of a set N , with a distinguished element 1
(one) arid map S: N + N , such that

(N1) S is injective, i.e. i f m , n E N and m + 12, then S(m) # S ( n ) ;
(N2) 1 e S ( N ) ;
(N3) i f M c N , 1~ M a r z d S ( M ) c _ M , t h e ~ z M = N .
The element S ( n ) of N is called the successor of 12. The axioms are satisfied by
{ l,2,3,...} if we take S(n) to be the element immediately following the element n.
It follows readily from the axioms that 1 is the only element of N which is not in S ( N ) .
For, if M = S ( N ) u { 1 }, then M G N, 1 E M and S ( M ) c M . Hence, by (N3), M = N .
It also follows from the axioms that S(n) # n for every n E N. For let M be the set of all
n E N such that S ( n ) # n. By (N2), 1 E M. If n E M and n' = S(n) then, by ( N l ) , S ( n f )# n'.
Thus S ( M ) c M and hence, by (N3), M = N .
The axioms (N1)-(N3) actually determine N up to 'isomosphism'. We will deduce this
as a corollary of the following general r-ecursion theorenz:

PROPOSITION 1 Given a set A, a n elenzent a l of A and u nzap T: A
exactly one map cp: N + A such that q ( 1 ) = al and
cp(S(n)) = Tcp(n) for every n

G

+ A,

there exists

N.

Proof We show first that there is at most one map with the required properties. Let cpl and cp2

be two such maps, and let M be the set of all rz E N such that

Evidently I

E

M. If

12 E

M , then also S(n) E M, since

Hence, by (N3), M = N . That is, cpl = cp2.
We now show that there exists such a map 9. Let % be the collection of all subsets C of
N x A such that ( 1 , ~E~ C) and such that, if ( n , a ) E C, then also (S(rz),T(a))E C. The
collection % is not empty, since it contains N x A. Moreover, since every set in % contains
( l a l ) ,the intersection D of all sets C E % is not empty. It is easily seen that actually D E %.
By its definition, however, no proper subset of D is in %.


I . Natural numbers

7

Let M be the set of all n E N such that @,a) E D for exactly one a E A and, for any
n E M, define cp(n) to be the unique a E A such that (n,a) E D. If M = N, then cp(1) = al and
cp(S(n))= Tcp(n) for all n E N. Thus we need only show that M = N. As usual, we do this by
showing that 1 E M and that n E M implies S(n) E M.
We have ( l , a l ) E D. Assume ( l , a f )E D for some a l g a l . I f D ' = D \ { ( l , ~ ' ) then
},

(1, a l ) E D '. Moreover, if (n,a) E D' then (S(n),T(a))E D', since (S(n),T(a))E D and
(S(n),T(a))# ( 1 , ~ ' )Hence
.
D' E %. But this is a contradiction, since D' is a proper subset of
D. We conclude that 1 E M.
Suppose now that n E M and let a be the unique element of A such that (n,a) E D. Then
(S(n),T(a))E D, since D E %. Assume that (S(n),aU)
E D for some a" # T ( a ) and put D" =
D \ { ( S ( n ) , a " ) ] .Then (S(n),T(a))E D" and ( l , a l ) E D M .For any (m,b) E D " we have
(S(m),T(b))E D. If (S(nz),T(b))= (S(n),an),then S(m) = S(n) and T ( b ) = a" + T ( a ) , which
implies m = n and b # a . Thus D contains both (n,b) and (n,a), which contradicts n E M.
Hence (S(m),T(b))# (S(n),aU),
and so (S(m),T(b))E D". But then D" E %, which is also a
contradiction, since D" is a proper subset of D. We conclude that S(n) E M. 0

COROLLARY2 If the axioms (N1)-(N3) are also satisfied by a set N' wth element 1'
and map S f : N' + N ', then there exists a bijective map cp of N onto N'such that cp(1) = 1'
and
cp(S(n)) = Sfcp(n)for every n E N .
Proof By taking A = N', al = l'and T = S' in Psoposition 1 , we see that there exists a unique
map cp: N + N'such that cp(1) = I' and
cp(S(n)) = S1cp(n) for every rz

E

N.

By interchanging N and N', we see also that there exists a unique map
~(1=

' )1 and

v(S1(n'))= Syf(n'j for every a'

x =y

E

v:N' -+ N such that

N'.

cp of N into N has the properties ~ ( 1=) 1 and x(S(n))= Sx(n) for
every n E N. But, by Proposition 1 again, is uniquely determined by these properties.
o cp is the identity map on N, and similarly cp 0 y! is the identity map on N'.
Hence
Consequently cp is a bijection.
The composite map

0

x

v

We can also use Proposition 1 to define addition and multiplication of natural numbers.
By Proposition 1, for each m E N there exists a unique map s,: N -+ N such that


I. The expanding universe of numbers

s,(S(n))

s,G) = S(m),
= Ss,(n) for every n E N.

We define the sum of m and n to be
nz

+n

= s,(n)

It is not difficult to deduce from this definition and the axioms (N1)-(N3) the usual rules
for addition: for all a,b,c E N,

(Al) i f a + c = b + c , t h e n a = h ;
(A2) a + b = b + a ;
(A3) ( a + b ) + c = a + ( b + c ) .

(cancellation law)
(commutative law)
(associative law)

By way of example, we prove the cancellation law. Let M be the set of all c E N such that
a + c = b + c only if a = b. Then 1 E M, since s,(l) = ~ ~ ( implies
1 )
S(a) = S(b) and hence
a = b. Suppose c E M. If a + S(c) = b + S(c), i.e. s,(S(c)) = sb(S(c)), then Ss,(c) = Ssb(c) and
hence, by (Nl), s,(c) = sb(c). Since c E M, this implies a = b. Thus also S(c) E M. Hence,

by (N3), M = N.
We now show that
m + n + n f o r a l l m , n ~N.
For a given m E N,let M be the set of all n E N such that m + n # n. Then 1 E M since, by
(N2), s,(l) = S(nz) # 1. If n E M, then s,(n) # n and hence, by (Nl),

Hence, by (N3), M = N.
By Proposition 1 again, for each nz E

N there exists a unique map p,: N + N such that

~ m ( l )= m,
for every n E N.
pn,(S(n)) = s,@,(n))
We define the product of m and n to be
n2.n = pm(n).

From this definition and the axioms (N1)-(N3) we may similarly deduce the usual rules
for multiplication: for all a,b,c E

N,

( M I ) i f a x = b.c, then a = b;

(cancellation law)

(M2) a.b = b.a;

(commutative law)

I . Natural numbers

(M3) (a.b).c = a.(b.c);
(M4) 0.1 = a .

(associative law)
(identity element)

Furthermore, addition and multiplication are connected by

(AM1) a.(b + c) = (a.b) + (a.c).

(distributive law)

As customary, we will often omit the dot when writing products and we will give
multiplication precedence over addition. With these conventions the distributive law becomes
simply
a(b + c) = a b + ac.
We show next how a relation of order may be defined on the set N. For any m,n E M, we
say that m is less than n, and write m < n, if

m + nz'

= n for some m ' N.
~

Evidently m < S(m) for evely nz E N, since S(m) = m + 1. Also, if m < n, then either
S(m) = n orS(m) < n . For suppose nz + m 1 = n . I f m f = 1, then S(m) = n . I f m ' + 1, then
m l = m " + 1 f o r s o m e m " ~Nand

S(m) + nz" = (nz + 1) + nz" = m + (1 + nz") = m + nz' = n.
Again, if n # 1, then 1 < rz, since the set consisting of 1 and all n
contains 1 and contains S(n) if it contains n.

E

N such that 1 < n

It will now be shown that the relation '<' induces a total order. on N, which is compatible
with both adhtion and multiplication: for all a,b,c E N,
(transitive law)
( 0 1 ) ifa < b and b < c, tlzerl a < c;
( 0 2 ) one and only one of the following alternatives holds:
a
(law of trichotomy)

(03) a + c < b + c i f a n d o l t l y i f a < b ;
( 0 4 ) a c < bc fi and only if a < b.
The relation ( 0 1 ) follows directly from the associative law for addition. We now prove
( 0 2 ) . If a < b then, for some a ' E N,

Together with (OI), this shows that at most one of the three alternatives in ( 0 2 ) holds.


10

I. The expanding universe of numbers

For a given a E N, let M be the set of all b E N such that at least one of the three

alternatives in ( 0 2 ) holds. Then 1 E M, since 1 < a if a # 1. Suppose now that b E M. If
a = b, then a < S(b). If a < b, then again a < S(b),by ( 0 1 ) . If b < a, then either S(b) = a or
S(b) < a. Hence also S(b)E M. Consequently, by (N3), M = N. This completes the proof of

(02).
It follows from the associative and commutative laws for addition that, if a < 6 , then
a + c < b + c. On the other hand, by using also the cancellation law we see that if a + c < b + c,
then a < 6.
It follows from the distributive law that, if a < b, then ac < be. Finally, suppose ac < bc.
Then a # b and hence, by (02), either a < b or b < a. Since b < a would imply hc < ac, by
what we have just proved, we must actually have a < b.
The law of trichotomy ( 0 2 )implies that, for given m,n E N, the equation
111

+X

= I2

has a solution x E N only if nz < n.
As customary, we write a I
b to denote either a < b or a = b. Also, it is sometimes
convenient to write b > a instead of a < b, and b 2 a instead of a I b.
A subset M of N is said to have a least element nz' if m' E M and m' I m for every

nz E M. The least element nz' is uniquely determined, if it exists, by ( 0 2 ) . By what we have
already proved, 1 is the least element of N.

PROPOSITION
3 Any nonempty subset M o f N has a least element.
Proof Assume that some nonempty subset M of N does not have a least element. Then

1 E M, since 1 is the least element of N. Let L be the set of all 1 E N such that 1 < nz for every
nz E M. Then L and M are disjoint and 1 E L. If 1 E L, then S(1)I
m for every m E M. Since
M does not have a least element, it follows that S(1)E M. Thus S(1)< n7 for evely m E M, and
so S(1) E L. Hence, by (N3), L = N. Since L n M = 0, this is a contradiction.
The method of proof by inductiou is a direct consequence of the axioms defining N.
Suppose that with each n E N there is associated a proposition P,. To show that P, is true for
is true if P, is true.
every 12 E N, we need only show that P I is true and that
Proposition 3 provides an alternative approach. To show that P, is tiue for every n

E

N,

we need only show that if P,, is false for some nz, then PI is false for some I < m. For then the
set of all n E N for which P, is false has no least element and consequently is empty.
For any n E N, we denote by In the set of all nz
S(n) E I,. It is easily seen that

E

N such that m I n. Thus I I = { 11 and


1. Natural numbers

Also, for any p E Is(,), there exists a bijective map fp of I, onto Is(,)\ {p}. For, if p = S(n) we
can take fp to be the identity map on In, and if p E IrLwe can take fp to be the map defined by

PROPOSITION
4 For any m,n E N, if a map f: I, -+ I, is injective and f(Im) # I,, then
m Proof The result certainly holds when m = 1, since I I = { 1 1. Let M be the set of all m E M
for which the result holds. We need only show that if m E M , then also S(m) E M.
Let$ Is(,) + I, be an injective map such that f(ls(,)) f I, and choose p E I, \ f(Is(,)).
The restriction g off to I, is also injective and g(Im)#I,. Since rn E M , it follows that m < n.
Assume S(nz) = n. Then there exists a bijective map gp of Is(,) \ [p} onto I,. The composite
map h = gp 0 f maps Is(,) into I, and is injective. Since m E M, we must have h(1,) =I,.
But, since h(S(nz))E I,, and h is injective, this is a contradiction. Hence S(m) < n and, since
this holds for every5 S(nz) E M. 0

PROPOSITION
5 For any m,n E N, if a map f: I,

-+I,

is not injective and f(I,) = I,,

then nz > n.
Proof The result holds vacuously when nz = 1, since any map$ I I + I, is injective. Let M
be the set of all m E N for which the result holds. We need only show that if nz E M, then also
S(m) E M.
Let$ Is(,) -+ I, be a map such that f(ls(,)) = I , which is not injective. Then there exist
p,q E Is(,) with p # q and f(p) = f(q). We may choose the notation so that q E I,. If fp is a
bijective map of I, onto Is(,) \ { p } ,then the composite map h =f 0 jb maps I, onto I,. If it is
not injective then m > n , since m E M, and hence also S(m) > n. If h is injective, then it is
bijective and has a bijective inverse h-l: I, -+ I,,. Since h-l(I,) is a proper subset of Is(,), it
follows from Proposition 4 that n < S(nz). Hence S(m) E M . 0
Propositions 4 and 5 immediately imply

COROLLARY6 For any n E N,a nmp f : I, + Ifl is injective if and only if it is surjective.
0

COROLLARY
7 If a map$ I,

+ I, is bijective, then rn = n.


12

I. The expandiq universe of immbers

Proof By Proposition 4, nz < S(n), i.e. nz 5 n. Replacing f by f l, we obtain in the same way
n 5 m. Hence m = n.
A set E is said to befinite if there exists a bijective map f: E + I,, for some n

N. Then
n is uniquely determined, by Corollary 7. We call it the calzlinality of E and denote it by #(E).
It is readily shown that if E is a finite set and F a proper subset of E, then F is also finite
and # ( F ) < #(El. Again, if E and F are disjoint finite sets, then their union E u F is also finite
and #(E u F) = #(E) + #(F). Furthermore, for any finite sets E and F, the product set E x F
is also finite and #(E x F) = #(E) . #(F).
Corollary 6 implies that, for any finite set E, a mapf: E + E is injective if and only if it is
surjective. This is a precise statement of the so-called pigeonholeprinciyle.
E

A set E is said to be countably infinite if there exists a bijective map f: E + N . Any
countably infinite set may be bijectively mapped onto a proper subset F, since N is bijectively

mapped onto a proper subset by the successor map S. Thus a map8 E + E of an infinite set E
may be in~ective,but not surjective. It may also be surjective, but not injective; an example is
the map$ N N defined by f(1) = 1 and, for 12 # 1,f(n) = m if S(m) = n.

2 Integers and rational numbers
The concept of number will now be extended. The natural numbers l,2,3, ... suffice for
counting purposes, but for bank balance purposes we require the larger set ...,- 2,-1,0,1,2, ...
of integers. (From this point of view, - 2 is not so 'unnatural'.) An important reason for
extending the concept of number is the greater freedom it gives us. In the realm of natural
numbers the equation a + x = b has a solution if and only if b > a ; in the extended realm of
integers it will always have a solution.
Rather than intsoduce a new set of axioms for the integers, we will define them in terms of
natural numbers. Intuitively, an integer is the difference nz - n of two natural numbers m,n,
with addltion and multiplication defined by

(m- n) + (p -q) = (nz + p ) - (n + q),
(nz - n) . ( p - q) = (nzp + nq) - (nlq + np).
However, two other natural numbers nz',nl may have the same difference as m,n, and anyway
what does nz - n mean if n~< n? To make things precise, we proceed in the following way.


2 Integers n~zdratio~~a111unzber.s

13

Consider the set W x N of all ordered pairs of natural numbers. For any two such ordered
pairs, (nz,n) and (nz',?~'),
we write

We will show that this is an equivakrm relation. It follows at once from the definition that

(nz,n) (nz,n) (reflexive law) and that (m,n) (nz1,n')implies (ml,n') (nz,n) (symmetric

-

-

-

law). It remains to prove the transitive law:
(nz,rz)

- (m',n')and (nzl,n')- (nzl',lz") imply

( n w ) - (mn,n").

This follows from the commutative, associative and cancellation laws for addition in N. For we
have
nz

+ 12' = nz' + 12, m' + n" = m" + IZ',

and hence

(m + n f )+ n" = (nl'+ 11)

+ 12"

= ( t n ' + t ~ "+) 12 = (nz" + 11') + I ? .

Thus

( n +~ I I " )

+ n' = (nz" + n ) + n',

and so m + n " = m "+ n .
The equivalence class containing (1,l) evidently consists of all pairs (n7,u)with m = n.
We define an integer. to be an equivalence class of ordered pairs of natural numbers and,
as is now customruy, we denote the set of all integers by
Addition of integers is defined comnponentwise:

Z.

(tn,12)+ (p,q) = (nz + y,r~+ q).
To justify this definition we must show that it does not depend on the choice of representatives
within an equivalence class, i.e. that
(n1,11) - (nz',n? and @,q) - @',q') imply (m+p,w + q) - (nz' +p',nl + q?.
However, if
nz

+ 11' = nz' + n, p + q' = p' + q,

then
( n z + p ) + ( n 1 + q ' )= ( m + n ' ) + O )+q')
= ( m 1 + n ) + ( y ' + q )= ( m l + p ' ) + ( r z + q ) .