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Springer Undergraduate Mathematics Series

Advisory Board

M. A. J. Chaplain, University of St. Andrews A. MacIntyre, Queen Mary University of London S. Scott, King’s College London N. Snashall, University of Leicester E. Süli, University of Oxford M. R. Tehranchi, University of Cambridge J. F. Toland, University of Cambridge

More information about this series at http://www.springer.com/series/3423

Jürg Kramer • Anna-Maria von Pippich

From Natural Numbers to Quaternions

123

Anna-Maria von Pippich Department of Mathematics Technische Universität Darmstadt Germany

Translation from the German language edition: Von den natürlichen Zahlen zu den Quaternionen by Jürg Kramer and Anna-Maria von Pippich, © Springer Spektrum 2013. All Rights Reserved.

ISSN 1615-2085 ISSN 2197-4144 (electronic) Springer Undergraduate Mathematics Series ISBN 978-3-319-69427-6 ISBN 978-3-319-69429-0 (eBook) https://doi.org/10.1007/978-3-319-69429-0

Library of Congress Control Number: 2017958024

Mathematics Subject Classification (2010): 08–01, 11–01, 12–01, 20–01

© Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface to the English Edition

This book on the construction of number systems first appeared in 2013 in a German edition with the same title. It can be seen from the following preface to that edition that the goal of this book is to present a basic and compre- hensive construction of number systems, beginning with the natural num- bers and ending with Hamilton’s quaternions, while providing relevant al- gebraic knowledge along the way. As a supplement to the German edition, an appendix has been added to each chapter in this English edition, which in contrast to the rigorous style of the rest of the book, presents in the more casual form of a survey some related aspects of the material of the chapter, including some recent developments.

We would like to offer our most heartfelt thanks to the translator, David Kramer, for his competent work, which has contributed significantly to this English version and in many places led to a more felicitous presentation of the material.

We hope that this book will help students and teachers of mathematics as well as all those with an interest in the subject to fill in any gaps in their mathematical education related to the construction of number systems and that the appendices will inspire some readers to pursue further mathemati- cal studies.

Berlin, September 2017 Jürg Kramer Anna-Maria von Pippich

Preface to the German Edition

The main topic of this book is an elementary introduction to the construc- tion of the number systems encountered by mathematics students in their first semesters of study. Beginning with the natural numbers, we succes- sively construct, along with the requisite algebraic machinery, all the num- ber fields containing the natural numbers, including the real numbers, com- plex numbers, and Hamiltonian quaternions. Our experience has shown us that time is frequently lacking in introductory mathematics courses for a well-founded construction of number systems; this book represents a con- tribution toward filling that gap.

The construction of number systems also represents an important compo- nent in the professional education of mathematics teachers. For this reason, this book offers a self-contained and compact construction of the number systems that are of relevance to different grade levels from a mathematical perspective with a view toward aspects of pedagogical content knowledge.

This book arose from a course in elementary abstract algebra and number theory given a number of times at the Humboldt University of Berlin. Parts of the first-named author’s book Zahlen für Einsteiger: Elemente der Algebra und Zahlentheorie (Vieweg Verlag, Wiesbaden, 2008) have been revised and expanded for inclusion in this newly conceived book on the construction of number systems. Numerous exercises with extensive solutions facilitate the reader’s engagement with the subject.

The completion of this book would not have been possible without the contributions of many individuals. Here we wish to thank first of all Christa Dobers and Matthias Fischmann for typing the first parts of the manuscript. In addition, we wish to thank all the students whose written course notes contributed to the text. We also wish to thank our colleagues, in particu- lar Andreas Filler and Wolfgang Schulz, for their numerous suggestions for improving early versions of the manuscript. A special word of thanks goes to Olaf Teschke for his work on creating the exercises, and we also thank Barbara Jung and André Henning for their work on writing up solutions to the exercises. Finally, we offer hearty thanks to Christoph Eyrich for his expert support in designing the layout of the book and to Ulrike Schmickler- Hirzebruch for her encouragement and support on behalf of the publisher, Springer Spektrum.

Berlin, February 2013 Jürg Kramer Anna-Maria von Pippich

Table of Contents

Introduction 1

I The Natural Numbers 9 1. The Peano Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. Divisibility and Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3. The Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . 22 4. Greatest Common Divisor, Least Common Multiple . . . . . . . . . . . . . 25 5. Division with Remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A. Prime Numbers: Facts and Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . 32

II The Integers 45 1. Semigroups and Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. Groups and Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3. Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4. Cosets and Normal Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5. Quotient Groups and the Homomorphism Theorem . . . . . . . . . . . . . 63 6. Construction of Groups from Regular Semigroups . . . . . . . . . . . . . . . 68 7. The Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 B. RSA Encryption: An Application of Number Theory . . . . . . . . . . . . . 77

III The Rational Numbers 93 1. The Integers and Divisibility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2. Rings and Subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3. Ring Homomorphisms, Ideals, and Quotient Rings . . . . . . . . . . . . . . 102 4. Fields and Skew Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5. Construction of Fields from Integral Domains . . . . . . . . . . . . . . . . . . . 112 6. The Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7. Unique Factorization Domains, Principal Ideal Domains, and

Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 C. Rational Solutions of Equations: A First Glimpse . . . . . . . . . . . . . . . . 129

IV The Real Numbers 141 1. Decimal Representation of Rational Numbers . . . . . . . . . . . . . . . . . . . 141 2. Construction of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3. The Decimal Expansion of a Real Number . . . . . . . . . . . . . . . . . . . . . . 155

x Table of Contents

4. Equivalent Characterizations of Completeness . . . . . . . . . . . . . . . . . . 159 5. The Real Numbers and the Real Number Line . . . . . . . . . . . . . . . . . . . 164 6. The Axiomatic Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 D. The p-adic Numbers: Another Completion ofQ . . . . . . . . . . . . . . . . . 171

V The Complex Numbers 183 1. The Complex Numbers as a Real Vector Space . . . . . . . . . . . . . . . . . . . 183 2. Complex Numbers of Modulus 1 and the Special Orthogonal

Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 3. The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4. Algebraic and Transcendental Numbers . . . . . . . . . . . . . . . . . . . . . . . . 193 5. The Transcendence of e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 E. Zeros of Polynomials: The Search for Solution Formulas . . . . . . . . . . 204

VI Hamilton’s Quaternions 219 1. Hamilton’s Quaternions as a Real Vector Space . . . . . . . . . . . . . . . . . . 219 2. Quaternions of Modulus 1 and the Special Unitary Group . . . . . . . . 223 3. Quaternions of Modulus 1 and the Special Orthogonal Group . . . . 227 F. Extensions of Number Systems: What Comes after the

Quaternions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

The Development of the Integers and Algebra

One of mankind’s earliest intellectual occupations was counting. The devel- opment of the concepts of numbers and the representation of numbers has therefore assumed a place of importance in the history of every civilization. The enormous effectiveness of our decimal system of numerical represen- tation is the culmination of centuries—indeed millennia—of earlier efforts that together represent a powerful cultural attainment.

The idea of counting objects, that is, of bringing a set of equivalent objects into a one-to-one correspondence with a fixed set of numbers, represents a significant intellectual process of abstraction.

In more advanced cultures, systems of symbolic notation for these num- bers—some more effective than others—were developed. We mention par- ticularly the cuneiform writing of the Babylonians, Egyptian hieroglyphics, Roman numerals, and the system of numerals developed in India. It was only in the thirteenth and fourteenth centuries that the Indian positional decimal system finally made its way via the Islamic world to Western Eu- rope, which to this day uses “Arabic” numerals.

The development of number systems goes relatively closely hand in hand with the development of methods of calculation. In this regard, the Babylo- nian and Indian number systems, for example, were far superior to those of the Egyptians and Romans. Nevertheless, until late in the fifteenth century, in both the ancient civilizations and Western Europe, numerical calculation was the province of a small group of specialists known as arithmeticians. It was not until the publication in the fifteenth century of Adam Ries’s books on calculation, which were based on the book Liber Abaci of Leonardo of Pisa, known as Fibonacci, that the usual methods of calculation that we use today became accessible to the “common people.” The diffusion of calcula- tional techniques is linked to a systemization of arithmetic in the academic world, which then led to the development of algebra. At first, algebra was viewed primarily as a practical tool, but it gradually took on a life of its own and eventually developed into the independent discipline that we know to- day. Algebra will therefore play a significant role in every rigorous scientifi- cally based construction of number systems.

2 Introduction

A First Look at Number Systems

We all recall from our schooldays how we first learned about the numbers 1, 2, 3, . . . , then the square roots of such numbers, for example

√ 2, and some-

what later became acquainted with the number π, associated with the cir- cumference of a circle, and perhaps Euler’s constant e. On our first encounter with these numbers, we had no idea that a powerful intellectual construct had to be developed before a number system could be created that could contain all these numbers and make possible a “sensible” way of calculating with them, namely the system of real numbers. The creation of this number system represents an outstanding achievement of the human intellect, and a fundamental objective of this book is to acquaint students with the con- struction of the real numbers so that they may become familiar with the fine structure of these objects.

It is astounding that the set of real numbers, which we denote by R, can be developed essentially from the single number 1 (one). Let us sketch briefly how this is done, for a thoroughgoing working out of this process is the main purpose of this book. We begin by identifying the number 1 with an object, and we then bring along another object of the same kind, so that we now have two objects and thereby have acquired the number 2. We may formalize this process by writing 2 = 1 + 1. Continuing in this manner, we obtain in sequence the numbers

3 = 2 + 1 = 1 + 1 + 1, 4 = 3 + 1 = 1 + 1 + 1 + 1, · · ·

that is, the set of natural numbers N except for the number 0 (zero), which we shall obtain momentarily, and append to the set of natural numbers. One might say that the number 1 generates additively every natural number. That is, the number 1 is, from an additive point of view, the atom from which every natural number is built.

We may picture the natural numbers 1, 2, 3, . . . as sitting equally spaced like pearls on a necklace beginning at the left with 1 and continuing sequen- tially off to the right. We might also represent these numbers geometrically, to which end we choose a unit length and mark it off on a horizontal line L by beginning at a point P and moving to the right. We denote by the symbol 1 the point on the line thereby constructed. Continuing, we obtain a second point, which we denote by 2, and so on:

L

P

1 For no reason other than symmetry we might wish to carry out a similar process by moving to the left. Of course, the new points that we thereby

3

obtain must be given new names. We denote the mirror image of 1 in the point P by −1, and so forth, obtaining

L

P

−3

We denote the reflection point P by 0. What we have obtained here in a very graphic way is the process of extending the system of natural numbersN to the system of integers Z. This can be interpreted algebraically by saying that…

Advisory Board

M. A. J. Chaplain, University of St. Andrews A. MacIntyre, Queen Mary University of London S. Scott, King’s College London N. Snashall, University of Leicester E. Süli, University of Oxford M. R. Tehranchi, University of Cambridge J. F. Toland, University of Cambridge

More information about this series at http://www.springer.com/series/3423

Jürg Kramer • Anna-Maria von Pippich

From Natural Numbers to Quaternions

123

Anna-Maria von Pippich Department of Mathematics Technische Universität Darmstadt Germany

Translation from the German language edition: Von den natürlichen Zahlen zu den Quaternionen by Jürg Kramer and Anna-Maria von Pippich, © Springer Spektrum 2013. All Rights Reserved.

ISSN 1615-2085 ISSN 2197-4144 (electronic) Springer Undergraduate Mathematics Series ISBN 978-3-319-69427-6 ISBN 978-3-319-69429-0 (eBook) https://doi.org/10.1007/978-3-319-69429-0

Library of Congress Control Number: 2017958024

Mathematics Subject Classification (2010): 08–01, 11–01, 12–01, 20–01

© Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface to the English Edition

This book on the construction of number systems first appeared in 2013 in a German edition with the same title. It can be seen from the following preface to that edition that the goal of this book is to present a basic and compre- hensive construction of number systems, beginning with the natural num- bers and ending with Hamilton’s quaternions, while providing relevant al- gebraic knowledge along the way. As a supplement to the German edition, an appendix has been added to each chapter in this English edition, which in contrast to the rigorous style of the rest of the book, presents in the more casual form of a survey some related aspects of the material of the chapter, including some recent developments.

We would like to offer our most heartfelt thanks to the translator, David Kramer, for his competent work, which has contributed significantly to this English version and in many places led to a more felicitous presentation of the material.

We hope that this book will help students and teachers of mathematics as well as all those with an interest in the subject to fill in any gaps in their mathematical education related to the construction of number systems and that the appendices will inspire some readers to pursue further mathemati- cal studies.

Berlin, September 2017 Jürg Kramer Anna-Maria von Pippich

Preface to the German Edition

The main topic of this book is an elementary introduction to the construc- tion of the number systems encountered by mathematics students in their first semesters of study. Beginning with the natural numbers, we succes- sively construct, along with the requisite algebraic machinery, all the num- ber fields containing the natural numbers, including the real numbers, com- plex numbers, and Hamiltonian quaternions. Our experience has shown us that time is frequently lacking in introductory mathematics courses for a well-founded construction of number systems; this book represents a con- tribution toward filling that gap.

The construction of number systems also represents an important compo- nent in the professional education of mathematics teachers. For this reason, this book offers a self-contained and compact construction of the number systems that are of relevance to different grade levels from a mathematical perspective with a view toward aspects of pedagogical content knowledge.

This book arose from a course in elementary abstract algebra and number theory given a number of times at the Humboldt University of Berlin. Parts of the first-named author’s book Zahlen für Einsteiger: Elemente der Algebra und Zahlentheorie (Vieweg Verlag, Wiesbaden, 2008) have been revised and expanded for inclusion in this newly conceived book on the construction of number systems. Numerous exercises with extensive solutions facilitate the reader’s engagement with the subject.

The completion of this book would not have been possible without the contributions of many individuals. Here we wish to thank first of all Christa Dobers and Matthias Fischmann for typing the first parts of the manuscript. In addition, we wish to thank all the students whose written course notes contributed to the text. We also wish to thank our colleagues, in particu- lar Andreas Filler and Wolfgang Schulz, for their numerous suggestions for improving early versions of the manuscript. A special word of thanks goes to Olaf Teschke for his work on creating the exercises, and we also thank Barbara Jung and André Henning for their work on writing up solutions to the exercises. Finally, we offer hearty thanks to Christoph Eyrich for his expert support in designing the layout of the book and to Ulrike Schmickler- Hirzebruch for her encouragement and support on behalf of the publisher, Springer Spektrum.

Berlin, February 2013 Jürg Kramer Anna-Maria von Pippich

Table of Contents

Introduction 1

I The Natural Numbers 9 1. The Peano Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. Divisibility and Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3. The Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . . . . . . . . . 22 4. Greatest Common Divisor, Least Common Multiple . . . . . . . . . . . . . 25 5. Division with Remainder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 A. Prime Numbers: Facts and Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . 32

II The Integers 45 1. Semigroups and Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. Groups and Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3. Group Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4. Cosets and Normal Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5. Quotient Groups and the Homomorphism Theorem . . . . . . . . . . . . . 63 6. Construction of Groups from Regular Semigroups . . . . . . . . . . . . . . . 68 7. The Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 B. RSA Encryption: An Application of Number Theory . . . . . . . . . . . . . 77

III The Rational Numbers 93 1. The Integers and Divisibility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2. Rings and Subrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 3. Ring Homomorphisms, Ideals, and Quotient Rings . . . . . . . . . . . . . . 102 4. Fields and Skew Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5. Construction of Fields from Integral Domains . . . . . . . . . . . . . . . . . . . 112 6. The Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7. Unique Factorization Domains, Principal Ideal Domains, and

Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 C. Rational Solutions of Equations: A First Glimpse . . . . . . . . . . . . . . . . 129

IV The Real Numbers 141 1. Decimal Representation of Rational Numbers . . . . . . . . . . . . . . . . . . . 141 2. Construction of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3. The Decimal Expansion of a Real Number . . . . . . . . . . . . . . . . . . . . . . 155

x Table of Contents

4. Equivalent Characterizations of Completeness . . . . . . . . . . . . . . . . . . 159 5. The Real Numbers and the Real Number Line . . . . . . . . . . . . . . . . . . . 164 6. The Axiomatic Point of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 D. The p-adic Numbers: Another Completion ofQ . . . . . . . . . . . . . . . . . 171

V The Complex Numbers 183 1. The Complex Numbers as a Real Vector Space . . . . . . . . . . . . . . . . . . . 183 2. Complex Numbers of Modulus 1 and the Special Orthogonal

Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 3. The Fundamental Theorem of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4. Algebraic and Transcendental Numbers . . . . . . . . . . . . . . . . . . . . . . . . 193 5. The Transcendence of e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 E. Zeros of Polynomials: The Search for Solution Formulas . . . . . . . . . . 204

VI Hamilton’s Quaternions 219 1. Hamilton’s Quaternions as a Real Vector Space . . . . . . . . . . . . . . . . . . 219 2. Quaternions of Modulus 1 and the Special Unitary Group . . . . . . . . 223 3. Quaternions of Modulus 1 and the Special Orthogonal Group . . . . 227 F. Extensions of Number Systems: What Comes after the

Quaternions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

The Development of the Integers and Algebra

One of mankind’s earliest intellectual occupations was counting. The devel- opment of the concepts of numbers and the representation of numbers has therefore assumed a place of importance in the history of every civilization. The enormous effectiveness of our decimal system of numerical represen- tation is the culmination of centuries—indeed millennia—of earlier efforts that together represent a powerful cultural attainment.

The idea of counting objects, that is, of bringing a set of equivalent objects into a one-to-one correspondence with a fixed set of numbers, represents a significant intellectual process of abstraction.

In more advanced cultures, systems of symbolic notation for these num- bers—some more effective than others—were developed. We mention par- ticularly the cuneiform writing of the Babylonians, Egyptian hieroglyphics, Roman numerals, and the system of numerals developed in India. It was only in the thirteenth and fourteenth centuries that the Indian positional decimal system finally made its way via the Islamic world to Western Eu- rope, which to this day uses “Arabic” numerals.

The development of number systems goes relatively closely hand in hand with the development of methods of calculation. In this regard, the Babylo- nian and Indian number systems, for example, were far superior to those of the Egyptians and Romans. Nevertheless, until late in the fifteenth century, in both the ancient civilizations and Western Europe, numerical calculation was the province of a small group of specialists known as arithmeticians. It was not until the publication in the fifteenth century of Adam Ries’s books on calculation, which were based on the book Liber Abaci of Leonardo of Pisa, known as Fibonacci, that the usual methods of calculation that we use today became accessible to the “common people.” The diffusion of calcula- tional techniques is linked to a systemization of arithmetic in the academic world, which then led to the development of algebra. At first, algebra was viewed primarily as a practical tool, but it gradually took on a life of its own and eventually developed into the independent discipline that we know to- day. Algebra will therefore play a significant role in every rigorous scientifi- cally based construction of number systems.

2 Introduction

A First Look at Number Systems

We all recall from our schooldays how we first learned about the numbers 1, 2, 3, . . . , then the square roots of such numbers, for example

√ 2, and some-

what later became acquainted with the number π, associated with the cir- cumference of a circle, and perhaps Euler’s constant e. On our first encounter with these numbers, we had no idea that a powerful intellectual construct had to be developed before a number system could be created that could contain all these numbers and make possible a “sensible” way of calculating with them, namely the system of real numbers. The creation of this number system represents an outstanding achievement of the human intellect, and a fundamental objective of this book is to acquaint students with the con- struction of the real numbers so that they may become familiar with the fine structure of these objects.

It is astounding that the set of real numbers, which we denote by R, can be developed essentially from the single number 1 (one). Let us sketch briefly how this is done, for a thoroughgoing working out of this process is the main purpose of this book. We begin by identifying the number 1 with an object, and we then bring along another object of the same kind, so that we now have two objects and thereby have acquired the number 2. We may formalize this process by writing 2 = 1 + 1. Continuing in this manner, we obtain in sequence the numbers

3 = 2 + 1 = 1 + 1 + 1, 4 = 3 + 1 = 1 + 1 + 1 + 1, · · ·

that is, the set of natural numbers N except for the number 0 (zero), which we shall obtain momentarily, and append to the set of natural numbers. One might say that the number 1 generates additively every natural number. That is, the number 1 is, from an additive point of view, the atom from which every natural number is built.

We may picture the natural numbers 1, 2, 3, . . . as sitting equally spaced like pearls on a necklace beginning at the left with 1 and continuing sequen- tially off to the right. We might also represent these numbers geometrically, to which end we choose a unit length and mark it off on a horizontal line L by beginning at a point P and moving to the right. We denote by the symbol 1 the point on the line thereby constructed. Continuing, we obtain a second point, which we denote by 2, and so on:

L

P

1 For no reason other than symmetry we might wish to carry out a similar process by moving to the left. Of course, the new points that we thereby

3

obtain must be given new names. We denote the mirror image of 1 in the point P by −1, and so forth, obtaining

L

P

−3

We denote the reflection point P by 0. What we have obtained here in a very graphic way is the process of extending the system of natural numbersN to the system of integers Z. This can be interpreted algebraically by saying that…