Table Of ContentThe Mathematical
Traveler
Exploring the
Grand History of Numbers
The Mathematical
Traveler
Exploring the
Grand History of Numbers
CALVIN C. CLAWSON
Springer Science+Business Media, LLC
Library of Congress Cataloging in PubJication Data
Clawson, Calvin C.
The mathematical traveler: exploring the grand history of numbers / Calvin C.
Oawson.
p. cm.
Includes bibJiographical references and index.
1. Numeration. 2. Counting. I. Title.
QAI41.C52 1994 94-2740
513.2-dc20 CIP
ISBN 978-0-306-44645-0 ISBN 978-1-4899-6014-6 (eBook)
DOI 10.1007/978-1-4899-6014-6
© 1994 Calvin C. Clawson
Originally published by Plenum US in 1994.
Softcover reprint of the hardcover 1s t edition 1994
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical, photocopying, microfilming,
recording, or otherwise, without written permission from the PubJisher
To my wife, Susan
Acknow ledgments
Many people helped to make this book a reality. Special thanks go to my
workshop friends who patiently reviewed the manuscript and offered
many excellent suggestions: Marie Edwards, Bruce Taylor, LinQa Shep
hard, Phyllis Lambert, and Brian Herbert. I also want to thank my
professors at the University of Utah in both the Departments of Philoso
phy and Mathematics for their care and dedication in teaching me to love
mathematics.
vii
Contents
INTRODUCTION 1
CHAPTER 1 How Do We Count? 5
CHAPTER 2 Early Counting 19
CHAPTER 3 Counting in Other Species 37
CHAPTER 4 Ancient Numbers 49
CHAPTER 5 Chinese and New World Numbers 77
CHAPTER 6 Problems in Paradise 95
CHAPTER 7 The Negative Numbers 121
CHAPTER 8 Dealing with the Infinite 135
CHAPTER 9 Dedekind's Cut: Irrational Numbers 161
ix
x CONTENTS
CHAPTER 10 Story of 'IT: Transcendental Numbers 181
CHAPTER 11 Expanding the Kingdom: Complex
Numbers 207
CHAPTER 12 Really Big: Transfinite Numbers 223
CHAPTER 13 The Genius Calculators 233
CHAPTER 14 What Does It All Mean? 247
CHAPTER 15 Numbers: Past, Present, and Future 263
End Notes 281
Glossary 289
Bibliography 299
Index 303
Introduction
This book is about numbers. However, it is not a book that addresses only
formal mathematics. While it is true that mathematics is the study of
numbers, the use of numbers is so interwoven with the fabric of human
experience that to talk of all aspects of numbers is to delve into many of
the very mysteries of human nature. While this book contains mathematics,
it also invokes anthropology, biology, psychology, anatomy, history, and
philosophy.
Human beings have been called "the tool-using apes," "the fire
using apes," and "the talking apes." However, it would be just as proper
to call us "the counting apes." Numbers and counting permeate our
culture. Try to imagine, if you will, our society without any numbers or
counting. We are so attuned to numbers that we use them to measure
every aspect of our lives. We mark where we live with street numbers. We
call each other on the telephone with numbers. Our money is based on
numbers, along with our calendars and clocks. When I woke up this
morning, I turned over and looked at my digital clock. It was 6:45. I
remembered today was the 17th, and that reminded me I had to pay a
$110 car payment. While I ate breakfast, I listened to 97.3 on the radio:
the stock market was up 14 points. Thus my day (and everyone else's)
continued. We rely on numbers in order to function in our modern
society.
The economy, our technology, and science all depend on our use of
numbers, and this dependency is reflected in how early we introduce our
children to numbers; one of the first things we teach them is how to count
2 INTRODUCTION
to ten. Yet even with this early introduction, many millions of Americans
shudder when someone mentions mathematics-the study of numbers.
"It's too dry," they complain. "It's too hard-all those symbols." But
some of these same individuals go to Las Vegas and sit up all night to play
a number adding game-blackjack. Or they playa dice number game
craps. Maybe they just stay home and play Monopoly, a game that
requires them to add together the dots on two dice and then count how
many places to move. If they land on someone else's property, they must
count out the rent. If the property is unowned, they count out enough to
buy it. The entire game is adding,! subtracting, and counting with
numbers. People love numbers. Sometimes they claim a disinterest in
formal mathematics, but in actuality, whether it's playing a game,
anticipating a check from the government, or "playing the stock mar
ket," people love numbers.
Given our secret delight in numbers, you will very likely enjoy this
book. We begin our adventure by looking at the natural numbers-those
numbers we first learned to count, namely one, two, three, four, and so
on. By "so on," we mean we can count on forever and never reach the last
number.
The natural numbers were the first numbers discovered by human
beings. The first thing they did with them was count. In Chapter 1 we will
define counting and then investigate that part of our brain responsible for
counting. Next we will consider how old counting and numbers might be
and how they were first used (Chapter 2). In Chapter 3 we will consider
whether other species can count. In Chapter 4 we will look at numbers as
used by the first farmers and then the first written records of numbers
from the Mesopotamians and Egyptians. We will then review the early
contributions to numbers by the Chinese and Native Americans (Chapter
5). Next we will learn of a great mystery about numbers that was first
encountered by the ancient Greeks but not solved for over two thousand
years-a mystery that eventually led to the strange irrational numbers
(Chapter 6). In Chapter 7 we will introduce the negative numbers and our
modern Hindu-Arabic number system, after which we will be ready to
look at the concept of the infinite and how this idea impacted our beliefs
about numbers (Chapter 8). By now we will have followed the evolution
of numbers sufficiently to solve the mystery of the irrational numbers
