Table Of Contentfile:///D|/temp/ifrah/ifrah/[nf]%20Ifrah,%20Georges%20-%20History%20of%20Numbers_SLUC%20[illus](htm).htm
PENGUIN BOOKS
FROM ONE TO ZERO
Born in Morocco and widely traveled, Georges Ifrah has for many years
been a teacher of mathematics in France, where he published many
scholarly articles and where From One to Zero first appeared to wide
acclaim.
Lowell Bair, who translated this work from the origi nal French in
consultation with the author, lives in Woodstock, New York, where he
has worked full time as a translator for many years.
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A Universal History of Numbers
GEORGES IFRAH
Translated by Lowell Bair
©
PENGUIN BOOKS
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PENGUIN BOOKS
Published by the Penguin Group
Viking Penguin Inc., 40 West 23rd Street,
New York, New York 10010, U.S.A.
Penguin Books Ltd, 27 Wrights Lane, London W8 5TZ, England
Penguin Books Australia Ltd, Ringwood, Victoria, Australia
Penguin Books Canada Ltd, 2801 John Street,
Markham, Ontario, Canada L3R 1B4
Penguin Books (N.Z.) Ltd, 182-190 Wairau Road,
Auckland 10, New Zealand
Penguin Books Ltd, Registered Offices: Harmondsworth, Middlesex,
England
Originally published in French under the title Histoire Univexselle des Chiffies by
Editions Seghers 1981
This English-language translation first published by
Viking Penguin Inc. 1985
Published in Penguin Books 1987
Reprinted 1988
English translation copyright © Viking Penguin Inc., 1985
Copyright © Editions Seghers, Paris, 1981
All rights reserved
All illustrations, including calligraphy, are by the author.
LIBRARY OF CONGRESS CATALOGING IN PUBLICATION DATA
Ifrah, Georges.
From one to zero.
Translation of: Histoire universelle des chiffres.
Bibliography: p.
Includes index.
1. Numeration — History. 2. Numerals — History.
I. Title.
QA141.2.I3613 1987 513'.5 86-19023
ISBN 0 14 00.9919 0
Printed in the United States of America by
R. R. Donnelley & Sons Company, Harrisonburg, Virginia
Set in Trump Medieval
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Except in the United States of America,
this book is sold subject to the condition
that it shall not, by way of trade or otherwise,
be lent, re-sold, hired out, or otherwise circulated
without the publisher's prior consent in any form of
binding or cover other than that in which it is
published and without a similar condition
including this condition being imposed
on the subsequent purchaser
For you, my wife,
who were a patient witness to the joys
and anxieties which this hard labor
gave me for several years.
For you, Anna,
to whom this book and its author owe so much.
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Contents
Introduction xi
/. AWARENESS OF NUMBERS 1
1 The Origin and Discovery of Numbers 3
Are animals able to count? 3
Natural faculties of number perception 5
Can one define a quantity without knowing how to count? 8
Rudimentary "accounting" techniques 14
Expressing number orally and by gestures 15
Counting: a human faculty 21
Symbolization of number 25
Ten fingers for counting 26
2 The Principle of the Base 31
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Decimal systems 31
Traces of the base 5 36
Twenty fingers and toes for counting 38
The sexagesimal system 46
The origin of the sexagesimal system 49
//. CONCRETE COUNTING 53
3 The First Calculating Machine: The Hand 55
A curious way of bargaining 59
Counting on the bones and joints of the fingers 61
Morra, a finger game 67
Elements of finger computation 70
An elaborate system of finger counting 72
History recounted on the fingers 76
Contents
4 Notching 81
5 From Pebbles to Calculation 88
6 Numbers on Strings 91
The archives of the Incas 91
Use of knotted strings in other parts of the world 94
7 Number, Value, Money 98
8 The Abacus 105
Greek and Roman abacuses 105
The first pocket calculator 111
Abacists versus algorists: the Renaissance 113
Rods on the Chinese counting board 117
The bead abacus 121
///. THE INVENTION OF NUMERALS 129
9 Roman Numerals: A Vestige of Primitive Origins? 131
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The numerals of Roman civilization 131
Questionable ancestors 134
The origin of Roman numerals . 136
Anthropological and historical evidence 143
10 Was Writing Invented by Accountants? 151
Five thousand years ago, the Sumerians and Elamites
learned to write 151
Precursors of written accounting 154
From clay envelopes to account tablets 155
The mnemonic function of Sumerian writing 165
11 Sumerian Numerals 170
The development of cuneiform numerals 173
Difficulties of the cuneiform numeration system 177
Continuance of the Sumerian system 179
How modern scholars deciphered Sumerian numerals 180
12 Deciphering a Forgotten Notation 182
13 Egyptian Numerals 200
The origin of hieroglyphs: Sumerian or Egyptian? 202
From drawings to numerals 203
Fractions and the dismembered god 207
14 Kindred Numeration Systems 213
Cretan numerals 213
The Hittite hieroglyphic system 218
The Aztec system 219
Greek acrophonic systems 225
Numerals of the Kingdom of Saba 229
A significant improvement 233
15 The Egyptian Scribes' Rapid Notation 240
Hieratic script 241
A remarkable simplification 244
Numerals in the time of the Hebrew kings 246
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IV. NUMERALS AND LETTERS 249
16 Numerical Use of the Hebrew Alphabet 251
17 Greek Alphabetic Numerals 261
18 The Phoenician Origin of Numeral Letters: A Legend 275
19 Syriac Numerals 279
20 Arabic Numeral Letters 282
21 Numerals, Letters, Magic, Mysticism, and Divination 291
The art and composition of chronograms 291
Interpretations and speculations of Gnostics, Cabalists,
magicians, and soothsayers 293
V. HYBRID NUMERATION SYSTEMS 311
22 The Drawbacks of the Additive Principle 313
Roman notations for large numbers 313
Appearance and extension of the multiplicative principle 320
23 The Common Mesopotamian Numeration System 325
24 Semitic Numerical Traditions 332
25 The Traditional Chinese Numeration System 341
The modern system 341
Different forms of Chinese numerals 343
x Contents
Origin of the Chinese system 347
Diffusion of the Chinese system in the Far East 353
Systems similar to the common Chinese system 356
A natural extension 361
Chinese notations of large numbers in scientific works 365
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VI. THE ULTIMATE STAGE OF NUMERICAL NOTATION 369
26 The First Place-Value Numeration System 371
The Babylonian place-value system 373
Difficulties of this system 375
The oldest zero in history 376
Survival of the Babylonian system 382
27 The Chinese Place-Value System 387
28 Amazing Achievements of a Vanished Civilization 397
Greatness and decline of Mayan civilization 397
Sources of our knowledge 398
Six centuries of intellectual and artistic achievement 400
Writing, arithmetic, and astronomy 403
The Mayan calendar 405
Chronology and numeration 412 Discovery of
the place-value principle and the zero in the
New World 420
Conclusion 426
29 The Origin of Hindu-Arabic Numerals 428
The keystone of our modern numeration system 428
A significant anecdote 434
The cradle of our modern system 437
The symbolic number words of Indian astronomers 445
Origin of the Indian place-value system 452
Worldwide propagation of the Indian system 459
Adoption of the Indian system by the Eastern Arabs 462
Numerals used by the Western Arabs 473
Hindu-Arabic numerals in Europe 475
Chronology 491
Bibliography of Works Quoted or Mentioned in This Book 499
Acknowledgments 503
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Introduction
T
his book began with a child's question. I was teaching mathematics at the time,
and like all good teachers I tried never to let any question go unanswered, no matter
how strange or naive it might seem. Intelligence is often nurtured by curiosity.
One day I was asked a question so simple that for a moment it left me speechless:
"Where do figures come from? When did people learn how to count? How did
numbers start?"
What is the origin of numbers? They are so familiar to us in our everyday lives that
they give the deceptive impression of being innate, and in adulthood we are inclined
to assume that they have always been part of our thoughts—like language, another
tool whose use must be learned.
When I was asked where figures came from, I could say only that they were
invented a long time ago, an answer that did not even disguise my ignorance.
Groping my way, I tried to go back through the centuries to a time before our Hindu-
Arabic and Roman numerals existed. Were there other numerals before them? If so,
what were they like? And would a search for the origin of figures reveal some
vestige of the brilliant in vention made by the first human being who conceived the
idea of count ing?
I am afraid my answers that day were sketchy, incomplete, and perhaps inaccurate. I
had an excuse. In the arithmetic books that were the tools of my trade, the question
was not even raised. In history books it was only touched on lightly, as if there were
a strange conspiracy to conceal one of humankind's most fantastic discoveries, and
perhaps the most fruitful, since it has enabled us to try to measure the world, un
derstand it a little better, and put some of its secrets to use. There were, as I later
learned, a few specialized works describing what was known on the subject; I will
mention them often in this book, and I owe a great deal to them. But they had been
written almost exclusively for specialists and were far from being complete, despite
the erudition shown in them.
My pupil's question stayed in my mind for years, to the point where I regretfully
gave up teaching to devote myself entirely to a quest that
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