We might take numbers and counting for granted, but we shouldn't. Our number literacy rests upon centuries of human effort, punctuated here and there by strokes of genius. In his successor and companion volume to Gnomon: From Pharaohs to Fractals, Midhat Gazalé takes us on a journey from the ancient worlds of the Egyptians, the Mesopotamians, the Mayas, the Greeks, the Hindus, up to the Arab invasion of Europe and the Renaissance. Our guide introduces us to some of the most fascinating and ingenious characters in mathematical history, from Ahmes the Egyptian scribe (whose efforts helped preserve some of the mathematical secrets of the architects of the pyramids) through the modern era of Georg Cantor (the great nineteenth-century inventor of transfinite numbers). As he deftly blends together history, mathematics, and even some computer science in his characteristically compelling style, we discover the fundamental notions underlying the acquisition and recording of "number," and what "number" truly means.
Gazalé tackles questions that will stimulate math enthusiasts in a highly accessible and inviting manner. What is a natural number? Are the decimal and binary systems the only legitimate ones? Did the Pythagorean theorem and the discovery of the unspeakable irrationals cost the unfortunate mathematician Hippasus his life? What was the Ladder of Theodorus of Cyrene and how did the ancient Greeks calculate square roots with such extraordinary proficiency? An original generalization of Euler's theorem is offered that explains the pattern of rational number representations. Later on, the field of Continued Fractions paves the way for another original contribution by Gazalé, that of cleavages, which sheds light on the mysterious nature of irrational numbers as it beautifully illustrates Dedekind's famous Schnitt. In the end the author introduces us to the Hilbert Hotel with its infinite number of rooms, guests, and an infinite number of people waiting to check in, where he sets the debate between Aristotle and Cantor about the true nature of infinity.
This abundantly illustrated book, remarkable for its coherency and simplicity, will fascinate all those who have an interest in the world of numbers. Number will be indispensable for all those who enjoy mathematical recreations and puzzles, and for those who delight in numeracy.
Preface xi
INTRODUCTION 3
CHAPTER 1 The Genesis of Number Systems 9
Foundations 9
Matching 10
Naming 11
Counting 12
Grouping 15
Archaic Number Systems 16
The Egyptians 17
The Mesopotamians 28
The Greeks 34
The Mayas 38
Two Current Number Systems 41
The Hindus 42
The Arabs 43
The Decimal Number System 45
Fractional Numbers 50
Uttering versus writing 52
Units 54
The Binary Number System 55
CHAPTER 2 Positional Number Systems 59
The Division Algorithm 59
Codes 61
Mixed-Base Positional Systems 64
Finding the Digits of an Integer 69
Addition 72
Uniform-Base Multiplication 75
Mixed-Base Multiplication 77
Construction 1: A Parallel Adder 78
Construction 2: A Digital-to-Analog Converter 80
Construction 3: A Reversible Binary-to-Analog Converter 80
Positional Representation of Fractional Numbers 82
Going to Infinity 89
How Precise Is a Mantissa? 93
Finding the Digits of a Fractional Number 96
Finding the Digits of a Real Number 98
Periodic Bases 99
A Triadic (Ternary) Yardstick 102
Marginalia 103
Unit Fractions Revisited 103
Appendix 2.1 105
Appendix 2.2 106
CHAPTER 3 Divisibility and Number Systems 107
The Fundamental Theorem of Arithmetic 107
Congruences 109
Pascal's Divisibility Test 111
Euler's Function and Theorem 114
Eulers Theorem 116
Exponents 118
Primitive Roots 120
A Generalization of Eulers Theorem 123
The Residue Sequence 124
Indices 126
Conjugates and Conformable Multiples 128
Positional Representation of Rational Numbers 131
Mixed Bases 134
Bases 2 and 10 137
Cyclic Numbers 140
Strings of Ones and Zeros 144
Marginalia 146
Mersenne Primes 146
On Dirichlets Distribution Principle 149
Appendix 150
Carmichael's Variation on Eulers Theorem 150
CHAPTER 4 Real Numbers 152
Rational Numbers 154
The Integral Domain 154
The Rational Numbers Field 155
Marginalia: On the Axiomatic Method 58
Commensurability 160
Irrational Numbers 162
Pythagoras's Theorem 162
Pythagorean Triples 164
The Plimpton 322 Tablet 167
The Ladder of Theodorus of Cyrene and Diophantine Equations 169
A Variation on the Ladder of Theodorus 171
Fermat's Last Theorem 172
The Irrationality of /2 173
A (Theoretically) Physical Impossibility 175
Dedekind 176
Eudoxus 178
Marginalia 182
Three Ancient Problems 182
Appendix 183
Proof of the Irrationality of e 183
CHAPTER 5 Continued Fractions 185
Euclid's Algorithm 185
Continued Fractions 187
Regular Continued Fractions 189
Convergents 190
Terminating Regular Continued Fractions191
Periodic Regular Continued Fractions 192
Spectra of Surds 194
Nonperiodic, Nonterminating Regular Continued Fractions 196
Two Celebrated Irregular Continued Fractions 197
Appendix 197
CHAPTER 6 Cleavages 199
The Number Lattice 200
Prime Nodes 202
Cleavages 203
Coherencc 211
A Definition of Real Numbers 215
Some Properties of Fractions 216
Contiguous Fractions 219
The Mediant 222
Affine Transformations 224
The Stern-Brocot Tree 226
Pencils and Ladders 229
Cleavages and Continued Fractions 233
Klein's Construction 238
The Greatest Common Divisor Revisited 239
Marginalia 242
Cleavages and Positional Number Systems242
Cleavages and Automata 246
Cleaving Crystals 248
Cleavages and Replicative Functions 249
Gaussian Primes 250
Appendix 6.1 252
Proof of Test (6.7) 252
Appendix 6.2 254
The Increment Sequence 254
CHAPTER 7 Infinity 257
Convergence 257
Paradoxes of Infinite Series 261
Further Paradoxes of Infinity 265
You Are Always Welcome at the Hilbert Hotel 268
Zeno's Paradoxes 269
Horror Infiniti? 270
Potential versus Actual Infinity 274
Cantor 275
The Power of the Continuum 278
Geometrical Metaphors 279
Transfinite Cardinal Numbers 281
Cantor Dust 281
Beyond Aleph 1 282
Postscript: The Balance Is Improbable but the Night Sky
Is Black 285
Subject Index 287
Name Index 295