Number Story (eBook)

Peter Higgins is a Professor of Mathematics at Essex University and inventor of Circular Sudoku. His previous books on mathematics include Mathematics for the Curious, Mathematics for the Imagination, and Nets, Puzzles and Postmen: An Exploration of Mathematical Connections.

Preface 9
The First Numbers 12
How Should We Think About Numbers? 16
The Structure of Numbers 19
Discovering Numbers 28
Counting and Its Consequences 34
Some Number Tricks 42
What Was the Domino? 45
Casting Out Nines 46
Divisibility Tests 50
Magical Arrays 60
Other Magic Number Arrays 68
Some Tricky Numbers 72
Catalan Numbers 76
Fibonacci Numbers 78
Stirling and Bell Numbers 83
Hailstone Numbers 86
The Primes 88
Lucky Numbers 95
Some Useful Numbers 96
Percentages, Ratios, and Odds 96
Scientific Notation 99
Meaning of Means 101
On the Trail of New Numbers 112
Pluses and Minuses 115
Fractions and Rationals 116
Glimpses of Infinity 128
The Hilbert Hotel 131
Cantor’s Comparisons 133
Structure of the Number Line 139
Infinity Plus One 144
Applications of Number: Chance 148
Some Examples 152
Some Collectable Problems on Chance 159
The Complex History of the Imaginary 176
Algebra and its History 179
Solution of the Cubic 185
From Imaginary to Complex 196
The Imaginary World Is Entered 200
The Polar System 206
Gaussian Integers 209
Glimpses of Further Consequences 211
The Number Line under the Microscope 220
Return to Egypt 223
Coin Problems, Sums, and Differences 227
Fibonacci and Fractions 232
Cantor’s Middle Third Set 236
Application of Number: Codes and Public Key Cryptography 240
Examples from History 241
Unbreakable Codes 249
New Codes for a New World of Coding 253
Simultaneous Key Creation 255
Opening the Trapdoor: Public Key Encryption 262
Alice and Bob Vanquish Eve with Modular Arithmetic 266
For Connoisseurs 274
Chapter 1 274
Chapter 3 279
Chapter 4 282
Chapter 5 292
Chapter 6 294
Chapter 7 300
Chapter 8 307
Chapter 9 311
Chapter 10 314
Chapter 11 320
Chapter 12 323
Further Reading 326
Index 330