The ancient Greeks discovered them, but it wasn't until the nineteenth century that irrational numbers were properly understood and rigorously defined, and even today not all their mysteries have been revealed. In The Irrationals, the first popular and comprehensive book on the subject, Julian Havil tells the story of irrational numbers and the mathematicians who have tackled their challenges, from antiquity to the twenty-first century. Along the way, he explains why irrational numbers are surprisingly difficult to define--and why so many questions still surround them.
That definition seems so simple: they are numbers that cannot be expressed as a ratio of two integers, or that have decimal expansions that are neither infinite nor recurring. But, as The Irrationals shows, these are the real "complex" numbers, and they have an equally complex and intriguing history, from Euclid's famous proof that the square root of 2 is irrational to Roger Apéry's proof of the irrationality of a number called Zeta(3), one of the greatest results of the twentieth century. In between, Havil explains other important results, such as the irrationality of e and pi. He also discusses the distinction between "ordinary" irrationals and transcendentals, as well as the appealing question of whether the decimal expansion of irrationals is "random".
Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.
Author(s): Julian Havil
Publisher: Princeton University Press
Year: 2012
Language: English
Pages: 312
Cover......Page 1
Contents......Page 8
Acknowledgments......Page 10
Introduction......Page 14
CHAPTER ONE: Greek Beginnings......Page 22
CHAPTER TWO: The Route to Germany......Page 65
CHAPTER THREE: Two New Irrationals......Page 105
CHAPTER FOUR: Irrationals, Old and New......Page 122
CHAPTER FIVE: A Very Special Irrational......Page 150
CHAPTER SIX: From the Rational to the Transcendental......Page 167
CHAPTER SEVEN: Transcendentals......Page 195
CHAPTER EIGHT: Continued Fractions Revisited......Page 224
CHAPTER NINE: The Question and Problem of Randomness......Page 238
CHAPTER TEN: One Question, Three Answers......Page 248
CHAPTER ELEVEN: Does Irrationality Matter?......Page 265
APPENDIX A: The Spiral of Theodorus......Page 285
APPENDIX B: Rational Parameterizations of the Circle......Page 291
APPENDIX C: Two Properties of Continued Fractions......Page 294
APPENDIX D: Finding the Tomb of Roger Apéry......Page 299
APPENDIX E: Equivalence Relations......Page 302
APPENDIX F: The Mean Value Theorem......Page 307
D......Page 308
I......Page 309
P......Page 310
Z......Page 311
