The irrationals : [a story of the numbers you can't count on

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. Annotation 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. Inthe 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, Asthe Irrationalsshows, 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.-- Greek Beginnings -- The Route To Germany -- Two New Irrationals -- Irrationals, Old And New -- A Very Special Irrational -- From The Rational To The Transcendental -- Transcendentals -- Continued Fractions Revisited -- The Question And Problem Of Randomness -- One Question, Three Answers -- Does Irrationality Matter? -- Appendix A: The Spiral Of Theodorus -- Appendix B: Rational Parameterizations Of The Circle -- Appendix C: Two Properties Of Continued Fractions -- Appendix D: Finding The Tomb Of Roger Apéry -- Appendix E: Equivalence Relations -- Appendix F: The Mean Value Theorem. Julian Havil. Includes Bibliographical References And Index. Annotation 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.-- Source other than Library of Congress 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 The first popular history of irrational numbers and their discoverers, from ancient Greece to the twenty-first century 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. Fascinating and illuminating, this is a book for everyone who loves math and the history behind it.