Pi] : a biography of the world's most mysterious number

This enlightening and stimulating approach to mathematics will entertain lay readers while improving their mathematical literacy. We all learned that the ratio of the circumference of a circle to its diameter is called pi and that the value of this algebraic symbol is roughly 3.14. What we weren't told, though, is that behind this seemingly mundane fact is a world of mystery, which has fascinated mathematicians from ancient times to the present. Simply put, pi is weird. Mathematicians call it a "transcendental number" because its value cannot be calculated by any combination of addition, subtraction, multiplication, division, and square root extraction. This elusive nature has led intrepid investigators over the years to attempt ever-closer approximations. In 2002, a Japanese professor using a supercomputer calculated the value to 1.24 trillion decimal places! Nonetheless, in this huge string of decimals there is no periodic repetition. In this delightful layperson's introduction to one of math's most interesting phenomena, Drs. Posamentier and Lehmann review pi's history from prebiblical times to the 21st century, the many amusing and mind-boggling ways of estimating p over the centuries, quirky examples of obsessing about pi (including an attempt to legislate its exact value), and useful applications of pi in everyday life, including statistics. Cover 1 Pi: A Biography of the World's Most Mysterious Number 4 Copyright 5 © 2004 by Alfred S.Posamentier and Ingmar Lehmann 5 ISBN 1591022002 5 QA484.P67 2004 512.7'3—dc22 5 Contents 6 Acknowledgments 8 Preface 10 Chapter 1: What Is Pi? 14 Introduction to \Pi 14 Aspects of \Pi 15 The Symbol \Pi 16 Recollections of \Pi 17 Formula for the Area of a Circle 18 The Square and the Circle 20 The Value of \Pi 21 \Pi Peculiarities 23 The Evolution of the Value of \Pi 24 Sharpening Our Intuition with \Pi 27 What the Bible Has as the Value of \Pi 28 Where the Symbol \Pi in Mathematics Came From 30 Euler 31 A \Pi Paradox 35 Legislating \Pi 37 \Pi in Probability 39 Chapter 2: The History of \Pi 42 In the Beginning 42 The Ancient Egyptians 43 Just Before the Common Era 45 Archimedes' Contributions 53 The Beginning of the Common Era 60 The Chinese Contributions 62 The Beginning of the Renaissance 63 The Sixteenth Century 64 The Seventeenth Century 65 The Eighteenth Century—When \Pi Gets Its Name 68 Approaching the Nineteenth Century 70 Entering the Twentieth Century 71 The Computer Enters the Story of \Pi 73 Chapter 3: Calculatingthe Value of \Pi ? 80 Archimedes' Method for Finding the Value of \Pi 81 A Reverse Method to Archimedes by Cusanus 92 Calculation of the Value of it by Counting Squares 100 Calculating the Value of it by Counting Lattice Points 102 Using Physical Properties to Calculate the Value of \Pi 104 The Monte-Carlo Method to Determine the Value of \Pi 106 Calculating \Pi from a Series of Numbers 109 A Better Series Calculation for the Value of \Pi 111 The Genius's Method for Finding the Value of \Pi 113 Chapter 4: \Pi Enthusiasts 118 Popularity of \Pi 118 \Pi Mnemonics 123 More Fascination with the Digits of \Pi 130 An Optical IlLusion 135 A \Pi Song 136 Chapter 5: \Pi Curiosities 138 \Pi Digit Curiosities 138 Probability's Use of \Pi 140 Using \Pi to Measure the Lengths of Rivers 140 Unexpected \Pi Coincidences 143 Continued Fractions and \Pi 147 Chapter 6: Applications of \Pi 158 \Pi When You Least Expect It 158 \Pi in Sports 171 A Spiral Formed by Semicircles 173 The Unique Seven-Circles Arrangement 176 A "Mushroom" Shape 179 A "Dolphin" Shape 187 The Yin and Yang 191 Using \Pi to Apportion a Pizza Pie Equally for Three People 198 Traditional Pieces 199 Concentric Circles 199 Fancy Pieces (Using the Teardrop Shape) 201 Parallel Pieces 203 The Constant Ring 205 The Constant Ring Extended 207 The Lost Circle Area 210 Unusual Circle Relationships 212 \Pi and the Imaginary Unit i 216 Chapter 7: Paradox in \Pi 218 Rolling Cylinders—\Pi Revolutions! 219 A Constant among Concentric Circles 221 A Rope around the Equator 223 Another Surprise 240 Epilogue 246 Afterword 276 Appendixes 292 Appendix A: A Three-Dimensional Example of a Rectilinear Equivalent to a Circular Measurement 294 Appendix B: Ramanujan's Work 298 Appendix C: Proof That exp(\Pi) > (\Pi)^e 302 Proof I 302 Proof II 303 Proof III 304 Proof IV 305 Appendix D: A Rope around the Regular Polygons 306 References 310 Index 314 Back Cover 327 "We all learned that the ratio of the circumference of a circle to its diameter is called pi and that the value of this algebraic symbol is roughly 3.14. What we weren't told, though, is that behind this seemingly mundane fact is a world of mystery, which has fascinated mathematicians from ancient times to the present. Simply put, pi is weird. Mathematicians call it a "transcendental number" because its value cannot be calculated by any combination of addition, subtraction, multiplication, division, and square root extraction. In this delightful layperson's introduction to one of math's most interesting phenomena, Drs. Posamentier and Lehmann review pi's history from prebiblical times to the 21st century, the many amusing and mind-boggling ways of estimating pi over the centuries, quirky examples of obsessing about pi (including an attempt to legislate its exact value), and useful applications of pi in everyday life, including statistics.This enlightening and stimulating approach to mathematics will entertain lay readers while improving their mathematical literacy."--Back cover We all learned that the ratio of the circumference of a circle to its diameter is called pi and that the value of this algebraic symbol is roughly 3.14. What we weren't told, though, is that behind this seemingly mundane fact is a world of mystery, which has fascinated mathematicians from ancient times to the present. Simply put, pi is weird. Mathematicians call it a "transcendental number" because its value cannot be calculated by any combination of addition, subtraction, multiplication, division, and square root extraction. In this delightful layperson's introduction to one of math's most interesting phenomena, Drs. Posamentier and Lehmann review pi's history from prebiblical times to the 21st century, the many amusing and mind-boggling ways of estimating pi over the centuries, quirky examples of obsessing about pi (including an attempt to legislate its exact value), and useful applications of pi in everyday life, including statistics. This enlightening and stimulating approach to mathematics will entertain lay readers while improving their mathematical literacy. Presenting an introduction to one of math's interesting phenomena, this work reviews p's history from pre-biblical times to the 21st century, the ways of estimating p over the centuries, examples of obsessing about p (including an attempt to legislate its exact value), and useful applications of p in everyday life, including statistics. The authors review Pi's history from prebiblical times to the twenty-first century and the many amusing and often mind-boggling attempts to estimate its precise value .. The symbol is the sixteenth letter of the Greek alphabet, yet it has gained fame because of its designation in mathematics.