The Joy of Mathematics: Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class

by Alfred S. Posamentier, Robert Geretschlager, Charles Li

Overview

Wouldn't it be great if all school teachers (from kindergarten through high school) would share the joy of mathematics with their students, rather than focus only on the prescribed curriculum that will subsequently be tested? This book promises to help teachers and all readers do just that by revealing some wonders of mathematics often missing from classrooms. Here's your chance to catch up with the math gems you may have missed in your school years.Using jargon-free language and many illustrations, the authors--all veteran math educators--explore five areas--arithmetic, algebra, geometry, probability, and the ways in which mathematics can reinforce common sense. Among other things, you'll learn "the rule of 72," which enables you to quickly determine how long it will take your bank account to double its value at a specific interest rate. Other handy techniques include an automatic algorithm for multiplying numbers mentally and a clever application that will allow you to convert from miles to kilometers (or the reverse) mentally. A delightful presentation of geometric novelties reveals relationships that could have made your study of geometry more fun and enlightening. In the area of probability there is a host of interesting examples: from the famous Monty-Hall problem to the counterintuitive probability of two people having the same birthday in a crowded room.Finally, the authors demonstrate how math will make you a better thinker by improving your organizing abilities and providing useful and surprising solutions to common mathematics problems. You'll come away with a grasp of math you never thought possible and a true appreciation for this "queen of the sciences."

Product Details

ISBN-13:	9781633882973
Publisher:	Rowman & Littlefield Publishers, Inc.
Publication date:	08/22/2017
Pages:	319
Sales rank:	405,339
Product dimensions:	5.90(w) x 8.90(h) x 0.90(d)

About the Author

Alfred S. Posamentier has published over sixty books in the area of mathematics and mathematics education, including, most recently, The Circle: A Mathematical Exploration beyond the Line (with Robert Geretschläger). After having been on the faculty of the City College of the City University of New York for forty years, where he was a professor of mathematics education and dean of the School of Education, he subsequently held the same positions at Mercy College, New York, for five years. He is currently executive director for internationalization and funded programs at Long Island University.

Robert Geretschläger is a mathematics teacher at Bundesrealgymnasium Keplerstrasse in Graz, Austria. He is the coauthor of The Circle (with Alfred S. Posamentier), Geometric Origami, and other works.

Charles Li is an Assistant Professor of Mathematics at Mercy College. He earned his Ph.D. in Mathematics from the CUNY Graduate Center, and his B.Eng. in Electrical Engineering from Cooper Union.

Christian Spreitzer teaches mathematics and physics at the University College of Teacher Education in Lower Austria.

Table of Contents

Introduction 9

Chapter 1 Arithmetic Novelties 13

When Is a Number Divisible by 3 or 9? 14

When Is a Number Divisible by 11? 15

Divisibility by Prime Numbers 17

Squaring Numbers Quickly 21

Squares and Sums 23

Using Squares to Multiply Arbitrary Numbers 24

An Alternative Method for Extracting a Square Root 25

Sensible Number Comparisons 26

Euclidean Algorithm to Find the GCD 28

Sums of Positive Integers 30

Sums of Odd Positive Integers 32

The Realm of Nonterminating Decimals 34

Atoms in the Universe of Numbers 37

Fun with Number Relationships 42

Friendly Numbers 45

Palindromic Numbers 47

Prime Numbers 52

Infinite Primes 55

The Neglected Triangular Numbers 56

Perfect Numbers 62

Making Mistaken Generalizations 66

The Fibonacci Numbers 69

Chapter 2 Algebraic Explanations of Accepted Concepts 77

Simple Algebra Helps Logical Reasoning 77

Division by Zero 78

Irrationality of the Square Root of 2 80

Bisection Method to Approximate Square Roots 83

Continued Fractions of Square Roots 86

Fermat's Method of Factoring 89

Comparing Means 91

Diophamine Equations 94

Falling Squares 98

Descartes's Rule of Signs 100

Horner's Method to Evaluate Polynomials 102

Generating Pythagorean Triples 105

The Frobenius Problem 110

Chapter 3 Geometric Curiosities 113

Parallelograms and Triangles 114

Using a Grid to Calculate Areas 115

The Center of a Quadrilateral 121

Beyond the Formula for the Area of a Triangle 124

Heronian Triangles 127

A New Formula for the Area of Isosceles Triangles 131

Pick's Theorem 135

When Intersecting Lines Meet a Circle 137

Origins of Trigonometry 140

Sines of Small Angles 142

An Unconventional View of the Sine 145

Surprising Proofs of the Pythagorean Theorem 147

Beyond the Pythagorean Theorem-Part I 153

Beyond the Pythagorean Theorem-Part II 159

Beyond the Pythagorean Theorem-Part III 163

The Pythagorean Theorem Extended to Three Dimensions 166

Polyhedra: Sides, Faces, and Vertices 168

Lunes and the Right Triangle 172

Concurrency 176

Similarity and the Golden Ratio 182

A Relation between Points and Circles 186

Constructions with Compasses Alone 191

The Sphere and the Cylinder 192

Regular Polygons and Stars 194

Platonic Solids and Star Polyhedra 200

Chapter 4 Probability Applied to Everyday Experiences 207

How the Theory of Probability Began 207

Benford's Law 209

The Birthday Phenomenon 212

The Monty Hall Problem 215

Bertrand's Box 218

The False Positive Paradox 221

Pascal's Triangle 224

Random Walks 228

The Poker Wild-Card Paradox 233

Chapter 5 Common Sense From a Mathematical Perspective 239

The Origins of Some Mathematics Symbols 239

The Counterintuitive 242

A Surprising Solution 244

Don't "Wine" over This Problem-A Problem-Solving Approach 246

Organized Thinking 247

Successive Percentages 251

Rule of 72 253

A Mathematical Conjecture 255

Unexpected Patterns 258

An Infinity Conundrum 261

The Concept of Infinity 263

Counting the Uncountable 266

Mathematics on a Bicycle 273

The Parabola: A Remarkable Curve 278

Acknowledgments 287

Appendix 289

Notes 293

Index 299

Editorial Reviews

"The Joy of Mathematics is aptly titled. The beauty, ingenuity, and mystery of this timeless discipline is laid out to captivate even a non-mathematician like myself. The history of major mathematical discoveries is noted as we are walked through the corridors of an unfolding digital universe, be it the science of probability, the intellectual beauty of Pascal's triangle, or the intricacies of Kepler's star shapes. The book should be a mandatory read for all who are mathematically literate and curious.”—Tom Breen, chairman, Breen Property, Sydney, Australia“Too many young people leave school without ever having acquired an enjoyment of [mathematics] or recognition of the importance of mathematical understanding in everyday life, from our bank accounts to politicians' use of statistics. This delightful book fills a vital role in making mathematics relevant, intelligible, and above all—fun! I recommend it unreservedly to pupils and teachers as a way of opening the door to joy in this key subject.”—Baroness Pauline Perry, former head of Lucy Cavendish College, Cambridge University, and formerly Her Majesty's Chief Inspector of Education“The Joy of Mathematics includes wonderful examples of often-overlooked gems in elementary mathematics. Even an expert can benefit a lot from it, but it is written in such a catchy style that everyone—layperson or expert—will enjoy each page.”—Rudolf Taschner, professor at the Technical University of Vienna, Austria's 2004 Scientist of the Year, and author of Game Changers“A pleasant and fascinating excursion through the amusements, byways, curiosities, and diversions of mathematics. It even gets up to public-key cryptography and several unsolved problems. This is just the book to give to anyone who doesn't like mathematics or who wonders, What is ‘recreational mathematics'?” —David Singmaster, author of Problems for Metagrobologists

From the Publisher