Arthur Benjamin - The Magic of Math: Solving for x and Figuring Out Why

Advance Praise for The Magic of Math

They say magicians should never reveal their secrets. Happily, Arthur Benjamin has ignored this silly adagefor in this small volume, Benjamin reveals to his audience the secrets of numbers and other mathematical illusions that have intrigued mathematicians for millennia.

Edward B. Burger, president, Southwestern University, and author of The 5 Elements of Effective Thinking

This book will be magical for my students, as it would have been for me throughout my school days. Theyll be able to revisit the book frequently as they learn more math, finding deeper appreciation and discovering new areas to explore with each visit.

Richard Rusczyk, founder, Art of Problem Solving, and director, USA Mathematical Talent Search

In The Magic of Math, Arthur Benjamin has pulled off a seemingly impossible trick. He has made higher mathematics appear so natural and engaging that you will wonder why you were ever bored and confused in math class. There are many books that attempt to popularize mathematics. This is one of the best. On virtually every page I found myself learning new things, or looking at familiar topics in novel ways.

Jason Rosenhouse, professor of Mathematics, James Madison University, and author of The Monty Hall Problem

In The Magic of Math, mathemagician Arthur Benjamin gives us an entertaining and enlightening tour of a wide swath of fundamental mathematical ideas, presented in a way that is accessible to a broad audience. A particularly appealing feature of the book is the frequent use of friendly, down-to-earth explanations of the concepts and connections between them.

Ronald Graham, president emeritus, American Mathematical Society, and coauthor of Magical Mathematics

This book is a whirlwind tour of mathematics from arithmetic and algebra all the way to calculus and infinity, and especially the number 9. Arthur Benjamins enthusiastic and engaging writing style makes The Magic of Math a great addition to any math enthusiasts bag of tricks.

Laura Taalman, professor of Mathematics and Statistics, James Madison University

Mathematics is full of surprisingly beautiful patterns, which Arthur Benjamins witty personality brings to life in The Magic of Math. You will not only discover many wonderful ideas, but you will also find some fun mathematical magic tricks that you will want to try out on your friends and family. Be prepared to learn that math is more entertaining than you may have thought.

George W. Hart, mathematical sculptor, research professor, Stony Brook University, and cofounder, The Museum of Mathematics

The Magic of Math is a delightful stroll through a garden filled with fascinating examples. Anyone with any interest in magic, puzzles, or math will have many hours of enjoyment in reading this book.

Maria M. Klawe, president, Harvey Mudd College

Arthur Benjamin has created an instant mathematical classic, by combining Isaac Asimovs clarity with Martin Gardners taste and adding his own sense of fun and adventure. I wish he wrote this book when I was a kid.

Paul A. Zeitz, professor and chair of Mathematics, University of San Francisco, and author of The Art and Craft of Problem Solving

Theres a playful joy to be found in this book, for readers at any level. Most magicians dont reveal their secrets, but in The Magic of Math, Arthur Benjamin shows how uncovering the mystery behind beautiful mathematical truths makes math even more marvelous to behold.

Francis Su, president, Mathematical Association of America

The Magic of Math offers an expansive, unforgettable journey through mathematics where numbers dance and mathematical secrets are revealed. Just open the book and start reading; youll be swept over by the magic of Benjamins writing. Luckily, there is no magicians code to these secrets as youll undoubtedly want to share and perform them with family and friends.

Tim Chartier, professor of Mathematics, Davidson College, and author of Math Bytes

Copyright 2015 by Arthur Benjamin

Published by Basic Books,

A Member of the Perseus Books Group

All rights reserved. Printed in the United States of America. No part of this book may be reproduced in any manner whatsoever without written permission except in the case of brief quotations embodied in critical articles and reviews. For information, address Basic Books, 250 West 57th Street, New York, NY 10107.

Books published by Basic Books are available at special discounts for bulk purchases in the United States by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA 19103, or call (800) 810-4145, ext. 5000, or e-mail .

Illustrations by Natalya St. Clair

A catalog record for this book is available from the Library of Congress.

Library of Congress Control Number: 2015936185

ISBN: 978-0-465-06162-4 (e-book)

10 9 8 7 6 5 4 3 2 1

I dedicate this book to my wife, Deena, and daughters, Laurel and Ariel.

Contents

Throughout my life, I have always had a passion for magic. Whether I was watching other magicians or performing magic myself, I was fascinated with the methods used to accomplish amazing and impressive feats, and I loved learning its secrets. With just a handful of simple principles, I could even invent tricks of my own.

I had the same experience with mathematics. From a very early age, I saw that numbers had a magic all their own. Heres a trick you might enjoy. Think of a number between 20 and 100. Got it? Now add your digits together. Now subtract the total from your original number. Finally, add the digits of the new number together. Are you thinking of the number 9? (If not, you might want to check your previous calculation.) Pretty cool, huh? Mathematics is filled with magic like this, but most of us are never exposed to it in school. In this book, you will see how numbers, shapes, and pure logic can yield delightful surprises. And with just a little bit of algebra or geometry, you can often discover the secrets behind the magic, and perhaps even discover some beautiful mathematics of your own.

This book covers the essential mathematical subjects like numbers, algebra, geometry, trigonometry, and calculus, but it also covers topics that are not so well represented, like Pascals triangle, infinity, and magical properties of numbers like 9, , e, i, Fibonacci numbers, and the golden ratio. And although none of the big mathematical subjects can be completely covered in just a few dozen pages, I hope you come away with an understanding of the major concepts, a better idea of why they work, and an appreciation of the elegance and relevance of each subject. Even if you have seen some of these topics before, I hope you will see them and enjoy them with new perspectives. And as we learn more mathematics, the magic becomes more sophisticated and fascinating. For example, here is one of my favorite equations: