Arthur Benjamin - Secrets of Mental Math

Copyright 2006 by Arthur Benjamin and Michael Shermer

All rights reserved.
Published in the United States by Three Rivers Press, an imprint of the Crown
Publishing Group, a division of Random House, Inc., New York.
www.crownpublishing.com

Originally published in different form as Mathemagics by Lowell House,
Los Angeles, in 1993.

Three Rivers Press and the Tugboat design are registered trademarks of Random House, Inc.

Library of Congress Cataloging-in-Publication Data

Benjamin, Arthur.
Secrets of mental math : the mathemagicians guide to lightning calculation and amazing math tricks / Arthur Benjamin and Michael Shermer. 1st ed.
p. cm.
Includes bibliographical references.
1. Mental arithmeticStudy and teaching. 2. Magic tricks in mathematics education. 3. Mental calculators. I. Shermer, Michael. II. Title.
QA111.B44 2006
510dc22

2005037289

eISBN: 978-0-307-34746-6

v3.1_r1

The authors wish to thank Steve Ross and Katie McHugh at Random House for their support of this book. Special thanks to Natalya St. Clair for typesetting the initial draft, which was partly supported by a grant from the Mellon Foundation.

Arthur Benjamin especially wants to acknowledge those who inspired him to become both a mathematician and a magiciancognitive psychologist William G. Chase, magicians Paul Gertner and James Randi, and mathematicians Alan J. Goldman and Edward R. Scheinerman. Finally, thanks to all of my colleagues and students at Harvey Mudd College, and to my wife, Deena, and daughters, Laurel and Ariel, for constant inspiration.

I like to think about the first humans, the people who came up with the idea to count things. They must have noticed right away that figuring on your fingertips works great. Perhaps Og (a typical ancient cave guy) or one of his pals or associates said, There are one, two, three, four, five of us here, so we need five pieces of fruit. Later, Hey, look, someone must have said (or grunted), you can count the number of people at the campfire, the number of birds on a tree, stones in a row, logs for a fire, or grapes in a bunch, just with your fingers. It was a great start. Its probably also how you came to first know numbers.

Youve probably heard that math is the language of science, or the language of Nature is mathematics. Well, its true. The more we understand the universe, the more we discover its mathematical connections. Flowers have spirals that line up with a special sequence of numbers (called Fibonacci numbers) that you can understand and generate yourself. Seashells form in perfect mathematical curves (logarithmic spirals) that come from a chemical balance. Star clusters tug on one another in a mathematical dance that we can observe and understand from millions and even billions of kilometers away.

We have spent centuries discovering the mathematical nature of Nature. With each discovery, someone had to go through the math and make sure the numbers were right. Well, Secrets of Mental Math can help you handle all kinds of numbers. Youll get comfortable with calculations in a way that will let you know some of Natures numerical secrets, and who knows where that might take you?

As you get to know numbers, the answer really is at your fingertips. Thats not a joke, because thats where it all begins. Almost everyone has ten fingers, so our system of mathematics started with 1 and went to 10. In fact, we call both our numbers and our fingers digits. Coincidence? Hardly. Pretty soon, though, our ancestors ran out of fingers. The same thing has probably happened to you. But we cant just ignore those big numbers and (this is a joke) throw up our hands.

We need numberstheyre part of our lives every day, and in ways we typically dont even notice. Think about a conversation you had with a friend. To call, you needed a phone number, and the time you spent on the phone was measured in numbers of hours and minutes. Every date in history, including an important one like your birthday, is reckoned with numbers. We even use numbers to represent ideas that have nothing to do with counting. Whats your 20? (I.e., Where are you? From the old police 10 codes, like 10-4 for yes.) Whats the 411 on that gal? (I.e., Whats her background; is she dating anyone? From the number for telephone information.) People describe one another in numbers representing height and weight. And, of course, we all like to know how much money we have or how much something costs in numbers: dollars, pesos, yuan, rupees, krona, euros, or yen. Additionally (another joke), this book has a time-saving section on remembering numbersand large numbers of numbers.

If, for some reason, youre not crazy about math, read a little further. Of course I, as the Science Guy, hope you do like math. Well, actually, I hope you love math. But no matter how you feel about math, hatred or love, Id bet that you often find yourself just wanting to know the answer right away, without having to write down everything carefully and work slowly and diligentlyor without even having to stop and grab a calculator. You want the answer, as we say, as if by magic. It turns out that you can solve or work many, many math problems almost magically. This book will show you how.

What makes any kind of magic so intriguing and fun is that the audience seldom knows how the trick is performed. How did she do that ? I dont know, but its cool. If you have an audience, the tricks and shortcuts in Secrets of Mental Math are a lot like magic. The audience seldom knows how a trick is performed; they just appreciate it. Notice, though, that in magic, its hardly worth doing if no one is watching. But with Secrets, knowing how it works doesnt subtract from the fun (or pun). When arithmetic is easy, you dont get bogged down in the calculating; you can concentrate on the wonderful nature of numbers. After all, math runs the universe.