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Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks
Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks
Arthur Benjamin, Michael Shermer
These simple math secrets and tricks will forever change how you look at the world of numbers.
Secrets of Mental Math will have you thinking like a math genius in no time. Get ready to amaze your friends—and yourself—with incredible calculations you never thought you could master, as renowned “mathemagician” Arthur Benjamin shares his techniques for lightningquick calculations and amazing number tricks. This book will teach you to do math in your head faster than you ever thought possible, dramatically improve your memory for numbers, and—maybe for the first time—make mathematics fun.
Yes, even you can learn to do seemingly complex equations in your head; all you need to learn are a few tricks. You’ll be able to quickly multiply and divide triple digits, compute with fractions, and determine squares, cubes, and roots without blinking an eye. No matter what your age or current math ability, Secrets of Mental Math will allow you to perform fantastic feats of the mind effortlessly. This is the math they never taught you in school.
Secrets of Mental Math will have you thinking like a math genius in no time. Get ready to amaze your friends—and yourself—with incredible calculations you never thought you could master, as renowned “mathemagician” Arthur Benjamin shares his techniques for lightningquick calculations and amazing number tricks. This book will teach you to do math in your head faster than you ever thought possible, dramatically improve your memory for numbers, and—maybe for the first time—make mathematics fun.
Yes, even you can learn to do seemingly complex equations in your head; all you need to learn are a few tricks. You’ll be able to quickly multiply and divide triple digits, compute with fractions, and determine squares, cubes, and roots without blinking an eye. No matter what your age or current math ability, Secrets of Mental Math will allow you to perform fantastic feats of the mind effortlessly. This is the math they never taught you in school.
Year:
2006
Publisher:
Three Rivers Press
Language:
english
Pages:
305
ISBN 10:
0307338401
ISBN 13:
9780307338402
File:
PDF, 1.53 MB
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Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page ii This book has been optimized for viewing at a monitor setting of 1024 x 768 pixels. Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page i SECRE+S OF MEN+AL MA+H Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page ii Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page iii SECRE+S OF MEN+AL MA+H The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks Arthur Benjamin and Michael Shermer Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page iv 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 CataloginginPublication Data Benjamin, Arthur. Secrets of mental math : the mathemagician’s guide to lightning calculation and amazing math tricks / Arthur Benjamin and Michael Shermer.— 1st ed. p. cm. Includes bibliographical references and index. 1. Mental arithmetic—Study and teaching. 2. Magic tricks in mathematics education. 3. Mental calculators. I. Shermer, Michael. II. Title. QA111.B44 2006 510—dc22 2005037289 eISBN: 9780307347466 v1.0 Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page v I dedicate this book to my wife, Deena, and daughters, Laurel and Ariel. —Arthur Benjamin My dedication is to my wife, Kim, for being my most trusted confidante and personal counselor. —Michael Shermer Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page vi Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page vii Acknowledgments 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 magician— cognitive 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. Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page viii Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page ix Contents Foreword by Bill Nye (the Science Guy®) xi Foreword by James Randi xvii Prologue by Michael Shermer xix Introduction by Arthur Benjamin xxiii Chapter 0 Quick Tricks: Easy (and Impressive) Calculations 1 Chapter 1 A Little Give and Take: Mental Addition and Subtraction 11 Chapter 2 Products of a Misspent Youth: Basic Multiplication 29 Chapter 3 New and Improved Products: Intermediate Multiplication 53 Chapter 4 Divide and Conquer: Mental Division 80 Chapter 5 Good Enough: The Art of “Guesstimation” 108 Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page x x Contents Chapter 6 Math for the Board: PencilandPaper Math 131 Chapter 7 A Memorable Chapter: Memorizing Numbers 151 Chapter 8 The Tough Stuff Made Easy: Advanced Multiplication 163 Chapter 9 Prestodigitation: The Art of Mathematical Magic 199 Chapter Epilogue by Michael Shermer: How Math Helps Us Think About Weird Things 222 Answers 233 Bibliography 271 Index 273 Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xi Foreword by Bill Nye (the Science Guy®) 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. It’s probably also how you came to first know numbers. You’ve probably heard that math is the language of science, or the language of Nature is mathematics. Well, it’s 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 Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xii xii Foreword math and make sure the numbers were right. Well, Secrets of Mental Math can help you handle all kinds of numbers. You’ll get comfortable with calculations in a way that will let you know some of Nature’s numerical secrets, and who knows where that might take you? As you get to know numbers, the answer really is at your fingertips. That’s not a joke, because that’s 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 can’t just ignore those big numbers and (this is a joke) throw up our hands. We need numbers—they’re part of our lives every day, and in ways we typically don’t 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. What’s your 20? (I.e., Where are you? From the old police “10” codes, like 104 for “yes.”) What’s the 411 on that gal? (I.e., What’s 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 timesaving section on remembering numbers—and large numbers of numbers. If, for some reason, you’re not crazy about math, read a little further. Of course I, as the Science Guy, hope you do like math. Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xiii Foreword Well, actually, I hope you love math. But no matter how you feel about math, hatred or love, I’d 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 diligently—or 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 don’t know, but it’s 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, it’s hardly worth doing if no one is watching. But with Secrets, knowing how it works doesn’t subtract from the fun (or pun). When arithmetic is easy, you don’t get bogged down in the calculating; you can concentrate on the wonderful nature of numbers. After all, math runs the universe. Dr. Benjamin got into this business of lightningfast calculating just for fun. We have to figure he impressed his teachers and classmates. Magicians might make some in their audience think that they have supernatural powers. Mathemagicians, at first, give the impression that they’re geniuses. Getting people to notice what you’re doing is an old part of sharing ideas. If they’re impressed, they’ll probably listen to what you have to say. So try some “mathemagics.” You may impress your friends, all right. But you’ll also find yourself performing just for yourself. You’ll find you’re able to do problems that you didn’t think you could. You’ll be impressed . . . with yourself. Now, counting on your fingers is one thing (one finger’s worth). But have you ever found yourself counting out loud or xiii Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xiv xiv Foreword whispering or making other sounds while you calculate? It almost always makes math easier. The problem, though, is that other people think you’re a little odd . . . not even (more math humor). Well, in Secrets of Mental Math, Dr. Benjamin helps you learn to use that “outloud” feature of the way your brain works to do math problems more easily, faster, and more accurately (which is surprising), all while your brain is thinking away—almost as if you’re thinking out loud. You’ll learn to move through math problems the same way we read in English, left to right. You’ll learn to handle big problems fast with good guesses, actually great guesses, within a percent or so. You will learn to do arithmetic fast; that way you can spend your time thinking about what the numbers mean. Og wondered, “Do we have enough fruit for each person sitting around the fire? If not, there might be trouble.” Now you might wonder, “Is there enough space on this computer to keep track of my music files . . . or my bank account? If not, there might be trouble.” There’s more to Secrets than just figuring. You can learn to take a day, month, and year, then compute what day of the week it was or will be. It’s fantastic, almost magical, to be able to tell someone what day of the week she or he was born. But, it’s really something to be able to figure that the United States had its first big Fourth of July on a Thursday in 1776. April 15, 1912, the day the Titanic sank, was a Monday. The first human to walk on the moon set foot there on July 20, 1969, a Sunday. You’ll probably never forget that the United States was attacked by terrorists on September 11, 2001. With Secrets of Mental Math, you’ll always be able to show it was a Tuesday. There are relationships in Nature that numbers describe better than any other way we know. There are simple numbers that you can count on your hands: one, two, three, and on up. But Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xv Foreword there are also an infinite number of numbers in between. There are fractions. There are numbers that never end. They get as big as you want and so small that they’re hard to imagine. You can know them. With Secrets of Mental Math, you can have even these inbetween numbers come so quickly to your mind that you’ll have a bit more space in your brain to think about why our world works this way. One way or another, this book will help you see that in Nature, it all adds up. xv Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xvi Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xvii Foreword by James Randi Mathematics is a wonderful, elegant, and exceedingly useful language. It has its own vocabulary and syntax, its own verbs, nouns, and modifiers, and its own dialects and patois. It is used brilliantly by some, poorly by others. Some of us fear to pursue its more esoteric uses, while a few of us wield it like a sword to attack and conquer income tax forms or masses of data that resist the less courageous. This book does not guarantee to turn you into a Leibniz, or put you on stage as a Professor Algebra, but it will, I hope, bring you a new, exciting, and even entertaining view of what can be done with that wonderful invention—numbers. We all think we know enough about arithmetic to get by, and we certainly feel no guilt about resorting to the handy pocket calculator that has become so much a part of our lives. But, just as photography may blind us to the beauty of a Vermeer painting, or an electronic keyboard may make us forget the magnificence of a Horowitz sonata, too much reliance on technology can deny us the pleasures that you will find in these pages. I remember the delight I experienced as a child when I was shown that I could multiply by 25 merely by adding two 0s to my number and dividing by 4. Casting out 9s to check multiplication came next, and when I found out about crossmultiplying I was hooked and became, for a short while, a generally unbearable Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xviii xviii Foreword math nut. Immunizations against such afflictions are not available. You have to recover all by yourself. Beware! This is a fun book. You wouldn’t have it in your hands right now if you didn’t have some interest either in improving your math skills or in satisfying a curiosity about this fascinating subject. As with all such instruction books, you may retain and use only a certain percentage of the varied tricks and methods described here, but that alone will make it worth the investment of your time. I know both the authors rather well. Art Benjamin is not only one of those whiz kids we used to groan about in school but also has been known to tread the boards at the Magic Castle in Hollywood, performing demonstrations of his skill, and on one occasion he traveled to Tokyo, Japan, to pit his math skills against a lady savant on live television. Michael Shermer, with his specialized knowledge of science, has an excellent overview of practical applications of math as it is used in the real world. If this is your first exposure to this kind of good math stuff, I envy you. You’ll discover, as you come upon each delicious new way to attack numbers, that you missed something in school. Mathematics, particularly arithmetic, is a powerful and dependable tool for daytoday use that enables us to handle our complicated lives with more assurance and accuracy. Let Art and Michael show you how to round a few of the corners and cut through some of the traffic. Remember these words of Dr. Samuel Johnson, an eminently practical soul in all respects: “Arithemetical inquiries give entertainment in solitude by the practice, and reputation in public by the effect.” Above all, enjoy the book. Let it entertain you, and have fun with it. That, with the occasional good deed, a slice of pizza (no anchovies!), and a selection of good friends is about all you can ask of life. Well, almost all. Maybe a Ferrari . . . Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xix Prologue by Michael Shermer My good friend Dr. Arthur Benjamin, mathematics professor at Harvey Mudd College in Claremont, California, takes the stage to a round of applause at the Magic Castle, a celebrated magic club in Hollywood, where he is about to perform “mathemagics,” or what he calls the art of rapid mental calculation. Art appears nothing like a mathematics professor from a prestigious college. Astonishingly quickwitted, he looks at home with the rest of the young magicians playing at the Castle—which he is. What makes Art so special is that he can play in front of any group, including professional mathematicians and magicians, because he can do something that almost no one else can. Art Benjamin can add, subtract, multiply, and divide numbers in his head faster than most people can with a calculator. He can square twodigit, threedigit, and fourdigit numbers, as well as find square roots and cube roots, without writing anything down on paper. And he can teach you how to perform your own mathematical magic. Traditionally, magicians refuse to disclose how they perform their tricks. If they did, everyone would know how they are done and the mystery and fascination of magic would be lost. But Art wants to get people excited about math. And he knows that one of the best ways to do so is to let you and other readers Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xx xx Prologue in on his secrets of “math genius.” With these skills, almost anyone can do what Art Benjamin does every time he gets on stage to perform his magic. This particular night at the Magic Castle, Art begins by asking if anyone in the audience has a calculator. A group of engineers raise their hands and join Art on the stage. Offering to test their calculators to make sure they work, Art asks a member of the audience to call out a twodigit number. “Fiftyseven,” shouts one. Art points to another who yells out, “Twentythree.” Directing his attention to those on stage, Art tells them: “Multiply 57 by 23 on the calculator and make sure you get 1311 or the calculators are not working correctly.” Art waits patiently while the volunteers finish inputting the numbers. As each participant indicates his calculator reads 1311, the audience lets out a collective gasp. The amazing Art has beaten the calculators at their own game! Art next informs the audience that he will square four twodigit numbers faster than his buttonpushers on stage can square them on their calculators. The audience asks him to square the numbers 24, 38, 67, and 97. Then, in large, bold writing for everyone to see, Art writes: 576, 1444, 4489, 9409. Art turns to his engineer volunteers, each of whom is computing a twodigit square, and asks them to call out their answers. Their response triggers gasps and then applause from the audience: “576, 1444, 4489, 9409.” The woman next to me sits with her mouth open in amazement. Art then offers to square threedigit numbers without even writing down the answer. “Five hundred and seventytwo,” a gentleman calls out. Art’s reply comes less than a second later: “572 squared is 327,184.” He immediately points to another member of the audience, who yells, “389,” followed by Art’s unblinking response: “389 squared will give you 151,321.” Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xxi Prologue Someone else blurts out, “262.” “That’ll give you 68,644.” Sensing he delayed just an instant on that last one, he promises to make up for it on the next number. The challenge comes—991. With no pause, Art squares the number, “982,081.” Several more threedigit numbers are given and Art responds perfectly. Members of the audience shake their heads in disbelief. With the audience in the palm of his hand, Art now declares that he will attempt to square a fourdigit number. A woman calls out, “1,036,” and Art instantly responds, “That’s 1,073,296.” The audience laughs and Art explains, “No, no, that’s much too easy a number. I’m not supposed to beat the calculators on these. Let’s try another one.” A man offers a challenging 2,843. Pausing briefly between digits, Art responds: “Let’s see, the square of that should be 8 million . . . 82 thousand . . . 649.” He is right, of course, and the audience roars their approval, as loudly as they did for the previous magician who sawed a woman in half and made a dog disappear. It is the same everywhere Art Benjamin goes, whether it is a high school auditorium, a college classroom, a professional conference, the Magic Castle, or a television studio. Professor Benjamin has performed his special brand of magic live all over the country and on numerous television talk shows. He has been the subject of investigation by a cognitive psychologist at Carnegie Mellon University and is featured in a scholarly book by Steven Smith called The Great Mental Calculators: The Psychology, Methods, and Lives of Calculating Prodigies, Past and Present. Art was born in Cleveland on March 19, 1961 (which he calculates was a Sunday, a skill he will teach you in Chapter 9). A hyperactive child, Art drove his teachers mad with his classroom antics, which included correcting the mathematical mistakes they occasionally made. Throughout this book when teaching you his mathematical secrets, Art recalls when and xxi Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xxii xxii Prologue where he learned these skills, so I will save the fascinating stories for him to tell you. Art Benjamin is an extraordinary individual with an extraordinary program to teach you rapid mental calculation. I offer these claims without hesitation, and ask only that you remember this does not come from a couple of guys promising miracles if you will only call our 800 hotline. Art and I are both credentialed in the most conservative of academic professions—Art in mathematics and I, myself, in the history of science—and we would never risk career embarrassment (or worse) by making such powerful claims if they were not true. To put it simply, this stuff works, and virtually everyone can do it because this art of “math genius” is a learned skill. So you can look forward to improving your math skills, impressing your friends, enhancing your memory, and, most of all, having fun! Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xxiii Introduction Ever since I was a child, I have loved playing with numbers, and in this book I hope to share my passion with you. I have always found numbers to have a certain magical appeal and spent countless hours entertaining myself and others with their beautiful properties. As a teenager, I performed as a magician, and subsequently combined my loves of math and magic into a fulllength show, called Mathemagics, where I would demonstrate and explain the secrets of rapid mental calculation to audiences of all ages. Since earning my PhD, I have taught mathematics at Harvey Mudd College, and I still enjoy sharing the joy of numbers with children and adults throughout the world. In this book, I will share all of my secrets for doing math in your head, quickly and easily. (I realize that magicians are not supposed to reveal their secrets, but mathemagicians have a different code of ethics. Mathematics should be awe inspiring, not mysterious.) What will you learn from this book? You will learn to do math in your head faster than you ever thought possible. After a little practice, you will dramatically improve your memory for numbers. You will learn feats of mind that will impress your friends, colleagues, and teachers. But you will also learn to view math as an activity that can actually be fun. Benj_0307338401_4p_fm_r1.r.qxd 5/4/06 1:37 PM Page xxiv xxiv Introduction Too often, math is taught as a set of rigid rules, leaving little room for creative thinking. But as you will learn from Secrets, there are often several ways to solve the same problem. Large problems can be broken down into smaller, more manageable components. We look for special features to make our problems easier to solve. These strike me as being valuable life lessons that we can use in approaching all kinds of problems, mathematical and otherwise. “But isn’t math talent something that you are born with?” I get this question all the time. Many people are convinced that lightning calculators are prodigiously gifted. Maybe I was born with some curiosity about how things work, whether it be a math problem or a magic trick. But I am convinced, based on many years of teaching experience, that rapid math is a skill that anyone can learn. And like any worthwhile skill, it takes practice and dedication if you wish to become an expert. But to achieve these results efficiently, it is important that you practice the right way. Let me show you the way! Mathemagically, Dr. Arthur Benjamin Claremont, California Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 1 Chapter 0 Quick Tricks: Easy (and Impressive) Calculations In the pages that follow, you will learn to do math in your head faster than you ever thought possible. After practicing the methods in this book for just a little while, your ability to work with numbers will increase dramatically. With even more practice, you will be able to perform many calculations faster than someone using a calculator. But in this chapter, my goal is to teach you some easy yet impressive calculations you can learn to do immediately. We’ll save some of the more serious stuff for later. INSTANT MULTIPLICATION Let’s begin with one of my favorite feats of mental math—how to multiply, in your head, any twodigit number by eleven. It’s very easy once you know the secret. Consider the problem: 32 11 , put To solve this problem, simply add the digits, 3 2 5 the 5 between the 3 and the 2, and there is your answer: Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 2 2 Secrets of Mental Math 35 2 What could be easier? Now you try: 53 11 Since 5 3 8, your answer is simply 583 One more. Without looking at the answer or writing anything down, what is 81 11? Did you get 891? Congratulations! Now before you get too excited, I have shown you only half of what you need to know. Suppose the problem is 85 11 Although 8 5 13, the answer is NOT 8135! 1 needs As before, the 3 goes in between the numbers, but the to be added to the 8 to get the correct answer: 935 Think of the problem this way: 1 835 935 Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 3 Quick Tricks: Easy (and Impressive) Calculations Here is another example. Try 57 11. Since 5 7 12, the answer is 1 527 627 Okay, now it’s your turn. As fast as you can, what is 77 11? If you got the answer 847, then give yourself a pat on the back. You are on your way to becoming a mathemagician. Now, I know from experience that if you tell a friend or teacher that you can multiply, in your head, any twodigit number by eleven, it won’t be long before they ask you to do 99 11. Let’s do that one now, so we are ready for it. Since 9 9 18, the answer is: 1 989 1089 Okay, take a moment to practice your new skill a few times, then start showing off. You will be amazed at the reaction you get. (Whether or not you decide to reveal the secret is up to you!) Welcome back. At this point, you probably have a few questions, such as: “Can we use this method for multiplying threedigit numbers (or larger) by eleven?” 3 Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 4 4 Secrets of Mental Math Absolutely. For instance, for the problem 314 11, the answer still begins with 3 and ends with 4. Since 3 1 4 , and 145 , the answer is 34 5 4. But we’ll save larger problems like this for later. More practically, you are probably saying to yourself, “Well, this is fine for multiplying by elevens, but what about larger numbers? How do I multiply numbers by twelve, or thirteen, or thirtysix?” My answer to that is, Patience! That’s what the rest of the book is all about. In Chapters 2, 3, 6, and 8, you will learn methods for multiplying together just about any two numbers. Better still, you don’t have to memorize special rules for every number. Just a handful of techniques is all that it takes to multiply numbers in your head, quickly and easily. SQUARING AND MORE Here is another quick trick. As you probably know, the square of a number is a number multiplied by itself. For example, the square of 7 is 7 7 49. Later, I will teach you a simple method that will enable you to easily calculate the square of any twodigit or threedigit (or higher) number. That method is especially simple when the number ends in 5, so let’s do that trick now. To square a twodigit number that ends in 5, you need to remember only two things. 1. The answer begins by multiplying the first digit by the next higher digit. 2. The answer ends in 25. Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 5 Quick Tricks: Easy (and Impressive) Calculations For example, to square the number 35, we simply multiply the first digit (3) by the next higher digit (4), then attach 25. Since 3 4 12, the answer is 1225. Therefore, 35 35 1225. Our steps can be illustrated this way: 35 35 3 4 12 25 5 5 Answer: 1225 How about the square of 85? Since 8 9 72, we immediately get 85 85 7225. 85 85 8 9 72 25 5 5 Answer: 7225 We can use a similar trick when multiplying twodigit numbers with the same first digit, and second digits that sum to 10. The answer begins the same way that it did before (the first digit multiplied by the next higher digit), followed by the product of the second digits. For example, let’s try 83 87. (Both numbers begin with 8, and the last digits sum to 3 7 10.) Since 8 9 72, and 3 7 21, the answer is 7221. 83 87 8 9 72 21 3 7 Answer: 7221 5 Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 6 6 Secrets of Mental Math Similarly, 84 86 7224. Now it’s your turn. Try 26 24 How does the answer begin? With 2 3 6. How does it end? With 6 4 24. Thus 26 24 624. Remember that to use this method, the first digits have to be the same, and the last digits must sum to 10. Thus, we can use this method to instantly determine that 31 39 1209 32 38 1216 33 37 1221 34 36 1224 35 35 1225 You may ask, “What if the last digits do not sum to ten? Can we use this method to multiply twentytwo and twentythree?” Well, not yet. But in Chapter 8, I will show you an easy way to do problems like this using the closetogether method. (For 22 23, you would do 20 25 plus 2 3, to get 500 6 506, but I’m getting ahead of myself!) Not only will you learn how to use these methods, but you will understand why these methods work, too. “Are there any tricks for doing mental addition and subtraction?” Definitely, and that is what the next chapter is all about. If I were forced to summarize my method in three words, I would say, “Left to right.” Here is a sneak preview. Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 7 Quick Tricks: Easy (and Impressive) Calculations Consider the subtraction problem 1241 587 Most people would not like to do this problem in their head (or even on paper!), but let’s simplify it. Instead of subtracting 587, subtract 600. Since 1200 600 600, we have that 1241 600 641 But we have subtracted 13 too much. (We will explain how to quickly determine the 13 in Chapter 1.) Thus, our painfullooking subtraction problem becomes the easy addition problem 641 13 654 which is not too hard to calculate in your head (especially from left to right). Thus, 1241 587 654. Using a little bit of mathematical magic, described in Chapter 9, you will be able to instantly compute the sum of the ten numbers below. 9 5 14 19 33 52 85 137 222 359 935 7 Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 8 8 Secrets of Mental Math Although I won’t reveal the magical secret right now, here is a hint. The answer, 935, has appeared elsewhere in this chapter. More tricks for doing math on paper will be found in Chapter 6. Furthermore, you will be able to quickly give the quotient of the last two numbers: 359 222 1.61 (first three digits) We will have much more to say about division (including decimals and fractions) in Chapter 4. MORE PRACTICAL TIPS Here’s a quick tip for calculating tips. Suppose your bill at a restaurant came to $42, and you wanted to leave a 15% tip. First we calculate 10% of $42, which is $4.20. If we cut that number in half, we get $2.10, which is 5% of the bill. Adding these numbers together gives us $6.30, which is exactly 15% of the bill. We will discuss strategies for calculating sales tax, discounts, compound interest, and other practical items in Chapter 5, along with strategies that you can use for quick mental estimation when an exact answer is not required. IMPROVE YOUR MEMORY In Chapter 7, you will learn a useful technique for memorizing numbers. This will be handy in and out of the classroom. Using an easytolearn system for turning numbers into words, you will be able to quickly and easily memorize any numbers: dates, phone numbers, whatever you want. Speaking of dates, how would you like to be able to figure out the day of the week of any date? You can use this to figure Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 9 Quick Tricks: Easy (and Impressive) Calculations out birth dates, historical dates, future appointments, and so on. I will show you this in more detail later, but here is a simple way to figure out the day of January 1 for any year in the twentyfirst century. First familiarize yourself with the following table. Monday Tuesday Wednesday Thursday Friday Saturday Sunday 1 2 3 4 5 6 7 or 0 For instance, let’s determine the day of the week of January 1, 2030. Take the last two digits of the year, and consider it to be your bill at a restaurant. (In this case, your bill would be $30.) Now add a 25% tip, but keep the change. (You can compute this by cutting the bill in half twice, and ignoring any change. Half of $30 is $15. Then half of $15 is $7.50. Keeping the change results in a $7 tip.) Hence your bill plus tip amounts to $37. To figure out the day of the week, subtract the biggest multiple of 7 (0, 7, 14, 21, 28, 35, 42, 49, . . .) from your total, and that will tell you the day of the week. In this case, 37 35 2, and so January 1, 2030, will occur on 2’s day, namely Tuesday: Bill: Tip: 30 7 37 subtract 7s: 3 5 2 Tuesday How about January 1, 2043: Bill: Tip: 43 10 53 subtract 7s: 4 9 4 Thursday 9 Benj_0307338401_4p_c00_r1.r.qxd 5/4/06 1:46 PM Page 10 10 Secrets of Mental Math Exception: If the year is a leap year, remove $1 from your tip, then proceed as before. For example, for January 1, 2032, a 25% tip of $32 would be $8. Removing one dollar gives a total of 32 7 39. Subtracting the largest multiple of 7 gives us 39 35 4. So January 1, 2032, will be on 4’s day, namely Thursday. For more details that will allow you to compute the day of the week of any date in history, see Chapter 9. (In fact, it’s perfectly okay to read that chapter first!) I know what you are wondering now: “Why didn’t they teach this to us in school?” I’m afraid that there are some questions that even I cannot answer. Are you ready to learn more magical math? Well, what are we waiting for? Let’s go! Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:45 PM Page 11 Chapter 1 A Little Give and Take: Mental Addition and Subtraction For as long as I can remember, I have always found it easier to add and subtract numbers from left to right instead of from right to left. By adding and subtracting numbers this way, I found that I could call out the answers to math problems in class well before my classmates put down their pencils. And I didn’t even need a pencil! In this chapter you will learn the lefttoright method of doing mental addition and subtraction for most numbers that you encounter on a daily basis. These mental skills are not only important for doing the tricks in this book but are also indispensable in school, at work, or any time you use numbers. Soon you will be able to retire your calculator and use the full capacity of your mind as you add and subtract twodigit, threedigit, and even fourdigit numbers with lightning speed. LEFTTORIGHT ADDITION Most of us are taught to do math on paper from right to left. And that’s fine for doing math on paper. But if you want to do Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:45 PM Page 12 12 Secrets of Mental Math math in your head (even faster than you can on paper) there are many good reasons why it is better to work from left to right. After all, you read numbers from left to right, you pronounce numbers from left to right, and so it’s just more natural to think about (and calculate) numbers from left to right. When you compute the answer from right to left (as you probably do on paper), you generate the answer backward. That’s what makes it so hard to do math in your head. Also, if you want to estimate your answer, it’s more important to know that your answer is “a little over 1200” than to know that your answer “ends in 8.” Thus, by working from left to right, you begin with the most significant digits of your problem. If you are used to working from right to left on paper, it may seem unnatural to work with numbers from left to right. But with practice you will find that it is the most natural and efficient way to do mental calculations. With the first set of problems—twodigit addition—the lefttoright method may not seem so advantageous. But be patient. If you stick with me, you will see that the only easy way to solve threedigit and larger addition problems, all subtraction problems, and most definitely all multiplication and division problems is from left to right. The sooner you get accustomed to computing this way, the better. TwoDigit Addition Our assumption in this chapter is that you know how to add and subtract onedigit numbers. We will begin with twodigit addition, something I suspect you can already do fairly well in your head. The following exercises are good practice, however, because the twodigit addition skills that you acquire here will be needed for larger addition problems, as well as virtually all Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:45 PM Page 13 A Little Give and Take: Mental Addition and Subtraction multiplication problems in later chapters. It also illustrates a fundamental principle of mental arithmetic—namely, to simplify your problem by breaking it into smaller, more manageable parts. This is the key to virtually every method you will learn in this book. To paraphrase an old saying, there are three components to success—simplify, simplify, simplify. The easiest twodigit addition problems are those that do not require you to carry any numbers, when the first digits sum to 9 or below and the last digits sum to 9 or below. For example: 47 32 (30 2) To solve 47 32, first add 30, then add 2. After adding 30, you have the simpler problem 77 2, which equals 79. We illustrate this as follows: 47 32 77 2 (first add 30) 79 (then add 2) The above diagram is simply a way of representing the mental processes involved in arriving at an answer using our method. While you need to be able to read and understand such diagrams as you work your way through this book, our method does not require you to write down anything yourself. Now let’s try a calculation that requires you to carry a number: 67 28 (20 8) Adding from left to right, you can simplify the problem by adding 67 20 87; then 87 8 95. 13 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:45 PM Page 14 14 Secrets of Mental Math 67 28 87 8 (first add 20) 95 (then add 8) Now try one on your own, mentally calculating from left to right, and then check below to see how we did it: 84 57 (50 7) How was that? You added 84 50 134 and added 134 7 141. 84 57 134 7 (first add 50) 141 (then add 7) If carrying numbers trips you up a bit, don’t worry about it. This is probably the first time you have ever made a systematic attempt at mental calculation, and if you’re like most people, it will take you time to get used to it. With practice, however, you will begin to see and hear these numbers in your mind, and carrying numbers when you add will come automatically. Try another problem for practice, again computing it in your mind first, then checking how we did it: 68 45 (40 5) You should have added 68 40 108, and then 108 5 113, the final answer. Was that easier? If you would like to try your hand at more twodigit addition problems, check out the set of exercises below. (The answers and computations are at the end of the book.) Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:45 PM Page 15 A Little Give and Take: Mental Addition and Subtraction EXERCISE: TWODIGIT ADDITION 1. 23 16 2. 64 43 3. 95 32 4. 34 26 5. 89 78 6. 73 58 7. 47 36 8. 19 17 9. 55 49 10. 39 38 ThreeDigit Addition The strategy for adding threedigit numbers is the same as for adding twodigit numbers: you add from left to right. After each step, you arrive at a new (and simpler) addition problem. Let’s try the following: 538 327 (300 20 7) Starting with 538, we add 300, then add 20, then add 7. After adding 300 (538 300 838), the problem becomes 838 27. After adding 20 (838 20 858), the problem simplifies to 858 7 865. This thought process can be diagrammed as follows: 538 327 300 838 27 20 858 7 7 865 All mental addition problems can be done by this method. The goal is to keep simplifying the problem until you are just adding a onedigit number. Notice that 538 327 requires you to hold on to six digits in your head, whereas 838 27 and 15 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 16 16 Secrets of Mental Math 858 7 require only five and four digits, respectively. As you simplify the problem, the problem gets easier! Try the following addition problem in your mind before looking to see how we did it: 623 159 (100 50 9) Did you reduce and simplify the problem by adding left to right? After adding the hundreds (623 100 723), you were left with 723 59. Next you should have added the tens (723 50 773), simplifying the problem to 773 9, which you then summed to get 782. Diagrammed, the problem looks like this: 623 159 100 723 59 50 773 9 9 782 When I do these problems mentally, I do not try to see the numbers in my mind—I try to hear them. I hear the problem 623 159 as six hundred twentythree plus one hundred fiftynine; by emphasizing the word hundred to myself, I know where to begin adding. Six plus one equals seven, so my next problem is seven hundred and twentythree plus fiftynine, and so on. When first doing these problems, practice them out loud. Reinforcing yourself verbally will help you learn the mental method much more quickly. Threedigit addition problems really do not get much harder than the following: 858 634 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 17 A Little Give and Take: Mental Addition and Subtraction Now look to see how we did it: 858 634 1458 34 1488 4 1492 600 30 4 At each step I hear (not see) a “new” addition problem. In my mind the problem sounds like this: 858 plus 634 is 1458 plus 34 is 1488 plus 4 is 1492. Your mindtalk may not sound exactly like mine (indeed, you might “see” the numbers instead of “hear” them), but whatever it is you say or visualize to yourself, the point is to reinforce the numbers along the way so that you don’t forget where you are and have to start the addition problem over again. Let’s try another one for practice: 759 496 (400 90 6) Do it in your mind first, then check our computation below: 759 496 1159 96 1249 6 1255 400 90 6 This addition problem is a little more difficult than the last one since it requires you to carry numbers in all three steps. However, with this particular problem you have the option of using an alternative method. I am sure you will agree that it is a 17 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 18 18 Secrets of Mental Math lot easier to add 500 to 759 than it is to add 496, so try adding 500 and then subtracting the difference: 759 496 (500 4) 759 496 1259 4 (first add 500) 1255 (then subtract 4) So far, you have consistently broken up the second number in any problem to add to the first. It really does not matter which number you choose to break up, but it is good to be consistent. That way, your mind will never have to waste time deciding which way to go. If the second number happens to be a lot simpler than the first, I sometimes switch them around, as in the following example: 207 528 207 528 (switch) 528 207 200 728 7 7 735 Let’s finish up by adding threedigit to fourdigit numbers. Since most human memory can hold only about seven or eight digits at a time, this is about as large a problem as you can handle without resorting to artificial memory devices, like fingers, calculators, or the mnemonics taught in Chapter 7. In many addition problems that arise in practice, especially within multiplication problems, one or both of the numbers will end in 0, so we shall emphasize those types of problems. We begin with an easy one: 2700 567 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 19 A Little Give and Take: Mental Addition and Subtraction Since 27 hundred 5 hundred is 32 hundred, we simply attach the 67 to get 32 hundred and 67, or 3267. The process is the same for the following problems: 3240 18 3240 72 Because 40 18 58, the first answer is 3258. For the second problem, since 40 72 exceeds 100, you know the answer will be 33 hundred and something. Because 40 72 112, the answer is 3312. These problems are easy because the (nonzero) digits overlap in only one place, and hence can be solved in a single step. Where digits overlap in two places, you require two steps. For instance: 4560 171 (100 71) This problem requires two steps, as diagrammed the following way: 4560 171 100 4660 71 71 4731 Practice the following threedigit addition exercises, and then add some (pun intended!) of your own if you like until you are comfortable doing them mentally without having to look down at the page. (Answers can be found in the back of the book.) 19 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 20 20 Secrets of Mental Math Carl Friedrich Gauss: Mathematical Prodigy prodigy is a highly talented child, usually called precocious or gifted, and almost always ahead of his peers.The German mathematician Carl Friedrich Gauss (1777–1855) was one such child. He often boasted that he could calculate before he could speak. By the ripe old age of three, before he had been taught any arithmetic, he corrected his father’s payroll by declaring “the reckoning is wrong.” A further check of the numbers proved young Carl correct. As a tenyearold student, Gauss was presented the following mathematical problem: What is the sum of numbers from 1 to 100? While his fellow students were frantically calculating with paper and pencil, Gauss immediately envisioned that if he spread out the numbers 1 through 50 from left to right, and the numbers 51 to 100 from right to left directly below the 1–50 numbers, each combination would add up to 101 (1 100, 2 99, 3 98, . . .). Since there were fifty sums, the answer would be 101 50 5050.To the astonishment of everyone, including the teacher, young Carl got the answer not only ahead of everyone else, but computed it entirely in his mind. He wrote out the answer on his slate, and flung it on the teacher’s desk with a defiant “There it lies.” The teacher was so impressed that he invested his own money to purchase the best available textbook on arithmetic and gave it to Gauss, stating, “He is beyond me, I can teach him nothing more.” Indeed, Gauss became the mathematics teacher of others, and eventually went on to become one of the greatest mathematicians in history, his theories still used today in the service of science. Gauss’s desire to better understand Nature through the language of mathematics was summed up in his motto, taken from Shakespeare’s King Lear (substituting “laws” for “law”): “Thou, nature, art my goddess; to thy laws/My services are bound.” A EXERCISE: THREEDIGIT ADDITION 1. 242 137 2. 312 256 3. 635 814 4. 457 241 5. 912 475 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 21 A Little Give and Take: Mental Addition and Subtraction 6. 852 378 11. 5400 252 7. 12. 457 269 1800 855 8. 13. 878 797 6120 136 9. 276 689 7830 348 14. 10. 15. 877 539 4240 371 LEFTTORIGHT SUBTRACTION For most of us, it is easier to add than to subtract. But if you continue to compute from left to right and to break down problems into simpler components, subtraction can become almost as easy as addition. TwoDigit Subtraction When subtracting twodigit numbers, your goal is to simplify the problem so that you are reduced to subtracting (or adding) a onedigit number. Let’s begin with a very simple subtraction problem: 86 25 (20 5) After each step, you arrive at a new and easier subtraction problem. Here, we first subtract 20 (86 20 66) then we subtract 5 to reach the simpler subtraction problem 66 5 for your final answer of 61. The problem can be diagrammed this way: 86 25 66 5 (first subtract 20) 61 (then subtract 5) 21 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 22 22 Secrets of Mental Math Of course, subtraction problems are considerably easier when there is no borrowing (which occurs when a larger digit is being subtracted from a smaller one). But the good news is that “hard” subtraction problems can usually be turned into “easy” addition problems. For example: 86 29 (20 9) or (30 1) There are two different ways to solve this problem mentally: 1. First subtract 20, then subtract 9: 86 29 66 9 (first subtract 20) 57 (then subtract 9) But for this problem, I would prefer the following strategy: 2. First subtract 30, then add back 1: 86 29 56 1 (first subtract 30) 57 (then add 1) Here is the rule for deciding which method to use: If a twodigit subtraction problem would require borrowing, then round the second number up (to a multiple of ten). Subtract the rounded number, then add back the difference. For example, the problem 54 28 would require borrowing (since 8 is greater than 4), so round 28 up to 30, compute 54 30 24, then add back 2 to get 26 as your final answer: 54 28 (30 2) 54 28 30 24 2 2 26 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 23 A Little Give and Take: Mental Addition and Subtraction Now try your hand (or head) at 81 37. Since 7 is greater than 1, we round 37 up to 40, subtract it from 81 (81 40 41), then add back the difference of 3 to arrive at the final answer: 81 37 40 41 3 3 44 With just a little bit of practice, you will become comfortable working subtraction problems both ways. Just use the rule above to decide which method will work best. EXERCISE: TWODIGIT SUBTRACTION 1. 38 23 2. 84 59 3. 92 34 6. 63 46 7. 51 27 8. 89 48 4. 9. 67 48 125 79 5. 10. 79 29 148 86 ThreeDigit Subtraction Now let’s try a threedigit subtraction problem: 958 417 (400 10 7) This particular problem does not require you to borrow any numbers (since every digit of the second number is less than the digit above it), so you should not find it too hard. Simply subtract one digit at a time, simplifying as you go. 23 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 24 24 Secrets of Mental Math 958 417 400 558 17 10 548 7 7 541 Now let’s look at a threedigit subtraction problem that requires you to borrow a number: 747 598 (600 2) At first glance this probably looks like a pretty tough problem, but if you first subtract 747 600 147, then add back 2, you reach your final answer of 147 2 149. 747 598 600 147 2 2 149 Now try one yourself: 853 692 Did you first subtract 700 from 853? If so, did you get 853 700 153? Since you subtracted by 8 too much, did you add back 8 to reach 161, the final answer? 853 692 700 153 8 8 161 Now, I admit we have been making life easier for you by subtracting numbers that were close to a multiple of 100. (Did you notice?) But what about other problems, like: 725 468 (400 60 8) or (500 ??) Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 25 A Little Give and Take: Mental Addition and Subtraction If you subtract one digit at a time, simplifying as you go, your sequence will look like this: 725 468 325 68 (first subtract 400) 265 8 (then subtract 60) 257 (then subtract 8) What happens if you round up to 500? 725 468 225 ?? (first subtract 500) ?? (then add ??) Subtracting 500 is easy: 725 500 225. But you have subtracted too much. The trick is to figure out exactly how much too much. At first glance, the answer is far from obvious. To find it, you need to know how far 468 is from 500. The answer can be found by using “complements,” a nifty technique that will make many threedigit subtraction problems a lot easier to do. Using Complements (You’re Welcome!) Quick, how far from 100 are each of these numbers? 57 68 49 21 79 Here are the answers: 57 43 100 68 32 100 49 51 100 21 79 100 79 21 100 Notice that for each pair of numbers that add to 100, the first digits (on the left) add to 9 and the last digits (on the right) add 25 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 26 26 Secrets of Mental Math to 10. We say that 43 is the complement of 57, 32 is the complement of 68, and so on. Now you find the complements of these twodigit numbers: 37 59 93 44 08 To find the complement of 37, first figure out what you need to add to 3 in order to get 9. (The answer is 6.) Then figure out what you need to add to 7 to get 10. (The answer is 3.) Hence, 63 is the complement of 37. The other complements are 41, 7, 56, 92. Notice that, like everything else you do as a mathemagician, the complements are determined from left to right. As we have seen, the first digits add to 9, and the second digits add to 10. (An exception occurs in numbers ending in 0—e.g., 30 70 100—but those complements are simple!) What do complements have to do with mental subtraction? Well, they allow you to convert difficult subtraction problems into straightforward addition problems. Let’s consider the last subtraction problem that gave us some trouble: 725 468 (500 32) To begin, you subtracted 500 instead of 468 to arrive at 225 (725 500 225). But then, having subtracted too much, you needed to figure out how much to add back. Using complements gives you the answer in a flash. How far is 468 from 500? The same distance as 68 is from 100. If you find the complement of 68 the way we have shown you, you will arrive at 32. Add 32 to 225, and you will arrive at 257, your final answer. Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 27 A Little Give and Take: Mental Addition and Subtraction 725 468 225 32 (first subtract 500) 257 (then add 32) Try another threedigit subtraction problem: 821 259 (300 41) To compute this mentally, subtract 300 from 821 to arrive at 521, then add back the complement of 59, which is 41, to arrive at 562, our final answer. The procedure looks like this: 821 259 300 521 41 41 562 Here is another problem for you to try: 645 372 (400 28) Check your answer and the procedure for solving the problem below: 645 372 400 245 28 20 265 8 8 273 Subtracting a threedigit number from a fourdigit number is not much harder, as the next example illustrates: 1246 579 (600 21) 27 Benj_0307338401_4p_c01_r1.r.qxd 5/4/06 1:46 PM Page 28 28 Secrets of Mental Math By rounding up, you subtract 600 from 1246, leaving 646, then add back the complement of 79, which is 21. Your final answer is 646 21 667. 1246 579 600 646 21 21 667 Try the threedigit subtraction exercises below, and then create more of your own for additional (or should that be subtractional?) practice. EXERCISE: THREEDIGIT SUBTRACTION 1. 583 271 2. 936 725 3. 587 298 4. 763 486 5. 204 185 6. 793 402 7. 219 176 8. 978 784 9. 455 319 10. 772 596 11. 873 357 12. 564 228 13. 1428 571 14. 2345 678 15. 1776 987 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 29 Chapter 2 Products of a Misspent Youth: Basic Multiplication I probably spent too much time of my childhood thinking about faster and faster ways to perform mental multiplication; I was diagnosed as hyperactive and my parents were told that I had a short attention span and probably would not be successful in school. (Fortunately, my parents ignored that advice. I was also lucky to have some incredibly patient teachers in my first few years of school.) It might have been my short attention span that motivated me to develop quick ways to do arithmetic. I don’t think I had the patience to carry out math problems with pencil and paper. Once you have mastered the techniques described in this chapter, you won’t want to rely on pencil and paper again, either. In this chapter you will learn how to multiply in your head onedigit numbers by twodigit numbers and threedigit numbers. You will also learn a phenomenally fast way to square twodigit numbers. Even friends with calculators won’t be able to keep up with you. Believe me, virtually everyone will be dumbfounded by the fact that such problems can not only be done Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 30 30 Secrets of Mental Math mentally, but can be computed so quickly. I sometimes wonder whether we were not cheated in school; these methods are so simple once you learn them. There is one small prerequisite for mastering the skills in this chapter—you need to know the multiplication tables through ten. In fact, to really make headway, you need to know your multiplication tables backward and forward. For those of you who need to shake the cobwebs loose, consult the multiplication chart below. Once you’ve got your tables down, you are ready to begin. Multiplication Table of Numbers 1–10 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90 100 2BY1 MULTIPLICATION PROBLEMS If you worked your way through Chapter 1, you got into the habit of adding and subtracting from left to right. You will do virtually all the calculations in this chapter from left to right as well. This is undoubtedly the opposite of what you learned in school. Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 31 Products of a Misspent Youth: Basic Multiplication But you’ll soon see how much easier it is to think from left to right than from right to left. (For one thing, you can start to say your answer aloud before you have finished the calculation. That way you seem to be calculating even faster than you are!) Let’s tackle our first problem: 42 7 First, multiply 40 7 280. (Note that 40 7 is just like 4 7, with a friendly zero attached.) Next, multiply 2 7 14. Then add 280 plus 14 (left to right, of course) to arrive at 294, the correct answer. We illustrate this procedure below. 42 (40 2) 7 40 7 280 14 27 294 We have omitted diagramming the mental addition of 280 14, since you learned how to do that computation in the last chapter. At first you will need to look down at the problem while doing the calculation. With practice you will be able to forgo this step and compute the whole thing in your mind. Let’s try another example: 48 (40 8) 4 Your first step is to break down the problem into small multiplication tasks that you can perform mentally with ease. Since 31 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 32 32 Secrets of Mental Math 48 40 8, multiply 40 4 160, then add 8 4 32. The answer is 192. (Note: If you are wondering why this process works, see the Why These Tricks Work section at the end of the chapter.) 48 (40 8) 4 40 4 160 32 84 192 Here are two more mental multiplication problems that you should be able to solve fairly quickly. First calculate 62 3. Then do 71 9. Try doing them in your head before looking at how we did it. 62 (60 2) 3 60 3 180 23 6 186 71 (70 1) 9 70 9 630 19 9 639 These two examples are especially simple because the numbers being added essentially do not overlap at all. When doing 180 6, you can practically hear the answer: One hundred eighty . . . six! Another especially easy type of mental multiplication problem involves numbers that begin with five. When the five is multiplied by an even digit, the first product will be a multiple of 100, which makes the resulting addition problem a snap. Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 33 Products of a Misspent Youth: Basic Multiplication 58 (50 8) 4 50 4 200 32 84 232 Try your hand at the following problem: 87 (80 7) 5 80 5 400 35 75 435 Notice how much easier this problem is to do from left to right. It takes far less time to calculate “400 plus 35” mentally than it does to apply the pencilandpaper method of “putting down the 5 and carrying the 3.” The following two problems are a little harder. 38 (30 8) 9 30 9 270 8 9 72 342 67 (60 7) 8 60 8 480 7 8 56 536 As usual, we break these problems down into easier problems. For the one on the left, multiply 30 9 plus 8 9, giving you 270 72. The addition problem is slightly harder because it involves carrying a number. Here 270 70 2 340 2 342. With practice, you will become more adept at juggling 33 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 34 34 Secrets of Mental Math problems like these in your head, and those that require you to carry numbers will be almost as easy as those that don’t. Rounding Up You saw in the last chapter how useful rounding up can be when it comes to subtraction. The same goes for multiplication, especially when you are multiplying numbers that end in eight or nine. Let’s take the problem of 69 6, illustrated below. On the left we have calculated it the usual way, by adding 360 54. On the right, however, we have rounded 69 up to 70, and subtracted 420 6, which you might find easier to do. 69 (60 9) or 6 60 6 360 9 6 54 414 69 (70 1) 6 70 6 420 1 6 6 414 The following example also shows how much easier rounding up can be: 78 (70 8) or 9 70 9 630 8 9 72 702 78 (80 2) 9 80 9 720 2 9 18 702 The subtraction method works especially well for numbers that are just one or two digits away from a multiple of 10. It does not work so well when you need to round up more than two dig Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 35 Products of a Misspent Youth: Basic Multiplication its because the subtraction portion of the problem gets difficult. As it is, you may prefer to stick with the addition method. Personally, for problems of this size, I use only the addition method because in the time spent deciding which method to use, I could have already done the calculation! So that you can perfect your technique, I strongly recommend practicing more 2by1 multiplication problems. Below are twenty problems for you to tackle. I have supplied you with the answers in the back, including a breakdown of each component of the multiplication. If, after you’ve worked out these problems, you would like to practice more, make up your own. Calculate mentally, then check your answer with a calculator. Once you feel confident that you can perform these problems rapidly in your head, you are ready to move to the next level of mental calculation. EXERCISE: 2BY1 MULTIPLICATION 1. 82 9 2. 43 7 3. 67 5 4. 71 3 5. 93 8 6. 49 9 7. 28 4 8. 53 5 9. 84 5 10. 58 6 11. 97 4 12. 78 2 13. 96 9 14. 75 4 15. 57 7 16. 37 6 17. 46 2 18. 76 8 19. 29 3 20. 64 8 35 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 36 36 Secrets of Mental Math 3BY1 MULTIPLICATION PROBLEMS Now that you know how to do 2by1 multiplication problems in your head, you will find that multiplying three digits by a single digit is not much more difficult. You can get started with the following 3by1 problem (which is really just a 2by1 problem in disguise): 320 (300 20) 7 300 7 2100 20 7 1 0 4 2240 Was that easy for you? (If this problem gave you trouble, you might want to review the addition material in Chapter 1.) Let’s try another 3by1 problem similar to the one you just did, except we have replaced the 0 with a 6 so you have another step to perform: 326 (300 20 6) 7 300 7 2100 20 7 1 0 4 2240 2 6 7 4 2282 In this case, you simply add the product of 6 7, which you already know to be 42, to the first sum of 2240. Since you do not need to carry any numbers, it is easy to add 42 to 2240 to arrive at the total of 2282. In solving this and other 3by1 multiplication problems, the difficult part may be holding in memory the first sum (in this case, 2240) while doing the next multiplication problem (in this Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 37 Products of a Misspent Youth: Basic Multiplication case, 6 7). There is no magic secret to remembering that first number, but with practice I guarantee you will improve your concentration, and holding on to numbers while performing other functions will get easier. Let’s try another problem: 647 (600 40 7) 4 600 4 2400 40 4 1 0 6 2560 7 4 2 8 2588 Even if the numbers are large, the process is just as simple. For example: 987 (900 80 7) 9 900 9 8100 80 9 7 0 2 8820 3 7 9 6 8883 When first solving these problems, you may have to glance down at the page as you go along to remind yourself what the original problem is. This is okay at first. But try to break the habit so that eventually you are holding the problem entirely in memory. In the last section on 2by1 multiplication problems, we saw that problems involving numbers that begin with five are 37 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 38 38 Secrets of Mental Math sometimes especially easy to solve. The same is true for 3by1 problems: 563 (500 60 3) 6 500 6 3000 60 6 360 8 3 6 1 3378 Notice that whenever the first product is a multiple of 1000, the resulting addition problem is no problem at all. This is because you do not have to carry any numbers and the thousands digit does not change. If you were solving the problem above in front of someone else, you would be able to say your first product—“three thousand . . .”—out loud with complete confidence that a carried number would not change it to 4000. (As an added bonus, by quickly saying the first digit, it gives the illusion that you computed the entire answer immediately!) Even if you are practicing alone, saying your first product out loud frees up some memory space while you work on the remaining 2by1 problem, which you can say out loud as well—in this case, “. . . three hundred seventyeight.” Try the same approach in solving the next problem, where the multiplier is a 5: 663 (600 60 3) 5 600 5 3000 60 5 300 3 5 1 5 3315 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 39 Products of a Misspent Youth: Basic Multiplication Because the first two digits of the threedigit number are even, you can say the answer as you calculate it without having to add anything! Don’t you wish all multiplication problems were this easy? Let’s escalate the challenge by trying a couple of problems that require some carrying. 184 (100 80 4) 7 100 7 700 80 7 5 0 6 1260 4 7 2 8 1288 684 (600 80 4) 9 600 9 5400 720 80 9 6120 36 49 6156 In the next two problems you need to carry a number at the end of the problem instead of at the beginning: 648 (600 40 8) 9 600 9 5400 360 40 9 5760 72 89 5832 39 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 40 40 Secrets of Mental Math 376 (300 70 6) 4 300 4 1200 70 4 2 0 8 1480 6 4 2 4 1504 The first part of each of these problems is easy enough to compute mentally. The difficult part comes in holding the preliminary answer in your head while computing the final answer. In the case of the first problem, it is easy to add 5400 360 5760, but you may have to repeat 5760 to yourself several times while you multiply 8 9 72. Then add 5760 72. Sometimes at this stage I will start to say my answer aloud before finishing. Because I know I will have to carry when I add 60 72, I know that 5700 will become 5800, so I say “fiftyeight hundred and . . .” Then I pause to compute 60 72 132. Because I have already carried, I say only the last two digits, “. . . thirtytwo!” And there is the answer: 5832. The next two problems require you to carry two numbers each, so they may take you longer than those you have already done. But with practice you will get faster: 489 (400 80 9) 7 400 7 2800 80 7 5 0 6 3360 9 7 6 3 3423 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 41 Products of a Misspent Youth: Basic Multiplication 224 (200 20 4) 9 200 9 1800 20 9 1 0 8 1980 4 9 3 6 2016 When you are first tackling these problems, repeat the answers to each part out loud as you compute the rest. In the first problem, for example, start by saying, “Twentyeight hundred plus five hundred sixty” a couple of times out loud to reinforce the two numbers in memory while you add them together. Repeat the answer—“thirtythree hundred sixty”— several times while you multiply 9 7 63. Then repeat “thirtythree hundred sixty plus sixtythree” aloud until you compute the final answer of 3423. If you are thinking fast enough to recognize that adding 60 63 will require you to carry a 1, you can begin to give the final answer a split second before you know it—“thirtyfour hundred and . . . twentythree!” Let’s end this section on 3by1 multiplication problems with some special problems you can do in a flash because they require one addition step instead of two: 511 (500 11) 7 500 7 3500 11 7 7 7 3577 41 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 42 42 Secrets of Mental Math 925 (900 25) 8 900 8 7200 200 25 8 7400 825 (800 25) 3 800 3 2400 75 25 3 2475 In general, if the product of the last two digits of the first number and the multiplier is known to you without having to calculate it (for instance, you may know that 25 8 200 automatically since 8 quarters equals $2.00), you will get to the final answer much more quickly. For instance, if you know without calculating that 75 4 300, then it is easy to compute 975 4: 975 (900 75) 4 900 4 3600 300 75 4 3900 To reinforce what you have just learned, solve the following 3by1 multiplication problems in your head; then check your computations and answers with ours (in the back of the book). I can assure you from experience that doing mental calculations is just like riding a bicycle or typing. It might seem impossible at first, but once you’ve mastered it, you will never forget how to do it. Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 43 Products of a Misspent Youth: Basic Multiplication EXERCISE: 3BY1 MULTIPLICATION 1. 431 6 2. 637 5 3. 862 4 4. 957 6 5. 927 7 6. 728 2 7. 328 6 8. 529 9 9. 807 9 10. 587 4 11. 184 7 12. 214 8 13. 757 8 14. 259 7 15. 297 8 16. 751 9 17. 457 7 18. 339 8 19. 134 8 20. 611 3 21. 578 9 22. 247 5 23. 188 6 24. 968 6 25. 499 9 26. 670 4 27. 429 3 28. 862 5 29. 285 6 30. 488 9 31. 693 6 32. 722 9 33. 457 9 34. 767 3 35. 312 9 36. 691 3 43 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 44 44 Secrets of Mental Math BE THERE OR B2 : SQUARING TWODIGIT NUMBERS Squaring numbers in your head (multiplying a number by itself) is one of the easiest yet most impressive feats of mental calculation you can do. I can still recall where I was when I discovered how to do it. I was thirteen, sitting on a bus on the way to visit my father at work in downtown Cleveland. It was a trip I made often, so my mind began to wander. I’m not sure why, but I began thinking about the numbers that add up to 20, and I wondered, how large could the product of two such numbers get? I started in the middle with 10 10 (or 102), the product of which is 100. Next, I multiplied 9 11 99, 8 12 96, 7 13 91, 6 14 84, 5 15 75, 4 16 64, and so on. I noticed that the products were getting smaller, and their difference from 100 was 1, 4, 9, 16, 25, 36, . . . —or 12, 22, 32, 42, 52, 62, . . . (see table below). Numbers that add to 20 Distance from 10 Their product Product’s difference from 100 10 10 0 100 0 9 11 1 99 1 8 12 2 96 4 7 13 3 91 9 6 14 4 84 16 5 15 5 75 25 4 16 6 64 36 3 17 7 51 49 2 18 8 36 64 1 19 9 19 81 I found this pattern astonishing. Next I tried numbers that add to 26 and got similar results. First I worked out 132 169, then computed 12 14 168, 11 15 165, 10 16 160, Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 45 Products of a Misspent Youth: Basic Multiplication 9 17 153, and so on. Just as before, the distances these products were from 169 was 12, 22, 32, 42, and so on (see table below). There is actually a simple algebraic explanation for this phenomenon (see Why These Tricks Work, page 50). At the time, I didn’t know my algebra well enough to prove that this pattern would always occur, but I experimented with enough examples to become convinced of it. Then I realized that this pattern could help me square numbers more easily. Suppose I wanted to square the number 13. Instead of multiplying 13 13, Numbers that add to 26 Distance from 13 Their product Product’s difference from 169 13 13 0 169 0 12 14 1 168 1 11 15 2 165 4 10 16 3 160 9 9 17 4 153 16 8 18 5 144 25 why not get an approximate answer by using two numbers that are easier to multiply but also add to 26? I chose 10 16 160. To get the final answer, I just added 32 9 (since 10 and 16 are each 3 away from 13). Thus, 132 160 9 169. Neat! This method is diagrammed as follows: 3 16 160 32 169 2 13 3 10 Now let’s see how this works for another square: 45 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 46 46 Secrets of Mental Math 1 42 1680 12 1681 412 1 40 To square 41, subtract 1 to obtain 40 and add 1 to obtain 42. Next multiply 40 42. Don’t panic! This is simply a 2by1 multiplication problem (specifically, 4 42) in disguise. Since 4 42 168, 40 42 1680. Almost done! All you have to add is the square of 1 (the number by which you went up and down from 41), giving you 1680 1 1681. Can squaring a twodigit number be this easy? Yes, with this method and a little practice, it can. And it works whether you initially round down or round up. For example, let’s examine 772, working it out both by rounding up and by rounding down: 7 84 5880 72 5929 772 7 70 or 3 77 80 5920 32 5929 2 3 74 In this instance the advantage of rounding up is that you are virtually done as soon as you have completed the multiplication problem because it is simple to add 9 to a number ending in 0! In fact, for all twodigit squares, I always round up or down to the nearest multiple of 10. So if the number to be squared ends in 6, 7, 8, or 9, round up, and if the number to be squared ends in 1, 2, 3, or 4, round down. (If the number ends in 5, you Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 47 Products of a Misspent Youth: Basic Multiplication do both!) With this strategy you will add only the numbers 1, 4, 9, 16, or 25 to your first calculation. Let’s try another problem. Calculate 562 in your head before looking at how we did it, below: 4 56 60 3120 42 3136 2 4 52 Squaring numbers that end in 5 is even easier. Since you will always round up and down by 5, the numbers to be multiplied will both be multiples of 10. Hence, the multiplication and the addition are especially simple. We have worked out 852 and 352, below: 5 85 35 90 7200 52 7225 2 5 80 5 40 1200 52 1225 2 5 30 As you saw in Chapter 0, when you are squaring a number that ends in 5, rounding up and down allows you to blurt out the first part of the answer immediately and then finish it with 25. For example, if you want to compute 752, rounding up to 80 and down to 70 will give you “Fiftysix hundred and . . . twentyfive!” For numbers ending in 5, you should have no trouble beating someone with a calculator, and with a little practice with the 47 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 48 48 Secrets of Mental Math other squares, it won’t be long before you can beat the calculator with any twodigit square number. Even large numbers are not to be feared. You can ask someone to give you a really big twodigit number, something in the high 90s, and it will sound as though you’ve chosen an impossible problem to compute. But, in fact, these are even easier because they allow you to round up to 100. Let’s say your audience gives you 962. Try it yourself, and then check how we did it. 4 96 100 9200 42 9216 2 4 92 Wasn’t that easy? You should have rounded up by 4 to 100 and down by 4 to 92, and then multiplied 100 92 to get 9200. At this point you can say out loud, “Ninetytwo hundred,” and then finish up with “sixteen” and enjoy the applause! EXERCISE: TWODIGIT SQUARES Compute the following: 1. 142 2. 272 3. 652 4. 892 5. 982 6. 312 7. 412 8. 592 9. 262 10. 532 11. 212 12. 642 13. 422 14. 552 15. 752 16. 452 17. 842 18. 672 19. 1032 20. 2082 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 49 Products of a Misspent Youth: Basic Multiplication Zerah Colburn: Entertaining Calculations ne of the first lightning calculators to capitalize on his talent was Zerah Colburn (1804–1839), an American farmer’s son from Vermont who learned the multiplication tables to 100 before he could even read or write. By the age of six, young Zerah’s father took him on the road, where his performances generated enough capital to send him to school in Paris and London. By age eight he was internationally famous, performing lightning calculations in England, and was described in the Annual Register as “the most singular phenomenon in the history of the human mind that perhaps ever existed.” No less than Michael Faraday and Samuel Morse admired him. No matter where he went, Colburn met all challengers with speed and precision. He tells us in his autobiography of one set of problems he was given in New Hampshire in June 1811: “How many days and hours since the Christian Era commenced, 1811 years ago? Answered in twenty seconds: 661,015 days, 15,864,360 hours. How many seconds in eleven years? Answered in four seconds; 346,896,000.” Colburn used the same techniques described in this book to compute entirely in his head problems given to him. For example, he would factor large numbers into smaller numbers and then multiply: Colburn once multiplied 21,734 543 by factoring 543 into 181 3. He then multiplied 21,734 181 to arrive at 3,933,854, and finally multiplied that figure by 3, for a total of 11,801,562. As is often the case with lightning calculators, interest in Colburn’s amazing skills diminished with time, and by the age of twenty he had returned to America and become a Methodist preacher. He died at a youthful thirtyfive. In summarizing his skills as a lightning calculator, and the advantage such an ability affords, Colburn reflected, “True, the method . . . requires a much larger number of figures than the common Rule, but it will be remembered that pen, ink and paper cost Zerah very little when engaged in a sum.” O 49 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 50 50 Secrets of Mental Math WHY THESE TRICKS WORK This section is presented for teachers, students, math buffs, and anyone curious as to why our methods work. Some people may find the theory as interesting as the application. Fortunately, you need not understand why our methods work in order to understand how to apply them. All magic tricks have a rational explanation behind them, and mathemagical tricks are no different. It is here that the mathemagician reveals his deepest secrets! In this chapter on multiplication problems, the distributive law is what allows us to break down problems into their component parts. The distributive law states that for any numbers a, b, and c: (b c) a (b a) (c a) That is, the outside term, a, is distributed, or separately applied, to each of the inside terms, b and c. For example, in our first mental multiplication problem of 42 7, we arrived at the answer by treating 42 as 40 2, then distributing the 7 as follows: 42 7 (40 2) 7 (40 7) (2 7) 280 14 294 You may wonder why the distributive law works in the first place. To understand it intuitively, imagine having 7 bags, each containing 42 coins, 40 of which are gold and 2 of which are silver. How many coins do you have altogether? There are two ways to arrive at the answer. In the first place, by the very definition of multiplication, there are 42 7 coins. On the other hand, there are 40 7 gold coins and 2 7 silver coins. Hence, Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 51 Products of a Misspent Youth: Basic Multiplication we have (40 7) (2 7) coins altogether. By answering our question two ways, we have 42 7 (40 7) (2 7). Notice that the numbers 7, 40, and 2 could be replaced by any numbers (a, b, or c) and the same logic would apply. That’s why the distributive law works! Using similar reasoning with gold, silver, and copper coins we can derive: (b c d) a (b a) (c a) (d a) Hence, to do the problem 326 7, we break up 326 as 300 20 6, then distribute the 7, as follows: 326 7 (300 20 6) 7 (300 7) (20 7) (6 7), which we then add up to get our answer. As for squaring, the following algebra justifies my method. For any numbers A and d A2 (A d) (A d) d2 Here, A is the number being squared; d can be any number, but I choose it to be the distance from A to the nearest multiple of 10. Hence, for 772, I set d 3 and our formula tells us that 772 (77 3) (77 3) 32 (80 74) 9 5929. The following algebraic relationship also works to explain my squaring method: (z d)2 z2 2zd d2 z(z 2d) d2 Hence, to square 41, we set z 40 and d 1 to get: 412 (40 1)2 40 (40 2) 12 1681 51 Benj_0307338401_4p_c02_r1.r.qxd 5/4/06 1:45 PM Page 52 52 Secrets of Mental Math Similarly, (z d)2 z(z 2d) d2 To find 772 when z 80 and d 3, 772 (80 3)2 80 (80 6) 32 80 74 9 5929 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 53 Chapter 3 New and Improved Products: Intermediate Multiplication Mathemagics really gets exciting when you perform in front of an audience. I experienced my first public performance in eighth grade, at the fairly advanced age of thirteen. Many mathemagicians begin even earlier. Zerah Colburn (1804–1839), for example, reportedly could do lightning calculations before he could read or write, and he was entertaining audiences by the age of six! When I was thirteen, my algebra teacher did a problem on the board for which the answer was 1082. Not content to stop there, I blurted out, “108 squared is simply 11,664!” The teacher did the calculation on the board and arrived at the same answer. Looking a bit startled, she said, “Yes, that’s right. How did you do it?” So I told her, “I went down 8 to 100 and up 8 to 116. I then multiplied 116 100, which is 11,600, and just added the square of 8, to get 11,664.” She had never seen that method before. I was thrilled. Thoughts of “Benjamin’s Theorem” popped into my head. I actually believed I had discovered something new. When I finally ran across this method a few years later in a book by Martin Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 54 54 Secrets of Mental Math Gardner on recreational math, Mathematical Carnival (1965), it ruined my day! Still, the fact that I had discovered it for myself was very exciting to me. You, too, can impress your friends (or teachers) with some fairly amazing mental multiplication. At the end of the last chapter you learned how to multiply a twodigit number by itself. In this chapter you will learn how to multiply two different twodigit numbers, a challenging yet more creative task. You will then try your hand—or, more accurately, your brain—at threedigit squares. You do not have to know how to do 2by2 multiplication problems to tackle threedigit squares, so you can learn either skill first. 2BY2 MULTIPLICATION PROBLEMS When squaring twodigit numbers, the method is always the same. When multiplying twodigit numbers, however, you can use lots of different methods to arrive at the same answer. For me, this is where the fun begins. The first method you will learn is the “addition method,” which can be used to solve all 2by2 multiplication problems. The Addition Method To use the addition method to multiply any two twodigit numbers, all you need to do is perform two 2by1 multiplication problems and add the results together. For example: 46 42 (40 2) 40 46 1840 92 2 46 1932 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 55 New and Improved Products: Intermediate Multiplication Here you break up 42 into 40 and 2, two numbers that are easy to multiply. Then you multiply 40 46, which is just 4 46 with a 0 attached, or 1840. Then you multiply 2 46 92. Finally, you add 1840 92 1932, as diagrammed above. Here’s another way to do the same problem: 46 (40 6) 42 40 42 1680 6 42 252 1932 The catch here is that multiplying 6 42 is harder to do than multiplying 2 46, as in the first problem. Moreover, adding 1680 252 is more difficult than adding 1840 92. So how do you decide which number to break up? I try to choose the number that will produce the easier addition problem. In most cases—but not all—you will want to break up the number with the smaller last digit because it usually produces a smaller second number for you to add. Now try your hand at the following problems: 48 73 (70 3) 70 48 3360 144 3 48 3504 81 (80 1) 59 80 59 4720 59 1 59 4779 The last problem illustrates why numbers that end in 1 are especially attractive to break up. If both numbers end in the 55 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 56 56 Secrets of Mental Math same digit, you should break up the larger number as illustrated below: 84 (80 4) 34 80 34 2720 136 4 34 2856 If one number is much larger than the other, it often pays to break up the larger number, even if it has a larger last digit. You will see what I mean when you try the following problem two different ways: 74 (70 4) 13 70 13 910 4 13 52 962 74 13 (10 3) 10 74 740 3 74 222 962 Did you find the first method to be faster than the second? I did. Here’s another exception to the rule of breaking up the number with the smaller last digit. When you multiply a number in the fifties by an even number, you’ll want to break up the number in the fifties: 84 59 (50 9) 50 84 4200 756 9 84 4956 Benj_0307338401_4p_c09_r1.r.qxd 5/4/06 1:39 PM Page 216 216 Secrets of Mental Math rest of your life. Once all future dates are taken care of, we can look back into the past and determine the days of the week for any date in the 1900s or any other century. Every year is assigned a code number, and for 2006 that year code happens to be 0 (see page 218). Now, to calculate the day of the week, you simply add the month code plus the date code plus the year code. Thus for December 3, 2006, we compute Month Code Date Year Code 4 3 0 7 Hence, this date will be on Day 7, which is Sunday. How about November 18, 2006? Since November has a month code of 2, we have Month Code Date Year Code 2 18 0 20. Now since the week repeats every seven days, we can subtract any multiple of 7 from our answer (7, 14, 21, 28, 35, . . .) and this will not change the day of the week. So our final step is to subtract the biggest multiple of 7 to get 20 14 6. Hence November 18, 2006, occurs on Saturday. How about 2007? Well, what happens to your birthday as you go from one year to the next? For most years, there are 365 days, and since 364 is a multiple of 7 (7 52 364), then the day of the week of your birthday will shift forward by one day in most years. If there are 366 days between your birthdays, then it will shift forward by two days. Hence, for 2007 we calculate the day of the week just as before, but now we use a year code of 1. Next, 2008 is a leap year. (Leap years occur every four years, so 2000, 2004, 2008, 2012, . . . , 2096 are the leap years of the twentyfirst century.) Hence, for 2008, the year Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 57 New and Improved Products: Intermediate Multiplication The last digit of the number 84 is smaller than the last digit of 59, but if you break up 59, your product will be a multiple of 100, just as 4200 is, in the example above. This makes the subsequent addition problem much easier. Now try an easy problem of a different sort: 42 11 (10 1) 10 42 420 42 1 42 462 Though the calculation above is pretty simple, there is an even easier and faster way to multiply any twodigit number by 11. This is mathemagics at its best: you won’t believe your eyes when you see it (unless you remember it from Chapter 0)! Here’s how it works. Suppose you have a twodigit number whose digits add up to 9 or less. To multiply this number by 11, merely add the two digits together and insert the total between the original two digits. For example, to do 42 11, first do 4 2 6. If you place the 6 between the 4 and the 2, you get 462, the answer to the problem! 42 11 42 6 462 Try 54 11 by this method. 54 11 54 9 594 57 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 59 New and Improved Products: Intermediate Multiplication 89 72 (70 2) 70 89 6230 178 2 89 6408 If you got the right answer the first or second time, pat yourself on the back. The 2by2 multiplication problems really do not get any tougher than this. If you did not get the answer right away, don’t worry. In the next two sections, I’ll teach you some much easier strategies for dealing with problems like this. But before you read on, practice the addition method on the following multiplication problems. EXERCISE: 2BY2 ADDITIONMETHOD MULTIPLICATION PROBLEMS 1. 31 41 2. 27 18 3. 59 26 4. 53 58 5. 77 43 6. 23 84 7. 62 94 8. 88 76 9. 92 35 10. 34 11 11. 85 11 The Subtraction Method The subtraction method really comes in handy when one of the numbers you want to multiply ends in 8 or 9. The following problem illustrates what I mean: 59 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 60 60 Secrets of Mental Math 59 (60 1) 17 60 17 1020 1 17 17 1003 Although most people find addition easier than subtraction, it is usually easier to subtract a small number than to add a big number. (If we had done this problem by the addition method, we would have added 850 153 1003.) Now let’s do the challenging problem from the end of the last section: 89 (90 1) 72 90 72 6480 1 72 72 6408 Wasn’t that a whole lot easier? Now, here’s a problem where one number ends in 8: 88 (90 2) 23 90 23 2070 2 23 46 2024 In this case you should treat 88 as 90 2, then multiply 90 23 2070. But you multiplied by too much. How much? By 2 23, or 46 too much. So subtract 46 from 2070 to arrive at 2024, the final answer. Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 61 New and Improved Products: Intermediate Multiplication I want to emphasize here that it is important to work out these problems in your head and not simply look to see how we did it in the diagram. Go through them and say the steps to yourself or even out loud to reinforce your thoughts. Not only do I use the subtraction method with numbers that end in 8 or 9, but also for numbers in the high 90s because 100 is such a convenient number to multiply. For example, if someone asked me to multiply 96 73, I would immediately round up 96 to 100: 96 (100 4) 73 100 73 7300 4 73 292 7008 When the subtraction component of a multiplication problem requires you to borrow a number, using complements (as we learned in Chapter 1) can help you arrive at the answer more quickly. You’ll see what I mean as you work your way through the problems below. For example, subtract 340 78. We know the answer will be in the 200s. The difference between 40 and 78 is 38. Now take the complement of 38 to get 62. And that’s the answer, 262! 340 78 262 78 40 38 Complement of 38 62 Now let’s try another problem: 61 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 62 62 Secrets of Mental Math 88 (90 2) 76 90 76 6840 2 76 152 There are two ways to perform the subtraction component of this problem. The “long” way subtracts 200 and adds back 48: 6840 152 6640 48 (first subtract 200) 6688 (then add 48) The short way is to realize that the answer will be 66 hundred and something. To determine something, we subtract 52 40 12 and then find the complement of 12, which is 88. Hence the answer is 6688. Try this one. 67 59 (60 1) 60 67 4020 1 67 67 3953 Again, you can see that the answer will be 3900 and something. Because 67 20 47, the complement 53 means the answer is 3953. As you may have realized, you can use this method with any subtraction problem that requires you to borrow a number, not just those that are part of a multiplication problem. All of this is further proof, if you need it, that complements are a very powerful tool in mathemagics. Master this technique and, pretty soon, people will be complimenting you! Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 63 New and Improved Products: Intermediate Multiplication EXERCISE: 2BY2 SUBTRACTIONMETHOD MULTIPLICATION PROBLEMS 1. 29 45 2. 98 43 3. 47 59 4. 68 38 5. 96 29 6. 79 54 7. 37 19 8. 87 22 9. 85 38 10. 57 39 11. 88 49 The Factoring Method The factoring method is my favorite method of multiplying twodigit numbers since it involves no addition or subtraction at all. You use it when one of the numbers in a twodigit multiplication problem can be factored into onedigit numbers. To factor a number means to break it down into onedigit numbers that, when multiplied together, give the original number. For example, the number 24 can be factored into 8 3 or 6 4. (It can also be factored into 12 2, but we prefer to use only singledigit factors.) Here are some other examples of factored numbers: 42 7 6 63 9 7 84 7 6 2 or 7 4 3 63 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 64 64 Secrets of Mental Math To see how factoring makes multiplication easier, consider the following problem: 46 42 7 6 Previously we solved this problem by multiplying 46 40 and 46 2 and adding the products together. To use the factoring method, treat 42 as 7 6 and begin by multiplying 46 7, which is 322. Then multiply 322 6 for the final answer of 1932. You already know how to do 2by1 and 3by1 multiplication problems, so this should not be too hard: 46 42 46 (7 6) (46 7) 6 322 6 1932 Of course, this problem could also have been solved by reversing the factors of 42: 46 42 46 (6 7) (46 6) 7 276 7 1932 In this case, it is easier to multiply 322 6 than it is to multiply 276 7. In most cases, I like to use the larger factor in solving the initial 2by1 problem and to reserve the smaller factor for the 3by1 component of the problem. Factoring results in a 2by2 multiplication problem being simplified to an easier 3by1 (or sometimes 2by1) multiplication problem. The advantage of the factoring method in mental calculation is you do not have to hold much in memory. Let’s look at another example, 75 63: 75 63 75 (9 7) (75 9) 7 675 7 4725 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 65 New and Improved Products: Intermediate Multiplication As before, you simplify this 2by2 problem by factoring 63 into 9 7 and then multiplying 75 by these factors. (By the way, the reason we can shift parentheses in the second step is the associative law of multiplication.) 63 75 63 (5 5 3) (63 5) 5 3 315 5 3 1575 3 4725 Try the following problem for practice: 57 24 57 8 3 456 3 1368 You could have factored 24 as 6 4 for another easy computation: 57 24 57 6 4 342 4 1368 Compare this approach with the addition method: 57 24 (20 4) 20 57 1140 228 4 57 or 1368 With the addition method, you problems and then add. With the just two multiplication problems: then you are done. The factoring your memory. 57 (50 7) 24 50 24 1200 7 24 168 1368 have to perform two 2by1 factoring method, you have a 2by1 and a 3by1, and method is usually easier on 65 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 66 66 Secrets of Mental Math Remember that challenging multiplication problem earlier in this chapter? Here it is again: 89 72 We tackled that problem easily enough with the subtraction method, but factoring works even faster: 89 72 89 9 8 801 8 6408 The problem is especially easy because of the 0 in the middle of 801. Our next example illustrates that it sometimes pays to factor the numbers in an order that exploits this situation. Let’s look at two ways of computing 67 42: 67 42 67 7 6 469 6 2814 67 42 67 6 7 402 7 2814 Ordinarily you should factor 42 into 7 6, as in the first example, following the rule of using the larger factor first. But the problem is easier to solve if you factor 42 into 6 7 because it creates a number with a 0 in the center, which is easier to multiply. I call such numbers friendly products. Look for the friendly product in the problem done two ways below: 43 56 43 8 7 344 7 2408 43 56 43 7 8 301 8 2408 Did you think the second way was easier? When using the factoring method, it pays to find friendly Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 67 New and Improved Products: Intermediate Multiplication products whenever you can. The following list should help. I don’t expect you to memorize it so much as to familiarize yourself with it. With practice you will be able to nose out friendly products more often, and the list will become more meaningful. Numbers with Friendly Products 12: 12 9 108 13: 13 8 104 15: 15 7 105 17: 17 6 102 18: 18 6 108 21: 21 5 105 23: 23 9 207 25: 25 4 100, 25 8 200 26: 26 4 104, 26 8 208 27: 27 4 108 29: 29 7 203 34: 34 3 102, 34 6 204, 34 9 306 35: 35 3 105 36: 36 3 108 38: 38 8 304 41: 41 5 205 43: 43 7 301 44: 44 7 308 45: 45 9 405 51: 51 2 102, 51 4 204, 51 6 306, 51 8 408 52: 52 2 104, 52 4 208 53: 53 2 106 54: 54 2 108 56: 56 9 504 61: 61 5 305 63: 63 8 504 67 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 68 68 Secrets of Mental Math 67: 67 3 201, 67 6 402, 67 9 603 68: 68 3 204, 68 6 408 72: 72 7 504 76: 76 4 304, 76 8 608 77: 77 4 308 78: 78 9 702 81: 81 5 405 84: 84 6 504 88: 88 8 704 89: 89 9 801 Previously in this chapter you learned how easy it is to multiply numbers by 11. It usually pays to use the factoring method when one of the numbers is a multiple of 11, as in the examples below: 52 33 52 11 3 572 3 1716 83 66 83 11 6 913 6 5478 EXERCISE: 2BY2 FACTORINGMETHOD MULTIPLICATION PROBLEMS 1. 27 14 2. 86 28 3. 57 14 4. 81 48 5. 56 29 6. 83 18 7. 72 17 8. 85 42 9. 33 16 10. 62 77 11. 45 36 12. 48 37 Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 69 New and Improved Products: Intermediate Multiplication APPROACHING MULTIPLICATION CREATIVELY I mentioned at the beginning of the chapter that multiplication problems are fun because they can be solved any number of ways. Now that you know what I mean,