Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks

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 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 10-4 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 time-saving 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. 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 lightning-fast 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 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 out-loud 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 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 in-between 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.

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 cross-multiplying I was hooked and became, for a short while, a generally unbearable 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 day-to-day 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…

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 quick-witted, 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 two-digit, three-digit, and four-digit 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 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 two-digit number. Fifty-seven, shouts one. Art points to another who yells out, Twenty-three.

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 two-digit numbers faster than his button-pushers 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 two-digit 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 three-digit numbers without even writing down the answer. Five hundred and seventy-two, 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. Someone else blurts out, 262. That’ll give you 68,644. Sensing he delayed just an instant on