Arthur Brisbane was a run-of-the-mill newspaperman, no better and no worse than a he receives free publicity in newspa. His techniques are explained in his book Secrets of. Mental Math: The Mathemagician's Guide to Lightning Calculation and. Amazing Math Tricks. Prolific math. October 30, Secrets of Mental Math. Arthur T. Benjamin. Harvey Mudd College. T e x t. Multiplication and Squaring. Squaring numbers. 3.
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SECRETS OF MENTAL MATH The Mathemagicians Guide to Lightning Calculation and Amazing Math Tricks Arthur Benjamin and Michael Shermer This book. 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. Mathematics in your head – the secrets of mental math. 1. Fundamentals: mental addition, subtraction, multiplication and division, and. “gestimation”. Addition.
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. But you'll also find yourself performing just for your- self. You'll find you're able to do problems that you didn't think you could. You'll be impressed. Now, counting on your fingers is one thing one finger's worth. But have you ever found yourself counting out loud or 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. 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 accu- rately 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 prob- lems fast with good guesses, actually great guesses, within a per- cent 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. 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 April 15, , the day the Titanic sank, was a Monday. The first human to walk on the moon set foot there on July 20, , a Sunday.
You'll probably never forget that the United States was attacked by terrorists on September 11, With Secrets of Mental Math, you'll always be able to show it was a Tuesday.
There are relationships in Nature that numbers describe bet- ter than any other way we know. There are simple numbers that you can count on your hands: But Foreword xv 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 lan- guage. 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 eso- teric 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 Leib- niz, 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 paint- ing, or an electronic keyboard may make us forget the magnifi- cence 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 Os 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 xviii Foreword math nut. Immunizations against such afflictions are not avail- able. You have to recover all by yourself.
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 sub- ject. 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 depend- able tool for day-to-day use that enables us to handle our compli- cated 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: Let it entertain you, and have fun with it. That, with the occasional good deed, a slice of pizza no anchovies! 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 "mathemag- ics," 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 xx Prologue in on his secrets of "math genius.
This particular night at the Magic Castle, Art begins by ask- ing if anyone in the audience has a calculator. A group of engi- neers 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. Art points to another who yells out, "Twenty-three.
As each participant indicates his calculator reads , the audi- ence 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 Then, in large, bold writing for everyone to see, Art writes: 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 audi- ence: Art then offers to square three-digit numbers without even writing down the answer. Art's reply comes less than a second later: The challenge comes — With no pause, Art squares the number, ", 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 four-digit number. A woman calls out, "1,," and Art instantly responds, "That's 1,, I'm not supposed to beat the calculators on these. Let's try another one.
Pausing briefly between digits, Art responds: It is the same everywhere Art Benjamin goes, whether it is a high school auditorium, a college classroom, a professional con- ference, the Magic Castle, or a television studio.
Professor Ben- jamin 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: Art was born in Cleveland on March 19, which he calcu- lates was a Sunday, a skill he will teach you in Chapter 9.
A hyperactive child, Art drove his teachers mad with his class- room antics, which included correcting the mathematical mis- takes they occasionally made. Throughout this book when teaching you his mathematical secrets, Art recalls when and xxii Prologue where he learned these skills, so I will save the fascinating sto- ries for him to tell you. Art Benjamin is an extraordinary individual with an extraor- dinary program to teach you rapid mental calculation.
I offer these claims without hesitation, and ask only that you remem- ber this does not come from a couple of guys promising miracles if you will only call our hotline. Art and I are both creden- tialed 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!
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 full-length show, called Mathemagics, where I would demon- strate 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. 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. 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 prac- tice 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. 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 meth- ods 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 some- one 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.
It's very easy once you know the secret. Consider the problem: Now you try: Without looking at the answer or writing any- thing down, what is 81x11?
Did you get ? Now before you get too excited, I have shown you only half of what you need to know. As before, the 3 goes in between the numbers, but the 1 needs to be added to the 8 to get the correct answer: I QuickTricks: Easy and Impressive Calculations 3 Here is another example.
Try 57 x As fast as you can, what is 77 X II? If you got the answer , 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 two-digit num- ber by eleven, it won't be long before they ask you to do 99 x Let's do that one now, so we are ready for it. I 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 ques- tions, such as: For instance, for the problem x 11, the answer still begins with 3 and ends with 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 thirty-six?
That's what the rest of the book is all about. In Chapters 2, 3, 6, and 8, you will learn meth- ods 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 num- bers in your head, quickly and easily. As you probably know, the square of a number is a number multiplied by itself.
Later, I will teach you a simple method that will enable you to easily calculate the square of any two-digit or three-digit or higher number. That method is especially simple when the number ends in 5, so let's do that trick now.
To square a two-digit number that ends in 5, you need to remember only two things.
The answer begins by multiplying the first digit by the next higher digit. The answer ends in Easy and Impressive Calculations 5 For example, to square the number 35, we simply multiply the first digit 3 by the next higher digit 4 , then attach Our steps can be illustrated this way: The answer begins the same way that it did before the first digit multiplied by the next higher digit , followed by the prod- uct of the second digits. For example, let's try 83 x Now it's your turn. Try 26 X 24 How does the answer begin?
How does it end? Remember that to use this method, the first digits have to be the same, and the last digits must sum to Can we use this method to multiply twenty-two and twenty-three? But in Chapter 8, 1 will show you an easy way to do problems like this using the close-together method.
Not only will you learn how to use these methods, but you will understand why these methods work, too. If I were forced to summarize my method in three words, I would say, "Left to right. Quick Tricks: Easy and Impressive Calculations 7 Consider the subtraction problem - Most people would not like to do this problem in their head or even on paper! Instead of subtracting , subtract We will explain how to quickly determine the 13 in Chapter 1. Using a little bit of mathematical magic, described in Chapter 9, you will be able to instantly compute the sum of the ten num- bers below.
The answer, , 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: We will discuss strategies for calculating sales tax, dis- counts, compound interest, and other practical items in Chapter 5, along with strategies that you can use for quick mental esti- mation when an exact answer is not required.
This will be handy in and out of the classroom. Using an easy-to-learn system for turning numbers into words, you will be able to quickly and easily memorize any numbers: 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 Quick Tricks: Easy and Impressive Calculations 9 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 twenty-first century. First familiarize yourself with the following table. Take the last two digits of the year, and consider it to be your bill at a restaurant.
You can compute this by cutting the bill in half twice, and ignoring any change. To figure out the day of the week, subtract the biggest mul- tiple of 7 0, 7, 14, 21, 28, 35, 42, 49,. So January 1, , 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: Are you ready to learn more magical math? Well, what are we waiting for?
Let's go! 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 left-to-right 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 indis- pensable in school, at work, or any time you use numbers. Soon you will be able to retire your calculator and use the full capac- ity of your mind as you add and subtract two-digit, three-digit, and even four-digit numbers with lightning speed. And that's fine for doing math on paper.
But if you want to do 1 2 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 esti- mate your answer, it's more important to know that your answer is "a little over " than to know that your answer "ends in 8. 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 — two-digit addition — the left- to-right method may not seem so advantageous. But be patient. If you stick with me, you will see that the only easy way to solve three-digit and larger addition problems, all subtraction prob- lems, and most definitely all multiplication and division prob- lems is from left to right.
The sooner you get accustomed to computing this way, the better. Two-Digit Addition Our assumption in this chapter is that you know how to add and subtract one-digit numbers. We will begin with two-digit addition, something I suspect you can already do fairly well in your head. The following exercises are good practice, however, because the two-digit addition skills that you acquire here will be needed for larger addition problems, as well as virtually all A Little Give and Take: Mental Addition and Subtraction 1 3 multiplication problems in later chapters.
It also illustrates a fundamental principle of mental arithmetic — namely, to sim- plify your problem by breaking it into smaller, more manage- able 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 two-digit 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: We illustrate this as follows: 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: 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 car- rying 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: Was that easier?
If you would like to try your hand at more two-digit addition problems, check out the set of exercises below. The answers and computations are at the end of the book. A Little Give and Take: After each step, you arrive at a new and simpler addition problem. Let's try the following: This thought process can be diagrammed as follows: The goal is to keep simplifying the problem until you are just adding a one-digit number.
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: Diagrammed, the problem looks like this: Six plus one equals seven, so my next problem is seven hundred and twenty-three plus fifty-nine, and so on. When first doing these problems, practice them out loud.
Rein- forcing yourself verbally will help you learn the mental method much more quickly. Three-digit addition problems really do not get much harder than the following: Mental Addition and Subtraction 1 7 Now look to see how we did it: In my mind the problem sounds like this: Your mind-talk 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: However, with this particular problem you have the option of using an alternative method. I am sure you will agree that it is a 1 8 Secrets of Mental Math lot easier to add to than it is to add , so try adding and then subtracting the difference: 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 sim- pler than the first, I sometimes switch them around, as in the fol- lowing example: 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, calcu- lators, 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: The process is the same for the following problems: 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: Answers can be found in the back of the book. Mathematical Prodigy A prodigy is a highly talented child, usually called precocious or gifted, and almost always ahead of his peers. The German math- ematician Carl Friedrich Gauss 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.
As a ten-year-old student, Gauss was presented the following math- ematical problem: What is the sum of numbers from I to ? 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. Gauss's desire to better understand Nature through the language of mathe- matics was summed up in his motto, taken from Shakespeare's King Lear substituting "laws" for "law": Mental Addition and Subtraction 21 6 i- b.
But if you con- tinue to compute from left to right and to break down problems into simpler components, subtraction can become almost as easy as addition. Two-Digit Subtraction When subtracting two-digit numbers, your goal is to simplify the problem so that you are reduced to subtracting or adding a one-digit number. Let's begin with a very simple subtraction problem: The problem can be diagrammed this way: But the good news is that "hard" subtraction problems can usually be turned into "easy" addition problems.
First subtract 20, then subtract 9: First subtract 30, then add back 1: If a two- digit 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. Mental Addition and Subtraction 23 Now try your hand or head at 81 - Just use the rule above to decide which method will work best.
Simply sub- tract one digit at a time, simplifying as you go. Since you subtracted by 8 too much, did you add back 8 to reach , the final answer? Did you notice? But what about other problems, like: Mental Addition and Subtraction 25 If you subtract one digit at a time, simplifying as you go, your sequence will look like this: But you have sub- tracted 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 is from The answer can be found by using "complements," a nifty technique that will make many three-digit subtraction problems a lot easier to do.
Using Complements You're Welcome! Quick, how far from are each of these numbers? We say that 43 is the complement of 57, 32 is the com- plement of 68, and so on. Now you find the complements of these two-digit numbers: The answer is 6.
Then figure out what you need to add to 7 to get The answer is 3. Hence, 63 is the complement of The other complements are 41, 7, 56, Notice that, like everything else you do as a mathemagician, the complements are determined from left to right. As we have seen, the first dig- its add to 9, and the second digits add to An exception occurs in numbers ending in 0 — e. 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: 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 from ? The same distance as 68 is from If you find the complement of 68 the way we have shown you, you will arrive at Add 32 to , and you will arrive at , your final answer. The procedure looks like this: 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 one-digit numbers by two-digit numbers and three-digit num- bers. You will also learn a phenomenally fast way to square two-digit numbers.
Even friends with calculators won't be able to keep up with you. Believe me, virtually everyone will be dumb- founded by the fact that such problems can not only be done 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. You will do vir- tually all the calculations in this chapter from left to right as well.
Secrets of Mental Math - Benjamin A., Shermer M. 2006.pdf
This is undoubtedly the opposite of what you learned in school. Products of a Misspent Youth: Basic Multiplication 3 I 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: Note that 40 x 7 is just like 4x7, with a friendly zero attached. Then add plus 14 left to right, of course to arrive at , the correct answer.
We illustrate this procedure below. 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: The answer is If you are wondering why this process works, see the Why These Tricks Work section at the end of the chapter. First calculate 62 x 3. Then do 71 x 9. Try doing them in your head before looking at how we did it.
One hundred eighty.
PDF Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing
Another especially easy type of mental multipli- cation problem involves numbers that begin with five. When the five is multiplied by an even digit, the first product will be a mul- tiple of , which makes the resulting addition problem a snap. It takes far less time to calculate " plus 35" mentally than it does to apply the pencil-and-paper method of "putting down the 5 and carrying the 3.
The addition problem is slightly harder because it involves carrying a number. With practice, you will become more adept at juggling 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 x 6, illustrated below. On the right, however, we have rounded 69 up to 70, and sub- tracted - 6, which you might find easier to do. It does not work so well when you need to round up more than two dig- 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. Per- sonally, 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 2-by-l 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 prob- lems, you would like to practice more, make up your own. Cal- culate 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. You can get started with the following 3-by-l problem which is really just a 2-by-l problem in disguise: If this problem gave you trouble, you might want to review the addition material in Chapter 1.
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Let's try another 3-by-l problem similar to the one you just did, except we have replaced the 0 with a 6 so you have another step to perform: Since you do not need to carry any numbers, it is easy to add 42 to to arrive at the total of In solving this and other 3-by-l multiplication problems, the difficult part may be holding in memory the first sum in this case, while doing the next multiplication problem in this Products of a Misspent Youth: Basic Multiplication 37 case, 6x7.
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: 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 2-by-l multiplication problems, we saw that problems involving numbers that begin with five are 38 Secrets of Mental Math sometimes especially easy to solve. The same is true for 3-by-l problems: This is because you do not have to carry any numbers and the thou- sands 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. 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 2-by-l problem, which you can say out loud as well — in this case, " Basic Multiplication Because the first two digits of the three-digit 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.
The difficult part comes in holding the pre- liminary answer in your head while computing the final answer. Some- times at this stage I will start to say my answer aloud before fin- ishing.
Because I have already carried, I say only the last two digits, " 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: In the first problem, for example, start by saying, "Twenty-eight hun- dred plus five hundred sixty" a couple of times out loud to reinforce the two numbers in memory while you add them together. Then repeat "thirty-three hundred sixty plus sixty-three" aloud until you compute the final answer of 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. 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 won- dered, how large could the product of two such numbers get?
I started in the middle with 10 x 10 or 10 2 , the product of which is I noticed that the products were getting smaller, and their differ- ence from was 1, 4, 9, 16, 25, 36,. Numbers that Distance add to 20 from 1 0 10 10 0 9 1 1 1 8 12 2 7 13 3 6 14 4 5 15 5 4 16 6 3 17 7 2 18 8 1 19 9 Their Product's difference product from 0 99 I 96 4 91 9 84 16 75 25 64 36 51 49 36 64 19 81 I found this pattern astonishing.
Next I tried numbers that add to 26 and got similar results. Just as before, the distances these products were from was l 2 , 2 2 , 3 2 , 4 2 , and so on see table below.
There is actually a simple algebraic explanation for this phe- nomenon see Why These Tricks Work, page 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 num- bers more easily. Suppose I wanted to square the number Instead of multiplying 13 x 13, Numbers that Distance Their Product's difference add to 26 from 13 product from 13 13 0 0 12 14 [ 1 1 1 15 2 4 10 16 3 9 9 17 4 16 8 18 5 25 why not get an approximate answer by using two numbers that are easier to multiply but also add to 26?
This method is diagrammed as follows: Now let's see how this works for another square: Secrets of Mental Math To square 41, subtract 1 to obtain 40 and add 1 to obtain Next multiply 40 x Don't panic!
This is simply a 2-by-l multiplication problem specifically, 4 x 42 in disguise. Almost done! Can squaring a two-digit 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 77 2 , working it out both by rounding up and by rounding down: In fact, for all two-digit squares, I always round up or down to the nearest multiple of 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 Products of a Misspent Youth: Basic Multiplication 47 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 56 2 in your head before looking at how we did it, below: Since you will always round up and down by 5, the numbers to be multiplied will both be multiples of Hence, the multiplication and the addition are especially simple. We have worked out 85 2 and 35 2 , below: For example, if you want to compute 75 2 , rounding up to 80 and down to 70 will give you "Fifty-six hundred and.
Even large numbers are not to be feared. You can ask someone to give you a really big two-digit 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 Let's say your audience gives you 96 2. Try it yourself, and then check how we did it. Wasn't that easy? You should have rounded up by 4 to and down by 4 to 92, and then multiplied x 92 to get At this point you can say out loud, "Ninety-two hundred," and then finish up with "sixteen" and enjoy the applause!
Basic Multiplication Zerah Colburn: Entertaining Calculations One of the first lightning calculators to capitalize on his talent was Zerah Colburn , an American farmer's son from Vermont who learned the multiplication tables to 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 interna- tionally 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 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 I: Answered in twenty seconds: How many sec- onds in eleven years? Answered in four seconds; ,, For example, he would fac- tor large numbers into smaller numbers and then multiply: Colburn once multiplied 2 1 , X by factoring into X 3.
He then multiplied 21, X to arrive at 3,,, and finally multiplied that figure by 3, for a total of I 1 ,80 1 , 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 thirty-five. In summarizing his skills as a lightning calculator, and the advantage such an ability affords, Colburn reflected, "True, the method.
Some people may find the theory as interesting as the application. Fortunately, you need not understand why our methods work in order to under- stand how to apply them. All magic tricks have a rational expla- nation 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 com- ponent parts. The distributive law states that for any numbers a, b, and c: To understand it intuitively, imagine having 7 bags, each containing 42 coins, 40 of which are gold and 2 of which are sil- ver.
How many coins do you have altogether? There are two ways to arrive at the answer. In the first place, by the very def- inition of multiplication, there are 42 x 7 coins. On the other hand, there are 40 x 7 gold coins and 2x7 silver coins. Hence, Products of a Misspent Youth: 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: As for squaring, the following algebra justifies my method. The following algebraic relationship also works to explain my squar- ing method: 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 mathemagi- cians begin even earlier.
Zerah Colburn , for exam- ple, 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 2. Not content to stop there, I blurted out, " squared is simply 11,! Looking a bit startled, she said, "Yes, that's right. How did you do it? I then multiplied x , which is 11,, and just added the square of 8, to get 11, I was thrilled.
Thoughts of "Benjamin's Theorem" popped into my head. I actu- ally believed I had discovered something new. When I finally ran across this method a few years later in a book by Martin 54 Secrets of Mental Math Gardner on recreational math, Mathematical Carnival , 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 two-digit number by itself. In this chapter you will learn how to multiply two differ- ent two-digit numbers, a challenging yet more creative task. You will then try your hand — or, more accurately, your brain — at three-digit squares. You will need to the email address of your friend or family member. Proceed with the checkout process as usual. Q: Why do I need to specify the email of the recipient?
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Q: Oops! The recipient already owns the course I gifted. What now?Art appears nothing like a mathematics professor from a prestigious college. If you add or sub- tract a number that is a multiple of 7 to the number you are test- ing, and the resulting number is a multiple of 7, then the test is positive.
But I feel he has yet to write a good investigation on the subject. I was thrilled. The odds are now for your choice, right? Return To Top Related Interests.