Русская версия Mnemonic - Articles


"THIS IS A BOOK TO IGNATE MATHEMATICAL
CONFIDENCE AND CURIOSITY."
. . . . LIBRARY JOURNAL

MATHEMAGICS


H O W   T O
LOOK  LIKE
A   GENIUS
W I THOUT
R E A L L Y
T R Y I N G

SECOND     EDITION

ARTHUR   BENJAMIN, PH.D.,   AND
MICHAEL BRANT SHERMER, PH.D.

FOREWORD BY
JAMES "THE AMAZING" RANDI

Contents




FOREWORD BY JAMES RANDIix
PREFACE BY MICHAEL SHERMERxi
INTRODUCTION BY ARTHUR BENJAMINxvii
CHAPTER 1     A LITTLE GIVE AND TAKE:
MENTAL ADDITION AND SUBTRACTION1
CHAPTER 2PRODUCTS OF A MISSPENT YOUTH:
BASIC MULTIPLICATION15
CHAPTER 3NEW AND IMPROVED PRODUCTS:
INTERMEDIATE MULTIPLICATION33
CHAPTER 4DIVIDE AND CONQUER: MENTAL DIVISION   53
CHAPTER 5THE ART OF "GUESSTIMATION"75
CHAPTER 6MATH FOR THE BOARD:
PENCIL-AND-PAPER MATHEMATICS89
CHAPTER 7A MEMORABLE CHAPTER105
CHAPTER 8THE TOUCH STUFF MADE EASY:
ADVANCED MULTIPLICATION115
CHAPTER 9MATHEMATICAL MAGIC145
ANSWERS161
BIBLIOGRAPHY203
INDEX205


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1


A Little Give and Take:
Mental Addition and
Subtraction

I remember the day in third grade when I discovered that it was easier to add and subtract from left to right than from right to left, which was the way we had all been taught. Suddenly I was able to blurt out the answers to math problems in class well before my classmates put down their pencils. And I didn't even need a pencil! The method was so simple that I performed most calculations in my head Looking hack. I admit I did so as much to show off as for any muthematical reason. Most kids outgrow such behavior. Those who don't probably become either teachers or magicians.
In this chapter you will learn the left-to-righl method of doing mental addition and subtraction for numbers that range in size from two to four digits. These mental skills are not only important for doing the tricks in this hook but are also indispensable in school or at work, or any time you use numbers. Soon you will he able lo retire your calculator and use the full capacity of your mind as you add, subtract, multiply, and divide 2-digit, 3-digit, and even 4-digit numbers.

LEFT-TO-RIGHT ADDITION

There are many good reasons why adding left to right is a superior method for mental calculation. For one thing, you do not have to reverse the numbers (as you do when adding right to left). And if you want to estimate your answer, then adding only the leading digits will get you pretty close. If you are used to working from right to left on paper, it may seem unnatural to add and multiply from left to right.


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But with practice you will find that it is the must natural and efficient way to do mental calculations.
With the first set of problems - 2-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 3-digit 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.

2-DIGIT ADDITION

Our assumption in this chapter is that you know how to add and subtract 1-digit numbers. We will begin with 2-digit addition, something I suspect you can already do fairly well in your head. The following exercises are good practice, however, because you will use the 2-digit addition skills you polish here for larger addition problems, as well as in virtually all 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 components. This is the key to virtually every method you will learn in this book. To paraphrase an old adage, there are just three components to success - simplify, simplify, simplify.
The easiest 2-digit addition problems, of course, are those that do not require you to carry any numbers. For example:
47
+ 32		 
(30 + 2)
To add 32 to 47, you can simplify by treating 32 as 30 + 2, add 30 to 47 and then add 2. In this way the problem becomes 77 + 2, which equals 79:
47
+ 32		 + 30>
 		 77
+   2		 + 2>
 		 = 79
 
Keep in mind that the above diagram is simply a way of representing the mental processes involved in arriving at an answer using one 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.


- 3 -


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.
67
+ 28		 + 20>
 		 87
+   8		 + 8>
 		 = 95
 
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)
No problem, right? You added 84 + 50 = 134 and added 134 + 7 = 141.
84
+ 57		 + 50>
 		 154
+   7		 + 7>
 		 = 141
 
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, and then chocking how wo did it:
68
+ 45		 
(40 + 5)
You should have added 68 + 40 = 108. and them 108 + 5 = 113, the final answer. No sweat, right? If you would like to try your hand at more 2-digit addition problems, check out the set of exercises below. (The answers and computations are at the end of the book.)

Exercises: 2-Digit Addition
(1)
 
 
 23
+ 16
(2)
 
 
 64
+ 43
(3)
 
 
 95
+ 32
(4)
 
 
 34
+ 26
(5)
 
 
 89
+ 78


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(6)
 
 
 73
+ 58
(7)
 
 
 47
+ 36
(8)
 
 
 19
+ 17
(9)
 
 
 55
+ 49
(10)
 
 
 39
+ 38


3-DIGIT ADDITION

The strategy for adding 3-digit numbers is the same as for adding 2-digit numbers: you add loft to right. After each step, you arrive at a new (and smaller) addition problem. Let's try the following:
538
+ 327		 
(300 + 20 + 7)
After adding the hundreds digit of the second number to the first number (538 + 300 = 838). the problem becomes 838 + 27. Next add the tens digit (838 + 20 = 858). simplifying the problem 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 worked using this method. The goal is to keep simplifying the problem until you are left adding a 1-digit number. It is important to reduce the number of digits you are manipulating because human short-term memory is limited to about 7 digits. Notice that 538 + 327 requires you to hold on to 6 digits in your head, whereas 838 + 27 and 858 + 7 require only 5 and 4 digits, respectively. As you simplify the problems, the problems get 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 digit (623 + 100 = 723), you were left with 723 + 59. Next you should have added the tens digit (723 + 50 = 773), simplifying the problem to 773 + 9, which you


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easily summed to 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 twenty-three plus one hundred fifty-nine; 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 twenty-three plus fifty-nine, 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.
Addition problems really do not get much harder than the following:
858
+ 634
Now look to see how we did it. below:
858
+ 634		 + 600>
 		 1458
+   34		 + 30>
 		 1488
+     4		 + 4>
 		 = 1492
 
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 mind-talk may not sound exactly like mine, but whatever it is you say 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		 + 400>
 		 1159
+   96		 + 90>
 		 1249
+     6		 + 6>
 		 = 1255
 


- 6 -


This addition problem is a little more difficult than the last one since it requires you to carry numbers in all three stops. However, with this particular problem you have the option of using an alternative method. I am sure you will agree that it is a lot easier to add 500 to 759 than it is to add 496, so try adding 500 und then subtracting the difference:
759
+ 496		 
(500 - 4)
759
+ 496		 + 500>
 		 1259
-     4		 - 4>
 		 = 1255
 
So far, you have consistently broken up tho second number in any problem to add to the first. It really does not matter which number you choose to break up as long as you are 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 switch them around, as in the following example:
207
+ 528		 = 528
+ 207		 + 200>
 		 728
+     7		 + 7>
 		 = 735
 
Let's finish up by adding 3-digit to 4-digit numbers. Again, since most human memory can only hold about 7 digits at a time, this is about as large a problem as you can handle without resorting to artificial memory devices (described in Chapter 7). Often (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
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