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Secrets of Mental Math: The Mathemagician's Guide to Lightning Calculation and Amazing Math Tricks Paperback – August 8, 2006
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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 lightning-quick 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.
- Print length304 pages
- LanguageEnglish
- PublisherCrown
- Publication dateAugust 8, 2006
- Dimensions7.76 x 5.08 x 0.44 inches
- ISBN-109780307338402
- ISBN-13978-0307338402
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Editorial Reviews
Review
—Brian Greene, author of The Elegant Universe
“A magical mystery tour of mental mathematics! Fascinating and fun.”
—Joseph Gallian, president of the Mathematical Association of America
“The clearest, simplest, most entertaining, and best book yet on the art of calculating in your head.”
—Martin Gardner, author of Mathematical Magic Show and Mathematical Carnival
“This book can teach you mental math skills that will surprise you and your friends. Better, you will have fun and have valuable practical tools inside your head.”
—Dr. Edward O. Thorp, mathematician and author of Beat the Dealer and Beat the Market
About the Author
Michael Shermer is host of the Caltech public lecture series, a contributing editor to and monthly columnist of Scientific American, the publisher of Skeptic magazine, and the author of several science books. He lives in Altadena, California.
Excerpt. © Reprinted by permission. All rights reserved.
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 two-digit number by eleven. It’s very easy once you know the secret. Consider the problem:
32 3 11
To solve this problem, simply add the digits, 3 1 2 5 5}, put the 5 between the 3 and the 2, and there is your answer:
35}2
What could be easier? Now you try:
53 3 11
Since 5 1 3 5 8, your answer is simply
583
One more. Without looking at the answer or writing anything down, what is
81 3 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 3 11
Although 8 1 5 5 1}3}, the answer is NOT 81}3}5!
As before, the 3} goes in between the numbers, but the 1} needs to be added to the 8 to get the correct answer:
93}5
Think of the problem this way:
Here is another example. Try 57 3 11.
Since 5 1 7 5 12, the answer is
Okay, now it’s your turn. As fast as you can, what is
77 3 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 two-digit number by eleven, it won’t be long before they ask you to do 99 3 11. Let’s do that one now, so we are ready for it.
Since 9 1 9 5 18, the answer is:
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 three-digit numbers (or larger) by eleven?”
Absolutely. For instance, for the problem 314 3 11, the answer still begins with 3 and ends with 4. Since 3 1 1 5 4}, and 1 1 4 5 5}, 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 thirty-six?”
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 3 7 5 49. 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.
1.The answer begins by multiplying the first digit by the next higher digit.
2.The answer ends in 25.
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 3 4 5 12, the answer is 1225. Therefore, 35 3 35 5 1225. Our steps can be illustrated this way:
How about the square of 85? Since 8 3 9 5 72, we immediately get 85 3 85 5 7225.
We can use a similar trick when multiplying two-digit 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 3 87. (Both numbers begin with 8, and the last digits sum to 3 1 7 5 10.) Since 8 3 9 5 72, and 3 3 7 5 21, the answer is 7221.
Similarly, 84 3 86 5 7224.
Now it’s your turn. Try
26 3 24
How does the answer begin? With 2 3 3 5 6. How does it end? With 6 3 4 5 24. Thus 26 3 24 5 624.
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 3 39 5 1209
32 3 38 5 1216
33 3 37 5 1221
34 3 36 5 1224
35 3 35 5 1225
You may ask,
“What if the last digits do not sum to ten? Can we use this method to multiply twenty-two and twenty-three?”
Well, not yet. But in Chapter 8, I will show you an easy way to do problems like this using the close-together method. (For 22 3 23, you would do 20 3 25 plus 2 3 3, to get 500 1 6 5 506, 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.
Consider the subtraction problem
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 2 600 5 600, we have that
But we have subtracted 13 too much. (We will explain how to quickly determine the 13 in Chapter 1.) Thus, our painful-looking subtraction problem becomes the easy addition problem
which is not too hard to calculate in your head (especially from left to right). Thus, 1241 2 587 5 654.
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.
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 4 222 5 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 easy-to-learn 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 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.
MondayTuesdayWednesdayThursdayFridaySaturdaySunday
1234567 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 2 35 5 2, and so January 1, 2030, will occur on 2’s day, namely Tuesday:
Bill:30
Tip: 1} } }7}
37
subtract 7s: 2} }3}5}
2 5 Tuesday
How about January 1, 2043:
Bill:43
Tip: 1} }1}0}
53
subtract 7s: 2} }4}9}
4 5 Thursday
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 1 7 5 39. Subtracting the largest multiple of 7 gives us 39 2 35 5 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!
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 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 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 two-digit, three-digit, and even four-digit numbers with lightning speed.
Left-to-Right 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 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—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 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.
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 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 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:
(30 1 2)
To solve 47 1 32, first add 30, then add 2. After adding 30, you have the simpler problem 77 1 2, which equals 79. We illustrate this as follows:
47 1 32 5 77 1 2 5 79
(first add 30)(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.
Product details
- ASIN : 0307338401
- Publisher : Crown; 33935th edition (August 8, 2006)
- Language : English
- Paperback : 304 pages
- ISBN-10 : 9780307338402
- ISBN-13 : 978-0307338402
- Item Weight : 4.6 ounces
- Dimensions : 7.76 x 5.08 x 0.44 inches
- Best Sellers Rank: #14,092 in Books (See Top 100 in Books)
- Customer Reviews:
About the authors
Dr. Michael Shermer is the Founding Publisher of Skeptic magazine, the host of the Science Salon Podcast, and a Presidential Fellow at Chapman University where he teaches Skepticism 101. For 18 years he was a monthly columnist for Scientific American. He is the author of New York Times bestsellers Why People Believe Weird Things and The Believing Brain, Why Darwin Matters, The Science of Good and Evil, The Moral Arc, and Heavens on Earth. His new book is Giving the Devil His Due: Reflections of a Scientific Humanist.
(Photo by Jordi Play)
Arthur Benjamin holds a PhD from Johns Hopkins University and is a professor of mathematics at Harvey Mudd College, where he has taught since 1989. He is a noted “mathemagician,” known for being able to perform complicated computations in his head. He is the author, most recently, of The Secrets of Mental Math, and has appeared on The Today Show and The Colbert Report. Benjamin has been profiled in such publications as the New York Times, the Los Angeles Times, USA Today, Scientific American, Discover, and Wired.
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Fourth graders struggle initially, but as they learn their tables, they see the power of being a super brainiac. I have been using the methods in this book as they are learning, to reinforce what they are learning. Young minds are blown as they tie concepts together.
This isn't a comprehensive textbook, but is more entertaining. It raises the magicians curtain to expose some of the secrets of show exhibition math. After learning single digit multiplication is an ideal time for students to get this book.
A caveat, though, the algebraic proofs will be above their learning, but is also a soft introduction to algebraic concepts. It fits well with how math is now taught.
This book is not about mathematics but arithmetic. If you're in science or business, you are an arithmetician by trade. High speed, close enough mental arithmetic is critical to the trade. Simple arithmetic remains the great divider in critical thinking from those that can think and those that stumble. If you stumble, you know you stumble. I know mathematicians that can't balance a check book or conceptulize GAAP P&L or Balance Sheet mechanics. I've imagined myself a seasoned veteran in `close enough' mastery of columnar data, formula application, and simple arithmetic operations of a rote nature. I can determine the `range and unit measures' of the correct answer to a particular problem by the problem statement.
It takes a lifetime to develop this skill from the rote arithmetic I should have paid more attention to it in elementary school, but ... it is what it is and it has to mastered. If you need a `correct' answer to 2 decimal places, use a calculator or Excel. If you need a fast approximation in the heat of mentally exploring alternatives use significance and proportionality to estimate within 5-10% of the calculated answer. That's usually performing well enough for fast critical thinking purposes. Remember that the equipment or source that provided the initial data is doing pretty well if it's within 5% of ground truth.
In `Secrets of Mental Math' you can explore refining your arithmetician skill to 2 decimal places with only slight change ups in heuristic application. Why isn't this stuff taught in schools? I mean it is important to know why arithmetic works, but for gosh sakes ... can't education get to the chase and deliver the shortcuts (all mathematically based in the philosophy of mathematics) to get an answer using outside of the box methods? No one can be `damaged' by exposure to these methods.
"Secrets of Mental Math" is an excellent consolidation of shortcuts and tricks to tune up you arithmetician trade. Our kids would benefit greatly if this or a similar text was included in a mandatory high school course called `Critical Thinking'. It would be a powerful primer for those that will never do a long division by hand again and for those that will go on to calculate chemistry and physics methods and equations. The answer is in the problem. Arithmetic is a tyranny. Do your kid and yourself a favor and try this powerful little book.
This book makes math fun!
Recently I visited family in Chicago. My great nephew is inquisitive and is advanced in math. Our mutual love for math makes it very easy to communicate with this youngster. Although he is only 10, I was able to teach him one of the tricks in the book that most adults do not know.
If you want to square a two-digit number that ends in five, the last two digits always end in 25. To discover the digits that come before 25, multiply the first digit by itself and add the first digit to it--for example 35 x 35, 3 x 3 + 3 = 12, so 35 squared = 1,225. Another example is 95 squared. 9 x 9 +9 = 90, so 95 squared = 9,025.
There are 231 easy to follow pages in this engaging book with numerous tricks and problem solving techniques with answers that follow the problems or in the back of the book.
Though my great nephew is hampered by a medical condition that is not conducive to good communications with some people, his math skills are far beyond his age. I know that he will be intrigued and have a blast with this book and can hardly wait for him to use it.
This book will not only make you seem smarter, it will make others think you are a mathematical whiz and they will be correct. Math can be and is fun when you master books like this one.
Enjoy!
I recommend Secrets of Mental Math by Arthur Benjamin and Michael Shermer to friends, family, and others who would like to increase their mathematical skills in a fun and easy manner.