{"product_id":"secrets-of-mental-math-isbn-9780307338402","title":"Secrets of Mental Math","description":"\u003cb\u003eThese simple math secrets and tricks will forever change how you look at the world   of numbers.\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e \u003ci\u003eSecrets of Mental Math\u003c\/i\u003e 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.\u003cbr\u003e\u003cbr\u003e 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, \u003ci\u003eSecrets of Mental Math\u003c\/i\u003e will   allow you to perform fantastic feats of the mind effortlessly. This is the math they   never taught you in school.“A great introduction to the wonder of numbers, from two superb teachers.” \u003cbr\u003e\u003cb\u003e—Brian   Greene, author of \u003ci\u003eThe Elegant Universe \u003c\/i\u003e\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e “A magical mystery tour of mental mathematics!   Fascinating and fun.” \u003cbr\u003e\u003cb\u003e—Joseph Gallian, president of the Mathematical Association   of America \u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e “The clearest, simplest, most entertaining, and best book yet on the   art of calculating in your head.” \u003cbr\u003e\u003cb\u003e—Martin Gardner, author of \u003ci\u003eMathematical Magic Show\u003c\/i\u003e and \u003ci\u003eMathematical Carnival\u003c\/i\u003e\u003c\/b\u003e\u003cbr\u003e\u003cbr\u003e “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.” \u003cbr\u003e\u003cb\u003e—Dr. Edward O. Thorp, mathematician and author of \u003ci\u003eBeat the   Deale\u003c\/i\u003er and\u003ci\u003e Beat the Market\u003c\/i\u003e\u003c\/b\u003e\u003cb\u003eArthur Benjamin \u003c\/b\u003eis a professor of mathematics at Harvey Mudd College in Claremont,   California. He is also a professional magician and performs his mixture of math and   magic all over the world.\u003cbr\u003e\u003cbr\u003e \u003cb\u003eMichael Shermer \u003c\/b\u003eis host of the Caltech public lecture   series, a contributing editor to and monthly columnist of \u003ci\u003eScientific American\u003c\/i\u003e, the   publisher of \u003ci\u003eSkeptic\u003c\/i\u003e magazine, and the author of several science books. He lives   in Altadena, California.Chapter 0\u003cbr\u003e\u003cbr\u003e Quick Tricks: Easy (and Impressive) Calculations\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e Instant Multiplication\u003cbr\u003e\u003cbr\u003e 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:\u003cbr\u003e\u003cbr\u003e 32 3 11\u003cbr\u003e\u003cbr\u003e 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:\u003cbr\u003e\u003cbr\u003e 35}2\u003cbr\u003e\u003cbr\u003e What could be easier? Now you try:\u003cbr\u003e\u003cbr\u003e 53 3 11\u003cbr\u003e\u003cbr\u003e Since 5 1 3 5 8, your answer is simply\u003cbr\u003e\u003cbr\u003e 583\u003cbr\u003e\u003cbr\u003e One more. Without looking at the answer or writing anything down, what   is\u003cbr\u003e\u003cbr\u003e 81 3 11?\u003cbr\u003e\u003cbr\u003e Did you get 891? Congratulations!\u003cbr\u003e\u003cbr\u003e Now before you get too excited, I have shown you only half of what you   need to know. Suppose the problem is\u003cbr\u003e\u003cbr\u003e 85 3 11\u003cbr\u003e\u003cbr\u003e Although 8 1 5 5 1}3}, the answer is NOT 81}3}5!\u003cbr\u003e\u003cbr\u003e As before, the 3} goes in between the numbers, but the 1} needs to be   added to the 8 to get the correct answer:\u003cbr\u003e\u003cbr\u003e 93}5\u003cbr\u003e\u003cbr\u003e Think of the problem this way:\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e Here is another example. Try 57 3 11.\u003cbr\u003e\u003cbr\u003e Since 5 1 7 5 12, the answer is\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e Okay, now it’s your turn. As fast as you can, what is\u003cbr\u003e\u003cbr\u003e 77 3 11?\u003cbr\u003e\u003cbr\u003e If you got the answer 847, then give yourself a pat on the back. You   are on your way to becoming a mathemagician.\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e Since 9 1 9 5 18, the answer is:\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e 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!)\u003cbr\u003e\u003cbr\u003e Welcome back. At this point, you probably have a few questions, such as:\u003cbr\u003e\u003cbr\u003e “Can we use this method for multiplying three-digit numbers (or larger)   by eleven?”\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e More practically, you are probably saying to yourself,\u003cbr\u003e\u003cbr\u003e “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?”\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e Squaring and More\u003cbr\u003e\u003cbr\u003e Here is another quick trick.\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e To square a two-digit number that ends in 5, you need to remember only   two things.\u003cbr\u003e\u003cbr\u003e 1.The answer begins by multiplying the first digit by the next higher   digit.\u003cbr\u003e\u003cbr\u003e 2.The answer ends in 25.\u003cbr\u003e\u003cbr\u003e 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:\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e How about the square of 85? Since 8 3 9 5 72, we immediately get 85 3   85 5 7225.\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e Similarly, 84 3 86 5 7224.\u003cbr\u003e\u003cbr\u003e Now it’s your turn. Try\u003cbr\u003e\u003cbr\u003e 26 3 24\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e 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\u003cbr\u003e\u003cbr\u003e 31 3 39 5 1209\u003cbr\u003e\u003cbr\u003e 32 3 38 5 1216\u003cbr\u003e\u003cbr\u003e 33 3 37 5 1221\u003cbr\u003e\u003cbr\u003e 34 3 36 5 1224\u003cbr\u003e\u003cbr\u003e 35 3 35 5 1225\u003cbr\u003e\u003cbr\u003e You may ask,\u003cbr\u003e\u003cbr\u003e “What if the last digits do not sum to ten? Can we use this method to   multiply twenty-two and twenty-three?”\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e “Are there any tricks for doing mental addition and subtraction?”\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e Consider the subtraction problem\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e 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\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e 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\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e which is not too hard to calculate in your head (especially from left   to right). Thus, 1241 2 587 5 654.\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e \u003cbr\u003e\u003cbr\u003e 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:\u003cbr\u003e\u003cbr\u003e 359 4 222 5 1.61 (first three digits)\u003cbr\u003e\u003cbr\u003e We will have much more to say about division (including decimals and   fractions) in Chapter 4.\u003cbr\u003e\u003cbr\u003e More Practical Tips\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e Improve Your Memory\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e MondayTuesdayWednesdayThursdayFridaySaturdaySunday\u003cbr\u003e\u003cbr\u003e 1234567 or 0\u003cbr\u003e\u003cbr\u003e 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:\u003cbr\u003e\u003cbr\u003e Bill:30\u003cbr\u003e\u003cbr\u003e Tip: 1} } }7}\u003cbr\u003e\u003cbr\u003e 37\u003cbr\u003e\u003cbr\u003e subtract 7s: 2} }3}5}\u003cbr\u003e\u003cbr\u003e 2 5 Tuesday\u003cbr\u003e\u003cbr\u003e How about January 1, 2043:\u003cbr\u003e\u003cbr\u003e Bill:43\u003cbr\u003e\u003cbr\u003e Tip: 1} }1}0}\u003cbr\u003e\u003cbr\u003e 53\u003cbr\u003e\u003cbr\u003e subtract 7s: 2} }4}9}\u003cbr\u003e\u003cbr\u003e 4 5 Thursday\u003cbr\u003e\u003cbr\u003e 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\u003cbr\u003e\u003cbr\u003e 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!)\u003cbr\u003e\u003cbr\u003e I know what you are wondering now:\u003cbr\u003e\u003cbr\u003e “Why didn’t they teach this to us in school?”\u003cbr\u003e\u003cbr\u003e 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!\u003cbr\u003e\u003cbr\u003e Chapter 1\u003cbr\u003e\u003cbr\u003e A Little Give and Take:\u003cbr\u003e\u003cbr\u003e Mental Addition and Subtraction\u003cbr\u003e\u003cbr\u003e 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!\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e Left-to-Right Addition\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e Two-Digit Addition\u003cbr\u003e\u003cbr\u003e 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.\u003cbr\u003e\u003cbr\u003e 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:\u003cbr\u003e\u003cbr\u003e (30 1 2)\u003cbr\u003e\u003cbr\u003e 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:\u003cbr\u003e\u003cbr\u003e 47 1 32   5   77 1 2   5   79\u003cbr\u003e\u003cbr\u003e (first add 30)(then add 2)\u003cbr\u003e\u003cbr\u003e 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.","brand":"Crown","offers":[{"title":"Default Title","offer_id":46301785161957,"sku":"NP9780307338402","price":19.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9780307338402.jpg?v=1767736279","url":"https:\/\/k12savings.com\/products\/secrets-of-mental-math-isbn-9780307338402","provider":"K12savings","version":"1.0","type":"link"}