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SECRETS OF MENTAL MATH By: Julianne Donahue Jordan Stevens November 18, 2008 Mathematics is a language of its own and Arthur Benjamin has developed a book of secrets on how to become better at mental math. Arthur Benjamin is both a professor of mathematics at Harvey Mudd College in California and he also a magician. He combined his two loves of life into a presentation called mathemagics. He takes simple math and combines it with magic and puts on presentations for all ages. He demonstrates his secrets for performing math problems quickly in his head. Often times he is able to compute the problem in his head faster than somebody who does it on a calculator. Arthur Benjamin has appeared on many television and radio shows. He has also been featured in some magazines and newspapers. Arthur Benjamin has developed theories on how to do these problems in your head, which is his book The Secrets of Mental Math. Arthur Benjamin’s presentations consist of his ability to compute numbers in his head really quickly, usually faster than a calculator. He demonstrates and explains his abilities to pretty much any kind of audience. He mostly goes around to schools, but his presentation is also ideal for math clubs, general audiences, inspirational talks and corporate events. Among his presentation he multiplies extremely large numbers, memorizes 100 digit numbers, figures out the day of the week of anyone’s birthday, and many more things that he can do. It is truly an amazing show that he can put on. Mental math is often times something that everyone struggles with, but he has created a book that summarizes some important topics that everyone should know and he gives good advice on how to solve those hard problems that no one wants to sit down and calculate. His mental math secrets give us the opportunity to save a lot of time when computing simple things for a math class or any other daily event that happens. The first topic covered in his book is a simple way for multiplication of two digit numbers by eleven. Arthur Benjamin explains that all you have to do is add the two digits together and put that number in between the two numbers that you added together. For example if you have 32 X 11, you take 3+2, which is equal to 5. Then you put the 5 in between the 3 and the 2. Your answer will be 352. This is one of the easiest ways of multiplying numbers by 11 without having to use a calculator. But there is an exception to the rule. For example, if you have 85 X 11. The problem comes when you add 8+5=13. So what you have to do is put the 3 in the middle, but instead you add 1 to 8 to get the correct answer of 935. Some may ask if you can multiply 11 by a three-digit number? The answer is yes. For example if you have 314 X 11, the answer will begin with a 3 and end with a 4. For the middle you add 3+1=4 and 1+4=5.so for the answer you will get 3454. So now you can teach this to people of all ages and they will be amazed. Arthur Benjamin says that all you need is a handful of techniques to multiply numbers in your head and it takes patience for the larger numbers. Another topic in the book is squaring numbers. As you already know a square of a number is a number that is multiplied by itself. But in this section of the book Arthur Benjamin talks about squaring a number that ends in 5. He says you only need to remember two parts when squaring a number by 5. The fist part is the answer begins by multiplying the first digit by the next higher digit. And the other part to remember is the answer ends in 25. For example, to square the number 35, we simply just need to multiply the first digit by the next higher digit and then attach 25. So for: 35 X 35 3 X 4=12 5 X 5=25 Answer= 1225 An interesting topic that we found in the book was one that dealt with figuring out the day of the week of any date. You are able to use this for birth dates, historical dates, future appointments, and anything else you would like to know. So for example, if you want to know the day of the week that January 1, 2030. Here is what you have to do. Take the last two digits of the year and consider it to be a bill at a restaurant. Now add a 25% tip to it, but keep the change. To compute a 25% tip, take half of the bill twice and keep the change. To figure out the day of the week subtract the biggest multiple of 7 from your total and that will tell you the day of the week. Bill: 30 Tip: + 7 37 Subtract largest multiple of 7: - 35 2= Tuesday Monday=1 Tuesday=2 Wednesday=3 Thursday=4 Friday=5 Saturday=6 Sunday=7,0 This is another amazing thing that we found interesting. Arthur Benjamin has this idea of the magic number 1089. This is a trick that has been around for many years and here is how it goes. 1. Secretly write down a three-digit number where the digits are decreasing (like 851 or 983, etc.) 2. Reverse the number and subtract it from the first number. 3. Take that answer and add it to the reverse of itself At the end of the problem the magic answer of 1089 will appear. So let’s try it. 851 -158 693 + 396 1089 You are probably asking yourself why this problem works? Well, no matter what three- digit number you or anyone else chooses in this game, the final result will always be 1089. Why? Well let abc denote the unknown three-digit number. Algebraically this is equal to 100a +10b +c. When you reverse the number cba, algebraically you get 100c + 10b +a. When subtracting cba from abc, you get 100a+10b+c- (100c+10bb+a), which is equal to 99(a-c). In conclusion, Arthur Benjamin is a brilliant man who has discovered and or thought of ways to do math in your head at a faster rate. This is an important thing for everyone to understand because if you do not know how to solve something quick when it is needed than you will not be able to move on to the next problem. The information that we have gone over is how to multiply two digit numbers by 11, squaring numbers that end in 5, determining the day of the week, and the magic number 1089. These are all important concepts to know. The reason we chose these is to give everyone a better understanding of how to do some simple math in your head quickly. Being able to multiply by 11 in two steps cannot get any easier. Why these things were not taught to kids in grade school, no one will ever have an answer. Maybe it is our priority to teach them these simple things. But now having the ability to know how to do that, hopefully our class will be able to show people new ways of doing things. Also being able to square a number that ends in a 5 in two steps cannot get any easier. These quick tricks can save a lot of time if you know how to do them. These two concepts are more mathematical than the other ones we chose. We wanted to be able to teach the class a new way to do these kinds of problems to make their multiplication that much quicker, rather than having to type it into a calculator. The other two topics of determining the day of the week and the magic number 1089 are more of a magic concept. To determine the day of the week way in advance is something I would have never thought of to do. It involves simple math but it also just kind of predict the future. The magic number 1089 is something we never thought about. Who would have thought that picking a three digit decreasing number will always end up to add to 1089. These topics are fun things to do if you have some time. These are not necessarily things that will happen in everyday life, but we thought it would be fun to do this. With these few topics that we covered from his book we hope to have taught the class some new ways of calculating their daily math problems that occur. Bibliography 1. Benjamin, Arthur. Secrets of Mental Math. New York. Three Rivers Press, 2006. 2. Benjamin, Arthur. Arthur T. Benjamin, Mathematician. 8 March 2008. < http://www.math.hmc.edu/~benjamin/index.html>. 3. Youtube.com, Arthur Benjamin movie