Topic “Pure intellectual stimulation that can be popped into Science Subtopic the [audio or video player] anytime.” & Mathematics Mathematics —Harvard Magazine The Secrets The Secrets of Mental Math “Passionate, erudite, living legend lecturers. Academia’s best lecturers are being captured on tape.” —The Los Angeles Times “A serious force in American education.” —The Wall Street Journal of Mental Math Course Guidebook Professor Arthur T. Benjamin Harvey Mudd College Professor Arthur T. Benjamin is an engaging, entertaining, and insightful Professor of Mathematics at Harvey Mudd College.
He has been repeatedly honored by the Mathematical Association of America and has been featured in Scientific American, The New York Times, and Reader’s Digest—which named him “America’s Best Math Whiz.” THE GREAT COURSES ® Corporate Headquarters 4840 Westfields Boulevard, Suite 500 Chantilly, VA 20151-2299 Guidebook USA Phone: 1-800-832-2412 www.com Cover Image: © Carol & Mike Werner/age fotostock. 1406 © 2011 The Teaching Company. PB1406A PUBLISHED BY: THE GREAT COURSES Corporate Headquarters 4840 Westfields Boulevard, Suite 500 Chantilly, Virginia 20151-2299 Phone: 1-800-832-2412 Fax: 703-378-3819 www.com Copyright © The Teaching Company, 2011 Printed in the United States of America This book is in copyright. All rights reserved.
Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form, or by any means (electronic, mechanical, photocopying, recording, or otherwise), without the prior written permission of The Teaching Company. Professor of Mathematics Harvey Mudd College P rofessor Arthur T. Benjamin is a Professor of Mathematics at Harvey Mudd College. He graduated from Carnegie Mellon University in 1983, where he earned a B.
in Applied Mathematics with university honors. He received his Ph. in Mathematical Sciences in 1989 from Johns Hopkins University, where he was supported by a National Science Foundation graduate fellowship and a Rufus P. Since 1989, Professor Benjamin has been a faculty member of the Mathematics Department at Harvey Mudd College, where he has served as department chair.
He has spent sabbatical visits at Caltech, Brandeis University, and the University of New South Wales in Sydney, Australia. In 1999, Professor Benjamin received the Southern California Section of the Mathematical Association of America (MAA) Award for Distinguished College or University Teaching of Mathematics, and in 2000, he received the MAA Deborah and Franklin Tepper Haimo National Award for Distinguished College or University Teaching of Mathematics. He was also named the 2006–2008 George Pólya Lecturer by the MAA. Professor Benjamin’s research interests include combinatorics, game theory, and number theory, with a special fondness for Fibonacci numbers.
Many of these ideas appear in his book (coauthored with Jennifer Quinn) Proofs That Really Count: The Art of Combinatorial Proof, published by the MAA. In 2006, that book received the MAA’s Beckenbach Book Prize. From 2004 to 2008, Professors Benjamin and Quinn served as the coeditors of Math Horizons magazine, which is published by the MAA and enjoyed by more than 20,000 readers, mostly undergraduate math students and their teachers. In 2009, the MAA published Professor Benjamin’s latest book, Biscuits of Number Theory, coedited with Ezra Brown.com Professor Benjamin is also a professional magician.
He has given more than 1000 “mathemagics” shows to audiences all over the world (from primary schools to scienti¿c conferences), in which he demonstrates and explains his calculating talents. His techniques are explained in his book Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks. Proli¿c math and science writer Martin Gardner calls it “the clearest, simplest, most entertaining, and best book yet on the art of calculating in your head.” An avid game player, Professor Benjamin was winner of the American Backgammon Tour in 1997. Professor Benjamin has appeared on dozens of television and radio programs, including the Today show, The Colbert Report, CNN, and National Public Radio.
He has been featured in Scienti¿c American, Omni, Discover, People, Esquire, The New York Times, the Los Angeles Times, and Reader’s Digest. In 2005, Reader’s Digest called him “America’s Best Math Whiz.com Table of Contents INTRODUCTION Professor Biography .3 LECTURE GUIDES LECTURE 1 Math in Your Head! .4 LECTURE 2 Mental Addition and Subtraction. 11 LECTURE 3 Go Forth and Multiply .21 LECTURE 4 Divide and Conquer .30 LECTURE 5 The Art of Guesstimation .35 LECTURE 6 Mental Math and Paper .41 LECTURE 7 Intermediate Multiplication .46 LECTURE 8 The Speed of Vedic Division .52 LECTURE 9 Memorizing Numbers .com Table of Contents LECTURE 10 Calendar Calculating .63 LECTURE 11 Advanced Multiplication .69 LECTURE 12 Masters of Mental Math .76 SUPPLEMENTAL MATERIAL Solutions .com The Secrets of Mental Math Scope: M ost of the mathematics that we learn in school is taught to us on paper with the expectation that we will solve problems on paper. But there is joy and lifelong value in being able to do mathematics in your head.
In school, learning how to do math in your head quickly and accurately can be empowering. In this course, you will learn to solve many problems using multiple strategies that reinforce number sense, which can be helpful in all mathematics courses. Success at doing mental calculation and estimation can also lead to improvement on several standardized tests. We encounter numbers on a daily basis outside of school, including many situations in which it is just not practical to pull out a calculator, from buying groceries to reading the newspaper to negotiating a car payment.
And as we get older, research has shown that it is important to ¿nd activities that keep our minds active and sharp. Not only does mental math sharpen the mind, but it can also be a lot of fun. Our ¿rst four lectures will focus on the nuts and bolts of mental math: addition, subtraction, multiplication, and division. Often, we will see that there is more than one way to solve a problem, and we will motivate many of the problems with real-world applications.
Once we have mastery of the basics of mental math, we will branch out in interesting directions. Lecture 5 offers techniques for easily ¿nding approximate answers when we don’t need complete accuracy. Lecture 6 is devoted to pencil-and-paper mathematics but done in ways that are seldom taught in school; we’ll see that we can simply write down the answer to a multiplication, division, or square root problem without any intermediate results. This lecture also shows some interesting ways to verify an answer’s correctness.
In Lecture 7, we go beyond the basics to explore advanced multiplication techniques that allow many large multiplication problems to be dramatically simpli¿ed.com In Lecture 8, we explore long division, short division, and Vedic division, a fascinating technique that can be used to generate answers faster than any method you may have seen before. Lecture 9 will teach you how to improve your memory for numbers using a phonetic code. Applying this code allows us to perform even larger mental calculations, but it can also be used for memorizing dates, phone numbers, and your favorite mathematical constants. Speaking of dates, one of my favorite feats of mental calculation is being able to determine the day of the week of any date in history.
This is actually a very useful skill to possess. It’s not every day that someone asks you for the square root of a number, but you probably encounter dates every day of your life, and it is quite convenient to be able to ¿gure out days of the week. You will learn how to do this in Lecture 10. In Lecture 11, we venture into the world of advanced multiplication; here, we’ll see how to square 3- and 4-digit numbers, ¿nd approximate cubes of 2-digit numbers, and multiply 2- and 3-digit numbers together.
In our ¿nal lecture, you will learn how to do enormous calculations, such as multiplying two 5-digit numbers, and discuss the techniques used by other world- record lightning calculators. Even if you do not aspire to be a grandmaster mathemagician, you will still bene¿t tremendously by acquiring the skills taught in this course.com Acknowledgments P utting this course together has been extremely gratifying, and there are several people I wish to thank. It has been a pleasure working with the very professional staff of The Great Courses, including Lucinda Robb, Marcy MacDonald, Zachary Rhoades, and especially Jay Tate. Thanks to Professor Stephen Lucas, who provided me with valuable historical information, and to calculating protégés Ethan Brown and Adam Varney for proof-watching this course.
Several groups gave me the opportunity to practice these lectures for live audiences, who provided valuable feedback. In particular, I am grateful to the North Dakota Department of Public Instruction, Professor Sarah Rundell of Dennison University, Dr. Daniel Doak of Ohio Valley University, and Lisa Loop of the Claremont Graduate University Teacher Education Program. Finally, I wish to thank my daughters, Laurel and Ariel, for their patience and understanding and, most of all, my wife, Deena, for all her assistance and support during this project.
Arthur Benjamin Claremont, California 3 www.com Math in Your Head! Lecture 1 Just by watching this course, you will learn all the techniques that are required to become a fast mental calculator, but if you want to actually improve your calculating abilities, then just like with any skill, you need to practice. I n school, most of the math we learn is done with pencil and paper, yet in many situations, it makes more sense to do problems in your head. The ability to do rapid mental calculation can help students achieve higher scores on standardized tests and can keep the mind sharp as we age. One of the ¿rst mental math tips you can practice is to calculate from left to right, rather than right to left.
On paper, you might add 2300 + 45 from right to left, but in your head, it’s more natural and faster to add from left to right. These lectures assume that you know the multiplication table, but there are some tricks to memorizing it that may be of interest to parents and teachers. I teach students the multiples of 3, for example, by ¿rst having them practice counting by 3s, then giving them the multiplication problems in The ability to do rapid mental order (3 × 1, 3 × 2 …) so that they calculation can help students associate the problems with the counting sequence. Finally, I mix Lecture 1: Math in Your Head! achieve higher scores on up the problems so that the students standardized tests and can can practice them out of sequence.
keep the mind sharp as we age. There’s also a simple trick to multiplying by 9s: The multiples of 9 have the property that their digits add up to 9 (9 × 2 = 18 and 1 + 8 = 9). Also, the ¿rst digit of the answer when multiplying by 9 is 1 less than the multiplier (e.com In many ways, mental calculation is a process of simpli¿cation. For example, the problem 432 × 3 sounds hard, but it’s the sum of three easy problems: 3 × 400 = 1200, 3 × 30 = 90, and 3 × 2 = 6; 1200 + 90 + 6 = 1296.
Notice that when adding the numbers, it’s easier to add from largest to smallest, rather than smallest to largest. Again, doing mental calculations from left to right is also generally easier because that’s the way we read numbers. On paper, you might start by multiplying 7 × 4 to get 28, but when doing the problem mentally, it’s better to start with 7 × 50 (350) to get an estimate of the answer. To get the exact answer, add the product of 7 × 50 and the product of 7 × 4: 350 + 28 = 378.