Alfred S. Posamentier - Math Tricks: The Surprising Wonders of Shapes and Numbers

Math Tricks

The Surprising Wonders of Shapes
and Numbers

Alfred S.Posamentier

An imprint of The Rowman & Littlefield Publishing Group, Inc.

4501 Forbes Blvd., Ste. 200

Lanham, MD 20706

www.rowman.com

Distributed by NATIONAL BOOK NETWORK

Copyright 2021 by Alfred S. Posamentier

All rights reserved. No part of this book may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without written permission from the publisher, except by a reviewer who may quote passages in a review.

British Library Cataloguing in Publication Information Available

Library of Congress Cataloging-in-Publication Data

Name: Posamentier, Alfred S., author.

Title: Math tricks : the surprising wonders of shapes and numbers / Alfred S. Posamentier.

Description: Lanham, MD : Prometheus, [2021] | Includes bibliographical references and index. | Summary: In his latest book, mathematician Alfred S. Posamentier provides easily understandable, easily presentable and easily replicated tricks that one can do with mathematicsProvided by publisher.

Identifiers: LCCN 2020042779 (print) | LCCN 2020042780 (ebook) | ISBN 9781633886643 (cloth) | ISBN 9781633886650 (ebook)

Subjects: LCSH: Mathematical recreations. | Mental arithmetic. | Geometry.

Classification: LCC QA95 .P657 2021 (print) | LCC QA95 (ebook) | DDC 793.74dc23

LC record available at https://lccn.loc.gov/2020042779

LC ebook record available at https://lccn.loc.gov/2020042780

The paper used in this publication meets the minimum requirements of American National Standard for Information SciencesPermanence of Paper for Printed Library Materials, ANSI/NISO Z39.48-1992

In recent years people seem to have gravitated toward cute things said about important subjects. It is no secret that most of them avoid mathematics, largely because school teachers sadly have rarely done much to motivate a love of the subject matter. Moreover, in more recent years secondary school teachers have been inclined to teach to the test, since their professional teaching evaluation is based on their students performance on various standardized tests. This trend seems to have usurped time that teachers could have used to enrich their instruction and thereby motivate students to enjoy mathematics. For years the author has developed books intended to enchant readers with mathematical topics that mostly either have gone unnoticed or have not been presented at all. What has not been developed until now is a book that provides easily understandable, easily presentable, and easily replicated tricks within the realm of mathematics that one can perform to show another interesting dimension of the subject. All that this text requires is the ability to do arithmetic, understand the very basics of algebra and geometry, and have an open mind for probability. Nothing presented in this book exceeds the first few years of high school mathematics.

The authors hope is that the many mathematical tricks presented here will be easily understood and replicated, so that the reader can be awed and then impress friends and colleagues with the amazing wonders that mathematics holds. Until now many of these tricks have unfortunately been well-kept secrets. With that in mind, this text presents interesting ways that the reader can use these as a personal bag of tricks that could make good dinner talk, or even bring some levity to professional meetings. Perhaps the most important part of these tricks is presentation, since most people shy away from mathematics. Consequently, we explain not only what each trick is and why it is, but also how best to present it to a nonmathematics-oriented audience. Naturally, much rests on the style of presentation. Therefore, the trickster should spend some time contemplating how and when to best present the tricks most appropriate for a particular audiencethis could be a large group or merely one individual.

This book should attract readers of all ages. The key here is to have fun with mathematics and, as a byproduct, perhaps motivate readers to delve further into the subject matter, which could support further investigation. The underlying intention of presenting these mathematics tricks is to enable audiences to recognize and appreciate the power and beauty of mathematics.

When we begin our education in elementary school, our first experience of mathematics is through arithmetic. We learn arithmetic as an automatic mechanical procedure, which in recent years has been heavily supported by an electronic calculatoreither as a separate tool or as a smartphone app. Clearly arithmetic is the basis for our work with numbers. However, through learning the basics of arithmetic, especially when teachers fixate on making all students proficient in the four basic operations, much potential insight into our work with numbers is lost. But it is exactly here, in doing arithmetic, that one can find enjoyment and enlightenment in working with numbers. In this chapter, we showcase lots of shortcuts and insights into arithmetic processes, some of which will be most entertaining and ones by which a trickster can certainly impress his or her audience with marvels all too often hidden from the general populace.

Arithmetic tricks can not only be entertaining, but also can be quite useful. Before we embark on the entertaining aspects, we will show some useful tricks that help with arithmetic calculation but could also be somewhat intriguing. Some of these might even be more efficient than using a calculator. However, with a bit of practice with each one, you could firm up the procedure for ready reference when needed.

We know that multiplying by 10 in the base 10 system requires merely placing a zero at the end of the mutiplicand; and, of course, multiplying by 100 requires that we add two zeros. Taking this one step further, when we multiply by 5, we could multiply by 10 and then divide the resulting number by 2. Or in some cases it might be easier to first divide by 2 and then multiply by 10. This is rather simple and usually taught in elementary school. We can take this a step further by considering some methods to multiply numbers by divisors of powers of 10. This is where a trickster can begin to impress the audience.

Multiplying by factors (or divisors) of powers of 10 is more involved than merely multiplying by powers of 10. In many cases, however, it can also be done mentally. Lets begin by considering multiplication by 25 (a factor of 100). Perhaps we should begin with an example. Lets consider 1625 . Since 25=1004 , we can write this as 1625=161004=164100=4100=400 . Rather simple! All we did was divide the multiplicand by four and then multiply by 100, which means adding two zeros to the number. However, more practice can be helpful, so here are some additional examples:

3825canbewrittenthisas381004=384100=192100=9.5100=950.

1.725canbewrittenthisas17101004=17410010=4.2510=42.5.

(In the previous example, we converted as follows: