A Survey of Mathematics with Applications, Eighth Edition Allen R. Angel, Christine D. Abbott, and Dennis C. Runde Understanding mathematics means understanding how a great deal of the world works. Let us introduce you to the practical, interesting, accessible, and powerful world of mathematics today— the world of A Survey of Mathematics with Applications, Eighth Edition. 4 C H A P T E R Systems of Numeration Chapter Openers and WHAT YOU WILL LEARN Section Openers • Additive, multiplicative, and ciphered systems of numeration Interesting and motivational • Place-value systems of numeration • Egyptian, Hindu-Arabic, Roman, Chinese, Ionic applications introduce Greek, Babylonian, and Mayan numerals each chapter and section, • Converting base 10 numerals to numerals in other bases illustrating the real-world • Converting numerals in other bases to base 10 nature of the chapter topics. numerals • Performing addition, subtraction, multiplication, and division in other bases • Other computational methods such as duplation and mediation, the lattice method, and Napier's rods WHY IT IS IMPORTANT The number system most of the world uses today, called the Hindu-Arabic system, is only one way to communicate numerically. In this chapter, we will study other systems of numeration. We will also study how to perform basic arithmetic in other bases and with methods other than those most of AThere are many ways that we can communicate numerically us were taught when we were children. 4.5 EARLY COMPUTATIONAL METHODS Suppose that last week you worked at your job for 27 hours at a rate of $8 per hour How much money did you make? You may find the answer by multiplying 27 x 8 "by hand." X 8 216 Although most of us would use this method, it is not the only method for multiplying two numbers together. Early civilizations used various other methods. In this section, we will study three other methods of multiplication. aMultiplication can be used to calculate the amount of your paycheck. Profiles in Mathematics Brief historical sketches and PROFILE IN MATHEMATICS MATHEMATICS vignettes present the stories of TODAY people who have advanced the John Napier The New Old Math discipline of mathematics. Mathematics Today This feature relates mathematics © ometimes the new way to to everyday life, helping students solve a problem is to use an to recognize the need for math old method. Such is the case in © uring the seventeenth cen- some elementary schools with re¬ and gain an appreciation for 1 tury, the growth of scientific gard to multiplying whole num¬ fields such as astronomy required math in their lives. bers. In addition to teaching the the ability to perform often un¬ traditional multiplication algorithm, wieldy calculations. Scottish mathe¬ some schools are also teaching the matician John Napier (1550-1617) lattice method of multiplication. made great contributions toward Lattice multiplication, along with solving the problem of computing the Hindu-Arabic numerals, was thesr Did You Know? introduced to Europe in the thir¬ elude teenth century by Fibonacci. (See and the Profile in Mathematics? on plica DID YOU KNOW? These colorful and engaging page 298.) Lattice multiplication is Napii boxed features highlight recommended by the National oped Council of Teachers of Mathematics Speaking to Machines the connections between as a way to help students increase their conceptual understanding as mathematics and a wide they are developing computational ability. This method gained in popu¬ variety of other disciplines, larity when it was included in sev¬ including history, the arts and eral elementary school textbooks as a way to help students understand sciences, and technology. how the traditional multiplication algorithm works. On digital video discs (DVDs) and com¬ pact discs (CDs), video and sount digitally encoded on the undersit RECREATIONAL MATHEMATICS the disc in a binary system of pits "lands" (nonpits). To play the dis 74. Determine b, by trial and error, if 1304/, = 204. laser beam tracks along the spira is reflected when it hits a land (si 75. In a base 4 system, each of the four numerals is repre¬ sent = 1), but it is not reflected sented by one of the following colors: pits (no signal = 0). The binary s Exercise Sets quence is then converted into vie ©images and music. The diverse and numerous Determine the value of each color if the following addition or the past 600 year have used the Hindu-, is true in base 4. exercise sets include exercise system of numeration w types such as Concept/Writing, change. Our base 10 nume system seems so obvious i Practice the Skills, Problem perhaps because of our 10 f Solving, Challenge Problem/ and 10 toes, but it would b to think that numbers in Group Activity, Recreational bases are not useful. In fact, < INTERNET/RESEARCH ACTIVITIES the most significant nume Mathematics, and Internet/ 76. Write a report on how computers use the binary (base 2), systems is the binary syste octal (base 8), and hexadecimal (base 16) numeration Research Activities. base 2, This system, with il systems. mental simplicity, is what is u: computers to process inforr and "talk" to one another. \A computer receives a comma data, every character in the mand or data must first be verted into a binary numeral f computer to understand and A Survey of MATHEMATICS with Applications EIGHTH EDITION Digitized by the Internet Archive in 2018 https://archive.org/details/surveyofmathemaOOange EIGHTH EDITION A Survey of , t ■ ?t I * j$jfj§ v-, >''■' " ».v with Applications fmm ALLEN R. ANGEL Monroe Community College CHRISTINE D. ABBOTT Monroe Community College DENNIS C. RUNDE Manatee Community College Boston San Francisco New York London Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Paris Cape Town Hong Kong Montreal Publisher: Greg Tobin Executive Editor: Anne Kelly Acquisitions Editor: Marnie Greenhut Senior Project Editor: Rachel Reeve Associate Project Editor: Elizabeth Bernardi Editorial Assistant: Leah Goldberg Senior Managing Editor: Karen Wemholm Senior Production Supervisor: Peggy McMahon Design Supervisor: Barbara T. Atkinson Cover Designer: Barbara T. Atkinson Cover Image: Rows of speakers, © Getty Images, Inc. 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Library of Congress Cataloging-in-Publication Data Angel, Allen R., 1942— A survey of mathematics with applications.—8th ed. / Allen R. Angel, Christine D. Abbott, Dennis C. Runde. p. cm. Includes index. ISBN 0-321-50107-1-ISBN 0-321-50132-2 I. Mathematics-Textbooks. I. Abbott, Christine D. II. Runde, Dennis C. QA39.3.A54 2009 510—dc22 2007060062 Copyright © 2009 Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in 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 publisher. Printed in the United States of America. For information on obtaining permission for use of material in this work, please submit a written request to Pearson Education, Inc.. Rights and Contracts Department, 75 Arlington Street, Suite 300, Boston, MA 02116, fax your request to 617-848-7047, or e-mail at http://www.pearsoned.com/legal/permissions.htm. ISBN-13: 978-0-321-50107-3 ISBN-10: 0-321-50107-1 5 67 89 10—VHP—11 1009 To my wife, Kathy Angel (photo on page 60) A.R.A. To my husband Jason; and children, Matthew and Jake (photo on page 197) C. D.A. To my parents, Bud and Tina Runde D. C.R.