Options for NEW! Mathematical Ideas by Miller/Heeren/Hornsby/Heeren 1 Standard MyMathLab This option gives you maximum control over course organization and assignment creation. 2 MyMathLab with Integrated Review This option offers a complete liberal arts math course along with integrated review of select topics from developmental math. Able to support a co-requisite course model, or any course where students will benefit from review of prerequisite skills, this Ready to Go MyMathLab solution includes prebuilt and pre-assigned assignments, making startup even easier! Mathematical Ideas captures the interest of non-majors by inspiring them to see mathematics as something interesting, relevant, and utterly practical. With a new focus on jobs, professions and careers as the context in which to frame the math, Mathematical Ideas demonstrates the importance math can play in everyday life while drawing students into the content. All Chapter Openers in this edition have been updated to reflect career-based applications. This example from Chapter 2 relates the basic concepts of set theory to choosing a career path in the health care field. A brand-new feature, When Will I Ever Use This?, highlights how the mathematical concepts covered in the chapter might be used in a particular field. Career examples range from video game programming to nursing to forestry. Margin notes appear frequently throughout the text. These provide relevant examples from media and literature in addition to historical anecdotes and current research. ThirTeenTh ediTion MatheMatical iDeas charles D. Miller Vern e. heeren American River College John hornsby University of New Orleans christopher heeren American River College aND Margaret l. Morrow Pittsburgh State University of New York for the chapter on Graph Theory Jill Van Newenhizen Lake Forest College for the chapter on Voting and Apportionment Boston Columbus Hoboken Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montréal Toronto Delhi Mexico City São Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editorial Director: Chris Hoag Editor in Chief: Anne Kelly Senior Acquisitions Editor: Marnie Greenhut Editorial Assistant: Lucia Kim Program Manager: Patty Bergin Project Manager: Sherry Berg Program Management Team Lead: Karen Wernholm Project Management Team Lead: Peter Silvia Media Producer: Nicholas Sweeny TestGen Content Manager: John Flanagan MathXL Content Developer: Bob Carroll Marketing Manager: Alicia Frankel Marketing Assistant: Brooke Smith Senior Author Support/Technology Specialist: Joe Vetere Rights and Permissions Project Manager: Diahanne Lucas Senior Procurement Specialist: Carol Melville Associate Director of Design: Andrea Nix Program Design Lead: Beth Paquin Text Design, Production Coordination, Composition, and Illustrations: Cenveo Publisher Services Cover Design: Infiniti Cover Image: Sergey Nivens/Shutterstock Copyright © 2016, 2012, 2008 by Pearson Education, Inc. All Rights Reserved. Printed in the United States of America. This publication is protected by copyright, and per- mission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise. For information regarding permis- sions, request forms and the appropriate contacts within the Pearson Education Global Rights & Permissions department, please visit www.pearsoned.com/permissions/. Acknowledgments of third-party content appear on page C-1, which constitutes an extension of this copyright page. PEARSON, ALWAYS LEARNING, and MYMATHLAB are exclusive trademarks owned by Pearson Education, Inc. or its affiliates in the United States and/or other countries. Unless otherwise indicated herein, any third-party trademarks that may appear in this work are the property of their respective owners, and any references to third-party trademarks, logos, or other trade dress are for demonstrative or descriptive purposes only. Such references are not intended to imply any sponsorship, endorsement, autho- rization, or promotion of Pearson’s products by the owners of such marks, or any relationship between the owner and Pearson Education, Inc. or its affiliates, authors, licensees, or distributors. Library of Congress Cataloging-in-Publication Data Miller, Charles D. (Charles David), 1942-1986.   Mathematical ideas. – 13th edition / Charles D. Miller, Vern E. Heeren, American River College, John Hornsby, University of New Orleans, Christopher Heeren, American River College.     pages cm   ISBN 0-321-97707-6 (student edition)  1. Mathematics–Textbooks. I. Heeren, Vern E., author. II. Hornsby, John, 1949- III. Heeren, Christopher. IV. Title. QA39.3.M55 2015 510–dc23 2014032895 1 2 3 4 5 6 7 8 9 10—V011—18 17 16 15 14 ISBN 13: 978-0-321-97707-6 www.pearsonhighered.com ISBN 10: 0-321-97707-6 Phor my phriend Phlash Phelps, whose Phunny Pharm helps me get through my mornings—JOHNNY (1119) To my beloved wife, Carole, for decades of inspiration and support—VERN To Heather, for your undying love and encouragement—CHRIS This page intentionally left blank conTenTS Preface  xiii acknoWledgmenTS  xx aboUT The aUThorS  xxii 1 The Art of Problem Solving 1 1.1 Solving ProblemS by indUcTive reaSoning 2 characteristics of inductive and deductive reasoning • Pitfalls of inductive reasoning 1.2 an aPPlicaTion of indUcTive reaSoning: nUmber PaTTernS 9 number Sequences • Successive differences • number Patterns and Sum formulas • figurate numbers 1.3 STraTegieS for Problem Solving 19 a general Problem-Solving method • Using a Table or chart • Working backward • Using Trial and error • guessing and checking • considering a Similar, Simpler Problem • drawing a Sketch • Using common Sense 1.4 nUmeracy in Today’S World 32 calculation • estimation • interpretation of graphs • communicating mathematics through language Skills ChaPter 1 SUmmary  42 ChaPter 1 TeST  45 2 The Basic Concepts of Set Theory 47 2.1 SymbolS and Terminology 48 designating Sets • Sets of numbers and cardinality • finite and infinite Sets • equality of Sets 2.2 venn diagramS and SUbSeTS 54 venn diagrams • complement of a Set • Subsets of a Set • Proper Subsets • counting Subsets 2.3 SeT oPeraTionS 60 intersection of Sets • Union of Sets • difference of Sets • ordered Pairs • cartesian Product of Sets • more on venn diagrams • de morgan’s laws 2.4 SUrveyS and cardinal nUmberS 71 Surveys • cardinal number formula • Tables ChaPter 2 SUmmary 79 ChaPter 2 TeST 82 3 Introduction to Logic 83 3.1 STaTemenTS and QUanTifierS 84 Statements • negations • Symbols • Quantifiers • Quantifiers and number Sets v vi conTenTS 3.2 TrUTh TableS and eQUivalenT STaTemenTS 91 conjunctions • disjunctions • negations • mathematical Statements • Truth Tables • alternative method for constructing Truth Tables • equivalent Statements and de morgan’s laws 3.3 The condiTional and circUiTS 102 conditionals • Writing a conditional as a disjunction • circuits 3.4 The condiTional and relaTed STaTemenTS 111 converse, inverse, and contrapositive • alternative forms of “if p, then q” • biconditionals • Summary of Truth Tables 3.5 analyzing argUmenTS WiTh eUler diagramS 117 logical arguments • arguments with Universal Quantifiers • arguments with existential Quantifiers 3.6 analyzing argUmenTS WiTh TrUTh TableS 123 Using Truth Tables to determine validity • valid and invalid argument forms • arguments of lewis carroll ChaPter 3 SUmmary 132 ChaPter 3 TeST 137 4 Numeration Systems 139 4.1 hiSTorical nUmeraTion SySTemS 140 basics of numeration • ancient egyptian numeration • ancient roman numeration • classical chinese numeration 4.2 more hiSTorical nUmeraTion SySTemS 148 basics of Positional numeration • hindu-arabic numeration • babylonian numeration • mayan numeration • greek numeration 4.3 ariThmeTic in The hindU-arabic SySTem 153 expanded form • historical calculation devices 4.4 converSion beTWeen nUmber baSeS 160 general base conversions • computer mathematics ChaPter 4 SUmmary 172 ChaPter 4 TeST 176 5 Number Theory 177 5.1 Prime and comPoSiTe nUmberS 178 Primes, composites, and divisibility • The fundamental Theorem of arithmetic 5.2 large Prime nUmberS 184 The infinitude of Primes • The Search for large Primes 5.3 SelecTed ToPicS from nUmber Theory 192 Perfect numbers • deficient and abundant numbers • amicable (friendly) numbers • goldbach’s conjecture • Twin Primes • fermat’s last Theorem conTenTS vii 5.4 greaTeST common facTor and leaST common mUlTiPle 199 greatest common factor • least common multiple 5.5 The fibonacci SeQUence and The golden raTio 208 The fibonacci Sequence • The golden ratio 5.6 magic SQUareS (online) magic Square • magic Sum formula ChaPter 5 SUmmary 214 ChaPter 5 TeST 218 6 The Real Numbers and Their Representations 219 6.1 real nUmberS, order, and abSolUTe valUe 221 Sets of real numbers • order in the real numbers • additive inverses and absolute value • applications of real numbers 6.2 oPeraTionS, ProPerTieS, and aPPlicaTionS of real nUmberS 230 operations on real numbers • order of operations • Properties of addition and multiplication of real numbers • applications of Signed numbers 6.3 raTional nUmberS and decimal rePreSenTaTion 243 definition and the fundamental Property • operations with rational numbers • density and the arithmetic mean • decimal form of rational numbers 6.4 irraTional nUmberS and decimal rePreSenTaTion 258 definition and basic concepts • irrationality of 2 and Proof by contradiction • operations with Square roots • The irrational numbers P, F, and e ! 6.5 aPPlicaTionS of decimalS and PercenTS 268 operations with decimals • rounding methods • Percent ChaPter 6 SUmmary 282 ChaPter 6 TeST 288 7 The Basic Concepts of Algebra 291 7.1 linear eQUaTionS 293 Solving linear equations • Special kinds of linear equations • literal equations and formulas • models 7.2 aPPlicaTionS of linear eQUaTionS 301 Translating Words into Symbols • guidelines for applications • finding Unknown Quantities • mixture and interest Problems • monetary denomination Problems • motion Problems 7.3 raTio, ProPorTion, and variaTion 313 Writing ratios • Unit Pricing • Solving Proportions • direct variation • inverse variation • Joint and combined variation