Mathematical Ideas

950 Pages·1998·92.68 MB·English

by  Charles D. Miller

y-^^:
CHARLES D. MILLER
VERN E. HEEREN
E. JOHN HORNSBY, JR.
jcaj
^M^^
h..
I
wtktal
ideas
\
\
.
Meteorologistsuseavarietyoftechnologies,likethe
heliostat shown here, to predict the weather. But
technologywithouteffective problem-solving strate-
gies is no more than simply guessing what the
weatherwillbeliketomorrow.Seepage1
This photo from Cornell University
dimensional graphic ofa rotary motor^
useideasfromsettheorytoconstructcomplexsk
suchasthisfromsimplershapes.Seepage51.
Computers in the 1960scon-
tainedhundredsoflogicgates
made from bufky transistors.
Today, desktopcomputers
hold thousands of logic gates
on silicon chips and possess
more computing power than
mainframes of thirty years,
lago. Seepage98.
iJu V ''"-
patient cataloging of DNA
uences is but one tool used by ^e.1
earchers in the fight against
AlOS. For information about how
serves as another tool to
nbatAIDS,seepage318.
Inordertoavoidexcessiveexposure
to ultraviolet radiation, sunbathers
should be aw/are of the intensity of
the ultraviolet rays at their favorite
beach. Real numberscanbeusedto
quantify the intensity of sunligh)^,'
whichvarifesaccordingtothelatiUjtle
Bar code scanners not only
translate the bar code to a
sequenceofnumbers,theyare
also programmed to use prop-
ertiesofournumerationsystem
tocheckthatthecodeisvalid.
For another application of
properties of our numeration
system,seepage152.
Digitized by the Internet Archive
2010
in
http://www.archive.org/details/mathematicalideOOmill
W-\
iMM-
Mathematical Ideas
EIGHTH EDITION
CHARLES D. MILLER
VERN E. HEEREN
American RiverCollege
E. JOHN HORNSBY, JR.
University ofNew Orleans
T
ADDISON-WESLEY
AnimprintofAddisonWesleyLongman, Inc.
Reading,Massachusetts•MonloPark,California•NewYurk•Harlow,England
DonMills,Ontario•Sydney•MexicoCity•Madrid •Amsterdam
Zi? T ''ii'. IJKti-'
SpoiiMtrint; F.ditor: Aiinc Kelly
Di-vi'lopiiK-ntal F.ditor: Greg McRill
Project Kditor: Cathy Wacaser
I)esij;n Administrator: Jess Schaal
Text and Cover Design: Lesiak/Crampton Design Inc: Lucy Lesiak
Cover Photo: Dinig Armand/Tony Stone Images
Photo Researcher: Kaicn Kohlik
Production Administrator: Randee Wire
Compositor: Interactive Composition Corporation
Printerand Hinder: R.R. Donnelley & Sons Company
Cover Printer: Phoenix ColorCorporation
Forpermissiontousecopyrightedmaterial,gratefulacknowledgmentismadeto
the copyright holders on pages 909-910. which are hereby inade part ofthis
copyrightpage.
MathematicalIdeas. Eighth Edition
LibraryofCongressCataloging-in-Publication Data
Miller,Charles David.
Mathemati—cal ideas/Charles D. Miller. Vern E. Heeren. E. John
Hornsby. Jr. 8thed.
p. cm.
Includes index.
ISBN 0-673-99893-2 (Student Edition)
ISBN0-673-98241-6 (Instructor's AnnotatedExercises)
1. Mathematics. 1. Heeren. Vern E. II. Hornsby. E. John.
III. Title.
QA39.2.M55 1997 96-25478
510—dc20 CIP
Copyright © 1997 by Addison-Wesley Educational Publishers Inc.
Allrightsreserved. Nopartofthispublication maybereproduced, storedina
retrieval system,ortransmitted, in any formorby any means, electronic,
mechanical, photocopying, recording, or otherwise, without the prior written
permission ofthe publishers. Printed in the United States ofAmerica.
—DOW—
96 97 98 99 9 8 7 6 5 4 3 2 1
^^eface
Nearlythirtyyearsafteritsfirstappearance.MathematicalIdeasremainsoneof
the most popular texts in college mathematics. This eighth edition, like its
predecessors, has been designed with a variety ofstudents in mind. It is well-
suited for several types ofcourses, including mathematics for liberal arts stu-
dents,surveycoursesinmathematics,finitemathematics,andmathematicsfor
prospectiveandin-serviceelementaryandmiddleschoolteachers.Ampletopics
are included for a two-term course, yet the variety oftopics and flexibility of
sequence make the text suitableforshortercourses as well.
In this edition we have attempted to retain many of the features of the
seventhedition, andatthe sametime introducenewonesthatwill appeal toa
newgenerationofinstructorsandstudents. Ourmainobjectivescontinuetobe
comprehensivecoverageoftopicsappropriateforamathematicssurveycourse,
logical and flexible organization, clear exposition, an abundance ofexamples,
and well-plannedexercise sets with numerous applications.
Wehaveworkeddiligentlytoincorporateintothiseditiontherecommenda-
tions of the Cuniciiliim and Evaluation Standards for School Mathematics.
preparedbytheNationalCouncilofTeachersofMathematics(NCTM),andthe
StandardsforIntroductoryCollegeMathematicsBeforeCalculus, presentedby
theAmericanMathematicalAssociationofTwo-YearColleges(AMATYC).For
example, to address the issue ofcooperative learning we have introduced the
featuretitled"CollaborativeInvestigation"ineachchapter.Thefaceofmathe-
maticseducationischangingdrasticallyduetotheavailabilityoflow-costgraph-
ingtechnology,andwehaveincludedmanyscreensindicatinghowsuchtechnol-
ogy can be used to support results found analytically in the text. (It is not
essential,however,thatthestudenthaveagraphingcalculatortostudyfromthis
book: the technology component is totally optional.)
Theseventheditionincludedanabundanceofexercisesnewtothatedition.
We have retained many of them, while also adding many new applications
focusingonreal-lifedata.Theseincludegraphs,tables,andchartswhenappro-
priate.Chapters6.7.and8areespeciallyrichinthesenewexercises.Inaddition
totheusualdrillexerci.ses.weincludemanythattestconceptualunderstanding.
In lightofthecurrentfocuson writingacrossthecurriculum,mostsetsinclude
some exercises that require the student to answer by writing a few .sentences.
Thepopularmarginnotesthathaveappearedinpreviouseditionshavebeen
retained and updated when necessary. New ones have been added as well. We
hope that users continue to enjoy them as mucii as we enjoy researching and
composing them.
Sometopics in the firstsixchaptersrequireabasic knowledgeofequation
solving. Dependingon thebackgroundofthe students, the instructormayomit
topicsthatrequirethisskill.Ontheotherhand,thetwoalgebrachapters(7and
8) provide an excellent overviewofbasic algebra. Because ofthe flexibility of
the text, they may be coveredat almost any time.
Pedagogical Features
Several new or enhanced features, designed to assist students in the learning
processandaidinstructorsinteaching,havebeenintroducedandretainedinthis
edition. We continue to use full color, and the art program remainsone ofthe
highlights of the book. Numerous carefully selected examples illustrate the
concepts andskillsbeing introduced. Illustratedmargin notes present itemsof
interestthatexplorethehumandimensionandhistoricalcontextofmathemati-
cal ideas. "For Further Thought" boxes encourage students to share among
themselves their reasoning processes to gain a deeper understanding of key
mathematical concepts. Special paragraphs labeled "Problem Sohing" relate
the discussion ofstrategies to techniques that ha\e been learnedearlierorwill
beappliedlater. Most sectionsincludeafew challengingexercisesthatrequire
studentstoextendtheideaspresentedinthesection.The.seare identifiedinthe
Instructor'sAnnotatedExercises (M£) by the symbol Tocomplementthe
drillandapplicationexercises,mostexercisesetsincludeafewexercisesrequir-
ingadeeperunderstandingoftheconceptsintroduced.Inaddition,manyexer-
cisesrequire studentstorespondby writingafewsentences. Theseconceptual
and writina exercises are indicated in thelAEb\ the symbols © and *.
Overview of the Chapters
Chapter 1 (Approach to Problem Solving) introduces the student to inductive
reasoning,patternrecognition,andproblem-solvingtechniques.Chapters2and
3 (Sets and Logic) give brief overviews of set theor\ and elementary logic:
instructors wishing to do so maycoverChapter3 before Chapter2. Chapter4
(Numeration and Mathematical Systems) co\ers various types ofnumeration
systems,modulosystems,andgroups.InChapter5(NumberTheory)wepresent
anelementaryintroductiontotopicssuchasprimeandcompositenumbersand
theFibonacciSequence.InChapter6(TheRealNumberSystem),weintroduce
some ofthe basic concepts ofarithmetic ofthe real numbers in their various
forms.Chapters7and8(TheBasicConceptsof.AlgebraandFunctions.Graphs,
andSystemsofEquationsandInequalities)formthecoreofthealgebracompo-
nentofthetext,providingeitheranintroductiontoorare\iewofbasicfreshman
algebra.InChapter9(Geometry),thestudentisintroducedtothebasicconcepts
ofEuclideangeometry.Se\eralenrichmentsectionsareincludedaswell.Chap-
ter10(CountingMethods)focu.sesonelementarycountingtechniques,inprepa-
ration for the chapter to follow. Chapter 11 (Probability) covers the basics of
probability,odds,andexpected\alue.InChapter 12(Statistics),wegiveabrief
introduction to the various topics of elementary statistics. Chapter 13 (Con-
sumerMathematics)focusesonmathematicsofinterest,mortgages,andinvest-
ments.InChapter 14(MatricesandTheirApplications)weshow howmatrices
play animportantrole inboththeoreticalandappliedmathematics.
New to this Edition
In this edition wehaveaddedthe following newfeatures.
ChapterOpeners A chapteropenerfocusingon an application ofthe topic
tobeaddressedinthechapteris nowincluded.Theseopenerswereresearched
by Gary Rockswold ofMankato State University. The goal ofthese openers is
toshowthestudentthatreal-worldapplicationsofmathematicsdoindeedexist,
andhopefullytheywillenablethestudenttoappreciatetherelevanceofmath-
ematics inourworld.
Graphing Calculator Notes While the useofgraphingcalculators isnot
a requirement for using this text, we have included many typical screens that
illustratehowthismoderntechnologycanbeusedtosupplementandsupportthe
conceptspresentedinthetext.Weoftenaddressthelimitationsoftechnologyas
well, pointing out that the calculatorcan only assist us ifwe are aware ofthe
concepts first.
Collaborative Investigations At the end ofeach chapter we have pro-
vided an activity for group discussion, based on the material covered in the
chapter.Thisisinresponsetoinstructorswholiketoincorporatesuchactivities
in theirclasses.
Increased Emphasis on Graphs, Charts, and Real Data We have in-
cludedmanynewapplications,particularlyinChapters6,7,8, 12,and 13,that
employdatafromreal-life sources. Thedataisoftenprovided intheformofa
graph orachart.
A Chapter on Consumer Mathematics Chapter 13 is a newly written
chapteronconsumermathematics.Suchachapterhadbeenincludedforseveral
editions but was eliminated from the seventh edition. Due to overwhelming
response from users, it is back, completely rewritten forthe 1990s.
A Chapter on Matrices Again, a chapter that was eliminated from the
seventheditionisback.Chapter 14isanewlywrittenintroductiontothetheory
and application ofmatrices.
Course Outline Considerations
Forthe mostpart,thechaptersinthetextareindependentandmaybecovered
intheorderchosenbytheinstructor,withafewexceptions.Chapter6contains
some material dependent on ideas found in Chapter .'i. Chapter 6 should be
covered before Chapter7 ifstudent background so dictates. Chapters 7 and 8
shouldbeconsidereda"package"andcovered in sequential order. Athorough
coverageofChapter 11 dependsonknowledgeofChapter 10material,although
probability can be covered without learning extensive counting methods by
avoidingthemoredifficultexercises.ThelatterpartofChapter 12,oninferential
statistics, depends on an understanding ofprobability (Chapter 11).
Supplements
Thisedition isaccompanied by an extensive supplemental package withoffer-
ings forboth the instructorand the student.
Forthe Instructor
Instric'TOR'sAn.notatkdExkrcisks Thismanualcontainsalloftheexercises
fromthestudenttext,witheachanswerprintedincolornexttothecorresponding
exercise. In addition, challenging exercises, conceptual exercises, writing
exercises, and calculatorexercises are identified with colored symbols so that
instructors may assign these problems at theirdescretion. Each section ofthis
manualispage-referencedtothecorrespondingexercisesetinthestudenttext.
INSTRUCTOR'STESTMANUAL This manual includes fourdifferent test forms
for each chapter paralleling the chapter tests in the text, plus an additional
125-150testitemsforeachchapter.Italsocontainsteachingtipsandsuggested
course outlines.
INSTRUCTOR'S SOLUTION MANUAL This manual contains solutions to all
even-numberedexercises and alistofall conceptual, writing, andchallenging
exercises.
TestGener.ator/EditorforMathematicswithQuizMaster This is a
computerized testgeneratorthat lets instructors selecttestquestionsbyobjec-
tive or section, or use a ready-made test for each chapter. The software is
algorithmdrivensothatregeneratednumbervaluesmaintainproblemtypesand
providealargenumberoftestitemsinbothmultiple-choiceandopen-response
formats foroneormore testforms. The Editorlets instructors modifyexisting
questions or create their own including graphics and accurate math symbols.
Tests created with the Test Generator can be used with QuizMaster, which
records student scores as they take tests on a single computer network, and
printsreportsforstudents,classes,orcourses.CLASTandTASPversionsofthis
package are alsoavailable (IBM. DOS. and Macintosh).
Forthe Student
StudyGuideandSolutionManual This for-sale manual contains solutions
toall odd-numbered section, extension, and appendix exercises. Also, chapter
summariesreview keypointsinthetext,provideextraexamples,andenumerate
majortopicobjectives.
GuidetoFloridaCLASTMathematicalCompetency This special for-
salestudyguideforuseinFloridaoffershelpinpreparingfortheCollegeLevel
Academic Skills Test (CLAST). It includes a review ofarithmetic, a CLAST
pretest, supplementary exercises, and aCLASTposttest.
Computer-AssistedTutorials The tutorials offerself-paced, interactive re-
view in IBM and Macintosh formats. Solutions aregivenforall examplesand
exercises, as needed. Studentswishingtopurchasethetutorialsshouldcontact
theirbookstore.
GraphExplorer With this sophisticated software, available in IBM and
Macintoshversions,studentscangraphrectangular,conic,polar,andparamet-
ricequations:zoom:transformfunctions:andexperimentwithfamiliesofequa-
tions quickly and easily. Students wishing to purchase this software should
contact theirbookstore.
GeoExplorer Available forIBM and Macintosh, this software package en-
ablesstudentstodraw,measure,modify,andtransformgeometricshapesonthe
screen. Students wishing to purchase this software should contacttheirbook-