Thinking mathematically

1055 Pages·2019·50.464 MB·English

by  Robert Blitzer

dumperina
Thinking
Mathematically
Seventh Edition
Robert Blitzer
Miami Dade College
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Library of Congress Cataloging-in-Publication Data
Names: Blitzer, Robert, author.
Title: Thinking mathematically / Robert F. Blitzer.
Description: Seventh edition. | Boston : Pearson, [2019]
Identifiers: LCCN 2017046337 | ISBN 9780134683713 (alk. paper) | ISBN 0134683714 (alk. paper)
Subjects: LCSH: Mathematics–Textbooks.
Classification: LCC QA39.3 .B59 2019 | DDC 510–dc23
LC record available at https://lccn.loc.gov/2017046337
ISBN-13: 978-0-13-468371-3
ISBN-10: 0-13-468371-4
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Contents
About the Author vi
Preface vii 3 Logic 117
Resources for Success ix
To the Student xi 3.1 Statements, Negations,
Acknowledgments xii and Quantified Statements 118
Index of Applications xv 3.2 Compound Statements
and Connectives 126

3.3 Truth Tables for
1Problem Solving and Negation, Conjunction,
Critical Thinking 1 and Disjunction 139
3.4 Truth Tables for the Conditional
1.1 Inductive and Deductive
Reasoning 2 and the  Biconditional 154
3.5 Equivalent Statements and Variations
1.2 Estimation, Graphs,
and Mathematical of Conditional Statements 166
Models 14 3.6 Negations of Conditional Statements
1.3 Problem Solving 30 and De Morgan’s Laws 176
3.7 Arguments and Truth Tables 184
Chapter Summary,
Review, and Test 43 3.8 Arguments and Euler Diagrams 199
Chapter 1 Test 46 Chapter Summary, Review, and Test 209
Chapter 3 Test 213
2 Set Theory 49 4 Number Representation
and Calculation 215
2.1 Basic Set Concepts 50 4.1 Our Hindu-Arabic
2.2 Subsets 64 System and Early
2.3 Venn Diagrams and Positional Systems 216
Set Operations 73 4.2 Number Bases in
2.4 Set Operations and Positional Systems 224
Venn Diagrams with 4.3 Computation in
Three Sets 87 Positional Systems 231
2.5 Survey Problems 99 4.4 Looking Back at Early
Chapter Summary, Numeration Systems 240
Review, and Test 110 Chapter Summary,
Chapter 2 Test 114 Review, and Test 247
Chapter 4 Test 250
iii
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iv Contents
5 7
Number Theory and Algebra: Graphs,
the Real Number Functions,
System 251 and Linear
Systems 411
5.1 Number Theory:
Prime and Composite 7.1 Graphing and
Numbers 252 Functions 412
5.2 The Integers; Order 7.2 Linear Functions
of Operations 262 and Their Graphs 424
5.3 The Rational 7.3 Systems of Linear
Numbers 276 Equations in
5.4 The Irrational Two Variables 438
Numbers 291 7.4 Linear Inequalities
5.5 Real Numbers and in Two Variables 453
Their Properties; 7.5 Linear Programming 462
Clock Addition 304
7.6 Modeling Data: Exponential,
5.6 E xponents and Scientific Logarithmic, and Quadratic Functions 468
Notation 315
Chapter Summary, Review, and Test 484
5.7 A rithmetic and Geometric
Chapter 7 Test 490
Sequences 326
Chapter Summary, Review, and Test 336
Chapter 5 Test 341 8 Personal
Finance 493

6
Algebra: 8.1 Percent,
Equations and Sales Tax,
and Discounts 494
Inequalities 343
8.2 Income Tax 503
6.1 Algebraic 8.3 Simple
Expressions Interest 514
and Formulas 344
8.4 Compound Interest 519
6.2 L inear Equations
8.5 Annuities, Methods of Saving,
in One Variable and  Proportions 354
and Investments 529
6.3 A pplications of Linear Equations 369
8.6 Cars 545
6.4 Linear Inequalities in One Variable 380
8.7 The Cost of Home Ownership 554
6.5 Quadratic Equations 390
8.8 Credit Cards 563
Chapter Summary, Review, and Test 405
Chapter Summary, Review, and Test 572
Chapter 6 Test 409
Chapter 8 Test 578
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Contents v
9 12
Measurement 581 Statistics 771

9.1 Measuring Length; 12.1 Sampling, Frequency
The Metric System 582 Distributions, and
9.2 Measuring Area Graphs 772
and Volume 592 12.2 Measures of
9.3 Measuring Weight Central Tendency 786
and Temperature 602 12.3 Measures of Dispersion 800
Chapter Summary, Review, and Test 611 12.4 The Normal Distribution 808
Chapter 9 Test 614 12.5 Problem Solving with the Normal
Distribution 822
10 Geometry 615 12.6 Scatter Plots, Correlation, and
Regression Lines 827

Chapter Summary, Review, and Test 838
10.1 Points, Lines,
Planes, and Angles 616 Chapter 12 Test 843
10.2 Triangles 625
13
Voting and
10.3 Polygons, Perimeter, and Tessellations 637
Apportionment 845
10.4 Area and Circumference 646
10.5 Volume and Surface Area 657 13.1 Voting Methods 846
10.6 Right Triangle Trigonometry 666 13.2 Flaws of Voting
10.7 Beyond Euclidean Geometry 676 Methods 858
Chapter Summary, Review, and Test 685 13.3 Apportionment
Methods 869
Chapter 10 Test 691
13.4 Flaws of
Apportionment
11
Counting Methods
Methods 883
and Probability
Chapter Summary, Review, and Test 893
Theory 693
Chapter 13 Test 896

11.1 The Fundamental
Counting Principle 694 14 Graph Theory 897
11.2 Permutations 700
11.3 Combinations 708 14.1 Graphs, Paths,
11.4 Fundamentals of Probability 715 and Circuits 898
11.5 Probability with the Fundamental 14.2 Euler Paths and Euler Circuits 908
Counting Principle, Permutations,
14.3 Hamilton Paths and Hamilton Circuits 920
and  Combinations 724
14.4 Trees 930
11.6 Events Involving Not and Or; Odds 731
Chapter Summary, Review, and Test 939
11.7 Events Involving And; Conditional
Probability 744 Chapter 14 Test 944
11.8 Expected Value 756
Answers to Selected Exercises AA1
Chapter Summary, Review, and Test 763
Subject Index I1
Chapter 11 Test 769 Credits C1
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About the Author
Bob Blitzer
is a native of Manhattan and received a Bachelor of Arts degree
with dual majors in mathematics and psychology (minor: English literature) from
the City College of New York. His unusual combination of academic interests
led him toward a Master of Arts in mathematics from the University of Miami
and a doctorate in behavioral sciences from Nova University. Bob’s love for
teaching mathematics was nourished for nearly 30 years at Miami Dade College,
where he received numerous teaching awards, including Innovator of the Year
from the League for Innovations
in the Community College and an
endowed chair based on excellence in
the classroom. In addition to Thinking
Mathematically, Bob has w ritten
textbooks covering introductory
algebra, intermediate algebra, college
algebra, algebra and trigonometry,
precalculus, trigonometry, and liberal
arts mathematics for high school
students, all published by Pearson.
When not secluded in his Northern
California writer’s cabin, Bob can be
found hiking the beaches and trails
of Point Reyes National Seashore,
and tending to the chores required
by his beloved entourage of horses,
chickens, and irritable roosters.
vi
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Preface
Thinking Mathematically, Seventh Edition provides a New applications include student-loan debt (Exercise
general survey of mathematical topics that are useful Set 1.2), movie rental options (Exercise  Set  1.3),
in our contemporary world. My primary purpose in impediments to academic performance (Section 2.1),
writing the book was to show students how mathematics measuring racial prejudice, by age (Exercise Set 2.1),
can be applied to their lives in interesting, enjoyable, generational support for legalized adult marijuana
and meaningful ways. The book’s variety of topics and use (Exercise Set 2.3), different cultural values among
flexibility of sequence make it appropriate for a one- or nations (Exercise Set 2.5), episodes from the television
two-term course in liberal arts mathematics, quantitative series The Twilight Zone (Section 3.6) and the film
reasoning, finite mathematics, as well as for courses Midnight Express (Exercise Set 3.7), excuses by
specifically designed to meet state-mandated requirements college students for not meeting assignment deadlines
in mathematics. (Exercise Set 5.3), fraction of jobs requiring various
I wrote the book to help diverse students, with levels of education by 2020 (Exercise Set 5.3), average
different backgrounds and career plans, to succeed. earnings by college major (Exercise Set 6.5), the pay gap
Thinking Mathematically, Seventh Edition, has four (Exercise Set 7.2), inmates in federal prisons for drug
major goals: offenses and all other crimes (Exercise Set 7.3), time
breakdown for an average 90-minute NFL broadcast
1. To help students acquire knowledge of fundamental
(Section 11.6), Scrabble tiles (Exercise Set 11.5), and
mathematics.
are inventors born or made? (Section 12.2).
2. To show students how mathematics can solve
• New Blitzer Bonuses. The Seventh Edition contains a
authentic problems that apply to their lives.
variety of new but optional enrichment essays. There
3. To enable students to understand and reason with
are more new Blitzer Bonuses in this edition than in any
quantitative issues and mathematical ideas they are
previous revision of Thinking Mathematically. These
likely to encounter in college, career, and life.
include “Surprising Friends with Induction” (Section 1.1),
4. To enable students to develop problem-solving skills,
“Predicting Your Own Life Expectancy” (Section 1.2),
while fostering critical thinking, within an interesting
“Is College Worthwhile?” (Section 1.2), “Yogi-isms”
setting.
(Section 3.4), “Quantum Computers” (Section 4.3),
One major obstacle in the way of achieving these “Slope and Applauding Together” (Section  7.2),
goals is the fact that very few students actually read their “A Brief History of U.S. Income Tax” (Section 8.2)
textbook. This has been a regular source of frustration “Three Decades of Mortgages” (Section 8.7), “Up to
for me and my colleagues in the classroom. Anecdotal Our Ears in Debt” (Section 8.8), “The Best Financial
evidence gathered over years highlights two basic reasons Advice for College Graduates” (Section 8.8), “Three
why students do not take advantage of their textbook: Weird Units of Measure” (Section 9.1), “Screen Math”
(Section 10.2), “Senate Voting Power” (Section 13.3),
“I’ll never use this information.”
“Hamilton Mania” (Section 13.3), “Dirty Presidential
“I can’t follow the explanations.”
Elections” (Section 13.3), “Campaign Posters as Art”
I’ve written every page of the Seventh Edition with the (Section 13.4), and “The 2016 Presidential Election”
intent of eliminating these two objections. The ideas and (Section 13.4).
tools I’ve used to do so are described for the student in • New Graphing Calculator Screens. All screens have
“A Brief Guide to Getting the Most from This Book,” been updated using the TI-84 Plus C.
which appears inside the front cover. • Updated Tax Tables. Section 8.2 (Income Tax)
contains the most current federal marginal tax tables
What’s New in the Seventh Edition? and FICA tax rates available for the Seventh Edition.
• New MyLabTM Math. In addition to the new
• New and Updated Applications and Real-World Data. functionalities within an updated MyLab Math,
I’m on a constant search for real-world data that can the new items specific to Thinking Mathematically,
be used to illustrate unique mathematical applications. Seventh Edition MyLab Math include
I researched hundreds of books, magazines, All new objective-level videos with assessment
~
newspapers, almanacs, and online sites to prepare the Interactive concept videos with assessment
~
Seventh Edition. This edition contains 110 worked-out Animations with assessment
~
examples and exercises based on new data sets and StatCrunch integration.
~
104 examples and exercises based on updated data.
vii
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viii Preface
What Familiar Features Have Been Retained in the • Concept and Vocabulary Checks. The Seventh Edition
Seventh Edition? contains 653 short-answer exercises, mainly fill-in-
the blank and true/false items, that assess students’
• Chapter-Opening and Section-Opening Scenarios. understanding of the definitions and c oncepts
Every chapter and every section open with a scenario presented in each section. The Concept and Vocabulary
presenting a unique application of mathematics in Checks appear as separate features preceding the
students’ lives outside the classroom. These scenarios Exercise Sets. These are assignable in the MyLab Math
are revisited in the course of the chapter or section course.
in an example, discussion, or exercise. The often • Extensive and Varied Exercise Sets. An abundant
humorous tone of these openers is intended to help collection of exercises is included in an Exercise Set at
fearful and reluctant students overcome their negative the end of each section. Exercises are organized within
perceptions about math. A feature called “Here’s seven category types: Practice Exercises, Practice
Where You’ll Find These Applications” is included Plus Exercises, Application Exercises, Explaining the
with each chapter opener. Concepts, Critical Thinking Exercises, Technology
• Section Objectives (What Am I Supposed to Learn?). Exercises, and Group Exercises.
Learning objectives are clearly stated at the beginning • Practice Plus Problems. This category of exercises
of each section. These objectives help students contains practice problems that often require students
recognize and focus on the section’s most important to combine several skills or concepts, providing
ideas. The objectives are restated in the margin at their instructors the option of creating assignments that
point of use. take Practice Exercises to a more challenging level.
• Detailed Worked-Out Examples. Each example • Chapter Summaries. Each chapter contains a review
is titled, making the purpose of the example clear. chart that summarizes the definitions and concepts in
Examples are clearly written and provide students with every section of the chapter. Examples that illustrate
detailed step-by-step solutions. No steps are omitted these key concepts are also referenced in the chart.
and each step is thoroughly explained to the right of • End-of-Chapter Materials. A comprehensive collection
the mathematics. of review exercises for each of the chapter’s sections
• Explanatory Voice Balloons. Voice balloons are follows the Summary. This is followed by a Chapter
used in a variety of ways to demystify mathematics. Test that enables students to test their understanding of
They translate mathematical language into everyday the material covered in the chapter. Worked-out video
English, help clarify problem-solving procedures, solutions are available for every Chapter Test Prep
present alternative ways of understanding concepts, problem in the MyLab Math course or on YouTube.
and connect problem solving to concepts students • Learning Guide. This study aid is organized by
have already learned. objective  and provides support for note-taking,
• Check Point Examples. Each example is followed practice,  and video review. The Learning Guide
by a similar matched problem, called a Check Point, is available as PDFs in MyLab Math. It can also
offering students the opportunity to test for conceptual be packaged with the textbook and MyLab Math
understanding by working a similar exercise. The access code.
answers to the Check Points are provided in the answer
section in the back of the book. Worked-out video
I hope that my love for learning, as well as my respect
solutions for many Check Points are in the MyLab
for the diversity of students I have taught and learned from
Math course.
over the years, is apparent throughout this new edition.
• Great Question! This feature presents study tips in
By connecting mathematics to the whole spectrum of
the context of students’ questions. Answers to the
learning, it is my intent to show students that their world is
questions offer suggestions for problem solving, point
profoundly mathematical, and indeed, p is in the sky.
out common errors to avoid, and provide informal
hints and suggestions. As a secondary benefit, this
Robert Blitzer
feature should help students not to feel anxious or
threatened when asking questions in class.
• Brief Reviews. The book’s Brief Review boxes
summarize mathematical skills that students should
have learned previously, but which many students
still need to review. This feature appears whenever a
particular skill is first needed and eliminates the need
to reteach that skill.
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Resources for Success ix
Resources for Success
TM
MyLab Math Online Course for
Thinking Mathematically, Seventh Edition
by Robert Blitzer
(access code required)
MyLab Math is available to accompany Pearson’s market leading text offerings. To give
students a consistent tone, voice, and teaching method each text’s flavor and approach
are tightly integrated throughout the accompanying MyLab Math course, making
learning the material as seamless as possible.
NEW! Video Program
All new objective-level videos provide
a new level of coverage throughout the
text. Videos at the objective level allow
students to get support just where they
need it. Instructors can assign these as
media assignments or use the provided
assessment questions for each video.
NEW! Interactive NEW! Animations
Concept Videos New animations let students interact with
the math in a visual, tangible way. These
New Interactive Concept Videos are also
animations allow students to explore and
available in MyLab Math. After a brief
manipulate the mathematical concepts,
explanation, the video pauses to ask
leading to more durable understanding.
students to try a problem on their own.
Corresponding exercises in MyLab Math
Incorrect answers are followed by further
make these truly assignable.
explanation, taking into consideration what
may have led to the student selecting
that particular wrong answer. Incorrect
answer ‘A’ goes down one path while
incorrect answer ‘B’ provides a different
explanation based on why the student may
have selected that option.
StatCrunch
Newly integrated StatCrunch allows
students to harness technology to
perform complex analyses on data.
pearson.com/mylab/math
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