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x

g x

f g x

g

f

output

input

College Algebra
with

Student Solutions Manual

Sheldon Axler
San Francisco State University

JOHN WILEY & SONS, INC.

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ISBN-13
ISBN-13
ISBN-13

978-0470-47076-3 (hardcover)
978-0470-47077-0 (softcover)

978-0470-47078-7 (binder ready)

Printed in the United States of America
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About the Author

Sheldon Axler is Dean of
the College of Science & Engineering at San Francisco
State University, where he
joined the faculty as Chair of the Mathematics Department in 1997.
Axler was valedictorian of his high school in Miami, Florida. He received
his AB from Princeton University with highest honors, followed by a PhD in
Mathematics from the University of California at Berkeley.
As a postdoctoral Moore Instructor at MIT, Axler received a university-wide
teaching award. Axler was then an assistant professor, associate professor,
and professor at Michigan State University, where he received the first J.
Sutherland Frame Teaching Award and the Distinguished Faculty Award.
Axler received the Lester R. Ford Award for expository writing from the
Mathematical Association of America in 1996. In addition to publishing
numerous research papers, Axler is the author of Linear Algebra Done Right
(which has been adopted as a textbook at over 240 universities and colleges)
and Precalculus: A Prelude to Calculus and co-author of Harmonic Function
Theory (a graduate/research-level book).
Axler has served as Editor-in-Chief of the Mathematical Intelligencer and as
Associate Editor of the American Mathematical Monthly. He has been a member of the Council of the American Mathematical Society and a member of
the Board of Trustees of the Mathematical Sciences Research Institute. Axler

currently serves on the editorial board of Springer’s series Undergraduate
Texts in Mathematics, Graduate Texts in Mathematics, and Universitext.

v

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Contents

About the Author

v

Preface to the Instructor
WileyPLUS

xiii

xviii

Acknowledgments

xix

Preface to the Student
1 The Real Numbers

1


1.1 The Real Line

2

xxi

Construction of the Real Line

2

Is Every Real Number Rational?
Problems

3

6

1.2 Algebra of the Real Numbers

7

Commutativity and Associativity

7

The Order of Algebraic Operations
The Distributive Property

8


10

Additive Inverses and Subtraction

11

Multiplicative Inverses and the Algebra of Fractions
Symbolic Calculators

16

Exercises, Problems, and Worked-out Solutions

1.3 Inequalities, Intervals, and Absolute Value
Positive and Negative Numbers
Lesser and Greater
Intervals

13

19

24

24

25

27


Absolute Value

30

Exercises, Problems, and Worked-out Solutions

33

Chapter Summary and Chapter Review Questions
vi

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40


Contents vii

2 Combining Algebra and Geometry
2.1 The Coordinate Plane
Coordinates

41

42

42

Graphs of Equations


44

Distance Between Two Points

46

Length, Perimeter, and Circumference

48

Exercises, Problems, and Worked-out Solutions

2.2 Lines

57

Slope

57

The Equation of a Line
Parallel Lines
Midpoints

58

61

Perpendicular Lines


62

64

Exercises, Problems, and Worked-out Solutions

2.3 Quadratic Expressions and Conic Sections
Completing the Square

75

The Quadratic Formula

77

Circles

66

75

79

Ellipses

81

Parabolas
Hyperbolas


83
85

Exercises, Problems, and Worked-out Solutions

2.4 Area

50

88

98

Squares, Rectangles, and Parallelograms
Triangles and Trapezoids
Stretching

98

99

101

Circles and Ellipses

102

Exercises, Problems, and Worked-out Solutions

105


Chapter Summary and Chapter Review Questions
3 Functions and Their Graphs
3.1 Functions

117

118

Definition and Examples

118

The Graph of a Function

121

The Domain of a Function
The Range of a Function

124
126

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115


viii Contents


Functions via Tables

128

Exercises, Problems, and Worked-out Solutions

3.2 Function Transformations and Graphs

129

142

Vertical Transformations: Shifting, Stretching, and Flipping
Horizontal Transformations: Shifting, Stretching, Flipping
Combinations of Vertical Function Transformations
Even Functions

152

Odd Functions

153

Exercises, Problems, and Worked-out Solutions

3.3 Composition of Functions

165

Definition of Composition


166

Order Matters in Composition

154

169

170

Composing More than Two Functions

171

Function Transformations as Compositions

172

Exercises, Problems, and Worked-out Solutions

3.4 Inverse Functions

149

165

Combining Two Functions

Decomposing Functions


174

180

The Inverse Problem

180

One-to-one Functions

181

The Definition of an Inverse Function

182

The Domain and Range of an Inverse Function

184

The Composition of a Function and Its Inverse

185

Comments About Notation

187

Exercises, Problems, and Worked-out Solutions


3.5 A Graphical Approach to Inverse Functions
The Graph of an Inverse Function

Graphical Interpretation of One-to-One
Inverse Functions via Tables

189

197

197

Increasing and Decreasing Functions

199

200

203

Exercises, Problems, and Worked-out Solutions

204

Chapter Summary and Chapter Review Questions
4 Polynomial and Rational Functions
4.1 Integer Exponents

142

145

214

Positive Integer Exponents

214

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213

209


Contents ix

Properties of Exponents
Defining x

0

215

217

Negative Integer Exponents

218


Manipulations with Exponents

219

Exercises, Problems, and Worked-out Solutions

4.2 Polynomials

221

227

The Degree of a Polynomial

227

The Algebra of Polynomials

228

Zeros and Factorization of Polynomials

230

The Behavior of a Polynomial Near ±∞

234

Graphs of Polynomials


237

Exercises, Problems, and Worked-out Solutions

4.3 Rational Functions

239

245

Ratios of Polynomials

245

The Algebra of Rational Functions
Division of Polynomials

246

247

The Behavior of a Rational Function Near ±∞
Graphs of Rational Functions

250

253

Exercises, Problems, and Worked-out Solutions


4.4 Complex Numbers

255

262

The Complex Number System

262

Arithmetic with Complex Numbers

263

Complex Conjugates and Division of Complex Numbers
Zeros and Factorization of Polynomials, Revisited
Exercises, Problems, and Worked-out Solutions

271

Chapter Summary and Chapter Review Questions
5 Exponents and Logarithms

276

279

5.1 Exponents and Exponential Functions
Roots


264

268

280

280

Rational Exponents
Real Exponents

284

285

Exponential Functions

286

Exercises, Problems, and Worked-out Solutions

287

5.2 Logarithms as Inverses of Exponential Functions
Logarithms Base 2

293

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293


x Contents

Logarithms with Any Base

295

Common Logarithms and the Number of Digits
Logarithm of a Power

297

Radioactive Decay and Half-Life

299

Exercises, Problems, and Worked-out Solutions

5.3 Applications of Logarithms
Logarithm of a Product

301

310

310

Logarithm of a Quotient


311

Earthquakes and the Richter Scale
Sound Intensity and Decibels

312

313

Star Brightness and Apparent Magnitude
Change of Base

297

315

316

Exercises, Problems, and Worked-out Solutions

5.4 Exponential Growth

328

Functions with Exponential Growth
Population Growth

319


329

333

Compound Interest

335

Exercises, Problems, and Worked-out Solutions

340

Chapter Summary and Chapter Review Questions
6 e and the Natural Logarithm
6.1 Defining e and ln

349

350

Estimating Area Using Rectangles
Defining e

350

352

Defining the Natural Logarithm

355


Properties of the Exponential Function and ln
Exercises, Problems, and Worked-out Solutions

356
358

6.2 Approximations and area with e and ln 366
Approximation of the Natural Logarithm

366

Approximations with the Exponential Function
An Area Formula

368

369

Exercises, Problems, and Worked-out Solutions

6.3 Exponential Growth Revisited

376

Continuously Compounded Interest
Continuous Growth Rates
Doubling Your Money

377


378

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376

372

347


Contents xi

Exercises, Problems, and Worked-out Solutions

380

Chapter Summary and Chapter Review Questions
7 Systems of Equations

385

387

7.1 Equations and Systems of Equations
Solving an Equation

388


388

Solving a System of Equations Graphically

391

Solving a System of Equations by Substitution
Exercises, Problems, and Worked-out Solutions

7.2 Solving Systems of Linear Equations
Linear Equations: How Many Solutions?
Systems of Linear Equations
Gaussian Elimination

392
393

399
399

402

404

Exercises, Problems, and Worked-out Solutions

406

7.3 Solving Systems of Linear Equations Using Matrices
Representing Systems of Linear Equations by Matrices

Gaussian Elimination with Matrices

411

411

413

Systems of Linear Equations with No Solutions

415

Systems of Linear Equations with Infinitely Many Solutions
How Many Solutions, Revisited

418

Exercises, Problems, and Worked-out Solutions

7.4 Matrix Algebra
Matrix Size

419

424

424

Adding and Subtracting Matrices


426

Multiplying a Matrix by a Number
Multiplying Matrices

427

428

The Inverse of a Matrix

433

Exercises, Problems, and Worked-out Solutions

440

Chapter Summary and Chapter Review Questions
8 Sequences, Series, and Limits
8.1 Sequences

416

447

448

Introduction to Sequences

448


Arithmetic Sequences

450

Geometric Sequences

451

Recursively Defined Sequences

454

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445


xii Contents

Exercises, Problems, and Worked-out Solutions

8.2 Series

463

Sums of Sequences

463


Arithmetic Series

463

Geometric Series

466

Summation Notation
The Binomial Theorem

468
470

Exercises, Problems, and Worked-out Solutions

8.3 Limits

456

476

483

Introduction to Limits
Infinite Series

483

487


Decimals as Infinite Series
Special Infinite Series

489

491

Exercises, Problems, and Worked-out Solutions

493

Chapter Summary and Chapter Review Questions
Photo Credits
Index

497

499

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496


Preface to the Instructor

Goals
This book aims to provide college students with the algebraic skill and
understanding needed for other coursework and for participating as an

educated citizen in a complex society.
Mathematics faculty frequently complain that most students in lowerdivision mathematics courses do not read the textbook. When doing homework, a typical college algebra student looks only at the relevant section
of the textbook or the student solutions manual for an example similar to
the homework problem at hand. The student reads enough of that example to imitate the procedure and then does the homework problem. Little
understanding may take place.
In contrast, this book is designed to be read by students. The writing style
and layout are meant to induce students to read and understand the material.
Explanations are more plentiful than typically found in college algebra books.
Examples of the concepts make the ideas concrete whenever possible.

Exercises and Problems
Students learn mathematics by actively working on a wide range of exercises
and problems. Ideally, a student who reads and understands the material in
a section of this book should be able to do the exercises and problems in
that section without further help. However, some of the exercises require
application of the ideas in a context that students may not have seen before;
many students will need help with these exercises. This help is available
from the complete worked-out solutions to all the odd-numbered exercises
that appear at the end of each section.
Because the worked-out solutions were written solely by the author of
the textbook, students can expect a consistent approach to the material.
Furthermore, students will save money by not having to purchase a separate
student solutions manual.
The exercises (but not the problems) occur in pairs, so that an oddnumbered exercise is followed by an even-numbered exercise whose solution
uses the same ideas and techniques. A student stumped by an even-numbered
exercise should be able to tackle it after reading the worked-out solution to
the corresponding odd-numbered exercise. This arrangement allows the text
to focus more centrally on explanations of the material and examples of the
concepts.
xiii


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Each exercise has a
unique correct answer, usually a number or a function; each
problem has multiple
correct answers, usually explanations or
examples.

This book contains
what is usually a separate book called the
student solutions
manual.


xiv Preface to the Instructor

Most students will read the student solutions manual when they are
assigned homework, even though they are reluctant to read the main text.
The integration of the student solutions manual within this book should
encourage students to drift over and also read the main text. To reinforce
this tendency, the worked-out solutions to the odd-numbered exercises at
the end of each section are intentionally typeset with a slightly less appealing
style (smaller type, two-column format, and not right justified) than the main
text. The reader-friendly appearance of the main text might nudge students
to spend some time there.
Exercises and problems in this book vary greatly in difficulty and purpose.
Some exercises and problems are designed to hone algebraic manipulation
skills; other exercises and problems are designed to push students to genuine
understanding beyond rote algorithmic calculation.

Some exercises and problems intentionally reinforce material from earlier
in the book and require multiple steps. For example, Exercise 30 in Section 5.3
asks students to find all numbers x such that
log5 (x + 4) + log5 (x + 2) = 2.
To solve this exercise, students will need to use the formula for a sum of
logarithms as well as the quadratic formula; they will also need to eliminate
one of the potential solutions produced by the quadratic formula because it
would lead to the evaluation of the logarithm of a negative number. Although
such multi-step exercises require more thought than most exercises in the
book, they allow students to see crucial concepts more than once, sometimes
in unexpected contexts.

The Calculator Issue
To aid instructors
in presenting the
kind of course they
want, the symbol
appears with exercises and problems
that require students
to use a calculator.

The issue of whether and how calculators should be used by students has
generated immense controversy.
Some sections of this book have many exercises and problems designed
for calculators (for example Section 5.4 on exponential growth), but some
sections deal with material not as amenable to calculator use. The text seeks
to provide students with both understanding and skills. Thus the book
does not aim for an artificially predetermined percentage of exercises and
problems in each section requiring calculator use.
Some exercises and problems that require a calculator are intentionally

designed to make students realize that by understanding the material, they
can overcome the limitations of calculators. As one example among many,
Exercise 83 in Section 5.3 asks students to find the number of digits in the
decimal expansion of 74000 . Brute force with a calculator will not work with
this problem because the number involved has too many digits. However, a
few moments’ thought should show students that they can solve this problem
by using logarithms (and their calculators!).

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Preface to the Instructor xv

The calculator icon
can be interpreted for some exercises, depending
on the instructor’s preference, to mean that the solution should be a decimal
approximation rather than the exact answer.
For example, Exercise 3 in Section 6.3 asks how much would need to be
deposited in a bank account paying 4% interest compounded continuously
so that at the end of 10 years the account would contain $10,000. The exact
answer to this exercise is 10000/e0.4 dollars, but it may be more satisfying to
the student (after obtaining the exact answer) to use a calculator to see that
approximately $6,703 needs to be deposited. For such exercises, instructors
can decide whether to ask for exact answers or decimal approximations (the
worked-out solutions for the odd-numbered exercises will usually contain
both).
Symbolic processing programs such as Mathematica and Maple offer appealing alternatives to hand-held calculators because of their ability to solve
equations and deal with symbols as well as numbers. Furthermore, the larger
size, better resolution, and color on a computer screen make graphs produced by such software more informative than graphs on a typical hand-held
graphing calculator.

Your students may not use a symbolic processing program because of the
complexity or expense of such software. However, easy-to-use free web-based
symbolic programs are becoming available. Occasionally this book shows
how students can use Wolfram|Alpha, which has almost no learning curve, to
go beyond what can be done easily by hand.
Even if you do not tell your students about such free tools, knowledge
about such web-based homework aids is likely to spread rapidly among
students.

What to Cover
Different instructors will want to cover different sections of this book. Many
instructors will want to cover Chapter 1 (The Real Numbers), even though it
should be review, because it deals with familiar topics in a deeper fashion
than students may have previously seen.
Some instructors will cover Section 4.3 (Rational Functions) only lightly
because graphing rational functions, and in particular finding local minima
and maxima, is better done with calculus. Many instructors will prefer to skip
Chapter 8 (Sequences, Series, and Limits), leaving that material to a calculus
course.
The prerequisite for this book is the usual course in intermediate algebra.
The book is fairly self-contained, starting with a review of the real numbers
in Chapter 1.

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Regardless of what
level of calculator use
an instructor expects,
students should not
turn to a calculator to

compute something
like log 1, because
then log has become
just a button on the
calculator.


xvi Preface to the Instructor

Distinctive Approaches
Half-life and Exponential Growth
Almost all college algebra textbooks present radioactive decay as an example
of exponential decay. Amazingly, the typical college algebra textbook states
that if a radioactive isotope has half-life h, then the amount left at time t will
equal e−(t ln 2)/h times the amount present at time 0.
A much clearer formulation would state, as this textbook does, that the
amount left at time t will equal 2−t/h times the amount present at time 0. The
unnecessary use of e and ln 2 in this context may suggest to students that e
and natural logarithms have only contrived and artificial uses, which is not
the message a textbook should send. Using 2−t/h helps students understand
the concept of half-life, with a formula connected to the meaning of the
concept.
Similarly, many college algebra textbooks consider, for example, a colony
of bacteria doubling in size every 3 hours, with the textbook then producing
the formula e(t ln 2)/3 for the growth factor after t hours. The simpler and
more natural formula 2t/3 seems not to be mentioned in such books. This
book presents the more natural approach to such issues of exponential
growth and decay.
Algebraic Properties of Logarithms


The initial separation
of logarithms and e
should help students
master both concepts.

The base for logarithms in Chapter 5 is arbitrary. Most of the examples and
motivation use logarithms base 2 or logarithms base 10. Students will see
how the algebraic properties of logarithms follow easily from the properties
of exponents.
The crucial concepts of e and natural logarithms are saved for Chapter 6.
Thus students can concentrate in Chapter 5 on understanding logarithms
(arbitrary base) and their properties without at the same time worrying about
grasping concepts related to e. Similarly, when natural logarithms arise
naturally in Chapter 6, students should be able to concentrate on issues
surrounding e without at the same time learning properties of logarithms.
Area
Section 2.4 in this book builds the intuitive notion of area starting with
squares, and then quickly derives formulas for the area of rectangles, triangles, parallelograms, and trapezoids. A discussion of the effects of stretching
either horizontally or vertically easily leads to the familiar formula for the
area enclosed by a circle. Similar ideas are then used to find the formula for
the area inside an ellipse (without calculus!).
Section 6.1 deals with the question of estimating the area under parts of
1
the curve y = x by using rectangles. This easy nontechnical introduction,
with its emphasis on ideas without the clutter of the notation of Riemann
sums, gives students a taste of an important idea from calculus.

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Preface to the Instructor xvii

e, The Exponential Function, and the Natural Logarithm
Most college algebra textbooks either present no motivation for e or motivate
e via continuously compounding interest or through the limit of an indeterminate expression of the form 1∞ ; these concepts are difficult for students
at this level to understand.
Chapter 6 presents a clean and well-motivated approach to e and the
natural logarithm. We do this by looking at the area (intuitively defined)
1
under the curve y = x , above the x-axis, and between the lines x = 1 and
x = c.
A similar approach to e and the natural logarithm is common in calculus
courses. However, this approach is not usually adopted in college algebra
textbooks. Using basic properties of area, the simple presentation given here
shows how these ideas can come through clearly without the technicalities
of calculus or Riemann sums.
The approach taken here also has the advantage that it easily leads, as
we will see in Chapter 6, to the approximation ln(1 + h) ≈ h for |h| small.
Furthermore, the same methods show that if r is any number, then
1+

r x
x

≈ er

for large values of x. A final bonus of this approach is that the connection
between continuously compounding interest and e becomes a nice corollary
of natural considerations concerning area.


Comments Welcome
I seek your help in making this a better book. Please send me your comments
and your suggestions for improvements. Thanks!
Sheldon Axler
San Francisco State University
e-mail:
web site: algebra.axler.net
Twitter: @AxlerAlgebra

www.pdfgrip.com

The approach taken
here to the exponential function and the
natural logarithm
shows that a good
understanding of
these subjects need
not wait until a calculus course.


WileyPLUS

WileyPLUS is an innovative online environment
for effective teaching and learning.
A Research-based Design. WileyPLUS provides an online environment that integrates relevant resources, including the entire digital textbook, in an easy-to-navigate framework that helps
students study more effectively.

WileyPLUS provides reliable, customizable resources that reinforce course goals inside and
outside the classroom. Pre-created materials and
activities help instructors optimize their time.

Customizable Course Plan. WileyPLUS comes
with a pre-created Course Plan designed specifically for this book. Simple drag-and-drop tools
make it easy to assign the course plan as-is or
• WileyPLUS adds structure by organizing textmodify it to reflect your course syllabus.
book content into smaller, more manageable
Pre-created Activity Types Include:
chunks.
• questions;
• Related media, examples, and sample prac• readings and resources;
tice items reinforce learning objectives.
• Innovative features such as calendars, visual
progress tracking, and self-evaluation tools
improve time management and strengthen
areas of weakness.

One-on-one Engagement. With WileyPLUS for
College Algebra students receive 24/7 access to
resources that promote positive learning outcomes. Students engage with related examples
and sample practice items, including:
• videos;
• Guided Online (GO) tutorial exercises;
• concept questions.

• print tests;
• concept mastery;
• project.
Course Materials and Assessment Content:
• lecture slides;
• Instructor’s Solutions Manual;
• question assignments: selected exercises

coded algorithmically with hints, links to
text, whiteboard/show-work feature, and
instructor-controlled problem-solving help;
• proficiency exams;
• computerized testbank;

Measurable Outcomes.
Throughout each
study session, students can assess their progress
and gain immediate feedback. WileyPLUS provides precise reporting of strengths and weaknesses, as well as individualized quizzes, so that
students are confident they are spending their
time on the right things. With WileyPLUS, students always know the outcome of their efforts.

• printable testbank.
Gradebook. WileyPLUS provides instant access to reports on trends in class performance,
student use of course materials, and progress
towards learning objectives, helping inform decisions and drive classroom discussions.
Learn More: www.wileyplus.com

xviii

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Acknowledgments
As usual in a textbook, as opposed to a research article, little attempt has
been made to provide proper credit to the original creators of the ideas
presented in this book. Where possible, I have tried to improve on standard
approaches to this material. However, the absence of a reference does not
imply originality on my part. I thank the many mathematicians who have

created and refined our beautiful subject.
I chose Wiley as the publisher of this book because of the company’s
commitment to excellence. The people at Wiley have made outstanding
contributions to this project, providing astute editorial advice, superb design
expertise, high-level production skill, and insightful marketing savvy. I
am truly grateful to the following Wiley folks, all of whom helped make
this a better and more successful book than it would have been otherwise:
Jonathan Cottrell, Joanna Dingle, Melissa Edwards, Jessica Jacobs, Ellen
Keohane, Madelyn Lesure, Beth Pearson, Mary Ann Price, Laurie Rosatone,
Lisa Sabatini, Ken Santor, Anne Scanlan-Rohrer, Jennifer Wreyford.
Celeste Hernandez, the accuracy checker, and Katrina Avery, the copy
editor, excelled at catching mathematical and linguistic errors.
The instructors and students who used the earlier versions of this book
provided wonderfully useful feedback. Numerous reviewers gave me terrific
suggestions as the book progressed through various stages of development.
I am grateful to all the class testers and reviewers whose names are listed on
the following page, with special thanks to Michael Price.
Like most mathematicians, I owe thanks to Donald Knuth, who invented
TEX, and to Leslie Lamport, who invented LATEX, which I used to typeset this
book. I am grateful to the authors of the many open-source LATEX packages I
used to improve the appearance of the book, especially to Hàn Th´
ê Thành
A
for pdfL TEX, Robert Schlicht for microtype, and Frank Mittelbach for multicol.
Thanks also to Wolfram Research for producing Mathematica, which is the
software I used to create the graphics in this book.
My awesome partner Carrie Heeter deserves considerable credit for her
wise advice and continual encouragement throughout the long book-writing
process.
Many thanks to all of you!


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Most of the results in
this book belong to
the common heritage
of mathematics, created over thousands
of years by clever and
curious people.


xx Acknowledgments

Class Testers and Reviewers
• LaVerne Chambers Alan, Crichton College

• Max Hibbs, Blinn College Brenham

• Aaron Altose, Cuyahoga Community College

• Jada Hill, Richland College

• George Anastassiou, University of Memphis

• James Hilsenbeck, University of Texas at Brownsville

• Karen Anglin, Blinn College Brenham


• Sarah Holliday, Southern Polytechnic State University

• Jan Archibald, Ventura College

• Kerry Johnson, Missouri Southern State University

• Vinod Arya, Fayetteville State University

• Susan Jordan, Arkansas Tech University

• Carlos Barron, Mountain View College

• Brianna Kurtz, Daytona State College

• Jamey Bass, City College of San Francisco

• Grant Lathrom, Missouri Southern State University

• Jaromir J. Becan, University of Texas at San Antonio

• Kiseop Lee, University of Louisville

• Jeff Berg, Arapahoe Community College

• Max Lee, Westchester Community College

• Matt Bertens, City College of San Francisco

• Scott Lewis, Utah Valley University


• Andrea Blum, Suffolk County Community College

• Gary Lippman, California State University East Bay

• Valerie Bouagnon, DePaul University

• William Livingston, Missouri Southern State University

• Brian Brock, San Jacinto College

• Syrous Marivani, Louisiana State University at
Alexandria

• Connie Buller, Metropolitan Community College of
Omaha

• Mary Barone Martin, Middle Tennessee State University

• Michael Butros, Victor Valley College

• Eric Matsuoka, Leeward Community College

• Jennifer Cabaniss, Central Texas College

• Mike McCraith, Cuyahoga Community College

• Debananda Chakraborty, State University of New York
at Buffalo

• Margaret Michener, University of Nebraska Kearney


• Denise Chellsen, Cuesta College

• David Miller, West Virginia University
• Juan Carlos Molina, Austin Community College

• Sharon Christensen, Cameron University

• Hojin Moon, California State University Long Beach

• De Cook, Northwest Florida State College
• Kathy Cousins-Cooper, North Carolina A&T University
• Christopher Danielson, Minnesota State University
Mankato
• Hilary Davies, University of Alaska Anchorage

• Bette Nelson, Alvin Community College
• Priti Patel, Tarrant County Community College
Southeast
• Mary Beth Pattengale, Sierra College
• Vic Perera, Kent State University Trumbull

• Michelle DeDeo, University of North Florida

• Sandy Poinsett, College of Southern Maryland

• Luis Carlos Diaz, LaRoche College

• Michael Price, University of Oregon


• Deanna Dick, Alvin College

• Mike Rosenthal, Florida International University

• Gay Ellis, Missouri State University

• Daniel T. Russow, Arizona Western College

• Joan Evans, Texas Southern University

• Alan Saleski, Loyola University Chicago

• Mike Everett, Santa Ana College

• Rebecca Schantz, East Central College

• Don Faust, Northern Michigan University
• Judy Fethe, Pellissippi State Technical Community
College

• Mayada Shahroki, Lone Star College Cy-Fair
• Robert Shea, Central Texas College

• Anne Fine, East Central University

• Linda Snellings-Neal, Wright State University

• Patricia Foard, South Plains College

• Paul Sontag, University of Cincinnati


• Pari Ford, University of Nebraska Kearney

• Jacqueline Stone, University of Maryland

• Lee R. Gibson, University of Louisville

• Mary Ann Teel, University of North Texas

• Renu Gupta, Louisiana State University at Alexandria

• Jennie Thompson, Leeward Community College

• Daniel Harned, Lansing Community College

• Jean Thornton, Western Kentucky University

• Bud Hart, Oregon Institute of Technology

• Michael van Opstall, University of Utah

• Cheryl Hawker, Eastern Illinois University

• Sara Weiss, Richland College

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Preface to the Student
This book will help provide you with the algebraic skill and understanding

needed for other coursework and for participating as an educated citizen in
a complex society.
To learn this material well, you will need to spend serious time reading
this book. You cannot expect to absorb mathematics the way you devour a
novel. If you read through a section of this book in less than an hour, then
you are going too fast. You should pause to ponder and internalize each
definition, often by trying to invent some examples in addition to those given
in the book. For each result stated in the book, you should seek examples to
show why each hypothesis is necessary. When steps in a calculation are left
out in the book, you need to supply the missing pieces, which will require
some writing on your part. These activities can be difficult when attempted
alone; try to work with a group of a few other students.
You will need to spend several hours per section doing the exercises
and problems. Make sure that you can do all the exercises and most of
the problems, not just the ones assigned for homework. By the way, the
difference between an exercise and a problem in this book is that each
exercise has a unique correct answer that is a mathematical object such as a
number or a function. In contrast, the solutions to problems often consist of
explanations or examples; thus most problems have multiple correct answers.
Have fun, and best wishes in your studies!
Sheldon Axler
San Francisco State University
web site: algebra.axler.net
Twitter: @AxlerAlgebra

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Complete worked-out

solutions to the oddnumbered exercises
are given at the end of
each section.


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chapter

1

The Parthenon, built
in Athens over 2400
years ago. The ancient
Greeks developed
and used remarkably sophisticated
mathematics.

The Real Numbers
Success in this course will require a good understanding of the basic properties of the real number system. Thus this book begins with a review of the
real numbers.
The first section of this chapter starts with the construction of the real
line. This section contains as an optional highlight the ancient Greek proof
that no rational number has a square equal to 2. This beautiful result appears
here not because you will need it, but because it should be seen by everyone
at least once.
Although this chapter will be mostly review, a thorough grounding in the

real number system will serve you well throughout this course and then for
the rest of your life. You will need good algebraic manipulation skills; thus
the second section of this chapter reviews the fundamental algebra of the real
numbers. You will also need to feel comfortable working with inequalities
and absolute values, which are reviewed in the last section of this chapter.
Even if your instructor decides to skip this chapter, you may want to read
through it. Make sure you can do all the exercises.

1

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