Algebra-and-Trigonometry-Solution-by-Sheldon-Axler

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10 9 8 7 6 5 4 3 2 1

Sheldon Axler is Dean of the College of Science & En- gineering 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),

College Algebra, and Precalculus: A Prelude to Calculus; he is also 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

mem-ber of the Council of the American Mathematical Society and a memmem-ber 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.

About the Author v Preface to the Instructor xvi WileyPLUS xxii

Acknowledgments xxiii Preface to the Student xxvi

1.1 The Real Line 2 Construction of the Real Line 2

Is Every Real Number Rational? 3 Problems 6

1.2 Algebra of the Real Numbers 7 Commutativity and Associativity 7 The Order of Algebraic Operations 8 The Distributive Property 10

Additive Inverses and Subtraction 11 Multiplicative Inverses and the Algebra of Fractions 13 Symbolic Calculators 16

Exercises, Problems, and Worked-out Solutions 19 1.3 Inequalities, Intervals, and Absolute Value 24 Positive and Negative Numbers 24

Lesser and Greater 25 Intervals 27

Absolute Value 30 Exercises, Problems, and Worked-out Solutions 33 Chapter Summary and Chapter Review Questions 40

vi

2 Combining Algebra and Geometry 41

2.1 The Coordinate Plane 42

Coordinates 42

Graphs of Equations 44

Distance Between Two Points 46

Length, Perimeter, and Circumference 48

Exercises, Problems, and Worked-out Solutions 50

Exercises, Problems, and Worked-out Solutions 66

2.3 Quadratic Expressions and Conic Sections 75

Completing the Square 75

The Quadratic Formula 77

Squares, Rectangles, and Parallelograms 98

Triangles and Trapezoids 99

Stretching 101

Circles and Ellipses 102

Exercises, Problems, and Worked-out Solutions 105

Chapter Summary and Chapter Review Questions 115

3.1 Functions 118

Definition and Examples 118

The Graph of a Function 121

The Domain of a Function 124

The Range of a Function 126

Functions via Tables 128 Exercises, Problems, and Worked-out Solutions 129 3.2 Function Transformations and Graphs 142 Vertical Transformations: Shifting, Stretching, and Flipping 142 Horizontal Transformations: Shifting, Stretching, Flipping 145 Combinations of Vertical Function Transformations 149 Even Functions 152

Odd Functions 153 Exercises, Problems, and Worked-out Solutions 154 3.3 Composition of Functions 165

Combining Two Functions 165 Definition of Composition 166 Order Matters in Composition 169 Decomposing Functions 170 Composing More than Two Functions 171 Function Transformations as Compositions 172 Exercises, Problems, and Worked-out Solutions 174 3.4 Inverse Functions 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 189 3.5 A Graphical Approach to Inverse Functions 197 The Graph of an Inverse Function 197

Graphical Interpretation of One-to-One 199 Increasing and Decreasing Functions 200 Inverse Functions via Tables 203

Exercises, Problems, and Worked-out Solutions 204 Chapter Summary and Chapter Review Questions 209

4 Polynomial and Rational Functions 213

4.1 Integer Exponents 214 Positive Integer Exponents 214

Properties of Exponents 215

Defining x0 217

Negative Integer Exponents 218

Manipulations with Exponents 219

Exercises, Problems, and Worked-out Solutions 221

4.2 Polynomials 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

The Behavior of a Rational Function Near ±∞ 250

Graphs of Rational Functions 253

Exercises, Problems, and Worked-out Solutions 255

4.4 Complex Numbers 262

The Complex Number System 262

Arithmetic with Complex Numbers 263

Complex Conjugates and Division of Complex Numbers 264

Zeros and Factorization of Polynomials, Revisited 268

Exercises, Problems, and Worked-out Solutions 271

Chapter Summary and Chapter Review Questions 276

5.1 Exponents and Exponential Functions 280

Roots 280

Rational Exponents 284

Real Exponents 285

Exponential Functions 286

Exercises, Problems, and Worked-out Solutions 287

5.2 Logarithms as Inverses of Exponential Functions 293

Logarithms Base 2 293

Logarithms with Any Base 295 Common Logarithms and the Number of Digits 297 Logarithm of a Power 297

Radioactive Decay and Half-Life 299 Exercises, Problems, and Worked-out Solutions 301 5.3 Applications of Logarithms 310

Logarithm of a Product 310 Logarithm of a Quotient 311 Earthquakes and the Richter Scale 312 Sound Intensity and Decibels 313 Star Brightness and Apparent Magnitude 315 Change of Base 316

Exercises, Problems, and Worked-out Solutions 319 5.4 Exponential Growth 328

Functions with Exponential Growth 329 Population Growth 333

Compound Interest 335 Exercises, Problems, and Worked-out Solutions 340 Chapter Summary and Chapter Review Questions 347

Approximation of the Natural Logarithm 366 Approximations with the Exponential Function 368

An Area Formula 369 Exercises, Problems, and Worked-out Solutions 372 6.3 Exponential Growth Revisited 376

Continuously Compounded Interest 376 Continuous Growth Rates 377

Doubling Your Money 378

Exercises, Problems, and Worked-out Solutions 380

Chapter Summary and Chapter Review Questions 385

7 Systems of Equations 387

7.1 Equations and Systems of Equations 388

Solving an Equation 388

Solving a System of Equations Graphically 391

Solving a System of Equations by Substitution 392

Exercises, Problems, and Worked-out Solutions 393

7.2 Solving Systems of Linear Equations 399

Linear Equations: How Many Solutions? 399

Systems of Linear Equations 402

Gaussian Elimination 404

Exercises, Problems, and Worked-out Solutions 406

7.3 Solving Systems of Linear Equations Using Matrices 411

Representing Systems of Linear Equations by Matrices 411

Gaussian Elimination with Matrices 413

Systems of Linear Equations with No Solutions 415

Systems of Linear Equations with Infinitely Many Solutions 416

How Many Solutions, Revisited 418

Exercises, Problems, and Worked-out Solutions 419

7.4 Matrix Algebra 424

Matrix Size 424

Adding and Subtracting Matrices 426

Multiplying a Matrix by a Number 427

Multiplying Matrices 428

The Inverse of a Matrix 433

Exercises, Problems, and Worked-out Solutions 440

Chapter Summary and Chapter Review Questions 445

8 Sequences, Series, and Limits 447

Exercises, Problems, and Worked-out Solutions 456 8.2 Series 463

Sums of Sequences 463 Arithmetic Series 463 Geometric Series 466 Summation Notation 468 The Binomial Theorem 470 Exercises, Problems, and Worked-out Solutions 476 8.3 Limits 483

Introduction to Limits 483 Infinite Series 487

Decimals as Infinite Series 489 Special Infinite Series 491 Exercises, Problems, and Worked-out Solutions 493 Chapter Summary and Chapter Review Questions 496

9.1 The Unit Circle 498 The Equation of the Unit Circle 498 Angles in the Unit Circle 499 Negative Angles 501

Angles Greater Than 360◦ 502 Length of a Circular Arc 503 Special Points on the Unit Circle 504 Exercises, Problems, and Worked-out Solutions 506 9.2 Radians 514

A Natural Unit of Measurement for Angles 514 The Radius Corresponding to an Angle 517 Length of a Circular Arc 520

Area of a Slice 521 Special Points on the Unit Circle 522 Exercises, Problems, and Worked-out Solutions 523 9.3 Cosine and Sine 529

Definition of Cosine and Sine 529 The Signs of Cosine and Sine 532 The Key Equation Connecting Cosine and Sine 534

The Graphs of Cosine and Sine 535

Exercises, Problems, and Worked-out Solutions 537

9.4 More Trigonometric Functions 542

Definition of Tangent 542

The Sign of Tangent 544

Connections Among Cosine, Sine, and Tangent 545

The Graph of Tangent 545

Three More Trigonometric Functions 547

Exercises, Problems, and Worked-out Solutions 549

9.5 Trigonometry in Right Triangles 555

Trigonometric Functions via Right Triangles 555

Two Sides of a Right Triangle 557

One Side and One Angle of a Right Triangle 558

Exercises, Problems, and Worked-out Solutions 559

9.6 Trigonometric Identities 566

The Relationship Among Cosine, Sine, and Tangent 566

Trigonometric Identities for the Negative of an Angle 568

Trigonometric Identities with π2 570

Trigonometric Identities Involving a Multiple of π 572

Exercises, Problems, and Worked-out Solutions 575

Chapter Summary and Chapter Review Questions 580

10.1 Inverse Trigonometric Functions 584

The Arccosine Function 584

The Arcsine Function 587

The Arctangent Function 590

Exercises, Problems, and Worked-out Solutions 593

10.2 Inverse Trigonometric Identities 599

Composition of Trigonometric Functions and Their Inverses 599

The Arccosine, Arcsine, and Arctangent of −t: Graphical

Exercises, Problems, and Worked-out Solutions 608 10.3 Using Trigonometry to Compute Area 613

The Area of a Triangle via Trigonometry 613 Ambiguous Angles 614

The Area of a Parallelogram via Trigonometry 616 The Area of a Polygon 617

Trigonometric Approximations 619 Exercises, Problems, and Worked-out Solutions 622 10.4 The Law of Sines and the Law of Cosines 628

The Law of Sines 628 Using the Law of Sines 629 The Law of Cosines 631 Using the Law of Cosines 632 When to Use Which Law 634 Exercises, Problems, and Worked-out Solutions 636 10.5 Double-Angle and Half-Angle Formulas 644

11.1 Parametric Curves 672

Curves in the Coordinate Plane 672 Graphing Inverse Functions as Parametric Curves 677 Shifting, Stretching, or Flipping a Parametric Curve 678 Exercises, Problems, and Worked-out Solutions 681

11.2 Transformations of Trigonometric Functions 687

Amplitude 687

Period 689

Phase Shift 692

Fitting Transformations of Trigonometric Functions to Data 694

Exercises, Problems, and Worked-out Solutions 696

11.3 Polar Coordinates 705

Defining Polar Coordinates 705

Converting from Polar to Rectangular Coordinates 706

Converting from Rectangular to Polar Coordinates 707

Graphs of Polar Equations 711

Exercises, Problems, and Worked-out Solutions 715

The Dot Product 726

Exercises, Problems, and Worked-out Solutions 728

11.5 The Complex Plane 732

Complex Numbers as Points in the Plane 732

Geometric Interpretation of Complex Multiplication and Division 734

De Moivre’s Theorem 737

Finding Complex Roots 738

Exercises, Problems, and Worked-out Solutions 739

Chapter Summary and Chapter Review Questions 741

Photo Credits 743

Index 745

Preface to the Instructor

Goals

This book aims to provide students with the algebraic and trigonometric skills and understanding needed for other coursework and for participating

as an educated citizen in a complex society.

Mathematics faculty frequently complain that many students do not read the textbook When doing homework, a typical student may look 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 algebra and trigonometry 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

Each exercise has a

unique correct

an-swer, usually a

num-ber or a function;

most problems have

multiple correct

an-swers, usually

expla-nations or examples.

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

odd-This book contains

to focus more centrally on explanations of the material and examples of the concepts.

xvi

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

The issue of whether and how calculators should be used by students has To aid instructors in

presenting the kind of course they want, the

with exercises and problems that require students to use a cal- culator.

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!).

The calculator icon can be interpreted for some exercises, depending

some-thing like log 1,

be-cause then log has

become just a

but-ton on the calculator.

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 /e0.4dollars, but it may be more satisfying to the student (after ing the exact answer) to use a calculator to see that approximately $6,703 needs to be deposited.

obtain-For exercises such as the one described in the paragraph above, instructors can decide whether to ask for exact answers or decimal approximations or both (the worked-out solutions for the odd-numbered exercises will usually contain both) If an instructor asks for only an exact answer, then a calculator may not be needed despite the presence of the calculator icon.

Symbolic processing programs such as Mathematica and Maple offer

ap-pealing 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 pro- duced 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.

Distinctive Approaches

Half-life and Exponential Growth Almost all algebra and trigonometry books present radioactive decay as an example of exponential decay Amazingly, the typical algebra and trigonom- etry textbook states that if a radioactive isotope has half-life h, then the

amount left at time t will equal e−(t ln 2)/htimes 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/htimes 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 algebra and trigonometry 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/3seems 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 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. The initial separation

of logarithms and e should help students master both concepts.

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,

trian-gles, 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

the curve y = 1

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.

e, The Exponential Function, and the Natural Logarithm

Most algebra and trigonometry 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 The approach taken

here to the tial function and the natural logarithm shows that a good understanding of these subjects need not wait until a calcu- lus course.

exponen-natural logarithm We do this by looking at the area (intuitively defined)

under the curve y = x1, 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

calcu-lus courses However, this approach is not usually adopted in algebra and

trigonometry textbooks Using basic properties of area, the simple

presenta-tion 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 +x r"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.

Inverse Functions The unifying concept of inverse functions is introduced in Section 3.4 This crucial idea has its first major use in this book in the definition of y1/mas the number x such that xm= y (in other words, the function y %→ y1/mis the inverse of the function x %→ xm; see Section 5.1) The second major use

of inverse functions occurs in the definition of logby as the number x such

that bx = y (in other words, the function y %→ logby is the inverse of the

function x %→ bx; see Section 5.2).

Thus students should be comfortable with using inverse functions by the time they reach the inverse trigonometric functions (arccosine, arcsine, and arctangent) in Section 10.1 For students who go on to calculus, this familiarity with inverse functions should help when dealing with inverse operations such as anti-differentiation.

This book emphasizes that f−1(y) = x means f (x) = y Thus this book

With the approach

taken in this book, the

statement “log b y = x

means b x = y” is

consistent with the

notation used for

inverse functions.

states that to find f−1(y), solve the equation f (x) = y for x.

In contrast, many books at this level unfortunately instruct the reader wanting to find f−1to start with the equation y = f (x), then “interchange

the variables x and y to obtain x = f (y)”, then solve for y in terms of

x This “interchange” method ends up with notation expressing f−1as a function of x.

However, the “interchange” method makes no sense when trying to find the value of an inverse function at a specific number instead of at a variable name Consider, for example, the problem of finding f−1(11) if f is the

function defined by f (x) = 2x + 3 The student mechanically following

the “interchange” method as it is stated in many books would start with the

equation 11 = 2x + 3 and then interchange x and 11, getting the equation

x = 2 · 11 + 3 This is, of course, completely wrong.

In contrast, this book does this problem by solving the equation 11 = 2x+3

for x, getting x = 4 and concluding that f−1(11) = 4.

The “interchange” method will also be confusing to students when the variables names have meaning For example, in an applied problem the variables might be t (for time) and d (for distance) rather than x and y, and

we might have a function that gives distance in terms of time: d = f (t).

The inverse function should then give time in terms of distance: t = f−1(d).

Interchanging the variable names here would be quite confusing.

This book defines cos θ and sin θ as the first and second coordinates of the

radius of the unit circle corresponding to θ (see Section 9.3) In contrast

to this definition using only one symbol, many books at this level require

students to juggle at least four symbols— θ (or t), x, y, and P—to parse the

definitions of the trigonometric functions These books define cos θ = x, and

students become accustomed to thinking of cos θ as the x-coordinate When

students encounter cos x, as often happens within a dozen pages of the

initial definition, they think that cos x is the x-coordinate of oops, that is

a different use of x No wonder so many students struggle with trigonometric

functions.

This book defines sine and cosine in one section, then defines the tangent This book

empha-sizes cos, sin, tan and places little emphasis

on sec, csc, cot.

function (and the other three trigonometric functions that have less

impor-tance) in another section This gentle approach contrasts with most books

that define all six trigonometric functions on the same page Students have

difficulty assimilating so many definitions simultaneously.

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 inverse trigonometric identities (Section 10.2) are given more space

in this book than in most books at this level This material is included not

so much for its intrinsic importance but as a way for students to obtain a

deeper understanding of the trigonometric functions Instructors can skip

this material or cover it lightly.

Comments Welcome

I seek your help in making this a better book Please send me your comments

and your suggestions for improvements Thanks!

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As usual in a textbook, as opposed to a research article, little attempt has Most of the results in

this book belong to the common heritage

of mathematics, ated over thousands

cre-of years by clever and curious people.

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

for pdfLATEX, 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!

Class Testers and Reviewers

• Vladimir Akis, California State University,

Los Angeles

• LaVerne Chambers Alan, Crichton College

• Aaron Altose, Cuyahoga Community College

• George Anastassiou, University of Memphis

• Karen Anglin, Blinn College Brenham

• Jan Archibald, Ventura College

• Vinod Arya, University of North Texas at

Dallas

• Carlos Barron, Mountain View College

• Jamey Bass, City College of San Francisco

• Jaromir J Becan, University of Texas at San

Antonio

• Jeff Berg, Arapahoe Community College

• Matt Bertens, City College of San Francisco

• Nicoleta Bila, Fayetteville State University

• Arlene Blasius, SUNY College at Old Westbury

• Andrea Blum, Suffolk County Community

College

• Valerie Bouagnon, DePaul University

• Brian Brock, San Jacinto College

• Connie Buller, Metropolitan Community

College of Omaha

• Michael Butros, Victor Valley College

• Jennifer Cabaniss, Central Texas College

• Debananda Chakraborty, State University of

New York at Buffalo

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• Sharon Christensen, Cameron University

• De Cook, Northwest Florida State College

• Kathy Cousins-Cooper, North Carolina A&T

• Michelle DeDeo, University of North Florida

• Luis Carlos Diaz, LaRoche College

• Deanna Dick, Alvin College

• David Dwiggins, University of Memphis

• Kathleen Ebert, Alfred State University

• Gay Ellis, Missouri State University

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• Don Faust, Northern Michigan University

• Judy Fethe, Pellissippi State Technical

Community College

• Anne Fine, East Central University

• Jack Flaherty, Brunswick Community College

• Patricia Foard, South Plains College

• Pari Ford, University of Nebraska Kearney

• Heng Fu, Thomas Nelson Community College

• Lee R Gibson, University of Louisville

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Alexandria

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• Daniel Harned, Lansing Community College

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• Max Hibbs, Blinn College Brenham

• Jada Hill, Richland College

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Brownsville

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University

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Technology

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University

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Community College

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• Grant Lathrom, Missouri Southern State

University

• Kiseop Lee, University of Louisville

• Max Lee, Westchester Community College

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East Bay

• William Livingston, Missouri Southern State

University

• Crepin M Mahop, Howard University

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Alexandria

• Mary Barone Martin, Middle Tennessee State

University

• Eric Matsuoka, Leeward Community College

• Mike McCraith, Cuyahoga Community

College

• Margaret Michener, University of Nebraska

Kearney

• David Miller, West Virginia University

• Juan Carlos Molina, Austin Community

College

• Hojin Moon, California State University Long

Beach

• Bette Nelson, Alvin Community College

• Ron Palcic, Johnson County Community

College

• Priti Patel, Tarrant County Community

College Southeast

• Mary Beth Pattengale, Sierra College

• Vic Perera, Kent State University Trumbull

• Sandy Poinsett, College of Southern Maryland

• Michael Price, University of Oregon

• Adrienne Riley, California Polytechnic State

University, San Luis Obispo

• Mike Rosenthal, Florida International

University

• Daniel T Russow, Arizona Western College

• Alan Saleski, Loyola University Chicago

• Rebecca Schantz, East Central College

• Jiajia Seffrood, Leeward Community College

• Mayada Shahroki, Lone Star College CyFair

• Sumanthi Shankar, Community College of

Denver

• Robert Shea, Central Texas College

• Linda Snellings-Neal, Wright State University

• Shannon Solis, San Jacinto College, Central

Campus

• Paul Sontag, University of Cincinnati

• Jacqueline Stone, University of Maryland

• Padma Tanniru, San Jose State University

• Marvelyn Tapp, University of Arkansas at

Little Rock

• Mary Ann Teel, University of North Texas

• Jennie Thompson, Leeward Community

College

• Jean Thornton, Western Kentucky University

• Michael van Opstall, University of Utah

• Sara Weiss, Richland College

• Nathanial Wiggins, San Jacinto College, North

Campus

Preface to the Student

This book will help provide you with the algebraic and trigonometric skills 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

Complete

worked-out solutions to the

odd-numbered

exer-cises are given at the

end of each section.

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: algebraTrig.axler.net Twitter: @AxlerAlgebra

xxvi

The Real Numbers

Success in this course will require a good understanding of the basic

proper-The Parthenon, built

in Athens over 2400 years ago The ancient Greeks developed and used remark- ably sophisticated mathematics.

ties 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.

1.1 The Real Line

learning objectives

By the end of this section you should be able toexplain the correspondence between the system of real numbers and thereal line;

show that some real numbers are not rational

The integers are the numbers

, −3, −2, −1, 0, 1, 2, 3, ;

here the dots indicate that the numbers continue without end in each tion The sum, difference, and product of any two integers are also integers The quotient of two integers is not necessarily an integer Thus we extend arithmetic to the rational numbers, which are numbers of the form

direc-The use of a

hori-zontal bar to

sepa-rate the numerator

where m and n are integers and n '= 0.

Division is the inverse of multiplication, in the sense that we want the equation

m

n · n = m

to hold In the equation above, if we take n = 0 and (for example) m = 1, we

get the nonsensical equation 10· 0 = 1 This equation is nonsensical because

multiplying anything by 0 should give 0, not 1 To get around this problem,

we leave expressions such as 10 undefined In other words, division by 0 is

However, geometry and algebra force us to consider an even richer system

of numbers—the real numbers To see why we need to go beyond the rational numbers, we will investigate the real line.

Construction of the Real Line

Imagine a horizontal line, extending without end in both directions Pick a point on this line and label it 0 Pick another point to the right of 0 and label

it 1, as in the figure below.

Two key points on the real line.

Once the points 0 and 1 have been chosen on the line, everything else is The symbol for zero

was invented in India more than 1100 years ago.

determined by thinking of the distance between 0 and 1 as one unit of length.

For example, 2 is one unit to the right of 1 Then 3 is one unit to the right

of 2, and so on The negative integers correspond to moving to the left of 0.

Thus −1 is one unit to the left of 0 Then −2 is one unit to the left of −1, and

so on.

Integers on the real line.

If n is a positive integer, then n1 is to the right of 0 by the length obtained

by dividing the segment from 0 to 1 into n segments of equal length Then

2

n is to the right of n1 by the same length, and n3 is to the right of n2 by the

same length again, and so on The negative rational numbers are placed on

the line similarly, but to the left of 0.

In this way, we associate with every rational number a point on the line.

No figure can show the labels of all the rational numbers, because we can

include only finitely many labels The figure below shows the line with labels

attached to a few of the points corresponding to rational numbers.

3 2 3

12 7 257 101

! 1 3

! 2 3

! 115 76

! 5

2

Some rational numbers on the real line.

We will use the intuitive notion that the line has no gaps and that every

conceivable distance can be represented by a point on the line With these

concepts in mind, we call the line shown above the real line We think

of each point on the real line as corresponding to a real number The

undefined intuitive notions (such as “no gaps”) can be made precise using

more advanced mathematics In this book, we let our intuitive notions of the

real line serve to define the system of real numbers.

Is Every Real Number Rational?

We know that every rational number corresponds to some point on the real

line Does every point on the real line correspond to some rational number?

In other words, is every real number rational?

If more and more labels of rational numbers were placed on the figure

above, the real line would look increasingly cluttered Probably the first

people to ponder these issues thought that the rational numbers fill up the

entire real line However, the ancient Greeks realized that this is not true To

see how they came to this conclusion, we make a brief detour into geometry.

Recall that for a right triangle, the sum of the squares of the lengths of

the two sides that form the right angle equals the square of the length of

the hypotenuse The figure below illustrates this result, which is called the

Pythagorean Theorem.

b c

The Pythagorean Theorem for right triangles: c2= a2+ b2.

Now consider the special case where both sides that form the right angle

This theorem is

named in honor of

the Greek

mathe-matician and

philoso-pher Pythagoras who

proved it over 2500

years ago The

Baby-lonians discovered

this result a thousand

years earlier than that.

have length 1, as in the figure below In this case, the Pythagorean Theorem states that the length c of the hypotenuse has a square equal to 2.

1

1

c

An isosceles right triangle The Pythagorean Theorem implies that c2= 2.

Because we have constructed a line segment whose length c satisfies the

equation c2= 2, a point to the right of 0 on the real line corresponds to c In

other words, there is a positive real number c whose square equals 2 This

raises the question of whether there exists a rational number whose square equals 2.

We could try to find a rational number whose square equals 2 by mentation One striking example is

experi-# 99 70

$2

= 9801 4900 ; here the numerator of the right side misses being twice the denominator by only 1 Although !99

70

"2is close to 2, it is not exactly equal to 2.

Another example is93693196625109 The square of this rational number is mately 1 .9999999999992, which is very close to 2 but again is not exactly

approxi-what we seek.

Because we have found rational numbers whose squares are very close

to 2, you might suspect that with further cleverness we could find a rational number whose square equals 2 However, the ancient Greeks proved this

is impossible This course does not focus much on proofs However, the Greek proof that there is no rational number whose square equals 2 is one of the great intellectual achievements of humanity It should be experienced

by every educated person Thus this proof is presented below for your enrichment.

What follows is a proof by contradiction We will start by assuming that there is a rational number whose square equals 2 Using that assumption, we

will arrive at a contradiction So our assumption must have been incorrect.

Thus there is no rational number whose square equals 2.

Understanding the logical pattern of thinking that goes into this proof can

be a valuable asset in dealing with complex issues.

No rational number has a square equal to 2.

Proof: Suppose there exist integers m and n such that

# m n

$2

= 2.

By canceling any common factors, we can choose m and n to have no

factors in common In other words,m n is reduced to lowest terms.

The equation above is equivalent to the equation

m2= 2n2.

This implies that m2 is even; hence m is even Thus m = 2k for some

integer k Substituting 2k for m in the equation above gives

4 k2= 2n2,

or equivalently

2 k2= n2.

This implies that n2is even; hence n is even.

We have now shown that both m and n are even, contradicting our

choice of m and n as having no factors in common This contradiction

means our original assumption that there is a rational number whose

square equals 2 must be incorrect Thus there do not exist integers m

and n such that !m n"2= 2.

The notation √

2 is used to denote the positive real number c such that

“When you have cluded the impossible, whatever remains, however improbable, must be the truth.”

The result above implies that √

2 is not a rational number Thus not every real number is a rational number In other words, not every point on the real

line corresponds to a rational number.

Irrational numbers

A real number that is not rational is called an irrational number.

We have just shown that √

2 is an irrational number The real numbers π

and e, which we will encounter in later chapters, are also irrational numbers.

Once we have found one irrational number, finding others is much easier,

as shown in the example below.

example 1 Show that 3 + √2 is an irrational number

solution Suppose 3 + √2 is a rational number Because

The attitude of the

ancient Greeks

to-ward irrational

num-bers persists in our

this implies that√

2 is the difference of two rational numbers, which implies that

√

2 is a rational number, which is not true Thus our assumption that 3 + √2 is a

rational number was incorrect In other words, 3 + √2 is an irrational number

The next example provides another illustration of how to use one irrational number to generate another irrational number.

this implies that√2 is the quotient of two rational numbers, which implies that√2

is a rational number, which is not true Thus our assumption that 8√2 is a rational

number was incorrect In other words, 8√

2 is an irrational number

problems

The problems in this section may be harder than

typical problems found in the rest of this book.

1 Show that 67+ √2 is an irrational number

2 Show that 5 − √2 is an irrational number

3 Show that 3√

2 is an irrational number

4 Show that 3√52is an irrational number

5 Show that 4 + 9 √2 is an irrational number

6 Explain why the sum of a rational number and

an irrational number is an irrational number

7 Explain why the product of a nonzero rational

number and an irrational number is an

irra-tional number

8 Supposet is an irrational number Explain why

1

t is also an irrational number

9 Give an example of two irrational numberswhose sum is an irrational number

10 Give an example of two irrational numberswhose sum is a rational number

11 Give an example of three irrational numberswhose sum is a rational number

12 Give an example of two irrational numberswhose product is an irrational number

13 Give an example of two irrational numberswhose product is a rational number

1.2 Algebra of the Real Numbers

learning objectives

By the end of this section you should be able to

manipulate algebraic expressions using the commutative, associative, and

distributive properties;

recognize the order of algebraic operations and the role of parentheses;

apply the crucial algebraic identities involving additive inverses and

fractions;

explain the importance of being careful about parentheses and the order

of operations when using a calculator or computer

The operations of addition, subtraction, multiplication, and division extend Exercises woven

throughout this book have been designed to sharpen your algebraic manipulation skills as

we cover other topics.

from the rational numbers to the real numbers We can add, subtract,

multiply, and divide any two real numbers and stay within the system of real

numbers, again with the exception that division by 0 is prohibited.

In this section we review the basic algebraic properties of the real numbers.

Because this material should indeed be review, no effort has been made to

show how some of these properties follow from others Instead, this section

focuses on highlighting key properties that should become so familiar to you

that you can use them comfortably and without effort.

Commutativity and Associativity

Commutativity is the formal name for the property stating that order does

not matter in addition and multiplication:

Commutativity

a + b = b + a and ab = ba

Here (and throughout this section) a, b, and other variables denote either

real numbers or expressions that take on values that are real numbers For

example, the commutativity of addition implies that x2+x5 = x5 + x2.

Neither subtraction nor division is commutative because order does matter

for those operations For example, 5 − 3 '= 3 − 5, and 62 '= 26.

Associativity is the formal name for the property stating that grouping

does not matter in addition and multiplication:

Associativity

(a + b) + c = a + (b + c) and (ab)c = a(bc)

Expressions inside parentheses should be calculated before further

com-putation For example, (a + b) + c should be calculated by first adding a

and b, and then adding that sum to c The associative property of addition

asserts that this number will be the same as a + (b + c), which should be

calculated by first adding b and c, and then adding that sum to a.

Because of the associativity of addition, we can dispense with parentheses when adding three or more numbers, writing expressions such as

a + b + c + d

without worrying about how the terms are grouped Similarly, because of the associative property of multiplication we do not need parentheses when multiplying together three or more numbers Thus we can write expressions such as abcd without specifying the order of multiplication or the grouping.

Neither subtraction nor division is associative because the grouping does matter for those operations For example,

(9 − 6) − 2 = 3 − 2 = 1,

but

9 − (6 − 2) = 9 − 4 = 5,

which shows that subtraction is not associative.

The standard practice is to evaluate subtractions from left to right unless

parentheses indicate otherwise For example, 9 − 6 − 2 should be interpreted

to mean (9 − 6) − 2, which equals 1.

The Order of Algebraic Operations

Consider the expression

2 + 3 · 7.

This expression contains no parentheses to guide us to which operation should be performed first Should we first add 2 and 3, and then multiply the result by 7? If so, we would interpret the expression above as

(2 + 3) · 7,

which equals 35.

Or to evaluate

2 + 3 · 7

should we first multiply together 3 and 7, and then add 2 to that result If so,

we would interpret the expression above as

Note that (2 + 3) · 7

does not equal

2 + (3 · 7) Thus the

order of these

oper-ations does matter.

2 + (3 · 7),

which equals 23.

So does 2 + 3 · 7 equal (2 + 3) · 7 or 2 + (3 · 7)? The answer to this

ques-tion depends on custom rather than anything inherent in the mathematical

situation Every mathematically literate person would interpret 2 + 3 · 7 to

mean 2 + (3 · 7) In other words, people in the modern era have adopted

the convention that multiplications should be performed before additions

unless parentheses dictate otherwise You need to become accustomed to

this convention:

Multiplication and division before addition and subtraction

Unless parentheses indicate otherwise, products and quotients are

calcu-lated before sums and differences.

Thus, for example, a+bc is interpreted to mean a+(bc), although almost

always we dispense with the parentheses and just write a + bc.

As another illustration of the principle above, consider the expression

4 m + 3n + 11(p + q).

The correct interpretation of this expression is that 4 should be multiplied

by m, 3 should be multiplied by n, 11 should be multiplied by p + q, and

then the three numbers 4 m, 3n, and 11(p + q) should be added together In

parenthe-ses is sometimes used

as an optional visual aid to indicate the order of operations Smaller parentheses should be used for more inner parenthe- ses Thus expressions enclosed in smaller parentheses should usually be evaluated before expressions en- closed in larger paren- theses.

(4m) + (3n) + ! 11 (p + q) " .

The three newly added sets of parentheses in the expression above are

unnecessary, although it is not incorrect to include them However, the

version of the same expression without the unnecessary parentheses is

cleaner and easier to read.

When parentheses are enclosed within parentheses, expressions in the

innermost parentheses are evaluated first.

Evaluate inner parentheses first

In an expression with parentheses inside parentheses, evaluate the

inner-most parentheses first and then work outward.

Evaluate the expression 2!

6 + 3(1 + 4)"

solution Here the innermost parentheses surround 1 + 4 Thus start by evaluating

that expression, getting 5:

Now to evaluate the expression 6 + 3 · 5, first evaluate 3 · 5, getting 15, then add that

to 6, getting 21 Multiplying by 2 completes our evaluation of this expression:

The Distributive Property

The distributive property connects addition and multiplication, converting a product with a sum into a sum of two products.

Distributive property

a(b + c) = ab + ac

Because multiplication is commutative, the distributive property can also

be written in the alternative form

(a + b)c = ac + bc.

Sometimes you will need to use the distributive property to transform

an expression of the form a(b + c) into ab + ac, and sometimes you will

need to use the distributive property in the opposite direction, transforming

an expression of the form ab + ac into a(b + c) Because the distributive

The distributive

prop-erty provides the

justification for

fac-toring expressions.

property is usually used to simplify an expression, the direction of the transformation depends on the context The next example shows the use of the distributive property in both directions.

example 2 Simplify the expression 2(3m + x) + 5x.

solution First use the distributive property to transform 2(3m + x) into 6m + 2x:

One of the most common algebraic manipulations involves expanding a

product of sums, as in the following example.

Expand(a + b)(c + d).

solution Think of(c + d) as a single number and then apply the distributive

property to the expression above, getting

(a + b)(c + d) = a(c + d) + b(c + d).

Now apply the distributive property twice more, getting After you use this

for-mula several times,

it will become so miliar that you can use it routinely with- out needing to pause Note that every term

fa-in the first set of parentheses is mul- tiplied by every term

in the second set of parentheses.

(a + b)(c + d) = ac + ad + bc + bd.

If you are comfortable with the distributive property, there is no need to

memorize the last formula from the example above, because you can always

derive it again Furthermore, by understanding how the identity above was

obtained, you should have no trouble finding formulas for more complicated

expressions such as (a + b)(c + d + t).

An important special case of the identity above occurs when c = a and

d = b In that case we have

(a + b)(a + b) = a2+ ab + ba + b2,

which, with a standard use of commutativity, becomes the identity

(a + b)2= a2+ 2ab + b2.

Additive Inverses and Subtraction

The additive inverseof a real number a is the number −a such that

a + (−a) = 0.

The connection between subtraction and additive inverses is captured by the

identity

a − b = a + (−b).

In fact, the equation above can be taken as the definition of subtraction.

You need to be comfortable using the following identities that involve

additive inverses and subtraction:

Identities involving additive inverses and subtraction

−(−a) = a

−(a + b) = −a − b (−a)(−b) = ab (−a)b = a(−b) = −(ab) (a − b)c = ac − bc a(b − c) = ab − ac

solution Start by thinking of(a + b) as a single number and applying the

distribu-tive property Then apply the distribudistribu-tive property twice more:

Be sure to distribute

the minus signs

cor-rectly when using

the distributive

prop-erty, as shown here.

(a + b)(a − b) = (a + b)a − (a + b)b

= a2+ ba − ab − b2

= a2− b2

You need to become sufficiently comfortable with the following identities

so that you can use them with ease.

Identities arising from the distributive property

(a + b)2= a2+ 2ab + b2(a − b)2= a2− 2ab + b2(a + b)(a − b) = a2− b2

example 5 Without using a calculator, evaluate 43 × 37.