Precalculus, 7th Edition

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Precalculus, 7th Edition

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Precalculus Seventh Edition

David Cohen Late of University of California, Los Angeles

Theodore B. Lee City College of San Francisco

David Sklar San Francisco State University

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Precalculus, Seventh Edition David Cohen, Theodore B. Lee, David Sklar Acquisitions Editor: Gary Whalen Developmental Editors: Carolyn Crockett, Leslie Lahr Assistant Editors: Stefanie Beeck, Cynthia Ashton Editorial Assistant: Naomi Dreyer

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Contents CHAPTER 1

Fundamentals 1.1 1.2 1.3 1.4 1.5 1.6 1.7



4.4 4.5 4.6 4.7


The Definition of a Function 127 The Graph of a Function 141 Shapes of Graphs. Average Rate of Change 154 Techniques in Graphing 166 Methods of Combining Functions. Iteration 181 Inverse Functions 194

Polynomial and Rational Functions. Applications to Optimization 4.1 4.2 4.3


Quadratic Equations: Theory and Examples 81 Other Types of Equations 92 Inequalities 103 More on Inequalities 112

Functions 3.1 3.2 3.3 3.4 3.5 3.6


Sets of Real Numbers 1 Absolute Value 6 Solving Equations (Review and Preview) 10 Rectangular Coordinates. Visualizing Data 19 Graphs and Graphing Utilities 31 Equations of Lines 42 Symmetry and Graphs. Circles 56

Equations and Inequalities 2.1 2.2 2.3 2.4



Linear Functions 215 Quadratic Functions 232 Using Iteration to Model Population Growth (Optional Section, Online) 247 Setting Up Equations That Define Functions 248 Maximum and Minimum Problems 259 Polynomial Functions 274 Rational Functions 296





Exponential and Logarithmic Functions 5.1 5.2 5.3 5.4 5.5 5.6 5.7






Trigonometric Functions of Real Numbers 531 Graphs of the Sine and Cosine Functions 543 Graphs of y  A sin(Bx  C) and y  A cos(Bx  C) 560 Simple Harmonic Motion 576 Graphs of the Tangent and the Reciprocal Functions 583

Analytical Trigonometry 9.1 9.2 9.3 9.4 9.5


Radian Measure 467 Trigonometric Functions of Angles 481 Evaluating the Trigonometric Functions 492 Algebra and the Trigonometric Functions 501 Right-Triangle Trigonometry 511

Graphs of the Trigonometric Functions 8.1 8.2 8.3 8.4 8.5


Trigonometric Functions of Acute Angles 420 Right-Triangle Applications 435 Trigonometric Functions of Angles 441 Trigonometric Identities 454

The Trigonometric Functions 7.1 7.2 7.3 7.4 7.5


Exponential Functions 325 The Exponential Function y  e x 335 Logarithmic Functions 346 Properties of Logarithms 361 Equations and Inequalities with Logs and Exponents 371 Compound Interest 382 Exponential Growth and Decay 392

An Introduction to Trigonometry via Right Triangles 6.1 6.2 6.3 6.4



The Addition Formulas 599 The Double-Angle Formulas 611 The Product-to-Sum and Sum-to-Product Formulas 620 Trigonometric Equations 628 The Inverse Trigonometric Functions 642

Additional Topics in Trigonometry 10.1 Right-Triangle Applications 663 10.2 The Law of Sines and the Law of Cosines 681



10.3 10.4 10.5 10.6 10.7 10.8


Systems of Equations 11.1 11.2 11.3 11.4 11.5 11.6 11.7



The Basic Equations 841 The Parabola 850 Tangents to Parabolas (Optional Section) 862 The Ellipse 864 The Hyperbola 879 The Focus–Directrix Property of Conics 889 The Conics in Polar Coordinates 898 Rotation of Axes 903

Roots of Polynomial Equations 13.1 13.2 13.3 13.4 13.5 13.6 13.7



Systems of Two Linear Equations in Two Unknowns 757 Gaussian Elimination 769 Matrices 782 The Inverse of a Square Matrix 795 Determinants and Cramer’s Rule 809 Nonlinear Systems of Equations 822 Systems of Inequalities 829

The Conic Sections 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8


Vectors in the Plane: A Geometric Approach 697 Vectors in the Plane: An Algebraic Approach 706 Parametric Equations 714 Introduction to Polar Coordinates 722 Curves in Polar Coordinates 732 Demoivre’s Theorem 740

Division of Polynomials 920 The Remainder Theorem and the Factor Theorem 926 The Fundamental Theorem of Algebra 935 Rational and Irrational Roots 945 Conjugate Roots and Descartes’ Rule of Signs 954 Introduction to Partial Fractions 960 More About Partial Fractions 967

Additional Topics in Algebra 14.1 14.2 14.3 14.4 14.5 14.6



Mathematical Induction 983 The Binomial Theorem 989 Introduction to Sequences and Series 999 Arithmetic Sequences and Series 1009 Geometric Sequences and Series 1014 An Introduction to Limits 1019




Appendix A A.1 A.2 A.3

Significant Digits A-1 12 Is Irrational A-4 The Complex Number System A-5

Appendix B (online)* B.1 B.2 B.3 B.4 B.5 B.6


Review of Integer Exponents B-1 Review of nth Roots B-7 Review of Rational Exponents B-14 Review of Factoring B-18 Review of Fractional Expressions B-23 Properties of the Real Numbers B-28

Appendix C (online)* Answers Index





*Appendix B and Appendix C can be accessed online at precalculus7e.

List of Projects All of the projects are available from the website at Section Title 1.4 1.5 1.6 1.7 2.1* 2.2 2.2 2.3* 2.4 3.1 3.2* 3.4 3.5* 3.6 4.1* 4.2*

4.2 4.4 4.5* 4.6* 4.7* 5.1 5.2* 5.3* 5.6 5.7 6.1* 6.2

Discuss, Compute, Reassess Drawing Conclusions from Visual Evidence Thinking About Slope Thinking About Symmetry Put the Quadratic Equation in Its Place! Flying the Flag Specific or General? Whichever Works! An Inequality for the Garden Wind Power A Prime Power Function Implicit Functions: Batteries Required? Correcting a Graphing Utility Display A Graphical Approach to Composition of Functions A Frequently Asked Question About Inverse Functions Who Are Better Runners, Men or Women? How Do You Know That the Graph of a Quadratic Function Is Always Symmetric About a Vertical Line? What’s Left in the Tank? Group Work on Functions of Time The Least-Squares Line Finding Some Maximum Values Without Using Calculus Finding Some Minimum Values Without Using Calculus Using Differences to Compare Exponential and Polynomial Growth Coffee Temperature More Coffee Loan Payments A Variable Growth Constant? Transits of Venus and the Scale of the Solar System Constructing a Regular Polygon

Section Title 7.3

A Linear Approximation for the Sine Function 7.4* Identities and Graphs 7.5* Constructing a Regular Pentagon 7.5** Transits of Venus and the Scale of the Solar System 8.2* Making Waves 8.3* Fourier Series 8.4* The Motion of a Piston 9.1* The Design of a Fresnel Lens 9.3* Superposition 9.4 Astigmatism and Eyeglass Lenses 9.5 Inverse Secant Functions 10.1* Snell’s Law and an Ancient Experiment 10.3 Vector Algebra Using Vector Geometry 10.4 Lines, Circles, and Ray Tracing 10.5 Parameterizations for Lines and Circles 11.1 Geometry Workbooks on the Euler Line and the Nine-Point Circle 11.2* The Leontief Input-Output Model 11.3 Communications and Matrices 11.4* The Leontief Model Revisited 12.2 A Bridge with a Parabolic Arch 12.2 Constructing a Parabola 12.4* The Circumference of an Ellipse 12.5 Using Hyperbolas to Determine a Location 13.3* A Geometric Interpretation of Complex Roots 13.3 Two Methods for Solving Certain Cubic Equations 13.6 Checking a Partial Fraction Decomposition 13.7* An Unusual Partial Fractions Problem 14.3 Perspective and Alternative Solutions for Population Growth Models 14.4 More on Sums

*denotes that the indicated project is included in the text at the end of the listed section. **This project is included in the text after Section 6.1.


Preface Welcome to the new edition of Precalculus. In this edition, we have combined two previous versions: Precalculus, A Problems-Oriented Approach, 6th ed., and Precalculus, with Unit-Circle Trigonometry, 4th ed. This single book now accommodates introducing trigonometry with either a right-triangle approach or a unitcircle approach. This text develops the elements of college algebra and trigonometry in a straightforward manner. As in the earlier editions, our goal has been to create a book that is accessible to the student. The presentation is student-oriented in three specific ways. First, we’ve tried to talk to, rather than lecture at, the student. Second, examples are used to introduce, to explain, and to motivate concepts. Third, most of the initial exercises for each section are carefully coordinated with the worked examples in that section. With all the changes that come with this new edition, please rest assured that we have made every effort to maintain the quality and style that users of previous versions of the book have come to expect. We take very seriously our ongoing responsibility to continue the legacy of David Cohen’s wonderful textbook, with his careful balancing of precalculus as a subject in its own right and as a stepping-stone to calculus. We hope you will be pleased with this new edition.

AUDIENCE In writing Precalculus, we have assumed that the students have been exposed to intermediate algebra but that they have not necessarily mastered the subject. Also, for many college algebra students, there may be a gap of several years between their last mathematics course and the present one. Appendix B, accessible online at, consists of review sections for such students, reviewing topics on integer exponents, nth roots, rational exponents, factoring, and fractional expressions. In Chapter 1, the reader is often referred to Appendix B for further practice.

CURRICULUM REFORM This new edition of Precalculus reflects several of the major themes that have developed in the curriculum reform movement of the past decade. Graphs, visualization of data, and functions are now introduced much earlier and receive greater emphasis. Many sections contain examples and exercises involving applications and real-life data. In addition to the Writing Mathematics sections from previous editions, there are Projects and Mini Projects. These, or references to them, appear at the ends of many sections. All of the Projects and Mini Projects are available online at x


xi Writing Mathematics, Projects, and Mini Projects give students additional opportunities to discuss, explore, learn, and explain mathematics, often using real-life data.

TECHNOLOGY In the following discussion and throughout this text, the term “graphing utility” refers to either a graphing calculator or a computer with software for graphing and analyzing functions. Over the past decade, all of us in the mathematics teaching community have become increasingly aware of the graphing utility and its potential for making a positive impact on our students’ learning. We are also aware of the limitations of the graphing utility as a sole analysis device. The role of the graphing utility has continued to expand in this edition. The existence of the graphing utility is taken for granted, and a number of examples make use of this technology. However, just as in the previous edition, this remains a text in which the central focus is on mathematics and its applications. If the instructor chooses, the text can be used without reference to the graphing utility, but a scientific calculator will be required for numerical calculations. Students already familiar with a graphing utility will, at a minimum, need to read the explanation in Section 1.5 on how to specify the dimensions of a viewing rectangle, since that notation accompanies some figures in the text. Graphing utility exercises (identified by the symbol ) are integrated into the regular exercise sets.

CHANGES IN THIS EDITION The main change in this new edition is that it accommodates introducing trigonometry with either a right-triangle approach or a unit-circle approach. A complete presentation of trigonometry is given in Chapters 6–10. •

To introduce trigonometry via right triangles, just follow Chapter 6–Chapter 10. Some right-triangle material in Chapter 7, Section 5, is clearly labeled as being repeated from Chapter 6 and may be skipped. To introduce trigonometry via the unit circle, merely skip Chapter 6, start with Chapter 7, and proceed through Chapter 10.

There are other major changes in this edition. Chapter 1. In Section 1.7, the presentation of symmetry has been largely rewritten, with increased emphasis on geometry. Symmetry to a point or to a line is based on pairs of points. We focus on the three basic symmetries of a curve: to the origin, to the x-axis, and to the y-axis. From a pair of points with a particular symmetry we build to symmetry of a curve as a set of pairs of points with that symmetry. Next we discuss symmetry of a curve given by an equation. Then we focus on symmetry of the graph of a function. Throughout, we take advantage of two very useful facts. • •

The graph of y as a function of x cannot have x-axis symmetry. If a graph has any two of the three basic symmetries, then it must have all three basic symmetries.


Preface Chapter 2. In Section 2.4 we complement our “table” presentation of nonlinear inequalities with a sign chart on a number line. Chapter 3. The material on graphing techniques in Section 3.4 has been entirely rewritten. In addition to vertical and horizontal shifts and reflections in the x-axis and y-axis, we cover vertical and horizontal scaling of graphs. Emphasis is on graphs of functions. There are two main types of problems. • •

Given the graph of a function given by an equation, if we apply a sequence of transformations to the graph, find the new equation of the resulting graph. Given the equation of a graph that has been obtained by applying a sequence of transformations to the graph of a function, find the sequence of transformations.

In the latter situation, determining the sequence of transformations on the original graph, especially when unsure of a correct order, we find it useful to follow order of operation. Students can draw on their previous experience with order of operation and, perhaps more importantly, it always works! We break down transformations one at a time as listed in the Property Summary table on page 173. The key idea is to determine how x or y changes at each step. Finally, we conclude this section with an introduction to even functions and odd functions. In Section 3.5, there is more discussion of finding the domain of a composition of two functions before finding the formula for that composition. We include new material of an abstract nature in discussing various combinations of two even functions, or two odd functions, or an even function and an odd function to determine whether these properties are “preserved.” The presentation of inverse functions in Section 3.6 has been almost completely rewritten, starting with the definition of inverse function. We now start with the concept of a one-to-one function and then define an inverse function. The emphasis throughout is on the connection between a one-to-one function and its inverse function. The theme is that any property of the inverse function is essentially a restatement of an already known property of the original function. In this way, domain and range, function values, and the graph of an inverse function follow immediately from corresponding information about the original function. Finally, using the fundamental relationships that the composition of a one-to-one function and its inverse function is an appropriate identity function, we can describe what an inverse function does and often find an explicit formula for the inverse function. Chapter 4. Section 4.6 has been extensively rewritten. There is more detailed discussion and analysis of limiting behavior without using limits or limit notation. We start with a detailed analysis of the squaring and cubing functions. Using only algebra and symmetry, we determine increasing and decreasing behavior and end behavior for these power functions, including a careful comparison of the two. We generalize our results to any power function with a positive integer power of at least two. Then we discuss polynomial functions, using these power functions to analyze a polynomial’s end behavior and approximate power function behavior near x-intercepts. In Section 4.7, we take an approach similar to our development of polynomial functions to discuss rational functions. We start with a detailed analysis of the reciprocal function and the reciprocal square function. Again, we use only algebra to



compare and determine important limiting properties of these two reciprocal functions. We generalize these properties to all reciprocal power functions for positive integer powers. Then we use these reciprocal power functions to help us analyze rational function behavior. Our discussion of the graph of a rational function with a horizontal asymptote includes determining whether the graph approaches the asymptote from above or below. Chapter 5. The enhanced discussion of limiting behavior developed in Sections 4.6 and 4.7 continues in Sections 5.1–5.3, with the analysis of asymptote and other end behaviors of graphs of exponential and logarithmic functions. Chapter 6. As mentioned, this chapter provides an introduction to the trigonometric functions through the study of right triangles. Those who prefer to introduce the trigonometric functions via the unit circle should skip this chapter. Chapters 7–10. These chapters contain a full development of trigonometry from a unit-circle point of view. Section 10.8 discusses the polar form of complex numbers and DeMoivre’s theorem. Chapters 11–14. These chapters largely coincide with Chapters 10–13 of the previous edition. There are two major changes. First, as mentioned, DeMoivre’s theorem has been moved to Chapter 10. Second, the book concludes with a brand new Section 14.6 on limits at infinity. Here we review our earlier work on limiting behavior in Sections 4.6 and 4.7, introduce limit notation, and use it in several examples. We then proceed to rigorous definitions of both infinite and finite limits at infinity and utilize them to prove a few familiar limits. We hope this final look at limits will help students in the transition to a calculus course.

Other Important Changes Many Projects and Mini Projects are no longer included in the text, but are referenced at appropriate places in the book and are now accessible online at Similarly, the only appendices included in the book are those on significant digits, the irrationality of the square root of 2, and the presentation of complex numbers. All other appendices are now accessible online at Homework problems from the previous edition are essentially intact except in sections that have been significantly changed, where new problems have been added and instructions for some old problems have been modified to reflect the new presentation. We have retained the division of the exercise sets into groups A, B, and C. Group A exercises are based fairly directly on the examples and definitions in that section of the text. Usually, these problems treat topics in the same order in which they appear in the text. Group B exercises are more difficult, involving some conceptual understanding, and tend to require the use of several different techniques or topics in a single solution. Group C problems are more challenging and may develop extensions of the material in the text and require more conceptual understanding.




Enhanced WebAssign. Exclusively from Cengage Learning, Enhanced WebAssign® offers an extensive online program for Precalculus to encourage the practice that is so critical for concept mastery. The meticulously crafted pedagogy and exercises in our proven texts become even more effective in Enhanced WebAssign, supplemented by multimedia tutorial support and immediate feedback as students complete their assignments. Additionally, students will have access to the Premium eBook, which offers an interactive version of the textbook with search features, highlighting and note-making tools, and direct links to videos or tutorials that elaborate on the text discussions. PowerLecture with ExamView (ISBN: 1-111-42879-4). This CD-ROM provides the instructor with dynamic media tools for teaching. Create, deliver, and customize tests (both print and online) in minutes with ExamView® Computerized Testing Featuring Algorithmic Equations. Easily build solution sets for homework or exams using Solution Builder’s online solutions manual. Microsoft® PowerPoint® lecture slides and figures from the book are also included on this CD-ROM. Companion Website ( The companion website provides you with Appendices, Projects, Supplemental readings, and other study tools. Visit to access additional course materials and companion resources. At the home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where free companion resources can be found. Complete Solutions Manual (ISBN: 1-111-42884-0, Ross Rueger, College of the Sequoias). The Complete Solutions Manual provides worked-out solutions to all of the problems in the text. Text-Specific Videos (ISBN: 1-111-42885-9). These DVDs cover all sections in the text. Ideal for promoting individual study and review, these comprehensive DVDs also support students in online courses or those who may have missed a lecture.


Enhanced WebAssign. Exclusively from Cengage Learning, Enhanced WebAssign® is designed for you to do your homework online. This proven and reliable system uses pedagogy and content found in the text, and then enhances it to help you learn precalculus more effectively. Automatically graded homework allows you to focus on your learning and get interactive study assistance outside of class. Companion Website ( The companion website provides you with Appendices, Projects, Supplemental readings, and other study tools. Visit to access additional course materials and companion resources. At the home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page. This will take you to the product page where free companion resources can be found.

Preface •


Student Solutions Manual (ISBN: 1-111-42824-7, Ross Rueger, College of the Sequoias). The Student Solutions Manual contains fully worked-out solutions to all of the odd-numbered end-of-section exercises as well as the complete workedout solutions to all of the exercises included at the end of each chapter in the text, giving students a way to check their answers and ensure that they took the correct steps to arrive at an answer.

REVIEWERS We are grateful for their many helpful suggestions about the text and exercises and would like to acknowledge the reviewers: Ignacio Alarcon, Santa Barbara City College Frank Bauerle, University of California, Santa Cruz Gregory Bell, North Carolina State University Veronica Burgdorff, Metropolitan State College of Denver Brenda Burns-Williams, North Carolina State University Miriam Castroconde, Irvine Valley College Paul Gunsul, Northern Illinois University Bill Hobbs, Santa Barbara City College Keith Howell, Southwestern Michigan College Larry Johnson, Metropolitan State College of Denver Serge Kruk, Oakland University Barbara Matthei, University of Michigan at Dearborn Scott Mortensen, Dixie State College Lauri Papay, Santa Clara University Kelsey Ryska, California Polytechnic State University Michael Semenoff, El Camino College Dragan Skropanic, Western Wyoming College Deidre Smith, University of Arizona Carol Warnes, University of Georgia We would also like to acknowledge the reviewers of the previous edition, many of whose suggestions are part of this edition as well: Donna J. Bailey, Truman State University Satish Bhatnagar, University of Nevada, Las Vegas M. Hilary Davies, University of Alaska, Anchorage Greg Dietrich, Florida Community College at Jacksonville John Gosselin, University of Georgia Johnny A. Johnson, University of Houston Richard Riggs, New Jersey City University Fred Schifando, Pennsylvania State University Jeffrey S. Snapp, Harvard-Westlake School Thomas J. Walters, University of California, Los Angeles (retired) Sandra Wray-McAfee, University of Michigan, Dearborn Loris I. Zucca, Kingwood College



ACKNOWLEDGMENTS A wonderful team of editors, accuracy checkers, and proofreaders has helped to eliminate many errors from the original manuscript. The remaining errors are those of the authors. Inspired by Donald Knuth, we would again like to offer a reward of $5 to the first person to inform us of each remaining error. Our e-mail addresses are [email protected] and [email protected]. Many students and teachers from both colleges and high schools have made constructive suggestions about the text and exercises, and we thank them for that. We would also like to thank David Cohen’s cousin, Bruce Cohen, for helpful discussions on using technology in the classroom. We are particularly indebted to Eric Barkan for numerous discussions on the text and the applications-oriented projects. Special thanks to Ross Rueger, who wrote the solutions manuals and prepared the answer section for the text. Ross worked with David Cohen on many of David’s textbooks, and it has been a continuing pleasure for us to work with him again. Thanks to Helen Medley for her outstanding work in checking the text and the exercise answers for accuracy and for her insightful suggestions. We have enjoyed working with Lynn Lustberg and Jennifer Risden on the production of the text, and we thank them for their extraordinary patience and ability to keep us on track. To Gary Whalen, Carolyn Crockett, Fred Dahl, Vernon Boes, and the staff at Cengage Learning, Brooks/Cole, thank you for all your work and help in bringing this manuscript into print. Finally we want to thank David Cohen’s sister, Susan Cohen, and David’s wife, Annie Cohen, for initially encouraging us to continue David’s work. Theodore Lee David Sklar




1.1 Sets of Real Numbers 1.2 Absolute Value 1.3 Solving Equations (Review and Preview) 1.4 Rectangular Coordinates. Visualizing Data 1.5 Graphs and Graphing Utilities 1.6 Equations of Lines 1.7 Symmetry and Graphs. Circles

1.1 Natural numbers have been used since time immemorial; fractions were employed by the ancient Egyptians as early as 1700 B.C.; and the Pythagoreans, in ancient Greece, about 400 B.C., discovered numbers, like 12, which cannot be fractions. —Stefan Drobot in Real Numbers (Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1964) What secrets lie hidden in decimals? —Stephan P. Richards in A Number for Your Thoughts (New Providence, N.J.: S. P. Richards, 1982)

Real numbers, equations, graphs—these topics set the stage for our work in precalculus. How much Photo by Nelson Ching/Bloomberg via Getty Images from previous courses should you remember about solving equations? Section 1.3 provides a review of the fundamentals. The rest of the chapter reviews and begins to extend what you’ve learned in previous courses about graphs and graphing. For example, we use graphs to visualize trends in • Spending by the television networks to broadcast the Olympic Games (Exercise 21 in Section 1.4.) • Internet usage (Exercise 23 in Section 1.4.) • Carbon dioxide levels in the atmosphere (Example 5 in Section 1.4.) • U.S. population growth (Exercises 7 and 8 in Section 1.6.)

SETS OF REAL NUMBERS Here, as in your previous mathematics courses, most of the numbers we deal with are real numbers. These are the numbers used in everyday life, in the sciences, in industry, and in business. Perhaps the simplest way to define a real number is this: A real number is any number that can be expressed in decimal form. Some examples of real numbers are 7 ( 7.000 . . .) 12 ( 1.4142 . . .) 23 ( 0.6) (Recall that the bar above the 6 in the decimal 0.6 indicates that the 6 repeats indefinitely.) Certain sets of real numbers are referred to often enough to be given special names. These are summarized in the box that follows. As you’ve seen in previous courses, the real numbers can be represented as points on a number line, as shown in Figure 1. As indicated in Figure 1, the point associated with the number zero is referred to as the origin. The fundamental fact here is that there is a one-to-one correspondence between the set of real numbers and the set of points on the line. This means that each real Origin

Figure 1












CHAPTER 1 Fundamentals


Sets of Real Numbers


Definition and Comments


Natural numbers

These are the ordinary counting numbers: 1, 2, 3, and so on.

1, 4, 29, 1066


These are the natural numbers along with their negatives and zero.

26, 0, 1, 1776

Rational numbers

As the name suggests, these are the real numbers that are ratios of two integers (with nonzero denominators). It can be proved that a real number is rational if and only if its decimal expansion terminates (for example, 3.15) or repeats (for example, 2.43).

Irrational numbers

These are the real numbers that are not rational. Section A.2 of the Appendix contains a proof of the fact that the number 12 is irrational. The proof that p is irrational is more difficult. The first person to prove that p is irrational was the Swiss mathematician J. H. Lambert (1728–1777).

œ„ 2 _1



π 2


Figure 2

_2π _π



2π 3π


2π 3π

(a) 1 _2π _π

0 (b)

Figure 3



(a) The open interval (a, b) contains all real numbers from a to b, excluding a and b.



(b) The closed interval [a, b] contains all real numbers from a to b, including a and b.

Figure 4

4 1 41 2,  23 , 1.7 1 17 10 2, 4.3, 4.173

12, 3  12, 312, p, 4  p, 4p

number is identified with exactly one point on the line; conversely, with each point on the line we identify exactly one real number. The real number associated with a given point is called the coordinate of the point. As a practical matter, we’re usually more interested in relative locations than precise locations on a number line. For instance, since p is approximately 3.1, we show p slightly to the right of 3 in Figure 2. Similarly, since 12 is approximately 1.4, we show 12 slightly less than halfway from 1 to 2 in Figure 2. It is often convenient to use number lines that show reference points other than the integers used in Figure 2. For instance, Figure 3(a) displays a number line with reference points that are multiples of p. In this case it is the integers that we then locate approximately. For example, in Figure 3(b) we show the approximate location of the number 1 on such a line. Two of the most basic relations for real numbers are less than and greater than, symbolized by  and , respectively. For ease of reference, we review these and two related symbols in the box on page 3. In general, relationships involving real numbers and any of the four symbols , , , and  are called inequalities. One of the simplest uses of inequalities is in defining certain sets of real numbers called intervals. Roughly speaking, any uninterrupted portion of the number line is referred to as an interval. In the definitions that follow, you’ll see notations such as a  x  b. This means that both of the inequalities a  x and x  b hold; in other words, the number x is between a and b. Definition Open Intervals and Closed Intervals The open interval (a, b) consists of all real numbers x such that a  x  b. See Figure 4(a). The closed interval [a, b] consists of all real numbers x such that a  x  b. See Figure 4(b).

1.1 Sets of Real Numbers



Notation for Less Than and Greater Than





a is less than b. On a number line, oriented as in Figure 1, 2, or 3, the point a lies to the left of b.

2  3; 3 2


a is less than or equal to b.

2  3; 3  3


b is greater than a. On a number line oriented as in Figure 1, 2, or 3, the point b lies to the right of a. (b  a is equivalent to a  b.)

3  2; 0  1


b is greater than or equal to a.

3  2; 3  3

Note that the brackets in Figure 4(b) are used to indicate that the numbers a and b are included in the interval [a, b], whereas the parentheses in Figure 4(a) indicate that a and b are excluded from the interval (a, b). At times you’ll see notation such as [a, b). This stands for the set of all real numbers x such that a  x  b. Similarly, (a, b] denotes the set of all numbers x such that a  x  b.


1 Understanding Interval Notation Show each interval on a number line, and specify inequalities describing the numbers x in each interval. [1, 2]


(1, 2)

(1, 2]

[1, 2)

See Figure 5. _1








[_1, 2]

(_1, 2)

(_1, 2]

[_1, 2)