Fundamentals of Mathematics PDF
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Denny Burzynski, Wade Ellis
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This textbook covers fundamental mathematics topics including whole numbers, fractions, decimals, ratios, and geometry. It includes examples, exercises, and practice tests for each subject. It's suitable for undergraduate-level learning.
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Fundamentals of Mathematics By: Denny Burzynski Wade Ellis Fundamentals of Mathematics By: Denny Burzynski Wade Ellis Online: < http://cnx.org/content/col10615/1.4/ > CONNEXIO...
Fundamentals of Mathematics By: Denny Burzynski Wade Ellis Fundamentals of Mathematics By: Denny Burzynski Wade Ellis Online: < http://cnx.org/content/col10615/1.4/ > CONNEXIONS Rice University, Houston, Texas This selection and arrangement of content as a collection is copyrighted by Denny Burzynski, Wade Ellis. It is licensed under the Creative Commons Attribution 2.0 license (http://creativecommons.org/licenses/by/2.0/). Collection structure revised: August 18, 2010 PDF generated: August 25, 2010 For copyright and attribution information for the modules contained in this collection, see p. 697. Table of Contents Preface............................................................................................... 1 Acknowledgements.................................................................................. 7 1 Addition and Subtraction of Whole Numbers 1.1 Objectives................................................................................... 9 1.2 Whole Numbers............................................................................ 10 1.3 Reading and Writing Whole Numbers...................................................... 15 1.4 Rounding Whole Numbers.................................................................. 21 1.5 Addition of Whole Numbers................................................................ 29 1.6 Subtraction of Whole Numbers............................................................. 44 1.7 Properties of Addition...................................................................... 62 1.8 Summary of Key Concepts................................................................. 67 1.9 Exercise Supplement....................................................................... 69 1.10 Pro ciency Exam......................................................................... 74 Solutions........................................................................................ 76 2 Multiplication and Division of Whole Numbers 2.1 Objectives.................................................................................. 91 2.2 Multiplication of Whole Numbers........................................................... 92 2.3 Concepts of Division of Whole Numbers................................................... 105 2.4 Division of Whole Numbers................................................................ 113 2.5 Some Interesting Facts about Division..................................................... 126 2.6 Properties of Multiplication............................................................... 130 2.7 Summary of Key Concepts................................................................ 135 2.8 Exercise Supplement...................................................................... 136 2.9 Pro ciency Exam.......................................................................... 139 Solutions....................................................................................... 141 3 Exponents, Roots, and Factorization of Whole Numbers 3.1 Objectives................................................................................. 151 3.2 Exponents and Roots...................................................................... 152 3.3 Grouping Symbols and the Order of Operations........................................... 160 3.4 Prime Factorization of Natural Numbers................................................... 170 3.5 The Greatest Common Factor............................................................. 178 3.6 The Least Common Multiple.............................................................. 182 3.7 Summary of Key Concepts................................................................ 188 3.8 Exercise Supplement...................................................................... 191 3.9 Pro ciency Exam.......................................................................... 196 Solutions....................................................................................... 198 4 Introduction to Fractions and Multiplication and Division of Fractions 4.1 Objectives................................................................................. 211 4.2 Fractions of Whole Numbers............................................................... 212 4.3 Proper Fractions, Improper Fractions, and Mixed Numbers................................ 221 4.4 Equivalent Fractions, Reducing Fractions to Lowest Terms, and Raising Fractions to Higher Terms........................................................................... 231 4.5 Multiplication of Fractions................................................................ 242 4.6 Division of Fractions...................................................................... 253 4.7 Applications Involving Fractions........................................................... 259 4.8 Summary of Key Concepts................................................................ 265 4.9 Exercise Supplement...................................................................... 268 iv 4.10 Pro ciency Exam........................................................................ 274 Solutions....................................................................................... 276 5 Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions 5.1 Objectives................................................................................. 293 5.2 Addition and Subtraction of Fractions with Like Denominators............................ 294 5.3 Addition and Subtraction of Fractions with Unlike Denominators.......................... 298 5.4 Addition and Subtraction of Mixed Numbers.............................................. 304 5.5 Comparing Fractions...................................................................... 308 5.6 Complex Fractions........................................................................ 312 5.7 Combinations of Operations with Fractions................................................ 316 5.8 Summary of Key Concepts................................................................ 320 5.9 Exercise Supplement...................................................................... 321 5.10 Pro ciency Exam........................................................................ 325 Solutions....................................................................................... 327 6 Decimals 6.1 Objectives................................................................................. 337 6.2 Reading and Writing Decimals............................................................ 338 6.3 Converting a Decimal to a Fraction........................................................ 344 6.4 Rounding Decimals........................................................................ 348 6.5 Addition and Subtraction of Decimals..................................................... 352 6.6 Multiplication of Decimals................................................................. 358 6.7 Division of Decimals....................................................................... 369 6.8 Nonterminating Divisions.................................................................. 381 6.9 Converting a Fraction to a Decimal........................................................ 386 6.10 Combinations of Operations with Decimals and Fractions................................. 392 6.11 Summary of Key Concepts............................................................... 395 6.12 Exercise Supplement..................................................................... 396 6.13 Pro ciency Exam........................................................................ 400 Solutions....................................................................................... 402 7 Ratios and Rates 7.1 Objectives................................................................................. 417 7.2 Ratios and Rates.......................................................................... 417 7.3 Proportions............................................................................... 423 7.4 Applications of Proportions................................................................ 429 7.5 Percent.................................................................................... 435 7.6 Fractions of One Percent.................................................................. 442 7.7 Applications of Percents................................................................... 446 7.8 Summary of Key Concepts................................................................ 456 7.9 Exercise Supplement...................................................................... 458 7.10 Pro ciency Exam........................................................................ 462 Solutions....................................................................................... 464 8 Techniques of Estimation 8.1 Objectives................................................................................. 475 8.2 Estimation by Rounding................................................................... 475 8.3 Estimation by Clustering.................................................................. 482 8.4 Mental Arithmetic-Using the Distributive Property........................................ 485 8.5 Estimation by Rounding Fractions......................................................... 491 8.6 Summary of Key Concepts................................................................ 493 8.7 Exercise Supplement...................................................................... 494 8.8 Pro ciency Exam.......................................................................... 500 v Solutions....................................................................................... 502 9 Measurement and Geometry 9.1 Objectives................................................................................. 511 9.2 Measurement and the United States System............................................... 512 9.3 The Metric System of Measurement....................................................... 517 9.4 Simpli cation of Denominate Numbers.................................................... 522 9.5 Perimeter and Circumference of Geometric Figures........................................ 530 9.6 Area and Volume of Geometric Figures and Objects....................................... 540 9.7 Summary of Key Concepts................................................................ 555 9.8 Exercise Supplement...................................................................... 557 9.9 Pro ciency Exam.......................................................................... 562 Solutions....................................................................................... 567 10 Signed Numbers 10.1 Objectives............................................................................... 575 10.2 Variables, Constants, and Real Numbers................................................. 576 10.3 Signed Numbers......................................................................... 582 10.4 Absolute Value........................................................................... 586 10.5 Addition of Signed Numbers............................................................. 590 10.6 Subtraction of Signed Numbers........................................................... 597 10.7 Multiplication and Division of Signed Numbers........................................... 602 10.8 Summary of Key Concepts............................................................... 611 10.9 Exercise Supplement..................................................................... 612 10.10 Pro ciency Exam....................................................................... 616 Solutions....................................................................................... 618 11 Algebraic Expressions and Equations 11.1 Objectives............................................................................... 629 11.2 Algebraic Expressions.................................................................... 630 11.3 Combining Like Terms Using Addition and Subtraction.................................. 636 11.4 Solving Equations of the Form x+a=b and x-a=b........................................ 639 11.5 Solving Equations of the Form ax=b and x/a=b......................................... 647 11.6 Applications I: Translating Words to Mathematical Symbols.............................. 655 11.7 Applications II: Solving Problems........................................................ 660 11.8 Summary of Key Concepts............................................................... 669 11.9 Exercise Supplement..................................................................... 671 11.10 Pro ciency Exam....................................................................... 678 Solutions....................................................................................... 680 Index............................................................................................... 692 Attributions........................................................................................697 vi 1 Preface To the next generation of explorers: Kristi, BreAnne, Lindsey, Randi, Piper, Meghan, Wyatt, Lara, Mason, and Sheanna. Fundamentals of Mathematics is a work text that covers the traditional topics studied in a modern prealgebra course, as well as the topics of estimation, elementary analytic geometry, and introductory algebra. It is intended for students who 1. have had a previous course in prealgebra, 2. wish to meet the prerequisite of a higher level course such as elementary algebra, and 3. need to review fundamental mathematical concepts and techniques. This text will help the student develop the insight and intuition necessary to master arithmetic techniques and manipulative skills. It was written with the following main objectives: 1. to provide the student with an understandable and usable source of information, 2. to provide the student with the maximum opportunity to see that arithmetic concepts and techniques are logically based, 3. to instill in the student the understanding and intuitive skills necessary to know how and when to use particular arithmetic concepts in subsequent material, courses, and nonclassroom situations, and 4. to give the student the ability to correctly interpret arithmetically obtained results. We have tried to meet these objectives by presenting material dynamically, much the way an instructor might present the material visually in a classroom. (See the development of the concept of addition and subtraction of fractions in Section 5.3, for example.) Intuition and understanding are some of the keys to creative thinking; we believe that the material presented in this text will help the student realize that mathematics is a creative subject. This text can be used in standard lecture or self-paced classes. To help meet our objectives and to make the study of prealgebra a pleasant and rewarding experience, Fundamentals of Mathematics is organized as follows. Pedagogical Features The work text format gives the student space to practice mathematical skills with ready reference to sample problems. The chapters are divided into sections, and each section is a complete treatment of a particular topic, which includes the following features: Section Overview Sample Sets Practice Sets 1 This content is available online at. 1 2 Section Exercises Exercises for Review Answers to Practice Sets The chapters begin with Objectives and end with a Summary of Key Concepts, an Exercise Supple- ment, and a Pro ciency Exam. Objectives Each chapter begins with a set of objectives identifying the material to be covered. Each section begins with an overview that repeats the objectives for that particular section. Sections are divided into subsections that correspond to the section objectives, which makes for easier reading. Sample Sets Fundamentals of Mathematics contains examples that are set o in boxes for easy reference. The examples are referred to as Sample Sets for two reasons: 1. They serve as a representation to be imitated, which we believe will foster understanding of mathe- matical concepts and provide experience with mathematical techniques. 2. Sample Sets also serve as a preliminary representation of problem-solving techniques that may be used to solve more general and more complicated problems. The examples have been carefully chosen to illustrate and develop concepts and techniques in the most instructive, easily remembered way. Concepts and techniques preceding the examples are introduced at a level below that normally used in similar texts and are thoroughly explained, assuming little previous knowledge. Practice Sets A parallel Practice Set follows each Sample Set, which reinforces the concepts just learned. There is adequate space for the student to work each problem directly on the page. Answers to Practice Sets The Answers to Practice Sets are given at the end of each section and can be easily located by referring to the page number, which appears after the last Practice Set in each section. Section Exercises The exercises at the end of each section are graded in terms of di culty, although they are not grouped into categories. There is an ample number of problems, and after working through the exercises, the student will be capable of solving a variety of challenging problems. The problems are paired so that the odd-numbered problems are equivalent in kind and di culty to the even-numbered problems. Answers to the odd-numbered problems are provided at the back of the book. Exercises for Review This section consists of ve problems that form a cumulative review of the material covered in the preceding sections of the text and is not limited to material in that chapter. The exercises are keyed by section for easy reference. Since these exercises are intended for review only, no work space is provided. Summary of Key Concepts A summary of the important ideas and formulas used throughout the chapter is included at the end of each chapter. More than just a list of terms, the summary is a valuable tool that reinforces concepts in preparation for the Pro ciency Exam at the end of the chapter, as well as future exams. The summary keys each item to the section of the text where it is discussed. 3 Exercise Supplement In addition to numerous section exercises, each chapter includes approximately 100 supplemental problems, which are referenced by section. Answers to the odd-numbered problems are included in the back of the book. Pro ciency Exam Each chapter ends with a Pro ciency Exam that can serve as a chapter review or evaluation. The Pro ciency Exam is keyed to sections, which enables the student to refer back to the text for assistance. Answers to all the problems are included in the Answer Section at the end of the book. Content The writing style used in Fundamentals of Mathematics is informal and friendly, oering a straightforward approach to prealgebra mathematics. We have made a deliberate eort not to write another text that mini- mizes the use of words because we believe that students can best study arithmetic concepts and understand arithmetic techniques by using words and symbols rather than symbols alone. It has been our experience that students at the prealgebra level are not nearly experienced enough with mathematics to understand symbolic explanations alone; they need literal explanations to guide them through the symbols. We have taken great care to present concepts and techniques so they are understandable and easily remem- bered. After concepts have been developed, students are warned about common pitfalls. We have tried to make the text an information source accessible to prealgebra students. Addition and Subtraction of Whole Numbers This chapter includes the study of whole numbers, including a discussion of the Hindu-Arabic numeration and the base ten number systems. Rounding whole numbers is also presented, as are the commutative and associative properties of addition. Multiplication and Division of Whole Numbers The operations of multiplication and division of whole numbers are explained in this chapter. Multiplication is described as repeated addition. Viewing multiplication in this way may provide students with a visualization of the meaning of algebraic terms such as 8x when they start learning algebra. The chapter also includes the commutative and associative properties of multiplication. Exponents, Roots, and Factorizations of Whole Numbers The concept and meaning of the word root is introduced in this chapter. A method of reading root notation and a method of determining some common roots, both mentally and by calculator, is then presented. We also present grouping symbols and the order of operations, prime factorization of whole numbers, and the greatest common factor and least common multiple of a collection of whole numbers. Introduction to Fractions and Multiplication and Division of Fractions We recognize that fractions constitute one of the foundations of problem solving. We have, therefore, given a detailed treatment of the operations of multiplication and division of fractions and the logic behind these operations. We believe that the logical treatment and many practice exercises will help students retain the information presented in this chapter and enable them to use it as a foundation for the study of rational expressions in an algebra course. 4 Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions A detailed treatment of the operations of addition and subtraction of fractions and the logic behind these operations is given in this chapter. Again, we believe that the logical treatment and many practice exercises will help students retain the information, thus enabling them to use it in the study of rational expressions in an algebra course. We have tried to make explanations dynamic. A method for comparing fractions is introduced, which gives the student another way of understanding the relationship between the words denominator and denomination. This method serves to show the student that it is sometimes possible to compare two dierent types of quantities. We also study a method of simplifying complex fractions and of combining operations with fractions. Decimals The student is introduced to decimals in terms of the base ten number system, fractions, and digits occurring to the right of the units position. A method of converting a fraction to a decimal is discussed. The logic behind the standard methods of operating on decimals is presented and many examples of how to apply the methods are given. The word of as related to the operation of multiplication is discussed. Nonterminating divisions are examined, as are combinations of operations with decimals and fractions. Ratios and Rates We begin by de ning and distinguishing the terms ratio and rate. The meaning of proportion and some applications of proportion problems are described. Proportion problems are solved using the "Five-Step Method." We hope that by using this method the student will discover the value of introducing a variable as a rst step in problem solving and the power of organization. The chapter concludes with discussions of percent, fractions of one percent, and some applications of percent. Techniques of Estimation One of the most powerful problem-solving tools is a knowledge of estimation techniques. We feel that estimation is so important that we devote an entire chapter to its study. We examine three estimation techniques: estimation by rounding, estimation by clustering, and estimation by rounding fractions. We also include a section on the distributive property, an important algebraic property. Measurement and Geometry This chapter presents some of the techniques of measurement in both the United States system and the metric system. Conversion from one unit to another (in a system) is examined in terms of unit fractions. A discussion of the simpli cation of denominate numbers is also included. This discussion helps the student understand more clearly the association between pure numbers and dimensions. The chapter concludes with a study of perimeter and circumference of geometric gures and area and volume of geometric gures and objects. Signed Numbers A look at algebraic concepts and techniques is begun in this chapter. Basic to the study of algebra is a working knowledge of signed numbers. De nitions of variables, constants, and real numbers are introduced. We then distinguish between positive and negative numbers, learn how to read signed numbers, and examine the origin and use of the double-negative property of real numbers. The concept of absolute value is presented both geometrically (using the number line) and algebraically. The algebraic de nition is followed by an interpretation of its meaning and several detailed examples of its use. Addition, subtraction, multiplication, and division of signed numbers are presented rst using the number line, then with absolute value. 5 Algebraic Expressions and Equations The student is introduced to some elementary algebraic concepts and techniques in this nal chapter. Alge- braic expressions and the process of combining like terms are discussed in Section 11.2 and Section 11.3. The method of combining like terms in an algebraic expression is explained by using the interpretation of multiplication as a description of repeated addition (as in Section 2.1). 6 2 Acknowledgements Many extraordinarily talented people are responsible for helping to create this text. We wish to acknowledge the eorts and skill of the following mathematicians. Their contributions have been invaluable. Barbara Conway, Berkshire Community College Bill Hajdukiewicz, Miami-Dade Community College Virginia Hamilton, Shawnee State University David Hares, El Centro College Norman Lee, Ball State University Ginger Y. Manchester, Hinds Junior College John R. Martin, Tarrant County Junior College Shelba Mormon, Northlake College Lou Ann Pate, Pima Community College Gus Pekara, Oklahoma City Community College David Price, Tarrant County Junior College David Schultz, Virginia Western Community College Sue S. Watkins, Lorain County Community College Elizabeth M. Wayt, Tennessee State University Prentice E. Whitlock, Jersey City State College Thomas E. Williamson, Montclair State College Special thanks to the following individuals for their careful accuracy reviews of manuscript, galleys, and page proofs: Steve Blasberg, West Valley College; Wade Ellis, Sr., University of Michigan; John R. Martin, Tarrant County Junior College; and Jane Ellis. We would also like to thank Amy Miller and Guy Sanders, Branham High School. Our sincere thanks to Debbie Wiedemann for her encouragement, suggestions concerning psychobiological examples, proofreading much of the manuscript, and typing many of the section exercises; Sandi Wiedemann for collating the annotated reviews, counting the examples and exercises, and untiring use of "white-out"; and Jane Ellis for solving and typing all of the exercise solutions. We thank the following people for their excellent work on the various ancillary items that accompany Fun- damentals of Mathematics : Steve Blasberg, West Valley College; Wade Ellis, Sr., University of Michigan; and Jane Ellis ( Instructor's Manual); John R. Martin, Tarrant County Junior College (Student Solutions Manual and Study Guide); Virginia Hamilton, Shawnee State University (Computerized Test Bank); Pa- tricia Morgan, San Diego State University (Prepared Tests); and George W. Bergeman, Northern Virginia Community College (Maxis Interactive Software). We also thank the talented people at Saunders College Publishing whose eorts made this text run smoothly and less painfully than we had imagined. Our particular thanks to Bob Stern, Mathematics Editor, Ellen Newman, Developmental Editor, and Janet Nuciforo, Project Editor. Their guidance, suggestions, open 2 This content is available online at. 7 8 minds to our suggestions and concerns, and encouragement have been extraordinarily helpful. Although there were times we thought we might be permanently damaged from rereading and rewriting, their eorts have improved this text immensely. It is a pleasure to work with such high-quality professionals. Denny Burzynski Wade Ellis, Jr. San Jose, California December 1988 I would like to thank Doug Campbell, Ed Lodi, and Guy Sanders for listening to my frustrations and encouraging me on. Thanks also go to my cousin, David Raety, who long ago in Sequoia National Forest told me what a dierential equation is. Particular thanks go to each of my colleagues at West Valley College. Our everyday conversations regarding mathematics instruction have been of the utmost importance to the development of this text and to my teaching career. D.B. Chapter 1 Addition and Subtraction of Whole Numbers 1.1 Objectives1 After completing this chapter, you should Whole Numbers (Section 1.2) know the dierence between numbers and numerals know why our number system is called the Hindu-Arabic numeration system understand the base ten positional number system be able to identify and graph whole numbers Reading and Writing Whole Numbers (Section 1.3) be able to read and write a whole number Rounding Whole Numbers (Section 1.4) understand that rounding is a method of approximation be able to round a whole number to a speci ed position Addition of Whole Numbers (Section 1.5) understand the addition process be able to add whole numbers be able to use the calculator to add one whole number to another Subtraction of Whole Numbers (Section 1.6) understand the subtraction process be able to subtract whole numbers be able to use a calculator to subtract one whole number from another whole number Properties of Addition (Section 1.7) understand the commutative and associative properties of addition understand why 0 is the additive identity 1 This content is available online at. 9 10 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 1.2 Whole Numbers2 1.2.1 Section Overview Numbers and Numerals The Hindu-Arabic Numeration System The Base Ten Positional Number System Whole Numbers Graphing Whole Numbers 1.2.2 Numbers and Numerals We begin our study of introductory mathematics by examining its most basic building block, the number. Number A number is a concept. It exists only in the mind. The earliest concept of a number was a thought that allowed people to mentally picture the size of some collection of objects. To write down the number being conceptualized, a numeral is used. Numeral A numeral is a symbol that represents a number. In common usage today we do not distinguish between a number and a numeral. In our study of introductory mathematics, we will follow this common usage. 1.2.2.1 Sample Set A The following are numerals. In each case, the rst represents the number four, the second represents the num- ber one hundred twenty-three, and the third, the number one thousand ve. These numbers are represented in dierent ways. Hindu-Arabic numerals 4, 123, 1005 Roman numerals IV, CXXIII, MV Egyptian numerals 1.2.2.2 Practice Set A Exercise 1.1 (Solution on p. 76.) Do the phrases "four," "one hundred twenty-three," and "one thousand ve" qualify as numerals? Yes or no? 2 This content is available online at. 11 1.2.3 The Hindu-Arabic Numeration System Hindu-Arabic Numeration System Our society uses the Hindu-Arabic numeration system. This system of numeration began shortly before the third century when the Hindus invented the numerals 0123456789 Leonardo Fibonacci About a thousand years later, in the thirteenth century, a mathematician named Leonardo Fibonacci of Pisa introduced the system into Europe. It was then popularized by the Arabs. Thus, the name, Hindu-Arabic numeration system. 1.2.4 The Base Ten Positional Number System Digits The Hindu-Arabic numerals 0 1 2 3 4 5 6 7 8 9 are called digits. We can form any number in the number system by selecting one or more digits and placing them in certain positions. Each position has a particular value. The Hindu mathematician who devised the system about A.D. 500 stated that "from place to place each is ten times the preceding." Base Ten Positional Systems It is for this reason that our number system is called a positional number system with base ten. Commas When numbers are composed of more than three digits, commas are sometimes used to separate the digits into groups of three. Periods These groups of three are called periods and they greatly simplify reading numbers. In the Hindu-Arabic numeration system, a period has a value assigned to each or its three positions, and the values are the same for each period. The position values are Thus, each period contains a position for the values of one, ten, and hundred. Notice that, in looking from right to left, the value of each position is ten times the preceding. Each period has a particular name. As we continue from right to left, there are more periods. The ve periods listed above are the most common, and in our study of introductory mathematics, they are su cient. The following diagram illustrates our positional number system to trillions. (There are, to be sure, other periods.) 12 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS In our positional number system, the value of a digit is determined by its position in the number. 1.2.4.1 Sample Set B Example 1.1 Find the value of 6 in the number 7,261. Since 6 is in the tens position of the units period, its value is 6 tens. 6 tens = 60 Example 1.2 Find the value of 9 in the number 86,932,106,005. Since 9 is in the hundreds position of the millions period, its value is 9 hundred millions. 9 hundred millions = 9 hundred million Example 1.3 Find the value of 2 in the number 102,001. Since 2 is in the ones position of the thousands period, its value is 2 one thousands. 2 one thousands = 2 thousand 1.2.4.2 Practice Set B Exercise 1.2 (Solution on p. 76.) Find the value of 5 in the number 65,000. Exercise 1.3 (Solution on p. 76.) Find the value of 4 in the number 439,997,007,010. Exercise 1.4 (Solution on p. 76.) Find the value of 0 in the number 108. 1.2.5 Whole Numbers Whole Numbers Numbers that are formed using only the digits 0123456789 are called whole numbers. They are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,... The three dots at the end mean "and so on in this same pattern." 13 1.2.6 Graphing Whole Numbers Number Line Whole numbers may be visualized by constructing a number line. To construct a number line, we simply draw a straight line and choose any point on the line and label it 0. Origin This point is called the origin. We then choose some convenient length, and moving to the right, mark o consecutive intervals (parts) along the line starting at 0. We label each new interval endpoint with the next whole number. Graphing We can visually display a whole number by drawing a closed circle at the point labeled with that whole number. Another phrase for visually displaying a whole number is graphing the whole number. The word graph means to "visually display." 1.2.6.1 Sample Set C Example 1.4 Graph the following whole numbers: 3, 5, 9. Example 1.5 Specify the whole numbers that are graphed on the following number line. The break in the number line indicates that we are aware of the whole numbers between 0 and 106, and 107 and 872, but we are not listing them due to space limitations. The numbers that have been graphed are 0, 106, 873, 874 1.2.6.2 Practice Set C Exercise 1.5 (Solution on p. 76.) Graph the following whole numbers: 46, 47, 48, 325, 327. Exercise 1.6 (Solution on p. 76.) Specify the whole numbers that are graphed on the following number line. A line is composed of an endless number of points. Notice that we have labeled only some of them. As we proceed, we will discover new types of numbers and determine their location on the number line. 14 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 1.2.7 Exercises Exercise 1.7 (Solution on p. 76.) What is a number? Exercise 1.8 What is a numeral? Exercise 1.9 (Solution on p. 76.) Does the word "eleven" qualify as a numeral? Exercise 1.10 How many dierent digits are there? Exercise 1.11 (Solution on p. 76.) Our number system, the Hindu-Arabic number system, is a number system with base. Exercise 1.12 Numbers composed of more than three digits are sometimes separated into groups of three by commas. These groups of three are called. Exercise 1.13 (Solution on p. 76.) In our number system, each period has three values assigned to it. These values are the same for each period. From right to left, what are they? Exercise 1.14 Each period has its own particular name. From right to left, what are the names of the rst four? Exercise 1.15 (Solution on p. 76.) In the number 841, how many tens are there? Exercise 1.16 In the number 3,392, how many ones are there? Exercise 1.17 (Solution on p. 76.) In the number 10,046, how many thousands are there? Exercise 1.18 In the number 779,844,205, how many ten millions are there? Exercise 1.19 (Solution on p. 76.) In the number 65,021, how many hundred thousands are there? For following problems, give the value of the indicated digit in the given number. Exercise 1.20 5 in 599 Exercise 1.21 (Solution on p. 76.) 1 in 310,406 Exercise 1.22 9 in 29,827 Exercise 1.23 (Solution on p. 76.) 6 in 52,561,001,100 Exercise 1.24 Write a two-digit number that has an eight in the tens position. Exercise 1.25 (Solution on p. 76.) Write a four-digit number that has a one in the thousands position and a zero in the ones position. Exercise 1.26 How many two-digit whole numbers are there? 15 Exercise 1.27 (Solution on p. 76.) How many three-digit whole numbers are there? Exercise 1.28 How many four-digit whole numbers are there? Exercise 1.29 (Solution on p. 76.) Is there a smallest whole number? If so, what is it? Exercise 1.30 Is there a largest whole number? If so, what is it? Exercise 1.31 (Solution on p. 76.) Another term for "visually displaying" is. Exercise 1.32 The whole numbers can be visually displayed on a. Exercise 1.33 (Solution on p. 76.) Graph (visually display) the following whole numbers on the number line below: 0, 1, 31, 34. Exercise 1.34 Construct a number line in the space provided below and graph (visually display) the following whole numbers: 84, 85, 901, 1006, 1007. Exercise 1.35 (Solution on p. 76.) Specify, if any, the whole numbers that are graphed on the following number line. Exercise 1.36 Specify, if any, the whole numbers that are graphed on the following number line. 1.3 Reading and Writing Whole Numbers3 1.3.1 Section Overview Reading Whole Numbers Writing Whole Numbers Because our number system is a positional number system, reading and writing whole numbers is quite simple. 3 This content is available online at. 16 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 1.3.2 Reading Whole Numbers To convert a number that is formed by digits into a verbal phrase, use the following method: 1. Beginning at the right and working right to left, separate the number into distinct periods by inserting commas every three digits. 2. Beginning at the left, read each period individually, saying the period name. 1.3.2.1 Sample Set A Write the following numbers as words. Example 1.6 Read 42958. 1. Beginning at the right, we can separate this number into distinct periods by inserting a comma between the 2 and 9. 42,958 2. Beginning at the left, we read each period individually: Forty-two thousand, nine hundred fty-eight. Example 1.7 Read 307991343. 1. Beginning at the right, we can separate this number into distinct periods by placing commas between the 1 and 3 and the 7 and 9. 307,991,343 2. Beginning at the left, we read each period individually. Three hundred seven million, nine hundred ninety-one thousand, three hundred forty-three. Example 1.8 Read 36000000000001. 1. Beginning at the right, we can separate this number into distinct periods by placing commas. 36,000,000,001 17 2. Beginning at the left, we read each period individually. Thirty-six trillion, one. 1.3.2.2 Practice Set A Write each number in words. Exercise 1.37 (Solution on p. 76.) 12,542 Exercise 1.38 (Solution on p. 76.) 101,074,003 Exercise 1.39 (Solution on p. 76.) 1,000,008 1.3.3 Writing Whole Numbers To express a number in digits that is expressed in words, use the following method: 1. Notice rst that a number expressed as a verbal phrase will have its periods set o by commas. 2. Starting at the beginning of the phrase, write each period of numbers individually. 3. Using commas to separate periods, combine the periods to form one number. 1.3.3.1 Sample Set B Write each number using digits. Example 1.9 Seven thousand, ninety-two. Using the comma as a period separator, we have 18 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 7,092 Example 1.10 Fifty billion, one million, two hundred thousand, fourteen. Using the commas as period separators, we have 50,001,200,014 Example 1.11 Ten million, ve hundred twelve. The comma sets o the periods. We notice that there is no thousands period. We'll have to insert this ourselves. 10,000,512 1.3.3.2 Practice Set B Express each number using digits. Exercise 1.40 (Solution on p. 77.) One hundred three thousand, twenty- ve. Exercise 1.41 (Solution on p. 77.) Six million, forty thousand, seven. Exercise 1.42 (Solution on p. 77.) Twenty trillion, three billion, eighty million, one hundred nine thousand, four hundred two. Exercise 1.43 (Solution on p. 77.) Eighty billion, thirty- ve. 1.3.4 Exercises For the following problems, write all numbers in words. Exercise 1.44 (Solution on p. 77.) 912 Exercise 1.45 84 19 Exercise 1.46 (Solution on p. 77.) 1491 Exercise 1.47 8601 Exercise 1.48 (Solution on p. 77.) 35,223 Exercise 1.49 71,006 Exercise 1.50 (Solution on p. 77.) 437,105 Exercise 1.51 201,040 Exercise 1.52 (Solution on p. 77.) 8,001,001 Exercise 1.53 16,000,053 Exercise 1.54 (Solution on p. 77.) 770,311,101 Exercise 1.55 83,000,000,007 Exercise 1.56 (Solution on p. 77.) 106,100,001,010 Exercise 1.57 3,333,444,777 Exercise 1.58 (Solution on p. 77.) 800,000,800,000 Exercise 1.59 A particular community college has 12,471 students enrolled. Exercise 1.60 (Solution on p. 77.) A person who watches 4 hours of television a day spends 1460 hours a year watching T.V. Exercise 1.61 Astronomers believe that the age of the earth is about 4,500,000,000 years. Exercise 1.62 (Solution on p. 77.) Astronomers believe that the age of the universe is about 20,000,000,000 years. Exercise 1.63 There are 9690 ways to choose four objects from a collection of 20. Exercise 1.64 (Solution on p. 77.) If a 412 page book has about 52 sentences per page, it will contain about 21,424 sentences. Exercise 1.65 In 1980, in the United States, there was $1,761,000,000,000 invested in life insurance. Exercise 1.66 (Solution on p. 77.) In 1979, there were 85,000 telephones in Alaska and 2,905,000 telephones in Indiana. Exercise 1.67 In 1975, in the United States, it is estimated that 52,294,000 people drove to work alone. Exercise 1.68 (Solution on p. 77.) In 1980, there were 217 prisoners under death sentence that were divorced. 20 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS Exercise 1.69 In 1979, the amount of money spent in the United States for regular-session college education was $50,721,000,000,000. Exercise 1.70 (Solution on p. 77.) In 1981, there were 1,956,000 students majoring in business in U.S. colleges. Exercise 1.71 In 1980, the average fee for initial and follow up visits to a medical doctors o ce was about $34. Exercise 1.72 (Solution on p. 77.) In 1980, there were approximately 13,100 smugglers of aliens apprehended by the Immigration border patrol. Exercise 1.73 In 1980, the state of West Virginia pumped 2,000,000 barrels of crude oil, whereas Texas pumped 975,000,000 barrels. Exercise 1.74 (Solution on p. 77.) The 1981 population of Uganda was 12,630,000 people. Exercise 1.75 In 1981, the average monthly salary oered to a person with a Master's degree in mathematics was $1,685. For the following problems, write each number using digits. Exercise 1.76 (Solution on p. 77.) Six hundred eighty-one Exercise 1.77 Four hundred ninety Exercise 1.78 (Solution on p. 77.) Seven thousand, two hundred one Exercise 1.79 Nineteen thousand, sixty- ve Exercise 1.80 (Solution on p. 77.) Five hundred twelve thousand, three Exercise 1.81 Two million, one hundred thirty-three thousand, eight hundred fty-nine Exercise 1.82 (Solution on p. 77.) Thirty- ve million, seven thousand, one hundred one Exercise 1.83 One hundred million, one thousand Exercise 1.84 (Solution on p. 77.) Sixteen billion, fty-nine thousand, four Exercise 1.85 Nine hundred twenty billion, four hundred seventeen million, twenty-one thousand Exercise 1.86 (Solution on p. 78.) Twenty-three billion Exercise 1.87 Fifteen trillion, four billion, nineteen thousand, three hundred ve Exercise 1.88 (Solution on p. 78.) One hundred trillion, one 21 1.3.4.1 Exercises for Review Exercise 1.89 (Section 1.2) How many digits are there? Exercise 1.90 (Solution on p. 78.) (Section 1.2) In the number 6,641, how many tens are there? Exercise 1.91 (Section 1.2) What is the value of 7 in 44,763? Exercise 1.92 (Solution on p. 78.) (Section 1.2) Is there a smallest whole number? If so, what is it? Exercise 1.93 (Section 1.2) Write a four-digit number with a 9 in the tens position. 1.4 Rounding Whole Numbers4 1.4.1 Section Overview Rounding as an Approximation The Method of Rounding Numbers 1.4.2 Rounding as an Approximation A primary use of whole numbers is to keep count of how many objects there are in a collection. Sometimes we're only interested in the approximate number of objects in the collection rather than the precise number. For example, there are approximately 20 symbols in the collection below. The precise number of symbols in the above collection is 18. Rounding We often approximate the number of objects in a collection by mentally seeing the collection as occurring in groups of tens, hundreds, thousands, etc. This process of approximation is called rounding. Rounding is very useful in estimation. We will study estimation in Chapter 8. When we think of a collection as occurring in groups of tens, we say we're rounding to the nearest ten. When we think of a collection as occurring in groups of hundreds, we say we're rounding to the nearest hundred. This idea of rounding continues through thousands, ten thousands, hundred thousands, millions, etc. The process of rounding whole numbers is illustrated in the following examples. Example 1.12 Round 67 to the nearest ten. 4 This content is available online at. 22 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS On the number line, 67 is more than halfway from 60 to 70. The digit immediately to the right of the tens digit, the round-o digit, is the indicator for this. Thus, 67, rounded to the near- est ten, is 70. Example 1.13 Round 4,329 to the nearest hundred. On the number line, 4,329 is less than halfway from 4,300 to 4,400. The digit to the immediate right of the hundreds digit, the round-o digit, is the indicator. Thus, 4,329, rounded to the nearest hundred is 4,300. Example 1.14 Round 16,500 to the nearest thousand. On the number line, 16,500 is exactly halfway from 16,000 to 17,000. By convention, when the number to be rounded is exactly halfway between two numbers, it is rounded to the higher number. Thus, 16,500, rounded to the nearest thousand, is 17,000. Example 1.15 A person whose salary is $41,450 per year might tell a friend that she makes $41,000 per year. She has rounded 41,450 to the nearest thousand. The number 41,450 is closer to 41,000 than it is to 42,000. 1.4.3 The Method of Rounding Whole Numbers From the observations made in the preceding examples, we can use the following method to round a whole number to a particular position. 1. Mark the position of the round-o digit. 2. Note the digit to the immediate right of the round-o digit. 23 a. If it is less than 5, replace it and all the digits to its right with zeros. Leave the round-o digit unchanged. b. If it is 5 or larger, replace it and all the digits to its right with zeros. Increase the round-o digit by 1. 1.4.3.1 Sample Set A Use the method of rounding whole numbers to solve the following problems. Example 1.16 Round 3,426 to the nearest ten. 1. We are rounding to the tens position. Mark the digit in the tens position 2. Observe the digit immediately to the right of the tens position. It is 6. Since 6 is greater than 5, we round up by replacing 6 with 0 and adding 1 to the digit in the tens position (the round-o position): 2 + 1 = 3. 3,430 Thus, 3,426 rounded to the nearest ten is 3,430. Example 1.17 Round 9,614,018,007 to the nearest ten million. 1. We are rounding to the nearest ten million. 2. Observe the digit immediately to the right of the ten millions position. It is 4. Since 4 is less than 5, we round down by replacing 4 and all the digits to its right with zeros. 9,610,000,000 Thus, 9,614,018,007 rounded to the nearest ten million is 9,610,000,000. Example 1.18 Round 148,422 to the nearest million. 1. Since we are rounding to the nearest million, we'll have to imagine a digit in the millions position. We'll write 148,422 as 0,148,422. 2. The digit immediately to the right is 1. Since 1 is less than 5, we'll round down by replacing it and all the digits to its right with zeros. 0,000,000 This number is 0. 24 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS Thus, 148,422 rounded to the nearest million is 0. Example 1.19 Round 397,000 to the nearest ten thousand. 1. We are rounding to the nearest ten thousand. 2. The digit immediately to the right of the ten thousand position is 7. Since 7 is greater than 5, we round up by replacing 7 and all the digits to its right with zeros and adding 1 to the digit in the ten thousands position. But 9 + 1 = 10 and we must carry the 1 to the next (the hundred thousands) position. 400,000 Thus, 397,000 rounded to the nearest ten thousand is 400,000. 1.4.3.2 Practice Set A Use the method of rounding whole numbers to solve each problem. Exercise 1.94 (Solution on p. 78.) Round 3387 to the nearest hundred. Exercise 1.95 (Solution on p. 78.) Round 26,515 to the nearest thousand. Exercise 1.96 (Solution on p. 78.) Round 30,852,900 to the nearest million. Exercise 1.97 (Solution on p. 78.) Round 39 to the nearest hundred. Exercise 1.98 (Solution on p. 78.) Round 59,600 to the nearest thousand. 1.4.4 Exercises For the following problems, complete the table by rounding each number to the indicated positions. Exercise 1.99 (Solution on p. 78.) 1,642 hundred thousand ten thousand million Table 1.1 Exercise 1.100 5,221 25 hundred thousand ten thousand million Table 1.2 Exercise 1.101 (Solution on p. 78.) 91,803 Hundred thousand ten thousand million Table 1.3 Exercise 1.102 106,007 hundred thousand ten thousand million Table 1.4 Exercise 1.103 (Solution on p. 78.) 208 hundred thousand ten thousand million Table 1.5 Exercise 1.104 199 hundred thousand ten thousand million Table 1.6 Exercise 1.105 (Solution on p. 78.) 863 hundred thousand ten thousand million Table 1.7 26 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS Exercise 1.106 794 hundred thousand ten thousand million Table 1.8 Exercise 1.107 (Solution on p. 79.) 925 hundred thousand ten thousand million Table 1.9 Exercise 1.108 909 hundred thousand ten thousand million Table 1.10 Exercise 1.109 (Solution on p. 79.) 981 hundred thousand ten thousand million Table 1.11 Exercise 1.110 965 hundred thousand ten thousand million Table 1.12 Exercise 1.111 (Solution on p. 79.) 551,061,285 hundred thousand ten thousand million 27 Table 1.13 Exercise 1.112 23,047,991,521 hundred thousand ten thousand million Table 1.14 Exercise 1.113 (Solution on p. 79.) 106,999,413,206 Hundred thousand ten thousand million Table 1.15 Exercise 1.114 5,000,000 hundred thousand ten thousand million Table 1.16 Exercise 1.115 (Solution on p. 79.) 8,006,001 hundred thousand ten thousand million Table 1.17 Exercise 1.116 94,312 hundred thousand ten thousand million Table 1.18 Exercise 1.117 (Solution on p. 79.) 33,486 28 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS hundred thousand ten thousand million Table 1.19 Exercise 1.118 560,669 hundred thousand ten thousand million Table 1.20 Exercise 1.119 (Solution on p. 80.) 388,551 hundred thousand ten thousand million Table 1.21 Exercise 1.120 4,752 hundred thousand ten thousand million Table 1.22 Exercise 1.121 (Solution on p. 80.) 8,209 hundred thousand ten thousand million Table 1.23 Exercise 1.122 In 1950, there were 5,796 cases of diphtheria reported in the United States. Round to the nearest hundred. Exercise 1.123 (Solution on p. 80.) In 1979, 19,309,000 people in the United States received federal food stamps. Round to the nearest ten thousand. Exercise 1.124 In 1980, there were 1,105,000 people between 30 and 34 years old enrolled in school. Round to the nearest million. 29 Exercise 1.125 (Solution on p. 80.) In 1980, there were 29,100,000 reports of aggravated assaults in the United States. Round to the nearest million. For the following problems, round the numbers to the position you think is most reasonable for the situation. Exercise 1.126 In 1980, for a city of one million or more, the average annual salary of police and re ghters was $16,096. Exercise 1.127 (Solution on p. 80.) The average percentage of possible sunshine in San Francisco, California, in June is 73%. Exercise 1.128 In 1980, in the state of Connecticut, $3,777,000,000 in defense contract payroll was awarded. Exercise 1.129 (Solution on p. 80.) In 1980, the federal government paid $5,463,000,000 to Viet Nam veterans and dependants. Exercise 1.130 In 1980, there were 3,377,000 salespeople employed in the United States. Exercise 1.131 (Solution on p. 80.) In 1948, in New Hampshire, 231,000 popular votes were cast for the president. Exercise 1.132 In 1970, the world production of cigarettes was 2,688,000,000,000. Exercise 1.133 (Solution on p. 80.) In 1979, the total number of motor vehicle registrations in Florida was 5,395,000. Exercise 1.134 In 1980, there were 1,302,000 registered nurses the United States. 1.4.4.1 Exercises for Review Exercise 1.135 (Solution on p. 80.) (Section 1.2) There is a term that describes the visual displaying of a number. What is the term? Exercise 1.136 (Section 1.2) What is the value of 5 in 26,518,206? Exercise 1.137 (Solution on p. 80.) (Section 1.3) Write 42,109 as you would read it. Exercise 1.138 (Section 1.3) Write "six hundred twelve" using digits. Exercise 1.139 (Solution on p. 80.) (Section 1.3) Write "four billion eight" using digits. 1.5 Addition of Whole Numbers5 1.5.1 Section Overview Addition Addition Visualized on the Number Line The Addition Process 5 This content is available online at. 30 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS Addition Involving Carrying Calculators 1.5.2 Addition Suppose we have two collections of objects that we combine together to form a third collection. For example, We are combining a collection of four objects with a collection of three objects to obtain a collection of seven objects. Addition The process of combining two or more objects (real or intuitive) to form a third, the total, is called addition. In addition, the numbers being added are called addends or terms, and the total is called the sum. The plus symbol (+) is used to indicate addition, and the equal symbol (=) is used to represent the word "equal." For example, 4 + 3 = 7 means "four added to three equals seven." 1.5.3 Addition Visualized on the Number Line Addition is easily visualized on the number line. Let's visualize the addition of 4 and 3 using the number line. To nd 4 + 3, 1. Start at 0. 2. Move to the right 4 units. We are now located at 4. 3. From 4, move to the right 3 units. We are now located at 7. Thus, 4 + 3 = 7. 1.5.4 The Addition Process We'll study the process of addition by considering the sum of 25 and 43. 25 means +43 We write this as 68. We can suggest the following procedure for adding whole numbers using this example. Example 1.20: The Process of Adding Whole Numbers To add whole numbers, The process: 31 1. Write the numbers vertically, placing corresponding positions in the same column. 25 +43 2. Add the digits in each column. Start at the right (in the ones position) and move to the left, placing the sum at the bottom. 25 +43 68 Caution: Confusion and incorrect sums can occur when the numbers are not aligned in columns properly. Avoid writing such additions as 25 +43 25 +43 1.5.4.1 Sample Set A Example 1.21 Add 276 and 103. 276 6+3=9. +103 7+0=7. 379 2+1=3. Example 1.22 Add 1459 and 130 9+0=9. 1459 5+3=8. +130 4+1=5. 1589 1+0=1. In each of these examples, each individual sum does not exceed 9. We will examine individual sums that exceed 9 in the next section. 1.5.4.2 Practice Set A Perform each addition. Show the expanded form in problems 1 and 2. Exercise 1.140 (Solution on p. 80.) Add 63 and 25. Exercise 1.141 (Solution on p. 80.) Add 4,026 and 1,501. Exercise 1.142 (Solution on p. 80.) Add 231,045 and 36,121. 32 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 1.5.5 Addition Involving Carrying It often happens in addition that the sum of the digits in a column will exceed 9. This happens when we add 18 and 34. We show this in expanded form as follows. Notice that when we add the 8 ones to the 4 ones we get 12 ones. We then convert the 12 ones to 1 ten and 2 ones. In vertical addition, we show this conversion by carrying the ten to the tens column. We write a 1 at the top of the tens column to indicate the carry. This same example is shown in a shorter form as follows: 8 + 4 = 12 Write 2, carry 1 ten to the top of the next column to the left. 1.5.5.1 Sample Set B Perform the following additions. Use the process of carrying when needed. Example 1.23 Add 1875 and 358. 5 + 8 = 13 Write 3, carry 1 ten. 1 + 7 + 5 = 13 Write 3, carry 1 hundred. 1 + 8 + 3 = 12 Write 2, carry 1 thousand. 1+1=2 The sum is 2233. Example 1.24 Add 89,208 and 4,946. 33 8 + 6 = 14 Write 4, carry 1 ten. 1+0+4=5 Write the 5 (nothing to carry). 2 + 9 = 11 Write 1, carry one thousand. 1 + 9 + 4 = 14 Write 4, carry one ten thousand. 1+8=9 The sum is 94,154. Example 1.25 Add 38 and 95. 8 + 5 = 13 Write 3, carry 1 ten. 1 + 3 + 9 = 13 Write 3, carry 1 hundred. 1+0=1 As you proceed with the addition, it is a good idea to keep in mind what is actually happening. The sum is 133. Example 1.26 Find the sum 2648, 1359, and 861. 8 + 9 + 1 = 18 Write 8, carry 1 ten. 1 + 4 + 5 + 6 = 16 Write 6, carry 1 hundred. 1 + 6 + 3 + 8 = 18 Write 8, carry 1 thousand. 1+2+1=4 The sum is 4,868. Numbers other than 1 can be carried as illustrated in Example 1.27. Example 1.27 Find the sum of the following numbers. 34 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 6 + 5 + 1 + 7 = 19 Write 9, carry the 1. 1+1+0+5+1=8 Write 8. 0 + 9 + 9 + 8 = 26 Write 6, carry the 2. 2 + 8 + 9 + 8 + 6 = 33 Write 3, carry the 3. 3 + 7 + 3 + 5 = 18 Write 8, carry the 1. 1+8=9 Write 9. The sum is 983,689. Example 1.28 The number of students enrolled at Riemann College in the years 1984, 1985, 1986, and 1987 was 10,406, 9,289, 10,108, and 11,412, respectively. What was the total number of students enrolled at Riemann College in the years 1985, 1986, and 1987? We can determine the total number of students enrolled by adding 9,289, 10,108, and 11,412, the number of students enrolled in the years 1985, 1986, and 1987. The total number of students enrolled at Riemann College in the years 1985, 1986, and 1987 was 30,809. 1.5.5.2 Practice Set B Perform each addition. For the next three problems, show the expanded form. Exercise 1.143 (Solution on p. 80.) Add 58 and 29. Exercise 1.144 (Solution on p. 81.) Add 476 and 85. Exercise 1.145 (Solution on p. 81.) Add 27 and 88. Exercise 1.146 (Solution on p. 81.) Add 67,898 and 85,627. For the next three problems, nd the sums. Exercise 1.147 (Solution on p. 81.) 57 26 84 35 Exercise 1.148 (Solution on p. 81.) 847 825 796 Exercise 1.149 (Solution on p. 81.) 16, 945 8, 472 387, 721 21, 059 629 1.5.6 Calculators Calculators provide a very simple and quick way to nd sums of whole numbers. For the two problems in Sample Set C, assume the use of a calculator that does not require the use of an ENTER key (such as many Hewlett-Packard calculators). 1.5.6.1 Sample Set C Use a calculator to nd each sum. Example 1.29 34 + 21 Display Reads Type 34 34 Press + 34 Type 21 21 Press = 55 Table 1.24 The sum is 55. Example 1.30 36 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 106 + 85 + 322 + 406 Display Reads Type 106 106 The calculator keeps a running subtotal Press + 106 Type 85 85 Press = 191 ← 106 + 85 Type 322 322 Press + 513 ← 191 + 322 Type 406 406 Press = 919 ← 513 + 406 Table 1.25 The sum is 919. 1.5.6.2 Practice Set C Use a calculator to nd the following sums. Exercise 1.150 (Solution on p. 81.) 62 + 81 + 12 Exercise 1.151 (Solution on p. 81.) 9, 261 + 8, 543 + 884 + 1, 062 Exercise 1.152 (Solution on p. 81.) 10, 221 + 9, 016 + 11, 445 1.5.7 Exercises For the following problems, perform the additions. If you can, check each sum with a calculator. Exercise 1.153 (Solution on p. 81.) 14 + 5 Exercise 1.154 12 + 7 Exercise 1.155 (Solution on p. 81.) 46 + 2 Exercise 1.156 83 + 16 Exercise 1.157 (Solution on p. 81.) 77 + 21 Exercise 1.158 321 + 42 37 Exercise 1.159 (Solution on p. 82.) 916 + 62 Exercise 1.160 104 +561 Exercise 1.161 (Solution on p. 82.) 265 +103 Exercise 1.162 552 + 237 Exercise 1.163 (Solution on p. 82.) 8, 521 + 4, 256 Exercise 1.164 16, 408 + 3, 101 Exercise 1.165 (Solution on p. 82.) 16, 515 +42, 223 Exercise 1.166 616, 702 + 101, 161 Exercise 1.167 (Solution on p. 82.) 43, 156, 219 + 2, 013, 520 Exercise 1.168 17 + 6 Exercise 1.169 (Solution on p. 82.) 25 + 8 Exercise 1.170 84 + 7 Exercise 1.171 (Solution on p. 82.) 75 + 6 Exercise 1.172 36 + 48 Exercise 1.173 (Solution on p. 82.) 74 + 17 Exercise 1.174 486 + 58 Exercise 1.175 (Solution on p. 82.) 743 + 66 Exercise 1.176 381 + 88 38 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS Exercise 1.177 (Solution on p. 82.) 687 +175 Exercise 1.178 931 +853 Exercise 1.179 (Solution on p. 82.) 1, 428 + 893 Exercise 1.180 12, 898 + 11, 925 Exercise 1.181 (Solution on p. 82.) 631, 464 +509, 740 Exercise 1.182 805, 996 + 98, 516 Exercise 1.183 (Solution on p. 82.) 38, 428, 106 +522, 936, 005 Exercise 1.184 5, 288, 423, 100 + 16, 934, 785, 995 Exercise 1.185 (Solution on p. 82.) 98, 876, 678, 521, 402 + 843, 425, 685, 685, 658 Exercise 1.186 41 + 61 + 85 + 62 Exercise 1.187 (Solution on p. 82.) 21 + 85 + 104 + 9 + 15 Exercise 1.188 116 27 110 110 + 8 Exercise 1.189 (Solution on p. 82.) 75, 206 4, 152 +16, 007 39 Exercise 1.190 8, 226 143 92, 015 8 487, 553 5, 218 Exercise 1.191 (Solution on p. 82.) 50, 006 1, 005 100, 300 20, 008 1, 000, 009 800, 800 Exercise 1.192 616 42, 018 1, 687 225 8, 623, 418 12, 506, 508 19 2, 121 195, 643 For the following problems, perform the additions and round to the nearest hundred. Exercise 1.193 (Solution on p. 82.) 1, 468 2, 183 Exercise 1.194 928, 725 15, 685 Exercise 1.195 (Solution on p. 82.) 82, 006 3, 019, 528 Exercise 1.196 18, 621 5, 059 40 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS Exercise 1.197 (Solution on p. 82.) 92 48 Exercise 1.198 16 37 Exercise 1.199 (Solution on p. 82.) 21 16 Exercise 1.200 11, 172 22, 749 12, 248 Exercise 1.201 (Solution on p. 82.) 240 280 210 310 Exercise 1.202 9, 573 101, 279 122, 581 For the next ve problems, replace the letter m with the whole number that will make the addition true. Exercise 1.203 (Solution on p. 82.) 62 + m 67 Exercise 1.204 106 + m 113 Exercise 1.205 (Solution on p. 82.) 432 + m 451 Exercise 1.206 803 + m 830 41 Exercise 1.207 (Solution on p. 82.) 1, 893 + m 1, 981 Exercise 1.208 The number of nursing and related care facilities in the United States in 1971 was 22,004. In 1978, the number was 18,722. What was the total number of facilities for both 1971 and 1978? Exercise 1.209 (Solution on p. 83.) The number of persons on food stamps in 1975, 1979, and 1980 was 19,179,000, 19,309,000, and 22,023,000, respectively. What was the total number of people on food stamps for the years 1975, 1979, and 1980? Exercise 1.210 The enrollment in public and nonpublic schools in the years 1965, 1970, 1975, and 1984 was 54,394,000, 59,899,000, 61,063,000, and 55,122,000, respectively. What was the total enrollment for those years? Exercise 1.211 (Solution on p. 83.) The area of New England is 3,618,770 square miles. The area of the Mountain states is 863,563 square miles. The area of the South Atlantic is 278,926 square miles. The area of the Paci c states is 921,392 square miles. What is the total area of these regions? Exercise 1.212 In 1960, the IRS received 1,188,000 corporate income tax returns. In 1965, 1,490,000 returns were received. In 1970, 1,747,000 returns were received. In 1972 1977, 1,890,000; 1,981,000; 2,043,000; 2,100,000; 2,159,000; and 2,329,000 returns were received, respectively. What was the total number of corporate tax returns received by the IRS during the years 1960, 1965, 1970, 1972 1977? Exercise 1.213 (Solution on p. 83.) Find the total number of scientists employed in 1974. Exercise 1.214 Find the total number of sales for space vehicle systems for the years 1965-1980. 42 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS Exercise 1.215 (Solution on p. 83.) Find the total baseball attendance for the years 1960-1980. Exercise 1.216 Find the number of prosecutions of federal o cials for 1970-1980. For the following problems, try to add the numbers mentally. 43 Exercise 1.217 (Solution on p. 83.) 5 5 3 7 Exercise 1.218 8 2 6 4 Exercise 1.219 (Solution on p. 83.) 9 1 8 5 2 Exercise 1.220 5 2 5 8 3 7 Exercise 1.221 (Solution on p. 83.) 6 4 3 1 6 7 9 4 Exercise 1.222 20 30 Exercise 1.223 (Solution on p. 83.) 15 35 44 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS Exercise 1.224 16 14 Exercise 1.225 (Solution on p. 83.) 23 27 Exercise 1.226 82 18 Exercise 1.227 (Solution on p. 83.) 36 14 1.5.7.1 Exercises for Review Exercise 1.228 (Section 1.2) Each period of numbers has its own name. From right to left, what is the name of the fourth period? Exercise 1.229 (Solution on p. 83.) (Section 1.2) In the number 610,467, how many thousands are there? Exercise 1.230 (Section 1.3) Write 8,840 as you would read it. Exercise 1.231 (Solution on p. 83.) (Section 1.4) Round 6,842 to the nearest hundred. Exercise 1.232 (Section 1.4) Round 431,046 to the nearest million. 1.6 Subtraction of Whole Numbers6 1.6.1 Section Overview Subtraction Subtraction as the Opposite of Addition The Subtraction Process Subtraction Involving Borrowing Borrowing From Zero Calculators 6 This content is available online at. 45 1.6.2 Subtraction Subtraction Subtraction is the process of determining the remainder when part of the total is removed. Suppose the sum of two whole numbers is 11, and from 11 we remove 4. Using the number line to help our visualization, we see that if we are located at 11 and move 4 units to the left, and thus remove 4 units, we will be located at 7. Thus, 7 units remain when we remove 4 units from 11 units. The Minus Symbol The minus symbol (-) is used to indicate subtraction. For example, 11 − 4 indicates that 4 is to be subtracted from 11. Minuend The number immediately in front of or the minus symbol is called the minuend, and it represents the original number of units. Subtrahend The number immediately following or below the minus symbol is called the subtrahend, and it represents the number of units to be removed. Dierence The result of the subtraction is called the dierence of the two numbers. For example, in 11 − 4 = 7, 11 is the minuend, 4 is the subtrahend, and 7 is the dierence. 1.6.3 Subtraction as the Opposite of Addition Subtraction can be thought of as the opposite of addition. We show this in the problems in Sample Set A. 1.6.3.1 Sample Set A Example 1.31 8 − 5 = 3 since 3 + 5 = 8. Example 1.32 9 − 3 = 6 since 6 + 3 = 9. 1.6.3.2 Practice Set A Complete the following statements. Exercise 1.233 (Solution on p. 83.) 7−5= since +5 = 7. Exercise 1.234 (Solution on p. 83.) 9−1= since +1 = 9. Exercise 1.235 (Solution on p. 83.) 17 − 8 = since +8 = 17. 46 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 1.6.4 The Subtraction Process We'll study the process of the subtraction of two whole numbers by considering the dierence between 48 and 35. which we write as 13. Example 1.33: The Process of Subtracting Whole Numbers To subtract two whole numbers, The process 1. Write the numbers vertically, placing corresponding positions in the same column. 48 −35 2. Subtract the digits in each column. Start at the right, in the ones position, and move to the left, placing the dierence at the bottom. 48 −35 13 1.6.4.1 Sample Set B Perform the following subtractions. Example 1.34 275 −142 133 5-2=3. 7-4=3. 2-1=1. Example 1.35 46, 042 − 1, 031 45, 011 2-1=1. 4-3=1. 0-0=0. 6-1=5. 4-0=4. 47 Example 1.36 Find the dierence between 977 and 235. Write the numbers vertically, placing the larger number on top. Line up the columns properly. 977 −235 742 The dierence between 977 and 235 is 742. Example 1.37 In Keys County in 1987, there were 809 cable television installations. In Flags County in 1987, there were 1,159 cable television installations. How many more cable television installations were there in Flags County than in Keys County in 1987? We need to determine the dierence between 1,159 and 809. There were 350 more cable television installations in Flags County than in Keys County in 1987. 1.6.4.2 Practice Set B Perform the following subtractions. Exercise 1.236 (Solution on p. 83.) 534 −203 Exercise 1.237 (Solution on p. 83.) 857 − 43 Exercise 1.238 (Solution on p. 83.) 95, 628 −34, 510 Exercise 1.239 (Solution on p. 83.) 11, 005 − 1, 005 Exercise 1.240 (Solution on p. 83.) Find the dierence between 88,526 and 26,412. In each of these problems, each bottom digit is less than the corresponding top digit. This may not always be the case. We will examine the case where the bottom digit is greater than the corresponding top digit in the next section. 48 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 1.6.5 Subtraction Involving Borrowing Minuend and Subtrahend It often happens in the subtraction of two whole numbers that a digit in the minuend (top number) will be less than the digit in the same position in the subtrahend (bottom number). This happens when we subtract 27 from 84. 84 −27 We do not have a name for 4 − 7. We need to rename 84 in order to continue. We'll do so as follows: Our new name for 84 is 7 tens + 14 ones. = 57 Notice that we converted 8 tens to 7 tens + 1 ten, and then we converted the 1 ten to 10 ones. We then had 14 ones and were able to perform the subtraction. Borrowing The process of borrowing (converting) is illustrated in the problems of Sample Set C. 1.6.5.1 Sample Set C Example 1.38 1. Borrow 1 ten from the 8 tens. This leaves 7 tens. 2. Convert the 1 ten to 10 ones. 3. Add 10 ones to 4 ones to get 14 ones. Example 1.39 1. Borrow 1 hundred from the 6 hundreds. This leaves 5 hundreds. 49 2. Convert the 1 hundred to 10 tens. 3. Add 10 tens to 7 tens to get 17 tens. 1.6.5.2 Practice Set C Perform the following subtractions. Show the expanded form for the rst three problems. Exercise 1.241 (Solution on p. 83.) 53 −35 Exercise 1.242 (Solution on p. 84.) 76 −28 Exercise 1.243 (Solution on p. 84.) 872 −565 Exercise 1.244 (Solution on p. 84.) 441 −356 Exercise 1.245 (Solution on p. 84.) 775 − 66 Exercise 1.246 (Solution on p. 84.) 5, 663 −2, 559 Borrowing More Than Once Sometimes it is necessary to borrow more than once. This is shown in the problems in Section 1.6.5.3 (Sample Set D). 1.6.5.3 Sample Set D Perform the Subtractions. Borrowing more than once if necessary Example 1.40 1. Borrow 1 ten from the 4 tens. This leaves 3 tens. 2. Convert the 1 ten to 10 ones. 3. Add 10 ones to 1 one to get 11 ones. We can now perform 11 − 8. 4. Borrow 1 hundred from the 6 hundreds. This leaves 5 hundreds. 50 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 5. Convert the 1 hundred to 10 tens. 6. Add 10 tens to 3 tens to get 13 tens. 7. Now 13 − 5 = 8. 8. 5 − 3 = 2. Example 1.41 1. Borrow 1 ten from the 3 tens. This leaves 2 tens. 2. Convert the 1 ten to 10 ones. 3. Add 10 ones to 4 ones to get 14 ones. We can now perform 14 − 5. 4. Borrow 1 hundred from the 5 hundreds. This leaves 4 hundreds. 5. Convert the 1 hundred to 10 tens. 6. Add 10 tens to 2 tens to get 12 tens. We can now perform 12 − 8 = 4. 7. Finally, 4 − 0 = 4. Example 1.42 71529 - 6952 After borrowing, we have 1.6.5.4 Practice Set D Perform the following subtractions. Exercise 1.247 (Solution on p. 84.) 526 −358 Exercise 1.248 (Solution on p. 84.) 63, 419 − 7, 779 Exercise 1.249 (Solution on p. 84.) 4, 312 −3, 123 51 1.6.6 Borrowing from Zero It often happens in a subtraction problem that we have to borrow from one or more zeros. This occurs in problems such as 503 1. − 37 and 5000 2. − 37 We'll examine each case. Example 1.43: Borrowing from a single zero. 503 Consider the problem − 37 Since we do not have a name for 3 − 7, we must borrow from 0. Since there are no tens to borrow, we must borrow 1 hundred. One hundred = 10 tens. We can now borrow 1 ten from 10 tens (leaving 9 tens). One ten = 10 ones and 10 ones + 3 ones = 13 ones. Now we can suggest the following method for borrowing from a single zero. Borrowing from a Single Zero To borrow from a single zero, 1. Decrease the digit to the immediate left of zero by one. 2. Draw a line through the zero and make it a 10. 3. Proceed to subtract as usual. 1.6.6.1 Sample Set E Example 1.44 Perform this subtraction. 503 − 37 The number 503 contains a single zero 52 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 1. The number to the immediate left of 0 is 5. Decrease 5 by 1. 5−1=4 2. Draw a line through the zero and make it a 10. 3. Borrow from the 10 and proceed. 1 ten + 10 ones 10 ones + 3 ones = 13 ones 1.6.6.2 Practice Set E Perform each subtraction. Exercise 1.250 (Solution on p. 84.) 906 − 18 Exercise 1.251 (Solution on p. 84.) 5102 − 559 Exercise 1.252 (Solution on p. 85.) 9055 − 386 Example 1.45: Borrowing from a group of zeros 5000 Consider the problem − 37 In this case, we have a group of zeros. Since we cannot borrow any tens or hundreds, we must borrow 1 thousand. One thousand = 10 hundreds. We can now borrow 1 hundred from 10 hundreds. One hundred = 10 tens. 53 We can now borrow 1 ten from 10 tens. One ten = 10 ones. From observations made in this procedure we can suggest the following method for borrowing from a group of zeros. Borrowing from a Group of zeros To borrow from a group of zeros, 1. Decrease the digit to the immediate left of the group of zeros by one. 2. Draw a line through each zero in the group and make it a 9, except the rightmost zero, make it 10. 3. Proceed to subtract as usual. 1.6.6.3 Sample Set F Perform each subtraction. Example 1.46 40, 000 − 125 The number 40,000 contains a group of zeros. 1. The number to the immediate left of the group is 4. Decrease 4 by 1. 4−1=3 2. Make each 0, except the rightmost one, 9. Make the rightmost 0 a 10. 3. Subtract as usual. Example 1.47 8, 000, 006 − 41, 107 The number 8,000,006 contains a group of zeros. 1. The number to the immediate left of the group is 8. Decrease 8 by 1. 8 − 1 = 7 2. Make each zero, except the rightmost one, 9. Make the rightmost 0 a 10. 54 CHAPTER 1. ADDITION AND SUBTRACTION OF WHOLE NUMBERS 3. To perform the subtraction, we'll need to borrow from the ten. 1 ten = 10 ones 10 ones + 6 ones = 16 ones 1.6.6.4 Practice Set F Perform each subtraction. Exercise 1.253 (Solution on p. 85.) 21, 007 − 4, 873 Exercise 1.254 (Solution on p. 85.) 10, 004 − 5, 165 Exercise 1.255 (Solution on p. 85.) 16, 000, 000 − 201, 060 1.6.7 Calculators In practice, calculators are used to nd the dierence between two whole numbers. 1.6.7.1 Sample Set G Find the dierence between 1006 and 284. Display Reads Type 1006 1006 Press − 1006 Type 284 284 Press = 722 Table 1.26 55 The dierence between 1006 and 284 is 722. (What happens if you type 284 rst and then 1006? We'll study such numbers in Chapter 10.) 1.6.7.2 Practice Set G Exercise 1.256 (Solution on p. 85.) Use a calculator to nd the dierence between 733
