1. Preface 2. Acknowledgements 3. Addition and Subtraction of Whole Numbers 1. Objectives . Whole Numbers . Reading and Writing Whole Numbers . Rounding Whole Numbers . Addition of Whole Numbers . Subtraction of Whole Numbers . Properties of Addition . Summary of Key Concepts . Exercise Supplement 10. Proficiency Exam 4. Multiplication and Division of Whole Numbers 1. Objectives 2. Multiplication of Whole Numbers 3. Concepts of Division of Whole Numbers 4. Division of Whole Numbers 5. Some Interesting Facts about Division 6 7. 8

WON DU BW WN

. Properties of Multiplication . Summary of Key Concepts . Exercise Supplement 9. Proficiency Exam 5. Exponents, Roots, and Factorization of Whole Numbers 1. Objectives . Exponents and Roots . Prime Factorization of Natural Numbers . The Greatest Common Factor . The Least Common Multiple . Summary of Key Concepts

NOD OB WN

8. Exercise Supplement 9. Proficiency Exam 6. Introduction to Fractions and Multiplication and Division of Fractions 1. Objectives . Fractions of Whole Numbers . Equivalent Fractions, Reducing Fractions to Lowest Terms, and Raising Fractions to Higher Terms 5. Multiplication of Fractions 6. Division of Fractions 7. Applications Involving Fractions 8. Summary of Key Concepts 9. Exercise Supplement 10. Proficiency Exam 7. Addition and Subtraction of Fractions, Comparing Fractions, and Complex Fractions 1. Objectives 2. Addition and Subtraction of Fractions with Like Denominators 3. Addition and Subtraction of Fractions with Unlike Denominators . Addition and Subtraction of Mixed Numbers . Comparing Fractions . Complex Fractions . Combinations of Operations with Fractions . Summary of Key Concepts . Exercise Supplement 10. Proficiency Exam 8. Decimals 1. Objectives 2. Reading and Writing Decimals

WON MD Uf

. Converting a Decimal to a Fraction

. Rounding Decimals

. Addition and Subtraction of Decimals

. Multiplication of Decimals

. Division of Decimals

. Nonterminating Divisions

. Converting a Fraction to a Decimal

. Combinations of Operations with Decimals and Fractions . Summary of Key Concepts

. Exercise Supplement

13.

Proficiency Exam

9. Ratios and Rates

. Objectives

. Ratios and Rates

. Proportions

. Applications of Proportions . Percent

. Fractions of One Percent

. Applications of Percents

. Summary of Key Concepts . Exercise Supplement

10.

Proficiency Exam

10. Techniques of Estimation

1. . Estimation by Rounding

. Estimation by Clustering

. Mental Arithmetic-Using the Distributive Property . Estimation by Rounding Fractions

. Summary of Key Concepts

. Exercise Supplement

8.

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Objectives

Proficiency Exam

11. Measurement and Geometry

CON DU BWN FE

. Objectives

. Measurement and the United States System

. The Metric System of Measurement

. Simplification of Denominate Numbers

. Perimeter and Circumference of Geometric Figures . Area and Volume of Geometric Figures and Objects . Summary of Key Concepts

. Exercise Supplement

9.

Proficiency Exam

12. Signed Numbers

., . Variables, Constants, and Real Numbers

. Signed Numbers

. Absolute Value

. Addition of Signed Numbers

. Subtraction of Signed Numbers

. Multiplication and Division of Signed Numbers . Summary of Key Concepts

. Exercise Supplement

10.

WON ADU BW WN

Objectives

Proficiency Exam

13. Algebraic Expressions and Equations

. Objectives

. Algebraic Expressions

. Combining Like Terms Using Addition and Subtraction . Solving Equations of the Form x+a=b and x-a=b

. Solving Equations of the Form ax=b and x/a=b

. Applications I: Translating Words to Mathematical

Symbols

. Applications Il: Solving Problems . Summary of Key Concepts

. Exercise Supplement

. Proficiency Exam

Preface This module contains the preface for Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr.

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 [link], 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:

e Section Overview

Â¢ Sample Sets

e Practice Sets

e Section Exercises

e Exercises for Review

e Answers to Practice Sets

The chapters begin with Objectives and end with a Summary _of Key. Concepts, an Exercise Supplement, and a Proficiency 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 off 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 mathematical 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 difficulty, 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 difficulty 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 five 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 Proficiency 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.

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.

Proficiency Exam

Each chapter ends with a Proficiency Exam that can serve as a chapter review or evaluation. The Proficiency 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, offering a straightforward approach to prealgebra mathematics. We have made a deliberate effort not to write another text that minimizes 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 remembered. 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 8a 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.

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 different 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 defining 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 first 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 simplification 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 figures and area and volume of geometric figures 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. Definitions 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 definition 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 first using the number line, then with absolute value.

Algebraic Expressions and Equations

The student is introduced to some elementary algebraic concepts and techniques in this final chapter. Algebraic expressions and the process of combining like terms are discussed in [link] and [link]. 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 [link]).

Acknowledgements This module contains the authors' acknowledgments and dedication of the book, Fundamentals of Mathematics by Denny Burzynski and Wade Ellis.

Many extraordinarily talented people are responsible for helping to create this text. We wish to acknowledge the efforts and skill of the following mathematicians. Their contributions have been invaluable.

e Barbara Conway, Berkshire Community College

e Bill Hajdukiewicz, Miami-Dade Community College e Virginia Hamilton, Shawnee State University

e David Hares, El Centro College

e Norman Lee, Ball State University

e Ginger Y. Manchester, Hinds Junior College

e John R. Martin, Tarrant County Junior College

e Shelba Mormon, Northlake College

e Lou Ann Pate, Pima Community College

e Gus Pekara, Oklahoma City Community College

e David Price, Tarrant County Junior College

Â¢ David Schultz, Virginia Western Community College e Sue S. Watkins, Lorain County Community College e Elizabeth M. Wayt, Tennessee State University

e Prentice E. Whitlock, Jersey City State College

Â¢ 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 Fundamentals 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); Patricia 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 efforts 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 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 efforts 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 Raffety, who long ago in Sequoia National Forest told me what a differential 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.

Objectives

This module contains the learning objectives for the chapter "Addition and Subtraction of Whole Numbers" from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, jr.

After completing this chapter, you should Whole Numbers ({link])

e know the difference between numbers and numerals

e know why our number system is called the Hindu-Arabic numeration system

e understand the base ten positional number system

e be able to identify and graph whole numbers

Reading and Writing Whole Numbers ([{link]) e be able to read and write a whole number Rounding Whole Numbers ((link])

Â¢ understand that rounding is a method of approximation e be able to round a whole number to a specified position

Addition of Whole Numbers ({link])

e understand the addition process e be able to add whole numbers e be able to use the calculator to add one whole number to another

Subtraction of Whole Numbers ({link])

e understand the subtraction process

e be able to subtract whole numbers

e be able to use a calculator to subtract one whole number from another whole number

Properties of Addition ({link])

e understand the commutative and associative properties of addition

Â¢ understand why 0 is the additive identity

Whole Numbers

This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses many of aspects of whole numbers, including the Hindu-Arabic numeration system, the base ten positional number system, and the graphing of whole numbers. By the end of this module students should be able to: know the difference between numbers and numerals, know why our number system is called the Hindu- Arabic numeration system, understand the base ten positional number system, and identify and graph whole numbers.

Section Overview

e Numbers and Numerals

e The Hindu-Arabic Numeration System

e The Base Ten Positional Number System Â¢ Whole Numbers

Â¢ Graphing Whole Numbers

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.

Sample Set A

The following are numerals. In each case, the first represents the number four, the second represents the number one hundred twenty-three, and the third, the number one thousand five. These numbers are represented in different ways.

e Hindu-Arabic numerals 4, 123, 1005

e Roman numerals IV, CX XIII, MV

e Egyptian numerals

li dd> ONNI11> biti

Strokes Coiled rope, Lotus flower heel bones, and strokes and strokes

Practice Set A Exercise:

Problem:

Wo

Do the phrases "four," "one hundred twenty-three," and "one thousand five" qualify as numerals? Yes or no?

Solution:

Yes. Letters are symbols. Taken as a collection (a written word), they represent a number.

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.

The Base Ten Positional Number System

Digits

The Hindu-Arabic numerals 01234567 89 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.

ae | eee | ee ee) ee | ee ee eee ee Oe | Trillions Billions Millions Thousands Units

As we continue from right to left, there are more periods. The five periods listed above are the most common, and in our study of introductory mathematics, they are sufficient.

The following diagram illustrates our positional number system to trillions. (There are, to be sure, other periods.)

In our positional number system, the value of a digit is determined by its position in the number.

Sample Set B

Example:

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:

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:

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

Practice Set B Exercise: Problem: Find the value of 5 in the number 65,000.

Solution:

five thousand

Exercise:

Problem: Find the value of 4 in the number 439,997,007,010.

Solution:

four hundred billion

Exercise:

Problem: Find the value of 0 in the number 108. Solution:

zero tens, Or Zero

Whole Numbers

Whole Numbers

Numbers that are formed using only the digits 0123456789

are called whole numbers. They are

On 4, 25354) 5,G,-750 79, 10.11. 120138. 14. Toya

The three dots at the end mean "and so on in this same pattern."

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 off 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."

Sample Set C

Example: Graph the following whole numbers: 3, 5, 9.

Example:

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.

0 106 107 872 873 874

The numbers that have been graphed are 0, 106, 873, 874

Practice Set C

Exercise:

Problem: Graph the following whole numbers: 46, 47, 48, 325, 327.

â€œ2 â€œ"/ ps â€”-

Solution: 0 46 47 48 325 326 327 Exercise: Problem:

Specify the whole numbers that are graphed on the following number line.

0 123 4 5 6 112 113 978 979 Solution: 4,5, 6, 113, 978

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.

Exercises

Exercise:

Problem: What is a number? Solution: concept

Exercise:

Problem: What is a numeral?

Exercise:

Problem: Does the word "eleven" qualify as a numeral?

Solution:

Yes, since it is a symbol that represents a number.

Exercise:

Problem: How many different digits are there? Exercise: Problem:

Our number system, the Hindu-Arabic number system, is a number system with base .

Solution:

positional; 10 Exercise: Problem:

Numbers composed of more than three digits are sometimes separated into groups of three by commas. These groups of three are called .

Exercise:

Problem:

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?

Solution:

ones, tens, hundreds Exercise:

Problem:

Each period has its own particular name. From right to left, what are the names of the first four?

Exercise:

Problem: In the number 841, how many tens are there?

Solution: 4

Exercise:

Problem: In the number 3,392, how many ones are there?

Exercise:

Problem: In the number 10,046, how many thousands are there?

Solution:

0

Exercise:

Problem:

In the number 779,844,205, how many ten millions are there? Exercise:

Problem: In the number 65,021, how many hundred thousands are there? Solution:

0

For following problems, give the value of the indicated digit in the given number. Exercise:

Problem: 5 in 599

Exercise:

Problem: 1 in 310,406

Solution:

ten thousand

Exercise:

Problem: 9 in 29,827

Exercise:

Problem: 6 in 52,561,001,100 Solution:

6 ten millions = 60 million

Exercise:

Problem

Write a two-digit number that has an eight in the tens position.

Exercise:

Problem

Write a four-digit number that has a one in the thousands position and

a zero in

the ones position.

Solution:

1,340 (answers may vary)

Exercise:

Problem

Exercise:

Problem

: How many two-digit whole numbers are there?

: How many three-digit whole numbers are there?

Solution:

900

Exercise:

Problem

Exercise:

Problem

: How many four-digit whole numbers are there?

: Is there a smallest whole number? If so, what is it?

Solution:

yes; Zero

Exercise:

Problem: Is there a largest whole number? If so, what is it?

Exercise:

Problem: Another term for "visually displaying" is .

Solution:

graphing

Exercise:

Problem: The whole numbers can be visually displayed ona. Exercise:

Problem:

Graph (visually display) the following whole numbers on the number line below: 0, 1, 31, 34.

0 12 8 4 #â€˜\%Y 29 30 31 32 33 34

Solution:

0 1 2 83 4 2 30 31 32 33 34

Exercise:

Problem:

Construct a number line in the space provided below and graph

(visually display) the following whole numbers: 84, 85, 901, 1006, 1007.

Exercise:

Problem:

Specify, if any, the whole numbers that are graphed on the following number line.

tJ tt MN ot

0 61 62 63 64 99 100 101 102

Solution:

61, 99, 100, 102 Exercise: Problem:

Specify, if any, the whole numbers that are graphed on the following number line.

+ â€”+-A\â€” ++ + + â€œ| -â€” + 4+ A +

01 8 9 10 11 73 74 85 86 87

Reading and Writing Whole Numbers

This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to read and write whole numbers. By the end of this module, students should be able to read and write whole numbers.

Section Overview

e Reading Whole Numbers e Writing Whole Numbers

Because our number system is a positional number system, reading and writing whole numbers is quite simple.

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.

Sample Set A

Write the following numbers as words.

Example: 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:

Lowa alt es , â€”â€” Forty-two thousand

Le cceaclnciceninsimeniianl Thousands period

[a 5) 8, â€”â€” nine hundred fifty-eight

Forty-two thousand, nine hundred fifty-eight.

Example: 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.

| â€˜aes L_jLojJt 1, â€”â€”Â» Three hundred seven million, eel

Millions period

bat ear hat Â» â€”â€” nine hundred ninety-one thousand, Li eenacnemainadianl Thousands period

3, el 3, â€”â€” three hundred forty-three Lf Units period

Three hundred seven million, nine hundred ninety-one thousand, three hundred forty-three.

Example: Read 36000000000001.

1. Beginning at the right, we can separate this number into distinct periods by placing commas. 36,000,000,001 2. Beginning at the left, we read each period individually.

a | 3 8, , â€”â€” Thirty-six trillion, LS Trillions period 0D: @ 6 aon L_jLUJL_J,â€”â€” zero billion, Le Billions period 0,0,,0 _ LJLLjLOJ,â€”- zero million, LS Millions period 0 ,0,,0 L-JL JL J,-â€”â€” zero thousand, â€”â€”â€”â€”â€”: | Thousands period "al el | ek LE Units period

Thirty-six trillion, one.

Practice Set A

Write each number in words. Exercise:

Problem: 12,542

Solution: Twelve thousand, five hundred forty-two Exercise:

Problem: 101,074,003

Solution: One hundred one million, seventy-four thousand, three Exercise:

Problem: 1,000,008

Solution:

One million, eight

Writing Whole Numbers

To express a number in digits that is expressed in words, use the following method:

1. Notice first that a number expressed as a verbal phrase will have its periods set off 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.

Sample Set B

Write each number using digits.

Example: Seven thousand, ninety-two. Using the comma as a period separator, we have

Seven thousand , â€”â€” 7, nearereete eae

ninety-two, â€”â€”> 092

7,092

Example: Fifty billion, one million, two hundred thousand, fourteen. Using the commas as period separators, we have

Fifty billion,, â€”â€” 50, one million ,â€”â€” 001,

two hundred thousand , â€”â€” 200,

fourteen, â€”â€” 014

50,001,200,014

Example: Ten million, five hundred twelve. The comma sets off the periods. We notice that there is no thousands

period. We'll have to insert this ourselves.

Ten million ,â€”â€” 10,

zero thousand , â€”â€” 000,

five hundred twelve, â€”â€” 512 ee

10,000,512

Practice Set B

Express each number using digits. Exercise:

Problem: One hundred three thousand, twenty-five.

Solution:

103,025

Exercise: Problem: Six million, forty thousand, seven.

Solution:

6,040,007

Exercise:

Problem:

Twenty trillion, three billion, eighty million, one hundred nine thousand, four hundred two.

Solution:

20,003,080, 109,402

Exercise:

Problem: Eighty billion, thirty-five. Solution:

80,000,000,035

Exercises

For the following problems, write all numbers in words. Exercise:

Problem: 912 Solution:

nine hundred twelve

Exercise:

Problem: 84

Exercise:

Problem: 1491

Solution:

one thousand, four hundred ninety-one

Exercise:

Problem: 8601

Exercise:

Problem: 35,223

Solution:

thirty-five thousand, two hundred twenty-three

Exercise:

Problem: 71,006

Exercise:

Problem: 437,105

Solution:

four hundred thirty-seven thousand, one hundred five

Exercise:

Problem: 201,040

Exercise:

Problem: 8,001,001 Solution:

eight million, one thousand, one

Exercise:

Problem: 16,000,053 Exercise: Problem: 770,311,101

Solution:

seven hundred seventy million, three hundred eleven thousand, one hundred one

Exercise:

Problem: 83,000,000,007

Exercise:

Problem: 106,100,001,010 Solution:

one hundred six billion, one hundred million, one thousand ten

Exercise:

Problem: 3,333,444,777 Exercise:

Problem: 800,000,800,000

Solution:

eight hundred billion, eight hundred thousand

Exercise:

Problem:

A particular community college has 12,471 students enrolled. Exercise:

Problem:

A person who watches 4 hours of television a day spends 1460 hours a

year watching T.V. Solution:

four; one thousand, four hundred sixty Exercise: Problem: Astronomers believe that the age of the earth is about 4,500,000,000 years. Exercise: Problem:

Astronomers believe that the age of the universe is about 20,000,000,000 years.

Solution:

twenty billion Exercise:

Problem:

There are 9690 ways to choose four objects from a collection of 20.

Exercise:

Problem:

If a 412 page book has about 52 sentences per page, it will contain about 21,424 sentences.

Solution: four hundred twelve; fifty-two; twenty-one thousand, four hundred twenty-four Exercise: Problem: In 1980, in the United States, there was $1,761,000,000,000 invested in life insurance. Exercise: Problem:

In 1979, there were 85,000 telephones in Alaska and 2,905,000 telephones in Indiana.

Solution: one thousand, nine hundred seventy-nine; eighty-five thousand; two million, nine hundred five thousand Exercise: Problem: In 1975, in the United States, it is estimated that 52,294,000 people drove to work alone. Exercise: Problem:

In 1980, there were 217 prisoners under death sentence that were divorced.

Solution:

one thousand, nine hundred eighty; two hundred seventeen Exercise: Problem: In 1979, the amount of money spent in the United States for regular- session college education was $50,721,000,000,000. Exercise: Problem:

In 1981, there were 1,956,000 students majoring in business in U.S. colleges.

Solution: one thousand, nine hundred eighty one; one million, nine hundred fifty-six thousand Exercise: Problem: In 1980, the average fee for initial and follow up visits to a medical doctors office was about $34. Exercise: Problem:

In 1980, there were approximately 13,100 smugglers of aliens apprehended by the Immigration border patrol.

Solution:

one thousand, nine hundred eighty; thirteen thousand, one hundred

Exercise:

Problem:

In 1980, the state of West Virginia pumped 2,000,000 barrels of crude oil, whereas Texas pumped 975,000,000 barrels.

Exercise:

Problem: The 1981 population of Uganda was 12,630,000 people.

Solution:

twelve million, six hundred thirty thousand Exercise:

Problem:

In 1981, the average monthly salary offered to a person with a Master's degree in mathematics was $1,685.

For the following problems, write each number using digits. Exercise:

Problem: Six hundred eighty-one

Solution:

681

Exercise:

Problem: Four hundred ninety

Exercise:

Problem: Seven thousand, two hundred one

Solution:

F201

Exercise:

Problem: Nineteen thousand, sixty-five

Exercise:

Problem: Five hundred twelve thousand, three

Solution:

512,003 Exercise:

Problem:

Two million, one hundred thirty-three thousand, eight hundred fifty- nine

Exercise: Problem: Thirty-five million, seven thousand, one hundred one

Solution:

35,007,101

Exercise:

Problem: One hundred million, one thousand

Exercise:

Problem: Sixteen billion, fifty-nine thousand, four

Solution:

16,000,059,004

Exercise:

Problem:

Nine hundred twenty billion, four hundred seventeen million, twenty- one thousand

Exercise: Problem: Twenty-three billion

Solution:

23,000,000,000 Exercise:

Problem:

Fifteen trillion, four billion, nineteen thousand, three hundred five

Exercise:

Problem: One hundred trillion, one

Solution:

100,000,000,000,001

Exercises for Review

Exercise:

Problem: ({link]) How many digits are there?

Exercise:

Problem: ({link]) In the number 6,641, how many tens are there?

Solution: 4

Exercise:

Problem: ({link]) What is the value of 7 in 44,763?

Exercise: Problem: ({link]) Is there a smallest whole number? If so, what is it?

Solution:

yes, Zero Exercise:

Problem:

({link]) Write a four-digit number with