Chapter 1 The Whole Numbers

Tips for Success in Mathematics 1.1 Tips for Success in Mathematics

Getting Ready for This Course Positive Attitude Believe you can succeed. Scheduling Make sure you have time for your classes. Be Prepared Have all the materials you need, like a lab manual, calculator, or other supplies. Objective A

General Tips for Success Details Get a contact person. Exchange names, phone numbers or e-mail addresses with at least one other person in class. Attend all class periods. Sit near the front of the classroom to make hearing the presentation, and participating easier. Do your homework. The more time you spend solving mathematics, the easier the process becomes. Check your work. Review your steps, fix errors, and compare answers with the selected answers in the back of the book. Learn from your mistakes. Find and understand your errors. Use them to become a better math student. Objective B Continued

General Tips for Success Details Get help if you need it. Ask for help when you don’t understand something. Know when your instructor’s office hours are, and whether tutoring services are available. Organize class materials. Organize your assignments, quizzes, tests, and notes for use as reference material throughout your course. Read your textbook. Review your section before class to help you understand its ideas more clearly. Ask questions. Speak up when you have a question. Other students may have the same one. Hand in assignments on time. Don’t lose points for being late. Show every step of a problem on your assignment. Objective B Continued

Using This Text Resource Details Continued Practice Problems. Try each Practice Problem after you’ve finished its corresponding example. Chapter Test Prep Video CD. Chapter Test exercises are worked out by the author, these are available off of the CD this book contains. Lecture Video CDs. Exercises marked with a CD symbol are worked out by the author on a video CD. Check with your instructor to see if these are available. Symbols before an exercise set. Symbols listed at the beginning of each exercise set will remind you of the available supplements. Objectives. The main section of exercises in an exercise set is referenced by an objective. Use these if you are having trouble with an assigned problem. Objective C Continued

Using This Text Resource Details Icons (Symbols). A CD symbol tells you the corresponding exercise may be viewed on a video segment. A pencil symbol means you should answer using complete sentences. Integrated Reviews. Reviews found in the middle of each chapter can be used to practice the previously learned concepts. End of Chapter Opportunities. Use Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews to help you understand chapter concepts. Study Skills Builder. Read and answer questions in the Study Skills Builder to increase your chance of success in this course. The Bigger Picture. This can help you make the transition from thinking “section by section” to thinking about how everything corresponds in the bigger picture. Objective C Continued

Get help as soon as you need it. Getting Help Tip Details Get help as soon as you need it. Material presented in one section builds on your understanding of the previous section. If you don’t understand a concept covered during a class period, there is a good chance you won’t understand the concepts covered in the next period. For help, try your instructor, a tutoring center, or a math lab. A study group can also help increase your understanding of covered materials. Objective D

Preparing for and Taking an Exam Steps for Preparing for a Test Review previous homework assignments. Review notes from class and section-level quizzes you have taken. Read the Highlights at the end of each chapter to review concepts and definitions. Complete the Chapter Review at the end of each chapter to practice the exercises. Take a sample test in conditions similar to your test conditions. Set aside plenty of time to arrive where you will be taking the exam. Objective E Continued

Preparing for and Taking an Exam Steps for Taking Your Test Read the directions on the test carefully. Read each problem carefully to make sure that you answer the question asked. Pace yourself so that you have enough time to attempt each problem on the test. Use extra time checking your work and answers. Don’t turn in your test early. Use extra time to double check your work. Objective E Continued

Tips for Making a Schedule Managing Your Time Tips for Making a Schedule Make a list of all of your weekly commitments for the term. Estimate the time needed and how often it will be performed, for each item. Block out a typical week on a schedule grid, start with items with fixed time slots. Next, fill in items with flexible time slots. Remember to leave time for eating, sleeping, and relaxing. Make changes to your workload, classload, or other areas to fit your needs. Objective F

Place Value, Names for Numbers, and Reading Tables 1.2 Place Value, Names for Numbers, and Reading Tables

Place Value The position of each digit in a number determines its place value. 3 5 6 8 9 4 0 2 Ones Hundred-thousands Hundred-billions Ten-billions Billions Hundred-millions Ten-millions Millions Ten-thousands Thousands Hundreds Tens

Writing a Number in Words A whole number such as 35,689,402 is written in standard form. The columns separate the digits into groups of threes. Each group of three digits is a period. Millions Thousands Billions Ones 3 5 6 8 9 4 0 2 Hundred-thousands Hundred-billions Ten-billions Hundred-millions Ten-millions Ten-thousands Hundreds Tens

Writing a Number in Words To write a whole number in words, write the number in each period followed by the name of the period. 3 5 6 8 9 4 0 2 Ones Hundred-thousands Hundred-billions Ten-billions Billions Hundred-millions Ten-millions Millions Ten-thousands Thousands Hundreds Tens thirty-five million, six hundred eighty-nine thousand, four hundred two

Helpful Hint The name of the ones period is not used when reading and writing whole numbers. Also, the word “and” is not used when reading and writing whole numbers. It is used when reading and writing mixed numbers and some decimal values as shown later.

Chapter 1 / Whole Numbers and Introduction to Algebra Expanded Form Standard Form Expanded Form 4,786 = 4000 + 700 + 80 + 6 The place value of a digit can be used to write a number in expanded form. The expanded form of a number shows each digit of the number with its place value.

Comparing Whole Numbers Chapter 1 / Whole Numbers and Introduction to Algebra Comparing Whole Numbers We can picture whole numbers as equally spaced points on a line called the number line. 1 2 3 4 5 A whole number is graphed by placing a dot on the number line. The graph of 4 is shown.

Comparing Numbers For any two numbers graphed on a number line, the number to the right is the greater number, and the number to the left is the smaller number. 5 4 1 2 3 2 is to the left of 5, so 2 is less than 5 5 is to the right of 2, so 5 is greater than 2

Chapter 1 / Whole Numbers and Introduction to Algebra Comparing Numbers . . . 2 is less than 5 can be written in symbols as 2 < 5 5 is greater than 2 is written as 5 > 2

are both true statements. Helpful Hint One way to remember the meaning of the inequality symbols < and > is to think of them as arrowheads “pointing” toward the smaller number. For example, 2 < 5 and 5 > 2 are both true statements.

Reading Tables Most Medals Olympic Winter (1924 – 2002) Games Gold Silver Bronze Total 107 104 86 297 113 83 78 274 94 92 74 260 69 71 51 191 41 57 64 162 Germany Russia Norway USA Austria Source: The Sydney Morning Herald, Flags courtesy of www.theodora.com/flags used with permission

Adding and Subtracting Whole Numbers, and Perimeter 1.3 Adding and Subtracting Whole Numbers, and Perimeter

The sum of 0 and any number is that number. Addition Property of 0 The sum of 0 and any number is that number. 8 + 0 = 8 and 0 + 8 = 8

Commutative Property of Addition Changing the order of two addends does not change their sum. 4 + 2 = 6 and 2 + 4 = 6

Associative Property of Addition Changing the grouping of addends does not change their sum. 3 + (4 + 2) = 3 + 6 = 9 and (3 + 4) + 2 = 7 + 2 = 9

Subtraction Properties of 0 Chapter 1 / Whole Numbers and Introduction to Algebra Subtraction Properties of 0 The difference of any number and that same number is 0. 9 – 9 = 0 The difference of any number and 0 is the same number. 7 – 0 = 7

Chapter 1 / Whole Numbers and Introduction to Algebra Polygons A polygon is a flat figure formed by line segments connected at their ends. Geometric figures such as triangles, squares, and rectangles are called polygons. triangle square rectangle

Perimeter The perimeter of a polygon is the distance around the polygon.

Addition Problems Descriptions of problems solved through addition may include any of these key words or phrases: Key Words Examples Symbols added to 3 added to 9 3 + 9 plus 5 plus 22 5 + 22 more than 7 more than 8 7 + 8 total total of 6 and 5 6 + 5 increased by 16 increased by 7 16 + 7 sum sum of 50 and 11 50 + 11

Chapter 1 / Whole Numbers and Introduction to Algebra Subtraction Problems Descriptions of problems solved by subtraction may include any of these key words or phrases: Key Words Examples Symbols subtract subtract 3 from 9 9 – 3 difference difference of 8 and 2 8 – 2 less 12 less 8 12 – 8 less than 2 less than 20 20 – 2 take away 14 take away 9 14 – 9 decreased by 16 decreased by 7 16 – 7 subtracted from 5 subtracted from 9 9 – 5

Helpful Hint Be careful when solving applications that suggest subtraction. Although order does not matter when adding, order does matter when subtracting. For example, 10 – 3 and 3 – 10 do not simplify to the same number.

Helpful Hint Since subtraction and addition are reverse operations, don’t forget that a subtraction problem can be checked by adding.

Number of Endangered Species Reading a Bar Graph The graph shows the number of endangered species in each country. Number of Endangered Species 146 89 83 73 Country 72 64 Source: The Top 10 of Everything, Russell Ash.

Rounding and Estimating 1.4 Rounding and Estimating

Rounding 20 30 23 23 rounded to the nearest ten is 20. 40 50 48 10 20 15 15 rounded to the nearest ten is 20.

Rounding Whole Numbers Chapter 1 / Whole Numbers and Introduction to Algebra Rounding Whole Numbers Step 1: Locate the digit to the right of the given place value. Step 2: If this digit is 5 or greater, add 1 to the digit in the given place value and replace each digit to its right by 0. Step 3: If this digit is less than 5, replace it and each digit to its right by 0.

Chapter 1 / Whole Numbers and Introduction to Algebra Estimates Making estimates is often the quickest way to solve real-life problems when their solutions do not need to be exact.

Helpful Hint Estimation is useful to check for incorrect answers when using a calculator. For example, pressing a key too hard may result in a double digit, while pressing a key too softly may result in the number not appearing in the display.

Multiplying Whole Numbers and Area 1.5 Multiplying Whole Numbers and Area

Multiplication Multiplication is repeated addition with a different notation. 4 + 4 + 4 + 4 + 4 = 5 ∙ 4 = 20 5 fours factor product

Multiplication Property of 0 Chapter 1 / Whole Numbers and Introduction to Algebra Multiplication Property of 0 The product of 0 and any number is 0. 9  0 = 0 0  6 = 0

Multiplication Property of 1 Chapter 1 / Whole Numbers and Introduction to Algebra Multiplication Property of 1 The product of 1 and any number is that same number. 9  1 = 9 1  6 = 6

Commutative Property of Multiplication Chapter 1 / Whole Numbers and Introduction to Algebra Commutative Property of Multiplication Changing the order of two factors does not change their product. 6 • 3 = 18 and 3 • 6 = 18

Associative Property of Multiplication Chapter 1 / Whole Numbers and Introduction to Algebra Associative Property of Multiplication Changing the grouping of factors does not change their product. 5 • ( 2 • 3) = 5 • 6 = 30 and (5 • 2) • 3 = 10 • 3 = 30

Distributive Property Chapter 1 / Whole Numbers and Introduction to Algebra Distributive Property Multiplication distributes over addition. 5(3 + 4) = 5 • 3 + 5 • 4

Area 1 square inch 1 5 inches 3 inches Area of a rectangle = length  width = (5 inches)(3 inches) = 15 square inches

Helpful Hint Remember that perimeter (distance around a plane figure) is measured in units. Area (space enclosed by a plane figure) is measured in square units. 5 inches 4 inches Rectangle 5 inches + 4 inches + 5 inches + 4 inches = 18 inches Perimeter = Area = (5 inches)(4 inches) = 20 square inches

Chapter 1 / Whole Numbers and Introduction to Algebra Multiplication Words There are several words or phrases that indicate the operation of multiplication. Some of these are as follows: Key Words Examples Symbols multiply multiply 4 by 3 4 • 3 product the product of 2 and 5 2 • 5 times 7 times 6 7 • 6

Dividing Whole Numbers 1.6 Dividing Whole Numbers

Division quotient divisor dividend The process of separating a quantity into equal parts is called division. quotient dividend divisor

Division Properties of 1 The quotient of any number, except 0, and that same number is 1. 6 1 5 7 = ÷

Division Properties of 1 The quotient of any number and 1 is that same number. 6 1 5 7 = ÷

Division Properties of 0 The quotient of 0 and any number (except 0) is 0. 6 7 = ÷ 5

Division Properties of 0 The quotient of any number and 0 is not a number. We say that 6 5 7 ÷ are undefined.

Helpful Hint Since division and multiplication are reverse operations, don’t forget that a division problem can be checked by multiplying.

Chapter 1 / Whole Numbers and Introduction to Algebra Division Words Here are some key words and phrases that indicate the operation of division. Key Words Examples Symbols divide divide 15 by 3 15  3 quotient quotient of 12 and 6 divided by 8 divided by 4 divided or shared equally $20 divided equally among five people 20  5

Chapter 1 / Whole Numbers and Introduction to Algebra Average How do you find an average? A student’s prealgebra grades at the end of the semester are: 90, 85, 95, 70, 80, 100, 98, 82, 90, 90. How do you find his average? Find the sum of the scores and then divide the sum by the number of scores. Sum = 880 Average = 880 ÷ 10 = 88

Exponents and Order of Operations 1.7 Exponents and Order of Operations

An exponent is a shorthand notation for repeated multiplication. Exponents An exponent is a shorthand notation for repeated multiplication. 3 • 3 • 3 • 3 • 3 3 is a factor 5 times Using an exponent, this product can be written as exponent base

Exponential Notation exponent base Read as “three to the fifth power” or “the fifth power of three.” This is called exponential notation. The exponent, 5, indicates how many times the base, 3, is a factor. 3 • 3 • 3 • 3 • 3 3 is a factor 5 times

Chapter 1 / Whole Numbers and Introduction to Algebra Reading Exponential Notation 4 = 41 is read as “four to the first power.” 4 • 4 = 42 is read as “four to the second power” or “four squared.”

Chapter 1 / Whole Numbers and Introduction to Algebra Reading Exponential Notation 4 • 4 • 4 = 43 is read as “four to the third power” or “four cubed.” 4 • 4 • 4 • 4 = 44 is read as “four to the fourth power.”

Helpful Hint Usually, an exponent of 1 is not written, so when no exponent appears, we assume that the exponent is 1. For example, 2 = 21 and 7 = 71.

Chapter 1 / Whole Numbers and Introduction to Algebra Evaluating Exponential Expressions To evaluate an exponential expression, we write the expression as a product and then find the value of the product. 35 = 3 • 3 • 3 • 3 • 3 = 243

Helpful Hint An exponent applies only to its base. For example, 4 • 23 means 4 • 2 • 2 • 2. Don’t forget that 24, for example, is not 2 • 4. 24 means repeated multiplication of the same factor. 24 = 2 • 2 • 2 • 2 = 16, whereas 2 • 4 = 8

Chapter 1 / Whole Numbers and Introduction to Algebra Order of Operations 1. Perform all operations within parentheses ( ), brackets [ ], or other grouping symbols such as fraction bars, starting with the innermost set. 2. Evaluate any expressions with exponents. 3. Multiply or divide in order from left to right. 4. Add or subtract in order from left to right.

Introduction to Variables, Algebraic Expressions, and Equations 1.8 Introduction to Variables, Algebraic Expressions, and Equations

Algebraic Expressions Chapter 1 / Whole Numbers and Introduction to Algebra Algebraic Expressions A combination of operations on letters (variables) and numbers is called an algebraic expression. Algebraic Expressions 5 + x 6 • y 3 • y – 4 + x 4x means 4 • x and xy means x • y

Algebraic Expressions Chapter 1 / Whole Numbers and Introduction to Algebra Algebraic Expressions Replacing a variable in an expression by a number and then finding the value of the expression is called evaluating the expression for the variable.

Evaluating Algebraic Expressions Chapter 1 / Whole Numbers and Introduction to Algebra Evaluate x + y for x = 5 and y = 2. Replace x with 5 and y with 2 in x + y. x + y = ( ) + ( ) 5 2 = 7

Chapter 1 / Whole Numbers and Introduction to Algebra Equation Statements like 5 + 2 = 7 are called equations. An equation is of the form expression = expression An equation can be labeled as Equal sign x + 5 = 9 left side right side

Chapter 1 / Whole Numbers and Introduction to Algebra Solutions When an equation contains a variable, deciding which values of the variable make an equation a true statement is called solving an equation for the variable. A solution of an equation is a value for the variable that makes an equation a true statement.

Chapter 1 / Whole Numbers and Introduction to Algebra Solutions Determine whether a number is a solution: Is –2 a solution of the equation 2y + 1 = –3? Replace y with –2 in the equation. 2y + 1 = –3 ? 2(–2) + 1 = –3 ? –4 + 1 = –3 –3 = –3 True Since –3 = –3 is a true statement, –2 is a solution of the equation.

Chapter 1 / Whole Numbers and Introduction to Algebra Solutions Determine whether a number is a solution: Is 6 a solution of the equation 5x – 1 = 30? Replace x with 6 in the equation. 5x – 1 = 30 ? 5(6) – 1 = 30 ? 30 – 1 = 30 29 = 30 False Since 29 = 30 is a false statement, 6 is not a solution of the equation.

Chapter 1 / Whole Numbers and Introduction to Algebra Solutions To solve an equation, we will use properties of equality to write simpler equations, all equivalent to the original equation, until the final equation has the form x = number or number = x Equivalent equations have the same solution. The word “number” above represents the solution of the original equation.

Chapter 1 / Whole Numbers and Introduction to Algebra Keywords and Phrases Keywords and phrases suggesting addition, subtraction, multiplication, division or equals. Addition Subtraction Multiplication Division Equal Sign sum difference product quotient equals plus minus times divide gives added to subtract multiply shared equally among is/was/will be more than less than multiply by per yields increased by decreased by of divided by amounts to total less double/triple divided into is equal to

Translating Word Phrases Chapter 1 / Whole Numbers and Introduction to Algebra Translating Word Phrases the product of 5 and a number 5x twice a number 2x a number decreased by 3 n – 3 a number increased by 2 z + 2 four times a number 4w

Additional Word Phrases Chapter 1 / Whole Numbers and Introduction to Algebra Additional Word Phrases x + 7 three times the sum of a number and 7 3(x + 7) the quotient of 5 and a number the sum of a number and 7

Helpful Hint Remember that order is important when subtracting. Study the order of numbers and variables below. Phrase Translation a number decreased by 5 x – 5 subtracted from 5 5 – x