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Elementary Algebra

1.8 The Real Numbers

Elementary Algebra1.8 The Real Numbers
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

Learning Objectives

By the end of this section, you will be able to:

  • Simplify expressions with square roots
  • Identify integers, rational numbers, irrational numbers, and real numbers
  • Locate fractions on the number line
  • Locate decimals on the number line
Be Prepared 1.8

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapters, Decimals and Properties of Real Numbers.

Simplify Expressions with Square Roots

Remember that when a number n is multiplied by itself, we write n2n2 and read it “n squared.” The result is called the square of n. For example,

82read8squared’6464is called thesquareof8.82read8squared’6464is called thesquareof8.

Similarly, 121 is the square of 11, because 112112 is 121.

Square of a Number

If n2=m,n2=m, then m is the square of n.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Square Numbers” will help you develop a better understanding of perfect square numbers.

Complete the following table to show the squares of the counting numbers 1 through 15.

There is a table with two rows and 17 columns. The first row reads from left to right Number, n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second row reads from left to right Square, n squared, blank, blank, blank, blank, blank, blank, blank, 64, blank, blank, 121, blank, blank, blank, and blank.

The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.

The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

(−3)2=9(−8)2=64(−11)2=121(−15)2=225(−3)2=9(−8)2=64(−11)2=121(−15)2=225

Did you notice that these squares are the same as the squares of the positive numbers?

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 102=100,102=100, we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is mm is called a square root of m.

Square Root of a Number

If n2=m,n2=m, then n is a square root of m.

Notice (−10)2=100(−10)2=100 also, so −10−10 is also a square root of 100. Therefore, both 10 and −10−10 are square roots of 100.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, m,m, denotes the positive square root. The positive square root is called the principal square root. When we use the radical sign that always means we want the principal square root.

We also use the radical sign for the square root of zero. Because 02=0,02=0, 0=0.0=0. Notice that zero has only one square root.

Square Root Notation

mm is read “the square root of m

A square root is given, with an arrow to the radical sign (it looks like a checkmark with a horizontal line extending from its long end) denoted radical sign and an arrow to the number under the radical sign, which is marked radicand.

If m=n2,m=n2, then m=n,m=n, for n0.n0.

The square root of m, m,m, is the positive number whose square is m.

Since 10 is the principal square root of 100, we write 100=10.100=10. You may want to complete the following table to help you recognize square roots.

There is a table with two rows and 15 columns. The first row reads from left to right square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second row consists of all blanks except for the tenth cell under the square root of 100, which reads 10.

Example 1.108

Simplify: 2525 121.121.

Try It 1.215

Simplify: 3636 169.169.

Try It 1.216

Simplify: 1616 196.196.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 100=10.100=10. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 100=−10.100=−10. We read 100100 as “the opposite of the square root of 10.”

Example 1.109

Simplify: 99 144.144.

Try It 1.217

Simplify: 44 225.225.

Try It 1.218

Simplify: 8181 100.100.

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as counting numbers, whole numbers, and integers. What is the difference between these types of numbers?

Counting numbers1,2,3,4,Whole numbers0,1,2,3,4,Integers−3,−2,−1,0,1,2,3,Counting numbers1,2,3,4,Whole numbers0,1,2,3,4,Integers−3,−2,−1,0,1,2,3,

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

Rational Number

A rational number is a number of the form pq,pq, where p and q are integers and q0.q0.

A rational number can be written as the ratio of two integers.

All signed fractions, such as 45,78,134,20345,78,134,203 are rational numbers. Each numerator and each denominator is an integer.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to 31,62,93,124,15531,62,93,124,155

An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.

3=31−8=810=013=31−8=810=01

Since any integer can be written as the ratio of two integers, all integers are rational numbers! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.



What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers.

We’ve already seen that integers are rational numbers. The integer −8−8 could be written as the decimal −8.0.−8.0. So, clearly, some decimals are rational.

Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means 7310,7310, we can write it as an improper fraction, 7310.7310. So 7.3 is the ratio of the integers 73 and 10. It is a rational number.

In general, any decimal that ends after a number of digits (such as 7.3 or −1.2684)−1.2684) is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.

Example 1.110

Write as the ratio of two integers: −27−27 7.31.

Try It 1.219

Write as the ratio of two integers: −24−24 3.57.

Try It 1.220

Write as the ratio of two integers: −19−19 8.41.

Let’s look at the decimal form of the numbers we know are rational.

We have seen that every integer is a rational number, since a=a1a=a1 for any integer, a. We can also change any integer to a decimal by adding a decimal point and a zero.

Integer−2−10123Decimal form−2.0−1.00.01.02.03.0Integer−2−10123Decimal form−2.0−1.00.01.02.03.0
These decimal numbers stop.These decimal numbers stop.

We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we considered above.

Ratio of integers4578134203The decimal form0.8−0.8753.256.6666.6Ratio of integers4578134203The decimal form0.8−0.8753.256.6666.6
These decimals either stop or repeat.These decimals either stop or repeat.

What do these examples tell us?

Every rational number can be written both as a ratio of integers, (pq,(pq, where p and q are integers and q0),q0), and as a decimal that either stops or repeats.

Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:

Fractions Integers
Number 4545 7878 134134 203203 −2−2 −1−1 00 11 22 33
Ratio of Integers 4545 7878 134134 203203 2121 1111 0101 1111 2121 3131
Decimal Form 0.80.8 −0.875−0.875 3.253.25 −6.6−6.6 −2.0−2.0 −1.0−1.0 0.00.0 1.01.0 2.02.0 3.03.0

Rational Number

A rational number is a number of the form pq,pq, where p and q are integers and q0.q0.

Its decimal form stops or repeats.

Are there any decimals that do not stop or repeat? Yes!

The number ππ (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.

π=3.141592654...π=3.141592654...

We can even create a decimal pattern that does not stop or repeat, such as

2.01001000100001…2.01001000100001…

Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational.

Irrational Number

An irrational number is a number that cannot be written as the ratio of two integers.

Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

Rational or Irrational?

If the decimal form of a number

  • repeats or stops, the number is rational.
  • does not repeat and does not stop, the number is irrational.

Example 1.111

Given the numbers 0.583,0.47,3.605551275...0.583,0.47,3.605551275... list the rational numbers irrational numbers.

Try It 1.221

For the given numbers list the rational numbers irrational numbers: 0.29,0.816,2.515115111.0.29,0.816,2.515115111.

Try It 1.222

For the given numbers list the rational numbers irrational numbers: 2.63,0.125,0.4183022.63,0.125,0.418302

Example 1.112

For each number given, identify whether it is rational or irrational: 3636 44.44.

Try It 1.223

For each number given, identify whether it is rational or irrational: 8181 17.17.

Try It 1.224

For each number given, identify whether it is rational or irrational: 116116 121.121.

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.

Real Number

A real number is a number that is either rational or irrational.

All the numbers we use in elementary algebra are real numbers. Figure 1.15 illustrates how the number sets we’ve discussed in this section fit together.

This figure consists of a Venn diagram. To start there is a large rectangle marked Real Numbers. The right half of the rectangle consists of Irrational Numbers. The left half consists of Rational Numbers. Within the Rational Numbers rectangle, there are Integers …, negative 2, negative 1, 0, 1, 2, …. Within the Integers rectangle, there are Whole Numbers 0, 1, 2, 3, … Within the Whole Numbers rectangle, there are Counting Numbers 1, 2, 3, …
Figure 1.15 This chart shows the number sets that make up the set of real numbers. Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be?

Can we simplify −25?−25? Is there a number whose square is −25?−25?

()2=−25?()2=−25?

None of the numbers that we have dealt with so far has a square that is −25.−25. Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to −25.−25.

The square root of a negative number is not a real number.

Example 1.113

For each number given, identify whether it is a real number or not a real number: −169−169 64.64.

Try It 1.225

For each number given, identify whether it is a real number or not a real number: −196−196 81.81.

Try It 1.226

For each number given, identify whether it is a real number or not a real number: 4949 −121.−121.

Example 1.114

Given the numbers −7,145,8,5,5.9,64,−7,145,8,5,5.9,64, list the whole numbers integers rational numbers irrational numbers real numbers.

Try It 1.227

For the given numbers, list the whole numbers integers rational numbers irrational numbers real numbers: −3,2,0.3,95,4,49.−3,2,0.3,95,4,49.

Try It 1.228

For the given numbers, list the whole numbers integers rational numbers irrational numbers real numbers: 25,38,−1,6,121,2.04197525,38,−1,6,121,2.041975

Locate Fractions on the Number Line

The last time we looked at the number line, it only had positive and negative integers on it. We now want to include fractions and decimals on it.

Manipulative Mathematics

Doing the Manipulative Mathematics activity “Number Line Part 3” will help you develop a better understanding of the location of fractions on the number line.

Let’s start with fractions and locate 15,45,3,74,92,−5,and8315,45,3,74,92,−5,and83 on the number line.

We’ll start with the whole numbers 33 and −5.−5. because they are the easiest to plot. See Figure 1.16.

The proper fractions listed are 15and45.15and45. We know the proper fraction 1515 has value less than one and so would be located between 0 and 1.0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts 15,25,35,45.15,25,35,45. We plot 15.15. See Figure 1.16.

Similarly, 4545 is between 0 and −1.−1. After dividing the unit into 5 equal parts we plot 45.45. See Figure 1.16.

Finally, look at the improper fractions 74,92,83.74,92,83. These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See Figure 1.16.

74=13492=−41283=22374=13492=−41283=223

Figure 1.16 shows the number line with all the points plotted.

There is a number line shown that runs from negative 6 to positive 6. From left to right, the numbers marked are negative 5, negative 9/2, negative 4/5, 1/5, 4/5, 8/3, and 3. The number negative 9/2 is halfway between negative 5 and negative 4. The number negative 4/5 is slightly to the right of negative 1. The number 1/5 is slightly to the right of 0. The number 4/5 is slightly to the left of 1. The number 8/3 is between 2 and 3, but a little closer to 3.
Figure 1.16

Example 1.115

Locate and label the following on a number line: 4,34,14,−3,65,52,and73.4,34,14,−3,65,52,and73.

Try It 1.229

Locate and label the following on a number line: −1,13,65,74,92,5,83.−1,13,65,74,92,5,83.

Try It 1.230

Locate and label the following on a number line: −2,23,75,74,72,3,73.−2,23,75,74,72,3,73.

In Example 1.116, we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.

  • a < ba is less than b” when a is to the left of b on the number line
  • a > ba is greater than b” when a is to the right of b on the number line

As we move from left to right on a number line, the values increase.

Example 1.116

Order each of the following pairs of numbers, using < or >. It may be helpful to refer Figure 1.17.

23___−123___−1 −312___−3−312___−3 34___1434___14 −2___83−2___83

There is a number line shown that runs from negative 4 to positive 4. From left to right, the numbers marked are negative 3 and 1/2, negative 3, negative 8/3, negative 2, negative 1, negative 3/4, negative 2/3, and negative 1/4. The number negative 3 and 1/2 is between negative 4 and negative 3 The number negative 8/3 is between negative 3 and negative 2, but closer to negative 3. The numbers negative 3/4, negative 2/3, and negative 1/4 are all between negative 1 and 0.
Figure 1.17
Try It 1.231

Order each of the following pairs of numbers, using < or >:

13___−113___−1 −112___−2−112___−2 23___1323___13 −3___73.−3___73.

Try It 1.232

Order each of the following pairs of numbers, using < or >:

−1___23−1___23 −214___−2−214___−2 35___4535___45 −4___103.−4___103.

Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Example 1.117

Locate 0.4 on the number line.

Try It 1.233

Locate on the number line: 0.6.

Try It 1.234

Locate on the number line: 0.9.

Example 1.118

Locate −0.74−0.74 on the number line.

Try It 1.235

Locate on the number line: −0.6.−0.6.

Try It 1.236

Locate on the number line: −0.7.−0.7.

Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So,

0.40>0.040.40>0.04

Again, we can use the number line to order numbers.

  • a < ba is less than b” when a is to the left of b on the number line
  • a > ba is greater than b” when a is to the right of b on the number line

Where are 0.04 and 0.40 located on the number line? See Figure 1.20.

There is a number line shown that runs from negative 0.0 to 1.0. From left to right, there are points 0.04 and 0.4 marked. The point 0.04 is between 0.0 and 0.1. The point 0.4 is between 0.3 and 0.5.
Figure 1.20

We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40 > 0.04.

How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.

0.31 0.308
Convert to fractions. 3110031100 30810003081000
We need a common denominator to compare them. . .
31010003101000 30810003081000

Because 310 > 308, we know that 3101000>3081000.3101000>3081000. Therefore, 0.31 > 0.308.

Notice what we did in converting 0.31 to a fraction—we started with the fraction 3110031100 and ended with the equivalent fraction 3101000.3101000. Converting 31010003101000 back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!

31100=3101000and0.31=0.31031100=3101000and0.31=0.310

We say 0.31 and 0.310 are equivalent decimals.

Equivalent Decimals

Two decimals are equivalent if they convert to equivalent fractions.

We use equivalent decimals when we order decimals.

The steps we take to order decimals are summarized here.

How To

Order Decimals.

  1. Step 1. Write the numbers one under the other, lining up the decimal points.
  2. Step 2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
  3. Step 3. Compare the numbers as if they were whole numbers.
  4. Step 4. Order the numbers using the appropriate inequality sign.

Example 1.119

Order 0.64___0.60.64___0.6 using << or >.>.

Try It 1.237

Order each of the following pairs of numbers, using <or>:0.42___0.4.<or>:0.42___0.4.

Try It 1.238

Order each of the following pairs of numbers, using <or>:0.18___0.1.<or>:0.18___0.1.

Example 1.120

Order 0.83___0.8030.83___0.803 using << or >.>.

Try It 1.239

Order the following pair of numbers, using <or>:0.76___0.706.<or>:0.76___0.706.

Try It 1.240

Order the following pair of numbers, using <or>:0.305___0.35.<or>:0.305___0.35.

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because −2−2 lies to the right of −3−3 on the number line, we know that −2>−3.−2>−3. Similarly, smaller numbers lie to the left on the number line. For example, because −9−9 lies to the left of −6−6 on the number line, we know that −9<−6.−9<−6. See Figure 1.21.

There is a number line shown that runs from negative 10 to 0. There are not points given and the hashmarks exist at every integer between negative 10 and 0.
Figure 1.21

If we zoomed in on the interval between 0 and −1,−1, as shown in Example 1.121, we would see in the same way that −0.2>−0.3and0.9<−0.6.−0.2>−0.3and0.9<−0.6.

Example 1.121

Use << or >> to order −0.1___−0.8.−0.1___−0.8.

Try It 1.241

Order the following pair of numbers, using < or >: −0.3___−0.5.−0.3___−0.5.

Try It 1.242

Order the following pair of numbers, using < or >: −0.6___−0.7.−0.6___−0.7.

Section 1.8 Exercises

Practice Makes Perfect

Simplify Expressions with Square Roots

In the following exercises, simplify.

659.

3636

660.

44

661.

6464

662.

169169

663.

99

664.

1616

665.

100100

666.

144144

667.

44

668.

100100

669.

11

670.

121121

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

In the following exercises, write as the ratio of two integers.

671.

5 3.19

672.

8 1.61

673.

1212 9.279

674.

1616 4.399

In the following exercises, list the rational numbers, irrational numbers

675.

0.75,0.223,1.391740.75,0.223,1.39174

676.

0.36,0.94729,2.5280.36,0.94729,2.528

677.

0.45,1.919293,3.590.45,1.919293,3.59

678.

0.13,0.42982,1.8750.13,0.42982,1.875

In the following exercises, identify whether each number is rational or irrational.

679.

2525 3030

680.

4444 4949

681.

164164 169169

682.

225225 216216

In the following exercises, identify whether each number is a real number or not a real number.

683.

8181 −121−121

684.

6464 −9−9

685.

−36−36 144144

686.

−49−49 144144

In the following exercises, list the whole numbers, integers, rational numbers, irrational numbers, real numbers for each set of numbers.

687.

−8,0,1.95286,125,36,9−8,0,1.95286,125,36,9

688.

−9,−349,9,0.409,116,7−9,−349,9,0.409,116,7

689.

100,−7,83,−1,0.77,314100,−7,83,−1,0.77,314

690.

−6,52,0,0.714285———,215,14−6,52,0,0.714285———,215,14

Locate Fractions on the Number Line

In the following exercises, locate the numbers on a number line.

691.

34,85,10334,85,103

692.

14,95,11314,95,113

693.

310,72,116,4310,72,116,4

694.

710,52,138,3710,52,138,3

695.

25,2525,25

696.

34,3434,34

697.

34,34,123,−123,52,5234,34,123,−123,52,52

698.

15,25,134,−134,83,8315,25,134,−134,83,83

In the following exercises, order each of the pairs of numbers, using < or >.

699.

−1___14−1___14

700.

−1___13−1___13

701.

−212___−3−212___−3

702.

−134___−2−134___−2

703.

512___712512___712

704.

910___310910___310

705.

−3___135−3___135

706.

−4___236−4___236

Locate Decimals on the Number Line In the following exercises, locate the number on the number line.

707.

0.8

708.

−0.9−0.9

709.

−1.6−1.6

710.

3.1

In the following exercises, order each pair of numbers, using < or >.

711.

0.37___0.630.37___0.63

712.

0.86___0.690.86___0.69

713.

0.91___0.9010.91___0.901

714.

0.415___0.410.415___0.41

715.

−0.5___−0.3−0.5___−0.3

716.

−0.1___−0.4−0.1___−0.4

717.

−0.62___−0.619−0.62___−0.619

718.

−7.31___−7.3−7.31___−7.3

Everyday Math

719.

Field trip All the 5th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people.

How many busses will be needed?
Why must the answer be a whole number?
Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

720.

Child care Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care center to serve 40 children.

How many teachers will be needed?
Why must the answer be a whole number?
Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

Writing Exercises

721.

In your own words, explain the difference between a rational number and an irrational number.

722.

Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.

This is a table that has five rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “simplify expressions with square roots,” “identify integers, rational numbers, irrational numbers and real numbers,” locate fractions on the number line,” and “locate decimals on the number line.” The rest of the cells are blank

On a scale of 110,110, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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