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Prealgebra 2e

7.1 Rational and Irrational Numbers

Prealgebra 2e7.1 Rational and Irrational Numbers
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. 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
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Identify rational numbers and irrational numbers
  • Classify different types of real numbers
Be Prepared 7.1

Before you get started, take this readiness quiz.

Write 3.193.19 as an improper fraction.
If you missed this problem, review Example 5.4.

Be Prepared 7.2

Write 511511 as a decimal.
If you missed this problem, review Example 5.30.

Be Prepared 7.3

Simplify: 144.144.
If you missed this problem, review Example 5.69.

Identify Rational Numbers and Irrational Numbers

Congratulations! You have completed the first six chapters of this book! It's time to take stock of what you have done so far in this course and think about what is ahead. You have learned how to add, subtract, multiply, and divide whole numbers, fractions, integers, and decimals. You have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. You have solved many different types of applications. You have established a good solid foundation that you need so you can be successful in algebra.

In this chapter, we'll make sure your skills are firmly set. We'll take another look at the kinds of numbers we have worked with in all previous chapters. We'll work with properties of numbers that will help you improve your number sense. And we'll practice using them in ways that we'll use when we solve equations and complete other procedures in algebra.

We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?

counting numbers 1,2,3,4…1,2,3,4…
whole numbers 0,1,2,3,4…0,1,2,3,4…
integers −3,−2,−1,0,1,2,3,4…−3,−2,−1,0,1,2,3,4…

Rational Numbers

What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you 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 Numbers

A rational number is a number that can be written in the form pq,pq, where pp and qq are integers and qo.qo.

All fractions, both positive and negative, are rational numbers. A few examples are

45,78,134,and20345,78,134,and203

Each numerator and each denominator is an integer.

We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this 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 all the counting numbers and all 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 7.3.7.3. Can we write it as a ratio of two integers? Because 7.37.3 means 7310,7310, we can write it as an improper fraction, 7310.7310. So 7.37.3 is the ratio of the integers 7373 and 10.10. It is a rational number.

In general, any decimal that ends after a number of digits (such as 7.37.3 or −1.2684)−1.2684) is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction.

Example 7.1

Write each as the ratio of two integers: −15−156.816.81−367.−367.

Try It 7.1

Write each as the ratio of two integers: −24−243.57.3.57.

Try It 7.2

Write each as the ratio of two integers: −19−19 8.41.8.41.

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.a. We can also change any integer to a decimal by adding a decimal point and a zero.

Integer−2,−1,0,1,2,3Decimal−2.0,−1.0,0.0,1.0,2.0,3.0These decimal numbers stop.Integer−2,−1,0,1,2,3Decimal−2.0,−1.0,0.0,1.0,2.0,3.0These decimal numbers stop.

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

Ratio of Integers 45, 78, 1314, 203 Decimal Forms 0.8, −0.875, 3.25, −6.666… These decimals either stop or repeat. −6.66 Ratio of Integers 45, 78, 1314, 203 Decimal Forms 0.8, −0.875, 3.25, −6.666… These decimals either stop or repeat. −6.66

What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.

Rational Numbers
Fractions Integers
Number 45,78,134,−20345,78,134,−203 −2,−1,0,1,2,3−2,−1,0,1,2,3
Ratio of Integer 45,−78,134,−20345,−78,134,−203 −21,−11,01,11,21,31−21,−11,01,11,21,31
Decimal number 0.8,−0.875,3.25,−6.6,0.8,−0.875,3.25,−6.6, −2.0,−1.0,0.0,1.0,2.0,3.0−2.0,−1.0,0.0,1.0,2.0,3.0

Irrational Numbers

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.......

Similarly, the decimal representations of square roots of whole numbers that are not perfect squares never stop and never repeat. For example,

5=2.236067978.....5=2.236067978.....

A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.

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.

If the decimal form of a number

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

Example 7.2

Identify each of the following as rational or irrational:

  1. 0.5830.583
  2. 0.4750.475
  3. 3.605551275…3.605551275…

Try It 7.3

Identify each of the following as rational or irrational:

0.290.290.8160.8162.515115111…2.515115111…

Try It 7.4

Identify each of the following as rational or irrational:

0.230.230.1250.1250.418302…0.418302…

Let's think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.

Example 7.3

Identify each of the following as rational or irrational:

  1. 3636
  2. 4444

Try It 7.5

Identify each of the following as rational or irrational:

  1. 8181
  2. 1717

Try It 7.6

Identify each of the following as rational or irrational:

  1. 116116

  2. 121121

Classify Real Numbers

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.

Figure 7.2 illustrates how the number sets are related.

The image shows a large rectangle labeled “Real Numbers”. The rectangle is split in half vertically. The right half is labeled “Irrational Numbers”. The left half is labeled “Rational Numbers” and contains three concentric rectangles. The outer most rectangle is labeled “Integers”, the next rectangle is “Whole Numbers” and the inner most rectangle is “Natural Numbers”.
Figure 7.2 This diagram illustrates the relationships between the different types of real numbers.

Real Numbers

Real numbers are numbers that are either rational or irrational.

Does the term “real numbers” seem strange to you? Are there any numbers that are not “real”, and, if so, what could they be? For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers. You won't encounter imaginary numbers in this course, but you will later on in your studies of algebra.

Example 7.4

Determine whether each of the numbers in the following list is a whole number, integer, rational number, irrational number, and real number.

−7,145,8,5,5.9,64−7,145,8,5,5.9,64
Try It 7.7

Determine whether each number is a whole number, integer, rational number, irrational number, and real number: −3,2,0.3,95,4,49.−3,2,0.3,95,4,49.

Try It 7.8

Determine whether each number is a whole number, integer, rational number, irrational number, and real number: 25,38,−1,6,121,2.041975…25,38,−1,6,121,2.041975…

Media Access Additional Online Resources

Section 7.1 Exercises

Practice Makes Perfect

Rational Numbers

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

1.
  1. 55
  2. 3.193.19
2.
  1. 88
  2. −1.61−1.61
3.
  1. −12−12
  2. 9.2799.279
4.
  1. −16−16
  2. 4.3994.399

In the following exercises, determine which of the given numbers are rational and which are irrational.

5.

0.750.75, 0.2230.223, 1.39174…1.39174…

6.

0.360.36, 0.94729…0.94729…, 2.5282.528

7.

0.450.45, 1.919293…1.919293…, 3.593.59

8.

0.13,0.42982…0.13,0.42982…, 1.8751.875

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

9.
  1. 2525
  2. 3030
10.
  1. 4444
  2. 4949
11.
  1. 164164
  2. 169169
12.
  1. 225225
  2. 216216

Classifying Real Numbers

In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.

13.

−8−8, 0,1.95286....0,1.95286...., 125125, 3636, 99

14.

−9−9, −349−349, 99, 0.4090.409,116116, 77

15.

100100, −7−7, 8383, −1−1, 0.770.77, 314314

Everyday Math

16.

Field trip All the 5th5th 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 147147 people. Each bus holds 4444 people.

How many buses will be needed?

Why must the answer be a whole number?

Why shouldn't you round the answer the usual way?

17.

Child care Serena wants to open a licensed child care center. Her state requires that there be no more than 1212 children for each teacher. She would like her child care center to serve 4040 children.

  1. How many teachers will be needed?

  2. Why must the answer be a whole number?

  3. Why shouldn't you round the answer the usual way?

Writing Exercises

18.

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

19.

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 objectives of this section.

.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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