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Contemporary Mathematics

3.6 Real Numbers

Contemporary Mathematics3.6 Real Numbers

Table of contents
  1. Preface
  2. 1 Sets
    1. Introduction
    2. 1.1 Basic Set Concepts
    3. 1.2 Subsets
    4. 1.3 Understanding Venn Diagrams
    5. 1.4 Set Operations with Two Sets
    6. 1.5 Set Operations with Three Sets
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  3. 2 Logic
    1. Introduction
    2. 2.1 Statements and Quantifiers
    3. 2.2 Compound Statements
    4. 2.3 Constructing Truth Tables
    5. 2.4 Truth Tables for the Conditional and Biconditional
    6. 2.5 Equivalent Statements
    7. 2.6 De Morgan’s Laws
    8. 2.7 Logical Arguments
    9. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  4. 3 Real Number Systems and Number Theory
    1. Introduction
    2. 3.1 Prime and Composite Numbers
    3. 3.2 The Integers
    4. 3.3 Order of Operations
    5. 3.4 Rational Numbers
    6. 3.5 Irrational Numbers
    7. 3.6 Real Numbers
    8. 3.7 Clock Arithmetic
    9. 3.8 Exponents
    10. 3.9 Scientific Notation
    11. 3.10 Arithmetic Sequences
    12. 3.11 Geometric Sequences
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  5. 4 Number Representation and Calculation
    1. Introduction
    2. 4.1 Hindu-Arabic Positional System
    3. 4.2 Early Numeration Systems
    4. 4.3 Converting with Base Systems
    5. 4.4 Addition and Subtraction in Base Systems
    6. 4.5 Multiplication and Division in Base Systems
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Projects
      5. Chapter Review
      6. Chapter Test
  6. 5 Algebra
    1. Introduction
    2. 5.1 Algebraic Expressions
    3. 5.2 Linear Equations in One Variable with Applications
    4. 5.3 Linear Inequalities in One Variable with Applications
    5. 5.4 Ratios and Proportions
    6. 5.5 Graphing Linear Equations and Inequalities
    7. 5.6 Quadratic Equations with Two Variables with Applications
    8. 5.7 Functions
    9. 5.8 Graphing Functions
    10. 5.9 Systems of Linear Equations in Two Variables
    11. 5.10 Systems of Linear Inequalities in Two Variables
    12. 5.11 Linear Programming
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  7. 6 Money Management
    1. Introduction
    2. 6.1 Understanding Percent
    3. 6.2 Discounts, Markups, and Sales Tax
    4. 6.3 Simple Interest
    5. 6.4 Compound Interest
    6. 6.5 Making a Personal Budget
    7. 6.6 Methods of Savings
    8. 6.7 Investments
    9. 6.8 The Basics of Loans
    10. 6.9 Understanding Student Loans
    11. 6.10 Credit Cards
    12. 6.11 Buying or Leasing a Car
    13. 6.12 Renting and Homeownership
    14. 6.13 Income Tax
    15. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  8. 7 Probability
    1. Introduction
    2. 7.1 The Multiplication Rule for Counting
    3. 7.2 Permutations
    4. 7.3 Combinations
    5. 7.4 Tree Diagrams, Tables, and Outcomes
    6. 7.5 Basic Concepts of Probability
    7. 7.6 Probability with Permutations and Combinations
    8. 7.7 What Are the Odds?
    9. 7.8 The Addition Rule for Probability
    10. 7.9 Conditional Probability and the Multiplication Rule
    11. 7.10 The Binomial Distribution
    12. 7.11 Expected Value
    13. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  9. 8 Statistics
    1. Introduction
    2. 8.1 Gathering and Organizing Data
    3. 8.2 Visualizing Data
    4. 8.3 Mean, Median and Mode
    5. 8.4 Range and Standard Deviation
    6. 8.5 Percentiles
    7. 8.6 The Normal Distribution
    8. 8.7 Applications of the Normal Distribution
    9. 8.8 Scatter Plots, Correlation, and Regression Lines
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  10. 9 Metric Measurement
    1. Introduction
    2. 9.1 The Metric System
    3. 9.2 Measuring Area
    4. 9.3 Measuring Volume
    5. 9.4 Measuring Weight
    6. 9.5 Measuring Temperature
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  11. 10 Geometry
    1. Introduction
    2. 10.1 Points, Lines, and Planes
    3. 10.2 Angles
    4. 10.3 Triangles
    5. 10.4 Polygons, Perimeter, and Circumference
    6. 10.5 Tessellations
    7. 10.6 Area
    8. 10.7 Volume and Surface Area
    9. 10.8 Right Triangle Trigonometry
    10. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  12. 11 Voting and Apportionment
    1. Introduction
    2. 11.1 Voting Methods
    3. 11.2 Fairness in Voting Methods
    4. 11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem
    5. 11.4 Apportionment Methods
    6. 11.5 Fairness in Apportionment Methods
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  13. 12 Graph Theory
    1. Introduction
    2. 12.1 Graph Basics
    3. 12.2 Graph Structures
    4. 12.3 Comparing Graphs
    5. 12.4 Navigating Graphs
    6. 12.5 Euler Circuits
    7. 12.6 Euler Trails
    8. 12.7 Hamilton Cycles
    9. 12.8 Hamilton Paths
    10. 12.9 Traveling Salesperson Problem
    11. 12.10 Trees
    12. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Videos
      4. Formula Review
      5. Projects
      6. Chapter Review
      7. Chapter Test
  14. 13 Math and...
    1. Introduction
    2. 13.1 Math and Art
    3. 13.2 Math and the Environment
    4. 13.3 Math and Medicine
    5. 13.4 Math and Music
    6. 13.5 Math and Sports
    7. Chapter Summary
      1. Key Terms
      2. Key Concepts
      3. Formula Review
      4. Projects
      5. Chapter Review
      6. Chapter Test
  15. A | Co-Req Appendix: Integer Powers of 10
  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
    12. Chapter 12
    13. Chapter 13
  17. Index
An illustration shows a man thinking about the equation 4 times 31 times 25 equals 4 times 25 times 31 equals 100 times 31 equals 3,100.
Figure 3.31 Quick mental math involves using the known properties of real numbers.

Learning Objectives

After completing this section, you should be able to:

  1. Define and identify numbers that are real numbers.
  2. Identify subsets of the real numbers.
  3. Recognize properties of real numbers.

Have you ever been impressed by the speed at which someone can do math in their head? Most of us at one time or another have witnessed a person speed through mental math, an impressive feat that often bests calculators. One such person is Neelkantha Bhanu Prakash. As of September 20, 2020, he is considered the world’s fastest human calculator. He currently holds four world records. How does someone do that, though? Have they memorized lots of arithmetic facts? Are they simply brilliant?

The answer isn't simple so much as it is about knowledge. Real numbers behave in some very regular ways, following rules that can be learned. In this section, those rules are explored.

Watch the video of Arthur Benjamin’s TED Talk to learn about another mathematician with remarkable mental abilities.

Defining and Identifying Real Numbers

Real numbers are the rational and irrational numbers combined. The real numbers represent the collection of all physical distances that exist, along with 0 and the negatives of those physical distances. For example, if you take a measure of three units, and divide that distance into eight (8) equal lengths, the distance you have formed is 3838 units. Also, if you draw a right triangle (a triangle with one angle equal to 90 degrees) with one side length of 1, and the other side length of 3, the long side of the triangle will have length 1010 units, as shown in Figure 3.32.

A right triangle. The legs measure 1 and 3. The hypotenuse measures square root of 10.
Figure 3.32 Right triangle

Of course, if we name something the real numbers, there must be numbers that aren't real. Otherwise, they’d just be called the numbers. One such not real number, one that cannot be a length, is 11. It is part of a collection of numbers called the complex numbers, it is denoted with the letter ii. As an extension, the square root of any negative number is not a real number, but instead a complex number.

To determine if a number is real, check to see if there are any negatives under a square root or any i'si's. If there are any present, the number is not real.

Example 3.96

Identifying Real Numbers

Determine if each of the following are real numbers:

  1. 437437
  2. 13.338113.3381
  3. 178178

Your Turn 3.96

Determine if each of the following are real numbers:
1.
8 15
2.
4 + 17 i
3.
17 46

Identifying Subsets of Real Numbers

The real numbers were built out of pieces, including integers, rational numbers, and irrational numbers. As such, the real numbers have named subsets, as shown in the table below.

Set Name Set Symbol Set Description
Natural Numbers The counting numbers
Whole Numbers The counting numbers and 0
Integers The natural numbers, their negatives, and 0
Rational Numbers Fractions of integers
Irrational Numbers Numbers that cannot be written as a fraction of integers
Real Numbers The union of the rational and irrational numbers, all possible physical lengths, and their negatives

When we categorize numbers using these sets, we use the smallest set that they belong to. For instance, −7 is an integer, and a rational number, and a real number. The smallest set to which −7 belongs is integer, so we’d say it belongs to the integers.

We can also represent the relationships between the different sets of real numbers using set notation. All natural numbers are integers, but there are integers that are not natural numbers, so . Similarly, every integer is a rational number, but there are rational numbers that are not integers, so . The same is true of the rational numbers and the real numbers, so .

There is no agreed-upon symbol for the irrational numbers. If we represent the irrationals as the set AA, we should note that the following are true: A=A= and A=A=. Recall that this means the irrationals are the complement of the rational numbers in the universal set of real numbers.

Example 3.97

Categorizing Numbers

Identify all subsets of the real numbers to which the following real numbers belong:

  1. 1414
  2. 14.22314.223
  3. 1717

Your Turn 3.97

Identify all subsets of the real numbers to which the following real numbers belong:
1.
14 3
2.
147
3.
37 150

Example 3.98

Categorizing Numbers within a Venn Diagram

Place the following numbers correctly in the Venn diagram (Figure 3.33).

421037150411204π421037150411204π
A Venn diagram shows four concentric ovals. The ovals are labeled from inner to outer as follows: N, Z, Q, and R.
Figure 3.33

Your Turn 3.98

1.
Place the following numbers correctly into the Venn diagram.
4 13.863 15 871 5 7 −3 π
A Venn diagram shows four concentric ovals. The ovals are labeled from inner to outer as follows: N, Z, Q, and R.

Recognizing Properties of Real Numbers

The real numbers behave in very regular ways. These behaviors are called the properties of the real numbers. Knowing these properties helps when evaluating formulas, working with equations, or performing algebra. Being familiar with these properties is helpful in all settings where numbers are used and manipulated. For example, when multiplying 4×13×254×13×25, you could multiply the 4 and 25 first. If you know that product is 100, it makes the multiplication easier.

The table below is a partial list of properties of real numbers.

Property Example In Words
Distributive property
a×(b+c)=a×b+a×ca×(b+c)=a×b+a×c
5×(3+4)=5×3+5×45×(3+4)=5×3+5×4 Multiplication distributes across addition
Commutative property of addition
a+b=b+aa+b=b+a
3+7=7+33+7=7+3 Numbers can be added in any order
Commutative property of multiplication
a×b=b×aa×b=b×a
10×4=4×1010×4=4×10 Numbers can be multiplied in any order
Associative property of addition
a+(b+c)=(a+b)+ca+(b+c)=(a+b)+c
4+(3+8)=(4+3)+84+(3+8)=(4+3)+8 Doesn't matter which pair of numbers is added first
Associative property of multiplication
a×(b×c)=(a×b)×ca×(b×c)=(a×b)×c
2×(5×7)=(2×5 )×72×(5×7)=(2×5)×7 Doesn't matter which pair of numbers is multiplied first
Additive identity property
a+0=aa+0=a
17+0=1717+0=17 Any number plus 0 is the number
Multiplicative identity property
a×1=aa×1=a
21×1=2121×1=21 Any number times one is the number
Additive inverse property
a+(a)=0a+(a)=0
14+(14)=014+(14)=0 Every number plus its negative is 0
Multiplicative inverse property
a×(1a)=1a×(1a)=1, provided a0a0
3×(13)=13×(13)=1 Every non-zero number times its reciprocal is 1

The names of the properties are suggestive. The commutative properties, for example, suggest commuting, or moving. Associative properties suggest which items are associated with others, or if order matters in the computation. The distributive property addresses how a number is distributed across parentheses.

Example 3.99

Identifying Properties of Real Numbers

In each of the following, identify which property of the real numbers is being applied.

  1. 4+(8+13)=(4+8)+134+(8+13)=(4+8)+13
  2. 34×(134)=134×(134)=1
  3. 14+27=27+1414+27=27+14

Your Turn 3.99

In each of the following, identify which property of the real numbers is being applied.
1.
5 × ( 6 + 19 ) = 5 × 6 + 5 × 19
2.
41.7 + ( 41.7 ) = 0

Using these properties to perform arithmetic quickly relies on spotting easy numbers to work with. Look for numbers that add to a multiple of 10, or multiply to a multiple of 10 or 100.

Example 3.100

Using Properties of Real Numbers in Calculations

Use properties of the real numbers and mental math to calculate the following:

  1. 2×13×502×13×50
  2. 13+84+2713+84+27
  3. 9×16×119×16×11

Your Turn 3.100

1.

Use properties of real numbers and mental math to calculate the following:

9 × 8 × 11

Check Your Understanding

33.
Which of the following are real numbers: 4 + i , 77 , −19 , 38.902 ?
34.
Indicate which of the sets are subsets of the others: Q , R , N , Z .
35.
Which property is demonstrated here: 3 × 6 + 3 × 15 = 3 × ( 6 + 15 ) ?

Section 3.6 Exercises

For the following exercises, identify each number as a natural number, an integer, a rational number, or a real number.
1 .
1 3
2 .
16
3 .
19
4 .
3 7
5 .
π
6 .
47
7 .
13
8 .
31 7
For the following exercises, correctly place the numbers in the Venn diagram.
A Venn diagram shows four concentric ovals. The ovals are labeled from inner to outer as follows: N, Z, Q, and R.
9 .
5.6 −7 26 41 13 50 −46
10 .
2 π 11 17 679 14 37 57 151
For the following exercises, identify the property of real numbers that is being illustrated.
11 .
14   +   38.9   =   38.9   +   14
12 .
37.12   +   98   =   98   +   37.12
13 .
38 × 16 = 16 × 38
14 .
6.3 × 14 = 14 × 6.3
15 .
4 + ( 8 + 7 ) = ( 4 + 8 ) + 7
16 .
13 × ( 6 11 × 14 ) = ( 13 × 6 11 ) × 14
17 .
( 5.6 × 8.7 ) × 6 = 5.6 × ( 8.7 × 6 )
18 .
( 3.8 + 5 ) + 13 = 3.8 + ( 5 + 13 )
19 .
4 × ( 5 + 3 ) = 4 × 5 + 4 × 3
20 .
74 + ( 74 ) = 0
21 .
3.4 π × ( 1 3.4 π ) = 1
22 .
17 × ( 1 17 ) = 1
23 .
4 35 + 0 = 4 35
24 .
17 × ( 5 + 3.6 ) = 17 × 5 + 17 × 3.6
25 .
( 10 π ) + 10 π = 0
26 .
3 7 × 1 = 3 7
27 .
3 π + 7 21 = 7 21 + 3 π
28 .
14 × ( 3 2 + 11.6 ) = 14 × 3 2 + 14 × 11.6
For the following exercises, use properties of real numbers and mental math to calculate the expression.
29 .
43 + 62 + 17
30 .
106 + 75 + 94
31 .
5 × 13 × 4
32 .
4 × 72 × 5
33 .
46 + 77 + 23 + 24 + 103
34 .
23 + 98 + 75 + 12 + 77
35 .
4 × 13 × 25
36 .
50 × 23 × 2
37 .
13 × 99
38 .
15 × 39
39 .
43 × 101
40 .
16 × 999
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