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Contemporary Mathematics

Chapter 4

Contemporary MathematicsChapter 4
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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

Your Turn

4.1
1.
416
2.
1,851
3.
17,488
4.2
1.
9 \times {10^2} + 2 \times {10^1} + 4 \times {10^0}
2.
1 \times {10^3} + 2 \times {10^2} + 7 \times {10^1} + 9 \times {10^0}
3.
4 \times {10^6} + 1 \times {10^5} + 3 \times {10^4} + 0 \times {10^3} + 0 \times {10^2} + 4 \times {10^1} + 5 \times {10^0}
4.3
1.
621
2.
3,203
3.
40,630,891
4.4
1.
1269
4.5
1.
42,136
4.6
1.
6,105,643
4.7
1.
257
4.8
1.
6,054
4.9
1.
1,248,073
4.10
1.
77
2.
240
3.
3,447
4.11
1.
XXVII
2.
XLIX
3.
DCCXXXIX
4.
MMMDCXLVII
4.12
1.
0, 1, 2, 3
4.13
1.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B
4.14
1.
157
4.15
1.
2,014
4.16
1.
851
4.17
1.
27
4.18
1.
0, 1, 2, 3
10, 11, 12, 13
20, 21, 22, 23
30, 31, 32, 33
100
4.19
1.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B

10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B

20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 2A, 2B

30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 3A, 3B

40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 4A, 4B

50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 5A, 5B

60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 6A, 6B

70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 7A, 7B

80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 8A, 8B

90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 9A, 9B

A0, A1, A2, A3, A4, A5, A6, A7, A8, A9, AA, AB

B0, B1, B2, B3, B4, B5, B6, B7, B8, B9, BA, BB

100

4.20
1.
0, 1, 2, 10, 11, 12, 20, 21, 22, 100
4.21
1.
20107
4.22
1.
554B12
4.23
1.
100010012
4.24
1.
Mayan numeral 17 is displayed.
Mayan numeral 6 is displayed.
2.
Mayan numeral 11 is displayed.
Mayan numeral 8 is displayed.
Mayan numeral 5 is displayed.
4.25
1.
The result has the digit 7 in it. In base 4, the 7 is an illegal symbol.
4.26
1.
The remainders include 10, which in base 6 is an illegal symbol.
4.27
1.
Since 12 is larger than 10, the base 10 number cannot have less digits than the base 12 number. Since it did, we know an error has been made.
4.28
1.
12426
4.29
1.
+ 0 1 2 3
0 0 1 2 3
1 1 2 3 10
2 2 3 10 11
3 3 10 11 12
Base 4 Addition Table
4.30
1.
6337
4.31
1.
+ 0 1 2 3 4 5 6 7 8 9 A B C D
0 0 1 2 3 4 5 6 7 8 9 A B C D
1 1 2 3 4 5 6 7 8 9 A B C D 10
2 2 3 4 5 6 7 8 9 A B C D 10 11
3 3 4 5 6 7 8 9 A B C D 10 11 12
4 4 5 6 7 8 9 A B C D 10 11 12 13
5 5 6 7 8 9 A B C D 10 11 12 13 14
6 6 7 8 9 A B C D 10 11 12 13 14 15
7 7 8 9 A B C D 10 11 12 13 14 15 16
8 8 9 A B C D 10 11 12 13 14 15 16 17
9 9 A B C D 10 11 12 13 14 15 16 17 18
A A B C D 10 11 12 13 14 15 16 17 18 19
B B C D 10 11 12 13 14 15 16 17 18 19 1A
C C D 10 11 12 13 14 15 16 17 18 19 1A 1B
D D 10 11 12 13 14 15 16 17 18 19 1A 1B 1C
Base 14 Addition Table
4.32
1.
13B912
4.33
1.
100100102
4.34
1.
326
4.35
1.
22712
4.36
1.
The symbols 4 and 5 are not legal symbols in base 4. Careful use of the base 4 addition table would correct this error.
1 3 3
+ 1 1 2
3 1 1
The correct answer is 3114.
4.37
1.
This is correct if the numbers are base 10 numbers, but these numbers are base 14 numbers. In base 14, 9 + 9 is not 18, but instead is 13. Careful use of the base 14 addition table generates the correct answer, {163_{14}}.
4.38
1.
* 0 1 2 3
0 0 0 0 0
1 0 1 2 3
2 0 2 10 12
3 0 3 12 21
Base 4 Multiplication Table
4.39
1.
* 0 1 2 3 4 5 6 7 8 9 A B C D
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 A B C D
2 0 2 4 6 8 A C 10 12 14 16 18 1A 1C
3 0 3 6 9 C 11 14 17 1A 1D 22 25 28 2B
4 0 4 8 C 12 16 1A 20 24 28 2C 32 36 3A
5 0 5 A 11 16 1B 22 27 2C 33 38 3D 44 49
6 0 6 C 14 1A 22 28 30 36 3C 44 4A 52 58
7 0 7 10 17 20 27 30 37 40 47 50 57 60 67
8 0 8 12 1A 24 2C 36 40 48 52 5A 64 6C 76
9 0 9 14 1D 28 33 3C 47 52 5B 66 71 7A 85
A 0 A 16 22 2C 38 44 50 5A 66 72 7C 88 94
B 0 B 18 25 32 3D 4A 57 64 71 7C 89 96 A3
C 0 C 1A 28 36 44 52 60 6C 7A 88 96 A4 B2
D 0 D 1C 2B 3A 49 58 67 76 85 94 A3 B2 C1
4.40
1.
40006
2.
10101112
4.41
1.
436912
4.42
1.
{10_6} \div {3_6} = {2_6}
2.
{50_{12}} \div {{\text{A}}_{12}} = {6_{12}}
4.43
1.
The symbols 4 and 5 are not legal symbols in base 4. Careful use of the base 4 multiplication table would correct this error. The correct answer is 3334.
4.44
1.
This is correct if the numbers are base 10 numbers, but these numbers are base 14 numbers. In base 14, {49_{14}} \times {9_{14}} = 2{\text{D}}{{\text{B}}_{14}} is not 81, but instead is 5B. Careful use of the base 14 addition table (Table 4.9) generates the correct answer, {49_{14}} \times {9_{14}} = 2{\text{D}}{{\text{B}}_{14}}.

Check Your Understanding

1.
A system in which the position of a numeral determines the value associated with that numeral.
2.
279
3.
4 \times {10^4} + 5 \times {10^3} + 2 \times {10^2} + 0 \times {10^1} + 9 \times {10^0}
4.
10
5.
A numeral is a symbol representing a number. A number is a quantity or amount.
6.
601,947
7.
60
8.
20
9.
Roman numerals do not use place value.
10.
341
11.
209
12.
247
13.
CDLXXIX
14.
A base 25 system would require 25 symbols.
15.
Since the 4 is the second digit, its place value is 181 times 4, or 72.
16.

329

17.
409
18.
5118
19.
2,126
20.
In base 4, 5 is not a valid symbol. So, a mistake has been made.
21.
+ 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 10
2 2 3 4 5 6 7 10 11
3 3 4 5 6 7 10 11 12
4 4 5 6 7 10 11 12 13
5 5 6 7 10 11 12 13 14
6 6 7 10 11 12 13 14 15
7 7 10 11 12 13 14 15 16
22.
1216
23.
78
24.
8214
25.
512
26.
In base 8, 8 is not a valid symbol. So, a mistake has been made.
27.
A common base 14 error is performing the operation in base 10.
28.
the addition table
29.
The process is the same, except the multiplication table for the base is used instead of the familiar base 10 rules.
30.
the multiplication table for the base
31.
22406
32.
B14
33.
The 8 is not a symbol used in base 5.
34.
A symbol that is not used in that base is present, or a base 10 rule is used.
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