Mathematics Beyond the Numbers
Edition: 2
Copyright: 2022
Edition: 2
Copyright: 2022
Edition: 2
Copyright: 2022
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Mathematics Beyond the Numbers is written in a conversational style and uses real-world data and applications to make the topics come to life for today’s students!
Designed for a one or two-semester liberal arts mathematics course, Mathematics Beyond the Numbers integrates a great deal of historical background so that students can see the development of mathematics over time. In addition, some of the topics included in Mathematics Beyond the Numbers are particularly applicable to a student’s field of study. For instance, voting methods and apportionment are of interest to a political science major
Mathematics Beyond the Numbers by George T. Gilbert and Rhonda L. Hatcher:
- Is Flexible! The chapters are independent of one another and can be presented in any order.
- Is Interactive! Students have access to an accompanying course website that includes online homework.
- Is Practical! Numerous exercises and examples teach students mathematical problem-solving skills so they can carry out what they have learned.
- Integrates Technology! Technology Tips guide instructors and students to the possibilities of using technology.
- Is Easy-to-Adopt! An Instructor’s Solutions Manual is provided to all adopting instructors.
PREFACE
CHAPTER 1 VOTING METHODS
1.1 Plurality and Runoff Methods
Runoff Elections
Preference Rankings
1.2 Borda’s Method: A Scoring System
1.3 Head-to-Head Comparisons
Single-Peaked Preference Rankings
1.4 Approval Voting
1.5 The Search for an Ideal Voting System
1.6 Weighted Voting Systems
Dictators, Dummies, and Veto Power
The Banzhaf Power Index
The Shapley-Shubik Power Index
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 2 APPORTIONMENT: SHARING WHAT CANNOT BE DIVIDED ARBITRARILY
2.1 Quota Methods
Hamilton’s Method
Lowndes’ Method
2.2 Early Divisor Methods
Jefferson’s Method
Webster’s Method
2.3 Apportionment in Today’s House of Representatives
The Hill–Huntington Method
Other Apportionment Methods
2.4 The Search for an Ideal Apportionment Method
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 3 THE MATHEMATICS OF MONEY
3.1 Powers, Roots, and Logarithms
3.2 Simple Interest
3.3 Compound Interest
Inflation
3.4 The Rewards of Systematic Savings
3.5 Amortized Loans
Finding a Loan Balance
Amortization Schedules
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 4 PROBABILITY
4.1 Elementary Probability
4.2 Odds
House Odds and Fair Bets
4.3 The Addition Rule
4.4 Conditional Probability and the Multiplication Rule
The Multiplication Rule
Independence and the Multiplication Rule
4.5 Counting Techniques
Permutations
Combinations
4.6 Probability Problems Using Counting Techniques
The Birthday Problem
4.7 Expected Value
4.8 Genetics
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 5 STATISTICS
5.1 Organizing and Presenting Data
Bar Graphs
Histograms
Pie Charts
5.2 Typical and Central Values
The Mode
The Median
The Mean
Estimating the Mean of Grouped Data
5.3 Measures of Spread
The Range
The Standard Deviation
Estimating the Standard Deviation of Grouped Data
5.4 The Normal Distribution
Percentiles
5.5 Estimating the Mean
Confidence Intervals
Sample Standard Deviation and Confidence Intervals
5.6 Polls and Margin of Error
5.7 Garbage In, Garbage Out: A Look at Misleading Uses of Statistics and at Sampling Techniques
The Source of the Data
The Data: Questions and Answers
The Conclusions Drawn from the Data
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 6 PATHS AND NETWORKS
6.1 Eulerian Paths and Circuits on Graphs
Eulerization
6.2 The Traveling Salesman Problem
The Nearest Neighbor Algorithm
The Greedy Algorithm
6.3 Efficient Networking: Minimal Spanning Trees
Prim’s Algorithm
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 7 TILINGS AND POLYHEDRA
7.1 Polygons
7.2 Tiling’s
Regular Tilings
Semiregular Tiling’s
Tiling’s with Nonregular Polygons
Tilings with Other Shapes
7.3 Polyhedra
Regular Polyhedra
Semiregular Polyhedra
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 8 NUMBER THEORY
8.1 Divisibility and Primes
Prime Numbers
The Division Algorithm
The Greatest Common Divisor
8.2 Modular Arithmetic
8.3 Divisibility Tests
8.4 Check Digits
8.5 Tournament Scheduling
8.6 Introduction to Cryptology
The Caesar Cipher
Affine Ciphers
8.7 Advanced Encryption Methods
The Hill Cipher
The RSA Public Key System
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
APPENDIX A
ANSWERS TO SELECTED EXERCISES
George Gilbert earned a B.A. in mathematics from Washington University in 1979 and a Ph.D. in mathematics from Harvard University in 1984, specializing in number theory. He has held teaching positions at the University of Texas in Austin, Saint Olaf College, and has been on the faculty at Texas Christian University for the past 20+ years. Professor Gilbert is active in the area of problem solving, having served on the William Lowell Putnam questions committee, the AIME contest committee, and as the editor of the Problems Section of Mathematics Magazine.
Rhonda Hatcher earned a B.A. in mathematics from the University of Colorado at Boulder in 1980 and a Ph.D. in mathematics from Harvard University in 1987. Her research specialty is number theory. She has taught at Saint Olaf College and is currently on the faculty of Texas Christian University. Professor Hatcher won the 1994 Deans’ Teaching Award, the 1997 Honors Professor of the Year Award, and the 2000 Chancellor’s Award for Distinguished Teaching at TCU. In 1998, the Mathematical Association of America awarded her the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics. In 2000, she was awarded the Texas Professor of the Year award from The Carnegie Foundation for Advancement of Teaching and Council for Advancement and Support of Education.
Mathematics Beyond the Numbers is written in a conversational style and uses real-world data and applications to make the topics come to life for today’s students!
Designed for a one or two-semester liberal arts mathematics course, Mathematics Beyond the Numbers integrates a great deal of historical background so that students can see the development of mathematics over time. In addition, some of the topics included in Mathematics Beyond the Numbers are particularly applicable to a student’s field of study. For instance, voting methods and apportionment are of interest to a political science major
Mathematics Beyond the Numbers by George T. Gilbert and Rhonda L. Hatcher:
- Is Flexible! The chapters are independent of one another and can be presented in any order.
- Is Interactive! Students have access to an accompanying course website that includes online homework.
- Is Practical! Numerous exercises and examples teach students mathematical problem-solving skills so they can carry out what they have learned.
- Integrates Technology! Technology Tips guide instructors and students to the possibilities of using technology.
- Is Easy-to-Adopt! An Instructor’s Solutions Manual is provided to all adopting instructors.
PREFACE
CHAPTER 1 VOTING METHODS
1.1 Plurality and Runoff Methods
Runoff Elections
Preference Rankings
1.2 Borda’s Method: A Scoring System
1.3 Head-to-Head Comparisons
Single-Peaked Preference Rankings
1.4 Approval Voting
1.5 The Search for an Ideal Voting System
1.6 Weighted Voting Systems
Dictators, Dummies, and Veto Power
The Banzhaf Power Index
The Shapley-Shubik Power Index
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 2 APPORTIONMENT: SHARING WHAT CANNOT BE DIVIDED ARBITRARILY
2.1 Quota Methods
Hamilton’s Method
Lowndes’ Method
2.2 Early Divisor Methods
Jefferson’s Method
Webster’s Method
2.3 Apportionment in Today’s House of Representatives
The Hill–Huntington Method
Other Apportionment Methods
2.4 The Search for an Ideal Apportionment Method
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 3 THE MATHEMATICS OF MONEY
3.1 Powers, Roots, and Logarithms
3.2 Simple Interest
3.3 Compound Interest
Inflation
3.4 The Rewards of Systematic Savings
3.5 Amortized Loans
Finding a Loan Balance
Amortization Schedules
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 4 PROBABILITY
4.1 Elementary Probability
4.2 Odds
House Odds and Fair Bets
4.3 The Addition Rule
4.4 Conditional Probability and the Multiplication Rule
The Multiplication Rule
Independence and the Multiplication Rule
4.5 Counting Techniques
Permutations
Combinations
4.6 Probability Problems Using Counting Techniques
The Birthday Problem
4.7 Expected Value
4.8 Genetics
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 5 STATISTICS
5.1 Organizing and Presenting Data
Bar Graphs
Histograms
Pie Charts
5.2 Typical and Central Values
The Mode
The Median
The Mean
Estimating the Mean of Grouped Data
5.3 Measures of Spread
The Range
The Standard Deviation
Estimating the Standard Deviation of Grouped Data
5.4 The Normal Distribution
Percentiles
5.5 Estimating the Mean
Confidence Intervals
Sample Standard Deviation and Confidence Intervals
5.6 Polls and Margin of Error
5.7 Garbage In, Garbage Out: A Look at Misleading Uses of Statistics and at Sampling Techniques
The Source of the Data
The Data: Questions and Answers
The Conclusions Drawn from the Data
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 6 PATHS AND NETWORKS
6.1 Eulerian Paths and Circuits on Graphs
Eulerization
6.2 The Traveling Salesman Problem
The Nearest Neighbor Algorithm
The Greedy Algorithm
6.3 Efficient Networking: Minimal Spanning Trees
Prim’s Algorithm
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 7 TILINGS AND POLYHEDRA
7.1 Polygons
7.2 Tiling’s
Regular Tilings
Semiregular Tiling’s
Tiling’s with Nonregular Polygons
Tilings with Other Shapes
7.3 Polyhedra
Regular Polyhedra
Semiregular Polyhedra
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
CHAPTER 8 NUMBER THEORY
8.1 Divisibility and Primes
Prime Numbers
The Division Algorithm
The Greatest Common Divisor
8.2 Modular Arithmetic
8.3 Divisibility Tests
8.4 Check Digits
8.5 Tournament Scheduling
8.6 Introduction to Cryptology
The Caesar Cipher
Affine Ciphers
8.7 Advanced Encryption Methods
The Hill Cipher
The RSA Public Key System
Writing Exercises
Projects
Key Terms
Review Test
Suggested Readings
APPENDIX A
ANSWERS TO SELECTED EXERCISES
George Gilbert earned a B.A. in mathematics from Washington University in 1979 and a Ph.D. in mathematics from Harvard University in 1984, specializing in number theory. He has held teaching positions at the University of Texas in Austin, Saint Olaf College, and has been on the faculty at Texas Christian University for the past 20+ years. Professor Gilbert is active in the area of problem solving, having served on the William Lowell Putnam questions committee, the AIME contest committee, and as the editor of the Problems Section of Mathematics Magazine.
Rhonda Hatcher earned a B.A. in mathematics from the University of Colorado at Boulder in 1980 and a Ph.D. in mathematics from Harvard University in 1987. Her research specialty is number theory. She has taught at Saint Olaf College and is currently on the faculty of Texas Christian University. Professor Hatcher won the 1994 Deans’ Teaching Award, the 1997 Honors Professor of the Year Award, and the 2000 Chancellor’s Award for Distinguished Teaching at TCU. In 1998, the Mathematical Association of America awarded her the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics. In 2000, she was awarded the Texas Professor of the Year award from The Carnegie Foundation for Advancement of Teaching and Council for Advancement and Support of Education.