Mathematics Beyond the Numbers

Edition: 3

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 T Gilbert

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 L Hatcher

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 T Gilbert

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 L Hatcher

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.