Part 1. The Mathematics of Social Choice
1. The Mathematics of Voting: The Paradox of Democracy
1.1 Preference Ballots and Preference Schedules
1.2 The Plurality Method
1.3 The Borda Count Method
1.4 The Plurality-with-Elimination Method (Instant Runoff Voting)
1.5 The Method of Piecewise Comparisons
1.6 Rankings
Profile: Kenneth J. Arrow
Key Concepts
Exercises
Projects and Papers
References and Further Readings
2. The Mathematics of Power: Weighted Voting
2.1 An Introduction to Weighted Voting
2.2 The Banzhaf Power Index
2.3 Applications of the Banzhaf Power Index
2.4 The Shapely-Shubik Power Index
2.5 Applications of the Shapely-Shubik Power Index
Profile: Lloyd S. Shapely
Key Concepts
Exercises
Projects and Papers
References and Further Readings
3. The Mathematics of Sharing: Fair-Division Games
3.1 Fair-Division Games
3.2 Two Players: The Divider-Chooser Method
3.3 The Lone-Divider Method
3.4 The Lone-Chooser Method
3.5 The Last-Diminisher Method
3.6 The Method of Sealed Bids
3.7 The Method of Markers
Profile: Hugo Steinhaus
Key Concepts
Exercises
Projects and Papers
References and Further Readings
4. The Mathematics of Apportionment: Making the Rounds
4.1 Apportionment Problems
4.2 Hamilton's Method and the Quota Rule
4.3 The Alabama and Other Paradoxes
4.4 Jefferson's Method
4.5 Adams's Method
4.6 Webster's Method
Historical Note: A Brief History of Apportionment in the United States
Key Concepts
Exercises
Projects and Papers
References and Further Readings
Mini-Excursion 1: Apportionment Today
Part 2. Management Science
5. The Mathematics of Getting Around: Euler Paths and Circuits
5.1 Euler Circuit Problems
5.2 What is a Graph?
5.3 Graph Concepts and Terminology
5.4 Graph Models
5.5 Euler's Theorems
5.6 Fleury's Algorithm
5.7 Eulerizing Graphs
Profile: Leonard Euler
Key Concepts
Exercises
Projects and Papers
References and Further Readings
6. The Mathematics of Touring: The Traveling Salesman Problem
6.1 Hamilton Circuits and Hamilton Paths
6.2 Complete Graphs
6.3 Traveling Salesman Problems
6.4 Simple Strategies for Solving TSPs
6.5 The Brute-Force and Nearest-Neighbor Algorithms
6.6 Approximate Algorithms
6.7 The Repetitive Nearest-Neighbor Algorithm
6.8 The Cheapest Link Algorithm
Profile: Sir William Rowan Hamilton
Key Concepts
Exercises
Projects and Papers
References and Further Readings
7. The Mathematics of Networks: The Cost of Being Connected
7.1 Trees
7.2 Spanning Trees
7.3 Kruskal's Algorithm
7.4 The Shortest Network Connecting Three Points
7.5 Shortest Networks for Four or More Points
Profile: Evangelista Torricelli
Key Concepts
Exercises
Projects and Papers
References and Further Readings
8. The Mathematics of Scheduling: Chasing the Critical Path
8.1 The Basic Elements of Scheduling
8.2 Directed Graphs (Digraphs)
8.3 Scheduling with Priority Lists
8.4 The Decreasing-Time Algorithm
8.5 Critical Paths
8.6 The Critical-Path Algorithm
8.7 Scheduling with Independent Tasks
Profile: Ronald L. Graham
Key Concepts
Exercises
Projects and Papers
References and Further Readings
Mini-Excursion 2: A Touch of Color
Part 3. Growth And Symmetry
9. The Mathematics of Spiral Growth: Fibonacci Numbers and the Golden Ratio
9.1 Fibonacci's Rabbits
9.2 Fibonacci Numbers
9.3 The Golden Ratio
9.4 Gnomons
9.5 Spiral Growth in Nature
Profile: Leonardo Fibonacci
Key Concepts
Exercises
Projects and Papers
References and Further Readings
10. The Mathematics of Money: Spending it, Saving It, and Growing It
10.1 Percentages
10.2 Simple Interest
10.3 Compound Interest
10.4 Geometric Sequences
10.5 Deferred Annuities: Planned Savings for the Future
Key Concepts
Exercises
Projects and Papers
References and Further Readings
11. The Mathematics of Symmetry: Beyond Reflection
11.1 Rigid Motions
11.2 Reflections
11.3 Rotations
11.4 Translations
11.5 Glide Reflections
11.6 Symmetry as a Rigid Motion
11.7 Patterns
Profile: Sir Roger Penrose
Key Concepts
Exercises
Projects and Papers
References and Further Readings
12. The Geometry of Fractal Shapes: Naturally Irregular
12.1 The Koch Snowflake
12.2 The Sierpinski Gasket
12.3 The Chaos Game
12.4 The Twisted Sierpinski Gasket
12.5 The Mandelbrot Set
Profile: Benoit Mandelbrot
Key Concepts
Exercises
Projects and Papers
References and Further Readings
Mini-Excursion 3: The Mathematics of Population Growth: There is Strength in Numbers
Part 4. Statistics
13. Collecting Statistical Data: Censuses, Surveys, and Clinical Studies
13.1 The Population
13.2 Sampling
13.3 Random Sampling
13.4 Sampling: Terminology and Key Concepts
13.5 The Capture-Recapture Method
13.6 Clinical Studies
Profile: George Gallup
Key Concepts
Exercises
Projects and Papers
References and Further Readings
14. Descriptive Statistics: Graphing and Summarizing Data
14.1 Graphical Descriptions of Data
14.2 Variables
14.3 Numerical Summaries of Data
14.4 Measures of Spread
Profile: W. Edwards Deming
Key Concepts
Exercises
Projects and Papers
References and Further Readings
15. Chances, Probabilities, and Odds: Measuring Uncertainty
15.1 Random Experiments and Sample Spaces
15.2 Counting Outcomes in Sample Spaces
15.3 Permutations and Combinations
15.4 Probability Spaces
15.5 Equiprobable Spaces
15.6 Odds
Profile: Persi Diaconis
Key Concepts
Exercises
Projects and Papers
References and Further Readings
16. The Mathematics of Normal Distributions: The Call of the Bell
16.1 Approximately Normal Distributions of Data
16.2 Normal Curves and Normal Distributions
16.3 Standardizing Normal Data
16.4 The 68-95-99.7 Rule
16.5 Normal Curves as Models of Real-Life Data Sets
16.6 Distributions of Random Events
16.7 Statistical Inference
Profile: Carl Friedrich Gauss
Key Concepts
Exercises
Projects and Papers
References and Further Readings
Mini-Excursion 4: The Mathematics of Managing Risk