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Table of Contents

1. Problem Solving: Strategies and Principles

1.1 Problem Solving

1.2 Inductive and Deductive Reasoning

1.3 Estimation

 

2. Set Theory: Using Mathematics to Classify Objects

2.1 The Language of Sets

2.2 Comparing Sets

2.3 Set Operations

2.4 Survey Problems

2.5 Looking Deeper: Infinite Sets

 

3. Logic: The Study of What's True or False or Somewhere in Between

3.1 Statements, Connectives, and Quantifiers

3.2 Truth Tables

3.3 The Conditional and Biconditional

3.4 Verifying Arguments

3.5 Using Euler Diagrams to Verify Syllogisms

3.6 Looking Deeper: Fuzzy Logic

 

4. Graph Theory (Networks): The Mathematics of Relationships

4.1 Graphs, Puzzles, and Map Coloring

4.2 The Traveling Salesperson Problem

4.3 Directed Graphs

4.4 Looking Deeper: Scheduling Projects Using PERT

 

5. Numeration Systems: Does It Matter How We Name Numbers?

5.1 The Evolution of Numeration Systems

5.2 Place Value Systems

5.3 Calculating in Other Bases

5.4 Looking Deeper: Modular Systems

 

6. Number Theory and the Real Number System: Understanding the Numbers All Around Us

6.1 Number Theory

6.2 The Integers

6.3 The Rational Numbers

6.4 The Real Number System

6.5 Exponents and Scientific Notation

6.6 Looking Deeper: Sequences

 

7. Algebraic Models: How Do We Approximate Reality?

7.1 Linear Equations

7.2 Modeling with Linear Equations

7.3 Modeling with Quadratic Equations

7.4 Exponential Equations and Growth

7.5 Proportions and Variation

7.6 Functions

7.7 Looking Deeper: Dynamical Systems

 

8. Modeling with Systems of Linear Equations and Inequalities: What's the Best Way to Do It?

8.1 Systems of Linear Equations

8.2 Systems of Linear Inequalities

8.3 Looking Deeper: Linear Programming

 

9. Consumer Mathematics: The Mathematics of Everyday Life

9.1 Percent

9.2 Interest

9.3 Consumer Loans

9.4 Annuities

9.5 Amortization

9.6 Personal Finance

9.7 Looking Deeper: The Annual Percentage Rate

 

10. Geometry: Ancient and Modern Mathematics Embrace

10.1 Lines, Angles, and Circles

10.2 Polygons

10.3 Perimeter and Area

10.4 Volume and Surface Area

10.5 The Metric System and Dimensional Analysis

10.6 Geometric Symmetry and Tessellations

10.7 Looking Deeper: Fractals

 

11. Apportionment: How Do We Measure Fairness?

11.1 Understanding Apportionment

11.2 The Huntington-Hill Apportionment Principle

11.3 Applications of the Apportionment Principle

11.4 Other Paradoxes and Apportionment Methods

11.5 Looking Deeper: Fair Division

 

12. Voting: Using Mathematics to Make Choices

12.1 Voting Methods

12.2 Defects in Voting Methods

12.3 Weighted Voting Systems

12.4 Looking Deeper: The Shapley-Shubik Index

 

13. Counting: Just How Many Are There?

13.1 Introduction to Counting Methods

13.2 The Fundamental Counting Principle

13.3 Permutations and Combinations

13.4 Looking Deeper: Counting and Gambling

 

14. Probability: What Are the Chances?

14.1 The Basics of Probability Theory

14.2 Complements and Unions of Events

14.3 Conditional Probability and Intersections of Events

14.4 Expected Value

14.5 Looking Deeper: Binomial Experiments

 

15. Descriptive Statistics: What a Data Set Tells Us

15.1 Organizing and Visualizing Data

15.2 Measures of Central Tendency

15.3 Measures of Dispersion

15.4 The Normal Distribution

15.5 Looking Deeper: Linear Correlation

 

Appendix A Basic Mathematics Review

 

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