# Mathematical Ideas

**by**Miller, Charles D.; Heeren, Vern E.; Hornsby, John

### 9780321693815

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## Summary

## Author Biography

**Charles Miller** has taught at America River College for many years.

**Vern Heeren** received his bachelor's degree from Occidental College and his master's degree from the University of California, Davis, both in mathematics. He is a retired professor of mathematics from American River College where he was active in all aspects of mathematics education and curriculum development for thirty-eight years. Teaming with Charles D. Miller in 1969 to write Mathematical Ideas, the pair later collaborated on Mathematics: An Everyday Experience; John Hornsby joined as co-author of Mathematical Ideas on the later six editions. Vern enjoys the support of his wife, three sons, three daughters in-law, and eight grandchildren.

**John Hornsby:**When a young John Hornsby enrolled in Lousiana State University, he was uncertain whether he wanted to study mathematics education or journalism. Ultimately, he decided to become a teacher. After twenty five years in high school and university classrooms, each of his goals has been realized. His passion for teaching and mathematics manifests itself in his dedicated work with students and teachers, while his penchant for writing has, for twenty five years, been exercised in the writing of mathematics textbooks. Devotion to his family (wife Gwen and sons Chris, Jack, and Josh), numismatics (the study of coins) and record collecting keep him busy when he is not involved in teaching or writing. He is also an avid fan of baseball and music of the 1960's. Instructors, students, and the 'general public' are raving about his recent Math Goes to Hollywood presentations across the country.

## Table of Contents

**1. The Art of Problem Solving**

1.1 Solving Problems by Inductive Reasoning

1.2 An Application of Inductive Reasoning: Number Patterns

1.3 Strategies for Problem Solving

1.4 Calculating, Estimating, and Reading Graphs

Extension: Using Writing to Learn about Mathematics

Collaborative Investigation: Discovering Patterns in Pascal's Triangle

Chapter 1 Test

**2. The Basic Concepts of Set Theory**

2.1 Symbols and Terminology

2.2 Venn Diagrams and Subsets

2.3 Set Operations and Cartesian Products

2.4 Surveys and Cardinal Numbers

Extension: Infinite Sets and Their Cardinalities

Collaborative Investigation: Surveying the Members of Your Class

Chapter 2 Test

**3. Introduction to Logic**

3.1 Statements and Quantifiers

3.2 Truth Tables and Equivalent Statements

3.3 The Conditional and Circuits

3.4 The Conditional and Related Statements

3.5 Analyzing Arguments with Euler Diagrams

Extension: Logic Problems and Sudoku

3.6 Analyzing Arguments with Truth Tables

Collaborative Investigation: Logic Problems and Sudoku Revisited

Chapter 3 Test

**4. Numeration Systems**

4.1 Historical Numeration Systems

4.2 More Historical Numeration Systems

4.3 Arithmetic in the Hindu-Arabic System

4.4 Conversion between Number Bases

Extension: Modular Systems

Collaborative Investigation: A Perpetual Calendar Algorithm

Chapter 4 Test

**5. Number Theory**

5.1 Prime and Composite Numbers

5.2 Large Prime Numbers

5.3 Selected Topics from Number Theory

5.4 Greatest Common Factor and Least Common Multiple

Extension: Modern Cryptography

5.5 The Fibonacci Sequence and the Golden Ratio

Collaborative Investigation: Investigating an Interesting Property of Number Squares

Chapter 5 Test

**6. The Real Numbers and Their Representations**

6.1 Real Numbers, Order, and Absolute Value

6.2 Operations, Properties, and Applications of Real Numbers

6.3 Rational Numbers and Decimal Representation

6.4 Irrational Numbers and Decimal Representation

6.5 Applications of Decimals and Percents

Extension: Complex Numbers

Collaborative Investigation: Budgeting to Buy a Car

Chapter 6 Test

**7. The Basic Concepts of Algebra**

7.1 Linear Equations

7.2 Applications of Linear Equations

7.3 Ratio, Proportion, and Variation

7.4 Linear Inequalities

7.5 Properties of Exponents and Scientific Notation

7.6 Polynomials and Factoring

7.7 Quadratic Equations and Applications

Extension: Complex Solutions of Quadratic Equations

Collaborative Investigation: How Does Your Walking Rate Compare to That of Olympic Race-walkers?

Chapter 7 Test

**8. Graphs, Functions, and Systems of Equations and Inequalities**

8.1 The Rectangular Coordinate System and Circles

8.2 Lines, Slope, and Average Rate of Change

8.3 Equations of Lines and Linear Models

8.4 An Introduction to Functions: Linear Functions, Applications, and Models

8.5 Quadratic Functions, Graphs, and Models

8.6 Exponential and Logarithmic Functions, Applications, and Models

8.7 Systems of Equations

8.8 Applications of Systems

Extension: Using Matrix Row Operations to Solve Systems

8.9 Linear Inequalities, Systems, and Linear Programming

Collaborative Investigation: Tracking an Epidemic

Chapter 8 Test

**9. Geometry**

9.1 Points, Lines, Planes, and Angles

9.2 Curves, Polygons, and Circles

Extension: Geometric Constructions

9.3 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem

9.4 Perimeter, Area, and Circumference

9.5 Volume and Surface Area

9.6 Transformational Geometry

9.7 Non-Euclidean Geometry, Topology, and Networks

9.8 Chaos and Fractal Geometry

Collaborative Investigation: Generalizing the Angle Sum Concept

Chapter 9 Test

**10. Counting Methods (formerly Chapter 11)**

10.1 Counting by Systematic Listing

10.2 Using the Fundamental Counting Principle

10.3 Using Permutations and Combinations

10.4 Using Pascal's Triangle

Extension: Magic Squares

10.5 Counting Problems Involving "Not" and "Or"

Collaborative Investigation: Solving a Traveling Salesman Problem

Chapter 10 Test

**11. Probability (formerly Chapter 12)**

11.1 Basic Concepts

11.2 Events Involving "Not" and "Or"

11.3 Conditional Probability; Events Involving "And"

11.4 Binomial Probability

11.5 Expected Value

Extension: Estimating Probabilities by Simulation

Collaborative Investigation: Finding Empirical Values of

Chapter 11 Test

**12. Statistics (formerly Chapter 13)**

12.1 Visual Displays of Data

12.2 Measures of Central Tendency

12.3 Measures of Dispersion

12.4 Measures of Position

12.5 The Normal Distribution

Extension: Regression and Correlation

Collaborative Investigation: Combining Sets of Data

Chapter 12 Test

**13. Personal Financial Management (formerly Chapter 14)**

13.1 The Time Value of Money

13.2 Consumer Credit

13.3 Truth in Lending

13.4 The Costs and Advantages of Home Ownership

13.5 Financial Investments

Extension: Ponzi Schemes and Other Investment Frauds

Collaborative Investigation: To Buy or to Rent?

Chapter 13 Test

**14. Trigonometry (formerly Ch. 10)**

14.1 Angles and Their Measures

14.2 Trigonometric Functions of Angles

14.3 Trigonometric Identities

14.4 Right Triangles and Function Values

14.5 Applications of Right Triangles

14.6 The Laws of Sines and Cosines

Extension: Area Formulas for Triangles

Collaborative Investigation: Making a Point about Trigonometric Function Values

Chapter 14 Test

**15. Graph Theory**

15.1 Basic Concepts

15.2 Euler Circuits

15.3 Hamilton Circuits and Algorithms

15.4 Minimum Spanning Trees

Extension: Route Planning

Collaborative Investigation: Finding the Number of Edges in a Complete Graph

Chapter 15 Test

**16. Voting and Apportionment**

16.1 The Possibilities of Voting

16.2 The Impossibilities of Voting

16.3 The Possibilities of Apportionment

Extension: Two Additional Apportionment Methods

16.4 The Impossibilities of Apportionment

Collaborative Investigation: Favorites, An Election Exploration

Chapter 16 Test

**Appendix: The Metric System**