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## Overview

Written for liberal arts students and based on the belief that learning to solve problems is the principal reason for studying mathematics, Karl Smith introduces students to Polya's problem-solving techniques and shows them how to use these techniques to solve unfamiliar problems that they encounter in their own lives. Through the emphasis on problem solving and estimation, along with numerous in-text study aids, students are assisted in understanding the concepts and mastering the techniques. In addition to the problem-solving emphasis, THE NATURE OF MATHEMATICS is renowned for its clear writing, coverage of historical topics, selection of topics, level, and excellent applications problems. Smith includes material on such practical real-world topics as finances (e.g. amortization, installment buying, annuities) and voting and apportionment. With the help of this text, thousands of students have "experienced" mathematics rather than just do problems--and benefited from a writing style that boosts their confidence and fosters their ability to use mathematics effectively in their everyday lives.

- The examples throughout the book have been redesigned to more effectively engage students. Each now includes a title and is presented in a fresh, easy-to-read format.
- A "Chapter Challenge" problem appears in each chapter, providing additional problem-solving practice. To encourage students to stretch their problem-solving skills, these new types of problems present contexts other than those presented in the chapter.
- "Historical Quest" problems, an overwhelming success in the previous edition, appear more frequently in the Twelfth Edition. These problems are designed to involve students in the historical development of the great ideas in mathematical history.
- Expanded and updated chapter openers include "What in the World?" introductions, which draw students into the material by relating the mathematics to common scenarios they may encounter in their everyday lives.
- The prologue--updated for this edition--asks the all-important question, "Why Math?". The prologue puts mathematics into a historical perspective for students, and is also designed to encourage them to begin thinking about problem solving.
- Updated for this edition, the problems accompanying the "Why Not Math?" epilogue show students how mathematics touches just about every course of study.

- Exercise Sets: Problems range from easy to challenging and are clearly marked by type. "Level 1" problems are mechanical or drill. "Level 2" problems require the understanding of concepts. "Level 3" problems require problem-solving skills or original thinking.
- Communication Skills: "In Your Own Words" problems provide students with practice in communicating the concepts they've learned. These problems are especially helpful for education majors who will go on to teach in secondary schools.
- Research: "Individual Research Projects" help students apply what they are learning to familiar situations. These projects require research or library work, further reinforcing for students the value of mathematical concepts in a variety of fields.
- Group Work: "Group Research Projects" require collaborative research work and are suitable for larger-scale class projects.
- Website: The text-specific website (www.mathnature.com) provides homework hints, a listing of essential ideas, projects, a list of important terms and types of problems (including section references), and links to related information on the Web.
- Real-World Problem-Solving Emphasis: Polya's method of problem solving is used throughout the text. Students learn how to apply these techniques to solve problems they encounter in their everyday lives in such areas as finance (installment buying, amortization, annuities) and voting and apportionment.
- Flexibility: The chapters are independent of one another and allow maximum teaching flexibility.
- Learning Aids: The book offers a variety of interesting and helpful learning aids. Icons like "Caution" and "Stop" guide students through concepts. "Historical Notes" place students' view of mathematics in a historical perspective and include a category of related problems entitled "Historical Quest". "Chapter Summaries" include chapter review exercises, group research projects, and individual research projects. Additionally, a list of important terms and types of problems, including section references, is available to students at www.mathnature.com.

1. THE NATURE OF PROBLEM SOLVING.

The Nature of Problem Solving. Problem Solving. Inductive and Deductive Reasoning. Scientific Notation and Estimation. Summary.

2. THE NATURE OF SETS.

Sets, Subsets, and Venn Diagrams. Operations with Sets. Applications of Sets. Finite and Infinite Sets. Summary.

3. THE NATURE OF LOGIC.

Deductive Reasoning. Truth Tables and the Conditional. Operators and Laws of Logic. The Nature of Proof. Problem Solving Using Logic. Logic Circuits (Optional). Summary.

4. THE NATURE OF NUMERATION SYSTEMS.

Early Numeration Systems. Hindu-Arabic Numeration System. Different Numeration Systems. Binary Numeration System. History of Calculating Devices (Optional). Summary.

5. THE NATURE OF NUMBERS.

Natural Numbers. Prime Numbers. Integers. Rational Numbers. Irrational Numbers. Groups, Fields, and Real Numbers. Discrete Mathematics. Cryptography (Optional). Summary.

6. THE NATURE OF ALGEBRA.

Polynomials. Factoring. Evaluation, Applications, and Spreadsheets. Equations.

GUEST ESSAY: "CHAOS". Inequalities. Algebra in Problem Solving. Ratios, Proportions, and Problem Solving. Percents. Modeling Uncategorized Problems. Summary.

7. THE NATURE OF GEOMETRY.

Geometry. Polygons and Angles. Triangles. Similar Triangles. Right Triangle Trigonometry (Optional). Mathematics, Art, and Non-Euclidean Geometries (Optional Sections). Summary.

8. THE NATURE OF NETWORKS AND GRAPH THEORY.

Euler Circuits and Hamiltonian Cycles. Trees and Minimum Spanning Trees. Topology and Fractals. GUEST ESSAY: "WHAT GOOD ARE FRACTALS?". Summary.

9. THE NATURE OF MEASUREMENT.

Perimeter. Area. Surface Area, Volume and Capacity. Miscellaneous Measurements (Optional). U.S.-Metric Conversions. Summary.

10. THE NATURE OF GROWTH.

Exponential Equations. Logarithmic Equations. Applications of Growth and Decay. Summary.

11. THE NATURE OF SEQUENCES, SERIES, AND FINANCIAL MANAGEMENT.

Interest. Installment Buying. Sequences. Series. Annuities. Amortization. Summary of Financial Formulas. Summary.

12. THE NATURE OF COUNTING.

Permutations. Combinations. Counting without Counting. Rubik''s Cube and Instant Insanity (Optional). Summary.

13. THE NATURE OF PROBABILITY.

Introduction to Probability. Mathematical Expectation. Probability Models. Calculated Probabilities. The Binomial Distribution (Optional). GUEST ESSAY: "EXTRASENSORY PERCEPTION". Summary.

14. THE NATURE OF STATISTICS.

Frequency Distributions and Graphs. Descriptive Statistics. The Normal Curve. Correlation and Regression. Sampling (Optional). Summary.

15. THE NATURE OF GRAPHS AND FUNCTIONS.

Cartesian Coordinates and Graphing Lines. Graphing Half-Planes. Graphing Curves. Conic Sections. Functions. Summary.

16. THE NATURE OF MATHEMATICAL SYSTEMS.

Systems of Linear Equations. Problem Solving with Systems. Matrix Solution of a System of Equations. Inverse Matrices. Modeling with Linear Programming (Optional). Summary.

17. THE NATURE OF VOTING AND APPORTIONMENT.

Voting. Voting Dilemmas. Apportionment. Apportionment Flaws. Summary.

18. THE NATURE OF CALCULUS (OPTIONAL).

What Is Calculus? Limits. Derivatives. Integrals. Summary.

Epilogue: Why Not Math? Mathematics in the Natural Sciences, Social Sciences, and in the Humanities.

Appendices.

A. Glossary.

B. Selected Answers.

C. Index.

Cengage provides a range of supplements that are updated in coordination with the main title selection. For more information about these supplements, contact your Learning Consultant.

### FOR INSTRUCTORS

#### Instructor's Manual

ISBN: 9780538495271

Written by author Karl Smith, the Instructor's Manual provides worked-out solutions to all of the problems in the text. For instructors only.

#### PowerLecture

ISBN: 9780840053305

This CD-ROM provides the instructor with dynamic media tools for teaching. Create, deliver, and customize tests (both print and online) in minutes with ExamView® Computerized Testing Featuring Algorithmic Equations. Easily build solution sets for homework or exams using Solution Builder's online solutions manual. Microsoft® PowerPoint® lecture slides and figures from the book are also included on this CD-ROM.

#### Annotated Instructor's Edition

ISBN: 9780538738699

The Annotated Instructor's Edition features an appendix containing the answers to all problems in the book as well as icons denoting which problems can be found in Enhanced WebAssign.

### FOR STUDENTS

#### Student Survival and Solutions Manual

ISBN: 9780538495288

The Student Survival and Solutions Manual provides helpful study aids and fully worked-out solutions to all of the odd-numbered exercises in the text. It's a great way to check your answers and ensure that you took the correct steps to arrive at an answer.