Thinking Mathematically
Preface  To the StudentI've written this book to give you control over the part of your life that involves numbers and mathematical ideas. Gaining an understanding and appreciation of mathematics will help you participate fully in the twenty-first century. In some ways, you cannot get along in life without the mathematics in this book. For example, if you do not understand the basic ideas of investment, you may find it impossible to achieve your financial goals. If you do not have at least a rudimentary understanding of set theory, you may not be able to meaningfully interpret the vast amount of survey data we are presented with almost daily.This book has been written so that you can learn about the power of mathematics directly from its pages. All concepts are carefully explained, important definitions and procedures are set off in boxes, and worked-out examples that present solutions in a step-by-step manner appear in every section. Study tip boxes offer hints and suggestions and often point out common errors to avoid. A great deal of attention has been given to show ways to apply mathematics to your life in order to make your learning experience both interesting and relevant.As you begin your studies, I would like to offer some specific suggestions for using this book and for being successful in this course:1. Attend all lectures. No book is intended to be a substitute for valuable insights and interactions that occur in the classroom. In addition to arriving for lecture on time and prepared, you will find it useful to read the section before it is covered in lecture. This will give you a clear idea of the new material that will be discussed.2. Read the book. Read each section with pen (or pencil) in hand. Move through the illustrative examples with great care. These worked-out examples provide a model for doing the exercises in the exercise sets. As you proceed through the reading, do not give up if you do not understand every single word. Things will become clearer as you read on and see how various procedures are applied to specific worked-out examples.3. Work problems every day and check your answers. The way to learn mathematics is by doing mathematics, which means working the assigned exercises in the exercise sets. The more exercises you work, the better you will understand the material.4. Prepare for chapter exams. After completing a chapter, study the summary, work the exercises in the chapter review, and work the exercises in the chapter test. Answers to all these exercises are given in the back of the book.5. Use the supplements available with this book. A solutions manual containing worked-out solutions to the book's odd-numbered exercises and all review exercises, a dynamic web page, and video tapes created for every section of the book are among the supplements created to help you tap into the power of mathematics. Ask your instructor or bookstore what supplements are available and where you can find them.It is my hope that you will enjoy the pages of this book, as you empower yourself with the mathematics needed to succeed in college, your career, and in your life.   To the InstructorThinking Mathematically provides a general survey of mathematical topics that are useful in our contemporary world. My primary purpose in writing the book was to show students how mathematics can be applied to their lives in interesting, enjoyable, and meaningful ways. The book's variety of topics and flexibility of sequence make it appropriate for a one- or two-term course in liberal arts mathematics, finite mathematics, mathematics for education majors, as well as for courses specifically designed to meet state-mandated requirements in mathematics.I wrote the book with three major goals: first, to help students acquire knowledge of fundamental mathematics; second, to show students how mathematics can solve authentic problems that apply to their lives; and third, to enable students to develop problem-solving skills, fostering critical thinking, within a varied, interesting, and contemporary setting.I am avidly interested in psychology, philosophy, and literature, as well as mathematics. I've worked as a musician, psychotherapist, and even attempted book, music, and lyrics to an original musical. (Compared to writing for musical theater, authoring mathematics textbooks is a breeze!) I hope that my love for learning, as well as my respect for the diversity of students I have taught and learned from over the years, is apparent In the hundreds of applications that appear throughout the book. By connecting mathematics to the whole spectrum of learning, it is my intent to show students that their world is profoundly mathematical and, indeed, "pi" is in the sky.   Robert Blitzer