The Macmillian Company

New York, 1973

**Preface**

This book is designed for a wide variety of college and university students, including those who have never had a course in high school mathematics. It is written in the hope that these students and others will want to become involved in mathematics so that they can learn to appreciate, understand, use, and enjoy it. A major objective is to present an interesting approach to mathematics that does not involve complicated algebraic manipulations. The topics were chosen on the basis of affirmative answers to each of the following questions: (1) Is it mathematically significant? (2) Is it easy to understand? (3) Is it interesting?

The chapter titles are as follows: (1) "Mathematical Recreations," (2) "What Is Mathematics?" (3) "Logic," (4) "Sets and Paradoxes," (5) "Geometry," (6) "Counting and Probability," (7) "Statistics," (8) "Linear Algebra," (9) "Game Theory," (10) "Calculus," (11) "Computers," and there is an appendix on the real number system. The first four chapters do not require any algebra; Chapter 5 introduces some basic rules of algebra via geometry; and the appendix presents some basic properties of the real numbers in an intuitive way. There is no attempt to present a formal development of the real number system.

The book can be used a s text in a variety of ways. For example, Chapters 1 through 4 can be used for a one-quarter course; Chapters 1 through 6 can be used for a one-semester course. the instructor may also want to have students present reports on the lives of mathematicians or on mathematical topics related to the course. The Suggestions for Further reading may help students select topics of real interest.

Students in the social, natural, and mathematical sciences may find some of the chapters to be of special interest. In particular, Chapters 6, 7, 8, 9, and 11 may be of special interest to those in the social sciences; Chapters 6, 7,8, 10, and 11 may be of special interest to those in the natural sciences; and Chapters 2, 3, 4, 10, and 11 may be of special interest to those in the mathematical sciences.

Although the book has been written for the novice, some of the problems may challenge students with a good background in high school mathematics. Those problems that have been rated "more challenging" or "optional" are preceded by the symbol *.

- Mathematical Recreations

1.1 Introduction

1.2 Mathematical Patterns

- What is Mathematics

2.1 Abstract Mathematical Structures

2.2 Boolean Arithmetic

- Logic

3.1 Introduction

3.2 AMS (Logic)

3.3 Methods for Proving Theorems in AMS (Logic)

3.4 Converse and Contrapositive

3.5 Concrete Logic

- Sets and Paradoxes

4.1 What Is a Set?

4.2 Some Relations for Sets

4.3 What Is a Number?

4.4 Some Operations for Sets

4.5 The Start Product

4.6 Addition of Cardinal Numbers

4.7 A Practical Application

4.8 The Relationship Between AMS (Sets) and AMS (Logic)

4.9 Some Paradoxes

4.10 Formalism vs. Intuitionism

- Geometry

5.1 The Origin of Geometry

5.2 Concrete Geometry in the Plane

5.3 Geometry and Algebra

5.4 Concrete Geometry in Space

5.5 Points and Numbers

5.6 The Straight Line

5.7 The Conic Sections

- Counting and Probability

6.1 Introduction

6.2 Counting Problems

6.3 Permutations

6.4 Combinations

6.5 Probability

6.6 Odds and Mathematical Expectation

- Statistics

7.1 What is Statistics?

7.2 Mean, Median, and Mode

7.3 Standard Deviation

7.4 The Normal Curve

- Linear Algebra

8.1 Equations

8.2 Linear Equations

8.3 What Is a Matrix?

8.4 Addition of Matrices

8.5 Multiplication of Matrices

8.6 Applications of Matrices

8.7 Linear Programming

- Game Theory

9.1 What is Game Theory?

9.2 Matrix Games with Saddle Points

9.3 Solution of the Matching Pennies Game

9.4 Matrix Games Without Saddle Points

9.5 An Application of Linear Programming to Game Theory

9.6 Nonzero-Sum Games

9.7 Voting Games

- Calculus

10.1 Introduction

10.2 Velocities and Slopes

10.3 Functions

10.4 The Derivative and Geometry

10.5 The Delta Process

10.6 Some Shortcuts

10.7 Maxima and Minima

10.8 The Antiderivative and the Intuitive Integral

10.9 Areas and Integrals

- Computers

11.1 Introduction

11.2 A Computer Language FORTRAN

11.3 Applications of Computers

11.4 History of Computers

11.5 Computers vs. Humans