Synopses & Reviews
Conceived by the author as an introduction to "why the calculus works" (otherwise known as "analysis"), this volume represents a critical reexamination of the infinite processes encountered in elementary mathematics. Part I presents a broad description of the coming parts, and Part II offers a detailed examination of the infinite processes arising in the realm of number--rational and irrational numbers and their representation as infinite decimals. Most of the text is devoted to analysis of specific examples. Part III explores the extent to which the familiar geometric notions of length, area, and volume depend on infinite processes. Part IV defines the evolution of the concept of functions by examining the most familiar examples--polynomial, rational, exponential, and trigonometric functions. Exercises form an integral part of the text, and the author has provided numerous opportunities for students to reinforce their newly acquired skills. Unabridged republication of
Infinite Processes as published by Springer-Verlag, New York, 1982. Preface. Advice to the Reader. Index.
Synopsis
Conceived by the author as an introduction to "why the calculus works," this volume offers a 4-part treatment: an overview; a detailed examination of the infinite processes arising in the realm of numbers; an exploration of the extent to which familiar geometric notions depend on infinite processes; and the evolution of the concept of functions. 1982 edition.
Table of Contents
PART I FROM CALCULUS TO ANALYSIS
and#160; CHAPTER I.1 What's Wrong with the Calculus?
and#160; CHAPTER I.2 Growth and Change in Mathematics
PART II NUMBER
and#160; CHAPTER II.1 Mathematics: Rational or Irrational?
and#160; CHAPTER II.2 Constructive and Non-constructive Methods in Mathematics
and#160; "CHAPTER II.3 Common Measures, Highest Common Factors and the Game of Euclid"
and#160; CHAPTER II.4 Sides and Diagonals of Regular Polygons
and#160; CHAPTER II.5 Numbers and Arithmetic?A Quick Review
and#160; CHAPTER II.6 Infinite Decimals (Part 1)
and#160; CHAPTER II.7 Infinite Decimals (Part 2)
and#160; CHAPTER II.8 Recurring Nines
and#160; CHAPTER II.9 Fractions and Recurring Decimals
and#160; CHAPTER II.10 The Fundamental Property of Real Numbers
and#160; CHAPTER II.11 The Arithmetic of Infinite Decimals
and#160; CHAPTER II.12 Reflections on Recurring Themes
and#160; CHAPTER II.13 Continued Fractions
PART III GEOMETRY
and#160; CHAPTER III.1 Numbers and Geometry
and#160; CHAPTER III.2 The Role of Geometrical Intuition
and#160; CHAPTER III.3 Comparing Areas
and#160; CHAPTER III.4 Comparing Volumes
and#160; CHAPTER III.5 Curves and Surfaces
PART IV FUNCTIONS
and#160; CHAPTER IV.1 What Is a Number?
and#160; CHAPTER IV.2 What Is a Function?
and#160; CHAPTER IV.3 What Is an Exponential Function?
Index
Errata