Understanding Infinity: The Mathematics of Infinite Processes

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Courier Corporation, 1982 - Mathematics - 308 pages
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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.
 

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Contents

FROM CALCULUS TO ANALYSIS
1
2 Growth and Change in Mathematics
13
NUMBER
25
2 Constructive and Nonconstructive Methods
37
4 Sides and Diagonals of Regular Polygons
51
5 Numbers and ArithmeticA Quick Review
59
6 Infinite Decimals Part 1
70
7 Infinite Decimals Part 2
81
GEOMETRY
155
2 The Role of Geometrical Intuition
162
4 Comparing Volumes
204
5 Curves and Surfaces
232
FUNCTIONS
245
1 What is a Number?
251
3 What Is an Exponential Function?
285
Index
301

8 Recurring Nines
96
10 The Fundamental Property of Real Numbers
118
12 Ref1ections on Recurring Themes
134
Errata
307
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