Understanding Infinity: The Mathematics of Infinite Processes
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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FROM CALCULUS TO ANALYSIS
2 Growth and Change in Mathematics
2 Constructive and Nonconstructive Methods
4 Sides and Diagonals of Regular Polygons
5 Numbers and ArithmeticA Quick Review
6 Infinite Decimals Part 1
7 Infinite Decimals Part 2
2 The Role of Geometrical Intuition
4 Comparing Volumes
5 Curves and Surfaces
1 What is a Number?
3 What Is an Exponential Function?
8 Recurring Nines
10 The Fundamental Property of Real Numbers
12 Ref1ections on Recurring Themes
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2-dimensions 3-dimensional algebraic approach area A'B'C'D area ABCD arithmetic base calculus Chapter column continued fraction corresponding decimal digits decimal fractions decimal places decimal representation defined diagonal differential dissected elementary endless sequence endless sum equal equation exactly example Exercise expression fact Figure finite sums formula Fourier series function function-concept fundamental property geometrical given graph greatest common measure Hence highest common factor idea inequalities infinite decimals infinite processes infinitesimal inner and outer integral interpreted intuitive irrational Leibniz length line segments mathematicians mathematics meaning method of inner n-gon naive numerical value obtain outer approximations pair particular polygonal polyhedron polynomial positive whole number precisely problem procedure proof property of real prove pyramid rational functions rational number real numbers recurring decimal remainder segment y sequence of finite shape shortest repeating block Show simply square standard cuboid standard rectangle step subtraction tetrahedron triangle vertices volume WXYZ zero