Infinite processes, background to analysis
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Infinite processes, background to analysis
Springer-Verlag, c1982
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Includes index
Description and Table of Contents
Description
What shall we say of this metamorphosis in passing from finite to infinite? Galileo, Two New Sciences As its title suggests, this book was conceived as a prologue to the study of "Why the calculus works"--otherwise known as analysis. It is in fact a critical reexamination of the infinite processes arising in elementary math- ematics: Part II reexamines rational and irrational numbers, and their representation as infinite decimals; Part III examines our ideas of length, area, and volume; and Part IV examines the evolution of the modern function-concept. The book may be used in a number of ways: firstly, as a genuine pro- logue to analysis; secondly, as a supplementary text within an analysis course, providing a source of elementary motivation, background and ex- amples; thirdly, as a kind of postscript to elementary analysis-as in a senior undergraduate course designed to reinforce students' understanding of elementary analysis and of elementary mathematics by considering the mathematical and historical connections between them.
But the contents of the book should be of interest to a much wider audience than this- including teachers, teachers in training, students in their last year at school, and others interested in mathematics.
Table of Contents
I From Calculus to Analysis.- I.1 What's Wrong with the Calculus?.- I.2 Growth and Change in Mathematics.- II Number.- II.1 Mathematics: Rational or Irrational?.- II.2 Constructive and Non-constructive Methods in Mathematics.- II.3 Common Measures, Highest Common Factors and the Game of Euclid.- II.4 Sides and Diagonals of Regular Polygons.- II.5 Numbers and Arithmetic-A Quick Review.- II.6 Infinite Decimals (Part 1).- II.7 Infinite Decimals (Part 2).- II.8 Recurring Nines.- II.9 Fractions and Recurring Decimals.- II.10 The Fundamental Property of Real Numbers.- II.11 The Arithmetic of Infinite Decimals.- II.12 Reflections on Recurring Themes.- II.13 Continued Fractions.- III Geometry.- III.1 Numbers and Geometry.- III.2 The Role of Geometrical Intuition.- III.3 Comparing Areas.- III.4 Comparing Volumes.- III.5 Curves and Surfaces.- IV Functions.- IV.1 What Is a Number?.- IV.2 What Is a Function?.- IV.3 What Is an Exponential Function?.
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