Infinite Processes

1. Front Matter

   Pages i-ix

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2. From Calculus to Analysis

   1. Front Matter

      Pages 1-1

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   2. What’s Wrong with the Calculus?

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      Pages 3-12

   3. Growth and Change in Mathematics

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      Pages 13-23

3. Number

   1. Front Matter

      Pages 25-25

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   2. Mathematics: Rational or Irrational?

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      Pages 27-36

   3. Constructive and Non-constructive Methods in Mathematics

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      Pages 37-39

   4. Common Measures, Highest Common Factors and the Game of Euclid

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      Pages 40-50

   5. Sides and Diagonals of Regular Polygons

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      Pages 51-58

   6. Numbers and Arithmetic— A Quick Review

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      Pages 59-69

   7. Infinite Decimals (Part 1)

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      Pages 70-80

   8. Infinite Decimals (Part 2)

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      Pages 81-95

   9. Recurring Nines

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      Pages 96-99

   10. Fractions and Recurring Decimals

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       Pages 100-117

   11. The Fundamental Property of Real Numbers

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       Pages 118-122

   12. The Arithmetic of Infinite Decimals

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       Pages 123-133

   13. Reflections on Recurring Themes

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       Pages 134-136

   14. Continued Fractions

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       Pages 137-154

4. Geometry

   1. Front Matter

      Pages 155-155

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   2. Numbers and Geometry

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      Pages 157-161

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.

• Analysis
• Area
• Calc
• Elementary Analysis
• Finite
• Mathematica
• Prolog
• Volume
• calculus
• design
• evolution
• function
• mathematics
• presentation
• training

• Department of Mathematics, University of Birmingham, Birmingham, England

  A. Gardiner

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