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AuthorGardiner, A. author
TitleInfinite Processes [electronic resource] : Background to Analysis / by A. Gardiner
ImprintNew York, NY : Springer New York, 1982
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Descript X, 306 p. online resource


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 mathยญ 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 proยญ logue to analysis; secondly, as a supplementary text within an analysis course, providing a source of elementary motivation, background and exยญ 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 thisยญ including teachers, teachers in training, students in their last year at school, and others interested in mathematics


I From Calculus to Analysis -- I.1 Whatโ{128}{153}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 Arithmeticโ{128}{148}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?

Mathematics Functions of real variables Mathematics Real Functions


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