Infinite Processes: Background to AnalysisWhat 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. |
Contents
| 2 | |
33 | 13 |
NUMBER | 24 |
2 | 37 |
4 | 51 |
5 | 59 |
6 | 70 |
8 | 96 |
GEOMETRY | 156 |
2 The Role of Geometrical Intuition | 162 |
4 | 204 |
5 | 232 |
FUNCTIONS | 246 |
2 | 254 |
3 | 285 |
10 | 118 |
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Common terms and phrases
2-dimensions 3-dimensional a₁ a₂ algebraic arithmetic b₁ b₂ calculus Chapter column continued fraction corresponding cuboid curve decimal digits decimal fractions decimal places decimal representation defined diagonal dissection elementary endless sequence endless sum equal exactly example Exercise expression fact Figure finite sums formula Fourier series function f(x function-concept fundamental property geometrical graph greatest common measure Hence highest common factor idea inequalities infinite decimals infinite processes infinitesimal inner and outer integral interpreted intuitive irrational K₁ Leibniz length line segments mathematics n-gon naive numerical value obtain outer approximations pair pinpoints polygonal polyhedron polynomial positive whole number precisely problem procedure proof property of real prove pyramid r₁ rational functions real numbers recurring decimal remainder segment y sequence of finite shape Show simply square standard cuboid standard rectangle step tetrahedron triangle vertices volume WXYZ