Mathematical Expeditions: Chronicles by the Explorers

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Springer Science & Business Media, 1999 - Mathematics - 275 pages
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This book contains the stories of five mathematical journeys into new realms, told through the writings of the explorers themselves. Some were guided by mere curiosity and the thrill of adventure, while others had more practical motives. In each case the outcome was a vast expansion of the known mathematical world and the realization that still greater vistas remained to be explored. The authors tell these stories by guiding the reader through the very words of the mathematicians at the heart of these events, and thereby provide insight into the art of approaching mathematical problems. The book can be used in a variety of ways. The five chapters are completely independent, each with varying levels of mathematical sophistication. The book will be enticing to students, to instructors, and to the intellectually curious reader. By working through some of the original sources and supplemental exercises, which discuss and solve - or attempt to solve - a great problem, this book helps the reader discover the roots of modern problems, ideas, and concepts, even whole subjects. Students will also see the obstacles that earlier thinkers had to clear in order to make their respective contributions to five central themes in the evolution of mathematics.
  

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Contents

Geometry The Parallel Postulate
1
12 Euclids Parallel Postulate
18
13 Legendres Attempts to Prove the Parallel Postulate
24
14 Lobachevskian Geometry
31
15 Poincares Euclidean Model for NonEuclidean Geometry
43
Set Theory Taming the Infinite
54
22 Bolzanos Paradoxes of the Infinite
69
23 Cantors Infinite Numbers
74
38 Appendix on Infinite Series
154
Number Theory Fermats Last Theorem
156
42 Euclids Classification of Pythagorean Triples
172
43 Eulers Solution for Exponent Four
179
44 Germains General Approach
185
45 Kummer and the Dawn of Algebraic Number Theory
193
46 Appendix on Congruences
199
Algebra The Search for an Elusive Formula
204

24 Zermelos Axiomatization
89
Analysis Calculating Areas and Volumes
95
32 Archimedes Quadrature of the Parabola
108
33 Archimedes Method
118
34 Cavalieri Calculates Areas of Higher Parabolas
123
35 Leibnizs Fundamental Theorem of Calculus
129
36 Cauchys Rigorization of Calculus
138
37 Robinson Resurrects Infinitesimals
150
52 Euclids Application of Areas and Quadratic Equations
219
53 Cardanos Solution of the Cubic
224
54 Lagranges Theory of Equations
233
55 Galois Ends the Story
247
References
259
Credits
269
Index
271
Copyright

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About the author (1999)

Laubenbacher of New Mexico State University, Las Cruces

Pengelley of New Mexico State University, Las Cruces

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