Mathematical expeditions : chronicles by the explorers
This book contains the stories of five mathematical journeys into new realms, told through the writings of the explorers themselves. 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.
x, 275 p. : illustrations ; 25 cm
9780387984346, 9780387984339, 0387984348, 038798433X, 0387984348
1023876601
1 Geometry: The Parallel Postulate.- 1.1 Introduction.- 1.2 Euclid’s Parallel Postulate.- 1.3 Legendre’s Attempts to Prove the Parallel Postulate.- 1.4 Lobachevskian Geometry.- 1.5 Poincaré’s Euclidean Model for Non-Euclidean Geometry..- 2 Set Theory: Taming the Infinite.- 2.1 Introduction.- 2.2 Bolzano’s Paradoxes of the Infinite.- 2.3 Cantor’s Infinite Numbers.- 2.4 Zermelo’s Axiomatization.- 3 Analysis: Calculating Areas and Volumes.- 3.1 Introduction.- 3.2 Archimedes’ Quadrature of the Parabola.- 3.3 Archimedes’ Method.- 3.4 Cavalieri Calculates Areas of Higher Parabolas.- 3.5 Leibniz’s Fundamental Theorem of Calculus.- 3.6 Cauchy’s Rigorization of Calculus.- 3.7 Robinson Resurrects Infinitesimals.- 3.8 Appendix on Infinite Series.- 4 Number Theory: Fermat’s Last Theorem.- 4.1 Introduction.- 4.2 Euclid’s Classification of Pythagorean Triples.- 4.3 Euler’s Solution for Exponent Four.- 4.4 Germain’s General Approach.- 4.5 Kummer and the Dawn of Algebraic Number Theory.- 4.6 Appendix on Congruences.- 5 Algebra: The Search for an Elusive Formula.- 5.1 Introduction.- 5.2 Euclid’s Application of Areas and Quadratic Equations.- 5.3 Cardano’s Solution of the Cubic.- 5.4 Lagrange’s Theory of Equations.- 5.5 Galois Ends the Story.- References.- Credits.