Mathematical Expeditions: Chronicles by the ExplorersSpringer Science & Business Media, 1. des. 2013 - 278 sider 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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Side 3
... through P. Euclid proves the existence of parallels in Proposition 31 of Book I , without the use of the parallel postulate . Uniqueness ' Greek mathematician , approx . 70 B.C.E. P " ' FIGURE 1.1 . Transport of angles . 1.1 Introduction 3.
... through P. Euclid proves the existence of parallels in Proposition 31 of Book I , without the use of the parallel postulate . Uniqueness ' Greek mathematician , approx . 70 B.C.E. P " ' FIGURE 1.1 . Transport of angles . 1.1 Introduction 3.
Side 4
... propositions can all be proved without using the parallel postulate . It is used , however , in Euclid's proof of Proposition 32. Much subsequent effort was focused on understanding the precise relationship between this result and the ...
... propositions can all be proved without using the parallel postulate . It is used , however , in Euclid's proof of Proposition 32. Much subsequent effort was focused on understanding the precise relationship between this result and the ...
Side 5
... propositions of the Elements , whose proof depended only on the first four postulates . Finally , using invalid reasoning , he convinced himself that he had found the elusive contradiction , and concluded that the parallel postulate was ...
... propositions of the Elements , whose proof depended only on the first four postulates . Finally , using invalid reasoning , he convinced himself that he had found the elusive contradiction , and concluded that the parallel postulate was ...
Side 7
... propositions but in the axioms with which he prefaced the first book [ 144 , pp . 99–101 ] . Specifically regarding the parallel postulate ( which Lambert calls the " 11th axiom " ) , he says : Undoubtedly , this basic assertion is far ...
... propositions but in the axioms with which he prefaced the first book [ 144 , pp . 99–101 ] . Specifically regarding the parallel postulate ( which Lambert calls the " 11th axiom " ) , he says : Undoubtedly , this basic assertion is far ...
Side 19
... propositions of Book I deal primarily with familiar properties of lines , angles , and triangles . Even when he begins the theory of parallels , Propositions 27 and 28 first prove what results he could deduce about parallels without ...
... propositions of Book I deal primarily with familiar properties of lines , angles , and triangles . Even when he begins the theory of parallels , Propositions 27 and 28 first prove what results he could deduce about parallels without ...
Innhold
1 | |
Taming the Infinite | 54 |
3 | 69 |
Calculating Areas and Volumes | 89 |
1 | 95 |
4 | 123 |
5 | 129 |
7 | 150 |
Fermats Last Theorem | 156 |
The Search for an Elusive Formula | 204 |
References | 259 |
Credits | 269 |
Andre utgaver - Vis alle
Mathematical Expeditions: Chronicles by the Explorers Reinhard Laubenbacher,David Pengelley Begrenset visning - 2000 |
Mathematical Expeditions Reinhard Laubenbacher,David Pengelley Ingen forhåndsvisning tilgjengelig - 2014 |
Mathematical Expeditions: Chronicles by the Explorers Reinhard Laubenbacher,David Pengelley Ingen forhåndsvisning tilgjengelig - 1998 |
Vanlige uttrykk og setninger
aggregate algebraic analysis angle sum Archimedes arithmetic Axiom Axiom of Choice called Cantor Cardano cardinal number Cauchy Cauchy's Cavalieri's century coefficients complex numbers Continuum Hypothesis cube curve definition divisor elements equal equations of degree equivalent Euclid Euclid's Euclid's Elements Euclidean Euclidean geometry Euler Exercise exponent factors Fermat equation Fermat's Last Theorem FIGURE finite follows formula functions Fundamental Theorem Galois Gauss Germain given Greek hyperbolic geometry Hypothesis indivisibles infinite sets infinitesimal Lagrange Legendre Leibniz Lemma Lobachevsky mathematicians mathematics method natural numbers non-Euclidean non-Euclidean geometry number theory one-to-one correspondence parabola parallel postulate perpendicular PHOTO Poincaré polynomial prime numbers problem proof proposed equation Proposition prove Pythagorean triples Quadrature rational numbers real numbers reduced equation relatively prime result right angles roots segment set theory sides solution solve square straight line tangent triangle FDC values variable Zermelo's