Quaternionen Und Andere Zahlbereiche Was Kommt Nach Den Komplexen Zahlen
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Quaternionen Und Andere Zahlbereiche Was Kommt Nach Den Komplexen Zahlen
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Author : Bastian Vincken
language : de
Publisher: GRIN Verlag
Release Date : 2004-11-29
Quaternionen Und Andere Zahlbereiche Was Kommt Nach Den Komplexen Zahlen written by Bastian Vincken and has been published by GRIN Verlag this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-11-29 with Mathematics categories.
Examensarbeit aus dem Jahr 2004 im Fachbereich Mathematik - Zahlentheorie, Note: 1,3, Rheinisch-Westfälische Technische Hochschule Aachen (Lehrstuhl A für Mathematik), Sprache: Deutsch, Abstract: Das traditionelle Zahlensystem gilt als wichtigste Grundlage in der Mathematik. Der Aufbau dieses Zahlensystems beginnt seit dem Ende des 19. Jahrhunderts bei den natürlichen Zahlen. Diese werden dann schrittweise zu den ganzen, den rationalen, den reellen bis hin zu den komplexen Zahlen erweitert. Die Schulmathematik umfasst im besten Fall das Zahlensystem bis hin zu den komplexen Zahlen. In dieser Arbeit wollen wir uns mit der Frage beschäftigen, ob es jenseits der komplexen Zahlen noch andere Zahlbereiche zu konstruieren gibt und inwieweit diese noch sinnvoll sind. Diese hyperkomplexen Zahlbereiche werden seit Beginn des 20. Jahrhunderts reelle Algebren genannt. Möchte man sich analog zu den komplexen Zahlen, die einen zweidimensionalen reellen Vektorraum bilden, höherdimensionale reelle Vektorräume zu hyperkomplexen Zahlbereichen machen, muss man entweder die Endlichkeit der Dimension aufgeben oder aber auf vertraute Körperaxiome wie die der Kommutativität oder der Assoziativität oder gar auf die Möglichkeit der Division verzichten. In dieser Arbeit werden wir uns auf die endlichdimensionalen Divisionsalgebren beschränken. Dies bedeutet, dass wir an der Endlichkeit der Dimension und der Möglichkeit der Division festhalten werden. Sollten wir diese Eigenschaften aufgeben, so würden wir von einer Masse neuer Zahlbereiche erschlagen werden. Diese neuen Zahlbereiche werden Eigenschaften aufweisen, die uns auf den ersten Blick merkwürdig vorkommen. Der _Vollständigkeitssatz_ der reellen Zahlen beinhaltet vomWort her schon eine gewisse _Vollständigkeit_ des Zahlbereichs. Wir werden feststellen, dass, je weiter man sich von den reellen Zahlen entfernt, immer mehr uns vertraute Eigenschaften verloren gehen und in diesem Zusammenhang deutlich machen, welche Kuriositäten mit deren Wegfall einhergehen. Hamilton schuf im Jahre 1843, nachdem er die komplexen Zahlen als erster rein arithmetisch begründet hatte, den vierdimensionalen Schiefkörper H der Quaternionen. Kurz darauf konstruierten Graves und Cayley die achtdimensionale Divisionsalgebra O der Oktonionen. Die Quaternionen sind bezüglich der Multiplikation nicht mehr kommutativ und bei den Oktonionen ist zusätzlich noch die Assoziativität verletzt. Bei beiden Zahlbereichen ist jedoch die Division noch eindeutig ausführbar.
Elementary Mathematics From An Advanced Standpoint
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Author : Felix Klein
language : en
Publisher: Cosimo, Inc.
Release Date : 2009-01-01
Elementary Mathematics From An Advanced Standpoint written by Felix Klein and has been published by Cosimo, Inc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-01-01 with Mathematics categories.
When the mathematician Felix Klein first went to university, he was surprised at just how little what he had learned up to that point was relevant to his new studies. Professors had their own interests, and these they conveyed without regard for the math students of the future that these prospective secondary schoolteachers would one day instruct. Elementary Mathematics from an Advanced Standpoint was written to help remedy that problem. Though highly regarded as one of the finest mathematical minds of his day, Professor Klein took a great deal of interest in guiding teachers and "reducing the gap between the school and the university." Readers will come away impressed at the clarity of Klein's writing, and the ease with which he conveys complex mathematical ideas. Divided into three parts-arithmetic, algebra, and analysis-and covering such topics as complex numbers, real equations, and logarithmic and exponential functions, Klein's classic is essential reading for math instructors and students planning to become math instructors. German mathematician FELIX KLEIN (1849-1925), a great teacher and scientific thinker, significantly advanced the field of mathematical physics and made a number of profound discoveries in the field of geometry. His published works include Elementary Mathematics from an Advanced Standpoint: Geometry and Famous Problems of Elementary Geometry.
Dtv Atlas Zur Mathematik
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Author : Fritz Reinhardt
language : de
Publisher:
Release Date : 1974
Dtv Atlas Zur Mathematik written by Fritz Reinhardt and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1974 with Algebra categories.
Kognitive Semantik Cognitive Semantics
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Author : Monika Schwarz
language : en
Publisher: Gunter Narr Verlag
Release Date : 1994
Kognitive Semantik Cognitive Semantics written by Monika Schwarz and has been published by Gunter Narr Verlag this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Cognitive grammar categories.
A Certain Ambiguity
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Author : Gaurav Suri
language : en
Publisher: Princeton University Press
Release Date : 2010-07-01
A Certain Ambiguity written by Gaurav Suri and has been published by Princeton University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-07-01 with Fiction categories.
While taking a class on infinity at Stanford in the late 1980s, Ravi Kapoor discovers that he is confronting the same mathematical and philosophical dilemmas that his mathematician grandfather had faced many decades earlier--and that had landed him in jail. Charged under an obscure blasphemy law in a small New Jersey town in 1919, Vijay Sahni is challenged by a skeptical judge to defend his belief that the certainty of mathematics can be extended to all human knowledge--including religion. Together, the two men discover the power--and the fallibility--of what has long been considered the pinnacle of human certainty, Euclidean geometry. As grandfather and grandson struggle with the question of whether there can ever be absolute certainty in mathematics or life, they are forced to reconsider their fundamental beliefs and choices. Their stories hinge on their explorations of parallel developments in the study of geometry and infinity--and the mathematics throughout is as rigorous and fascinating as the narrative and characters are compelling and complex. Moving and enlightening, A Certain Ambiguity is a story about what it means to face the extent--and the limits--of human knowledge.
Princess Neelam And Other Stories
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Author : Safia Nasir
language : en
Publisher:
Release Date : 2005
Princess Neelam And Other Stories written by Safia Nasir and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Juvenile Fiction categories.
The Words Of Mathematics
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Author : Steven Schwartzman
language : en
Publisher: MAA
Release Date : 1994
The Words Of Mathematics written by Steven Schwartzman and has been published by MAA this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994 with Mathematics categories.
This book explains the origins of over 1500 mathematical terms used in English.
Classical Topology And Combinatorial Group Theory
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Author : John Stillwell
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Classical Topology And Combinatorial Group Theory written by John Stillwell and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does nr~ understand the simplest topological facts, such as the rcason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical development where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recreations like the seven bridges; rather, it resulted from the l'isualization of problems from other parts of mathematics-complex analysis (Riemann), mechanics (Poincare), and group theory (Dehn). It is these connec tions to other parts of mathematics which make topology an important as well as a beautiful subject.
Numbers
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Author : Heinz-Dieter Ebbinghaus
language : en
Publisher: Springer Science & Business Media
Release Date : 1991
Numbers written by Heinz-Dieter Ebbinghaus and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991 with Mathematics categories.
This book is about all kinds of numbers, from rationals to octonians, reals to infinitesimals. It is a story about a major thread of mathematics over thousands of years, and it answers everything from why Hamilton was obsessed with quaternions to what the prospect was for quaternionic analysis in the 19th century. It glimpses the mystery surrounding imaginary numbers in the 17th century and views some major developments of the 20th century.
Galois Theory
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Author : Emil Artin
language : en
Publisher:
Release Date : 2020-02
Galois Theory written by Emil Artin and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-02 with Education categories.
The author Emil Artin is known as one of the greatest mathematicians of the 20th century. He is regarded as a man who gave a modern outlook to Galois theory. Original lectures by the master. This emended edition is with completely new typesetting and corrections. The free PDF file available on the publisher's website www.bowwowpress.org