Historical Aspects Of Complex Quaternions And Cayley Numbers
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Historical Aspects Of Complex Quaternions And Cayley Numbers
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Author : Marilyn M. Simmons
language : en
Publisher:
Release Date : 1992
Historical Aspects Of Complex Quaternions And Cayley Numbers written by Marilyn M. Simmons and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with Numbers, Complex categories.
On Quaternions And Octonions
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Author : John H. Conway
language : en
Publisher: CRC Press
Release Date : 2003-01-23
On Quaternions And Octonions written by John H. Conway and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-01-23 with Mathematics categories.
This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional Euclidean spaces using quaternions, leading to enumerations of the corresponding finite groups of symmetries. The second half of the book discusses the less f
Quaternions And Cayley Numbers
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Author : J.P. Ward
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Quaternions And Cayley Numbers written by J.P. Ward 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 essence, this text is written as a challenge to others, to discover significant uses for Cayley number algebra in physics. I freely admit that though the reading of some sections would benefit from previous experience of certain topics in physics - particularly relativity and electromagnetism - generally the mathematics is not sophisticated. In fact, the mathematically sophisticated reader, may well find that in many places, the rather deliberate progress too slow for their liking. This text had its origin in a 90-minute lecture on complex numbers given by the author to prospective university students in 1994. In my attempt to develop a novel approach to the subject matter I looked at complex numbers from an entirely geometric perspective and, no doubt in line with innumerable other mathematicians, re-traced steps first taken by Hamilton and others in the early years of the nineteenth century. I even enquired into the possibility of using an alternative multiplication rule for complex numbers (in which argzlz2 = argzl- argz2) other than the one which is normally accepted (argzlz2 = argzl + argz2). Of course, my alternative was rejected because it didn't lead to a 'product' which had properties that we now accept as fundamental (i. e.
Hurwitz S Lectures On The Number Theory Of Quaternions
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Author : Nicola Oswald
language : en
Publisher: European Mathematical Society
Release Date : 2023-05-15
Hurwitz S Lectures On The Number Theory Of Quaternions written by Nicola Oswald and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-05-15 with Mathematics categories.
Quaternions are non-commutative generalizations of the complex numbers, invented by William Rowan Hamilton in 1843. Their number-theoretical aspects were first investigated by Rudolf Lipschitz in the 1880s, and, in a streamlined form, by Adolf Hurwitz in 1896. This book contains an English translation of Hurwitz's 1919 textbook on this topic as well as his famous 1-2-3-4 theorem on composition algebras. In addition, the reader can find commentaries that shed historical light on the development of this number theory of quaternions, for example, the classical preparatory works of Fermat, Euler, Lagrange and Gauss, to name but a few, the different notions of quaternion integers in the works of Lipschitz and Hurwitz, analogies to the theory of algebraic numbers, and the further development (including Dickson's work in particular). The authors have implemented parts of the book in stand-alone courses, and they believe that the present book can also complement a course on algebraic number theory (with respect to a noncommutative extension of the rational numbers).
The Realm Of Sedenions
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Author : David R Ely
language : en
Publisher: David R Ely
Release Date : 2024-05-22
The Realm Of Sedenions written by David R Ely and has been published by David R Ely this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-05-22 with Mathematics categories.
"The Realm of Sedenions: A Look Into Higher-Complex Numbers" dives into the fascinating world of sedenions, a 16-dimensional number system with unique properties. The book explores the motivation behind studying sedenions, their formal definition with addition and non-commutative, non-associative multiplication, and how they differ from familiar number systems like real and complex numbers. While sedenions lack widespread practical applications yet, the book delves into potential areas of exploration in physics and computer science. It also highlights the challenges mathematicians face due to the complex multiplication rules and the lack of a straightforward geometric interpretation for these high-dimensional numbers. The book provides historical context by mentioning James Joseph Sylvester, who introduced sedenions. While directly visualizing 16 dimensions is difficult, the book discusses potential approaches like subspace projections and geometric analogies to gain a better understanding. Exercises tailored to the audience's level and a resource section for further exploration round out the book, making it a valuable resource for anyone curious about venturing beyond the realm of familiar numbers.
Turning Points In The History Of Mathematics
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Author : Hardy Grant
language : en
Publisher: Birkhäuser
Release Date : 2016-04-15
Turning Points In The History Of Mathematics written by Hardy Grant and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-04-15 with Mathematics categories.
This book explores some of the major turning points in the history of mathematics, ranging from ancient Greece to the present, demonstrating the drama that has often been a part of its evolution. Studying these breakthroughs, transitions, and revolutions, their stumbling-blocks and their triumphs, can help illuminate the importance of the history of mathematics for its teaching, learning, and appreciation. Some of the turning points considered are the rise of the axiomatic method (most famously in Euclid), and the subsequent major changes in it (for example, by David Hilbert); the “wedding,” via analytic geometry, of algebra and geometry; the “taming” of the infinitely small and the infinitely large; the passages from algebra to algebras, from geometry to geometries, and from arithmetic to arithmetics; and the revolutions in the late nineteenth and early twentieth centuries that resulted from Georg Cantor’s creation of transfinite set theory. The origin of each turning point is discussed, along with the mathematicians involved and some of the mathematics that resulted. Problems and projects are included in each chapter to extend and increase understanding of the material. Substantial reference lists are also provided. Turning Points in the History of Mathematics will be a valuable resource for teachers of, and students in, courses in mathematics or its history. The book should also be of interest to anyone with a background in mathematics who wishes to learn more about the important moments in its development.
A Conceptual History Of Space And Symmetry
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Author : Pietro Giuseppe Fré
language : en
Publisher: Springer
Release Date : 2018-09-14
A Conceptual History Of Space And Symmetry written by Pietro Giuseppe Fré and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-14 with Science categories.
This book presents the author’s personal historical perspective and conceptual analysis on symmetry and geometry. The author enlightens with modern views the historical process which led to the contemporary vision of space and symmetry that are used in theoretical physics and in particular in such abstract and advanced descriptions of the physical world as those provided by supergravity. The book is written intertwining storytelling and philosophical argumentation with some essential technical material. The author argues that symmetry and geometry are inextricably entangled and their current meaning is the result of a long process of abstraction which was determined through history and can be understood within the analytic system of thought of western civilization that started with the Ancient Greeks. The evolution of geometry and symmetry theory in the last forty years has been deeply and constructively influenced by supersymmetry/supergravity and the allied constructions of strings and branes. Further advances in theoretical physics cannot be based simply on the Galilean method of interrogating nature and then formulating a testable theory to explain the observed phenomena. One ought to interrogate human thought, meaning frontier-line mathematics concerned with geometry and symmetry in order to find there the threads of so far unobserved correspondences, reinterpretations and renewed conceptions.
Quaternionic Quantum Mechanics And Quantum Fields
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Author : Stephen L. Adler
language : en
Publisher: Oxford University Press
Release Date : 1995-04-27
Quaternionic Quantum Mechanics And Quantum Fields written by Stephen L. Adler and has been published by Oxford University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995-04-27 with Science categories.
It has been known since the 1930s that quantum mechanics can be formulated in quaternionic as well as complex Hilbert space. But systematic work on the quaternionic extension of standard quantum mechanics has scarcely begun. Authored by a world-renowned theoretical physicist, this book signals a major conceptual advance and gives a detailed development and exposition of quaternionic quantum mechanics for the purpose of determining whether quaternionic Hilbert space is the appropriate arena for the long sought-after unification of the standard model forces with gravitation. Significant results from earlier literature, together with many new results obtained by the author, are integrated to give a coherent picture of the subject. The book also provides an introduction to the problem of formulating quantum field theories in quaternionic Hilbert space. The book concludes with a chapter devoted to discussions on where quaternionic quantum mechanics may fit into the physics of unification, experimental and measurement theory issues, and the many open questions that still challenge the field. This well-written treatise is a very significant contribution to theoretical physics. It will be eagerly read by a wide range of physicists.
Unknown Quantity
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Author : John Derbyshire
language : en
Publisher: National Academies Press
Release Date : 2006-05-02
Unknown Quantity written by John Derbyshire and has been published by National Academies Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-05-02 with Science categories.
Prime Obsession taught us not to be afraid to put the math in a math book. Unknown Quantity heeds the lesson well. So grab your graphing calculators, slip out the slide rules, and buckle up! John Derbyshire is introducing us to algebra through the ages-and it promises to be just what his die-hard fans have been waiting for. "Here is the story of algebra." With this deceptively simple introduction, we begin our journey. Flanked by formulae, shadowed by roots and radicals, escorted by an expert who navigates unerringly on our behalf, we are guaranteed safe passage through even the most treacherous mathematical terrain. Our first encounter with algebraic arithmetic takes us back 38 centuries to the time of Abraham and Isaac, Jacob and Joseph, Ur and Haran, Sodom and Gomorrah. Moving deftly from Abel's proof to the higher levels of abstraction developed by Galois, we are eventually introduced to what algebraists have been focusing on during the last century. As we travel through the ages, it becomes apparent that the invention of algebra was more than the start of a specific discipline of mathematics-it was also the birth of a new way of thinking that clarified both basic numeric concepts as well as our perception of the world around us. Algebraists broke new ground when they discarded the simple search for solutions to equations and concentrated instead on abstract groups. This dramatic shift in thinking revolutionized mathematics. Written for those among us who are unencumbered by a fear of formulae, Unknown Quantity delivers on its promise to present a history of algebra. Astonishing in its bold presentation of the math and graced with narrative authority, our journey through the world of algebra is at once intellectually satisfying and pleasantly challenging.
From Natural Numbers To Quaternions
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Author : Jürg Kramer
language : en
Publisher: Springer
Release Date : 2017-11-15
From Natural Numbers To Quaternions written by Jürg Kramer and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-11-15 with Mathematics categories.
This textbook offers an invitation to modern algebra through number systems of increasing complexity, beginning with the natural numbers and culminating with Hamilton's quaternions. Along the way, the authors carefully develop the necessary concepts and methods from abstract algebra: monoids, groups, rings, fields, and skew fields. Each chapter ends with an appendix discussing related topics from algebra and number theory, including recent developments reflecting the relevance of the material to current research. The present volume is intended for undergraduate courses in abstract algebra or elementary number theory. The inclusion of exercises with solutions also makes it suitable for self-study and accessible to anyone with an interest in modern algebra and number theory.