Historical Aspects Of Complex Quaternions And Cayley Numbers
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| Author | : J.P. Ward |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 252 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 9401157685 |
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.
| Author | : John H. Conway |
| Publisher | : CRC Press |
| Total Pages | : 172 |
| Release | : 2003-01-23 |
| Genre | : Mathematics |
| ISBN | : 1439864187 |
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
| Author | : Stephen L. Adler |
| Publisher | : Oxford University Press |
| Total Pages | : 599 |
| Release | : 1995-04-27 |
| Genre | : Science |
| ISBN | : 0195345061 |
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.
| Author | : Tevian Dray |
| Publisher | : World Scientific |
| Total Pages | : 229 |
| Release | : 2015-04-08 |
| Genre | : Mathematics |
| ISBN | : 9814401838 |
There are precisely two further generalizations of the real and complex numbers, namely, the quaternions and the octonions. The quaternions naturally describe rotations in three dimensions. In fact, all (continuous) symmetry groups are based on one of these four number systems. This book provides an elementary introduction to the properties of the octonions, with emphasis on their geometric structure. Elementary applications covered include the rotation groups and their spacetime generalization, the Lorentz group, as well as the eigenvalue problem for Hermitian matrices. In addition, more sophisticated applications include the exceptional Lie groups, octonionic projective spaces, and applications to particle physics including the remarkable fact that classical supersymmetry only exists in particular spacetime dimensions.
| Author | : Pietro Giuseppe Fré |
| Publisher | : Springer |
| Total Pages | : 323 |
| Release | : 2018-09-14 |
| Genre | : Science |
| ISBN | : 3319980238 |
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.
| Author | : Boris A. Rosenfeld |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 481 |
| Release | : 2012-09-08 |
| Genre | : Mathematics |
| ISBN | : 1441986804 |
The Russian edition of this book appeared in 1976 on the hundred-and-fiftieth anniversary of the historic day of February 23, 1826, when LobaeevskiI delivered his famous lecture on his discovery of non-Euclidean geometry. The importance of the discovery of non-Euclidean geometry goes far beyond the limits of geometry itself. It is safe to say that it was a turning point in the history of all mathematics. The scientific revolution of the seventeenth century marked the transition from "mathematics of constant magnitudes" to "mathematics of variable magnitudes. " During the seventies of the last century there occurred another scientific revolution. By that time mathematicians had become familiar with the ideas of non-Euclidean geometry and the algebraic ideas of group and field (all of which appeared at about the same time), and the (later) ideas of set theory. This gave rise to many geometries in addition to the Euclidean geometry previously regarded as the only conceivable possibility, to the arithmetics and algebras of many groups and fields in addition to the arith metic and algebra of real and complex numbers, and, finally, to new mathe matical systems, i. e. , sets furnished with various structures having no classical analogues. Thus in the 1870's there began a new mathematical era usually called, until the middle of the twentieth century, the era of modern mathe matics.
| Author | : Hardy Grant |
| Publisher | : Birkhäuser |
| Total Pages | : 112 |
| Release | : 2016-04-15 |
| Genre | : Mathematics |
| ISBN | : 1493932640 |
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.
| Author | : Heinz-Dieter Ebbinghaus |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 424 |
| Release | : 1991 |
| Genre | : Mathematics |
| ISBN | : 9780387974972 |
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.
| Author | : Jeremy J. Gray |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 346 |
| Release | : 2011-08-31 |
| Genre | : Mathematics |
| ISBN | : 0821869043 |
Algebra, as a subdiscipline of mathematics, arguably has a history going back some 4000 years to ancient Mesopotamia. The history, however, of what is recognized today as high school algebra is much shorter, extending back to the sixteenth century, while the history of what practicing mathematicians call "modern algebra" is even shorter still. The present volume provides a glimpse into the complicated and often convoluted history of this latter conception of algebra by juxtaposing twelve episodes in the evolution of modern algebra from the early nineteenth-century work of Charles Babbage on functional equations to Alexandre Grothendieck's mid-twentieth-century metaphor of a ``rising sea'' in his categorical approach to algebraic geometry. In addition to considering the technical development of various aspects of algebraic thought, the historians of modern algebra whose work is united in this volume explore such themes as the changing aims and organization of the subject as well as the often complex lines of mathematical communication within and across national boundaries. Among the specific algebraic ideas considered are the concept of divisibility and the introduction of non-commutative algebras into the study of number theory and the emergence of algebraic geometry in the twentieth century. The resulting volume is essential reading for anyone interested in the history of modern mathematics in general and modern algebra in particular. It will be of particular interest to mathematicians and historians of mathematics.
| Author | : Charles C Pinter |
| Publisher | : Courier Corporation |
| Total Pages | : 402 |
| Release | : 2010-01-14 |
| Genre | : Mathematics |
| ISBN | : 0486474178 |
Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications. 1990 edition.
