Quaternions and Cayley Numbers
Algebra and Applications
Published on 30. April 1997
Book
Hardback
XI, 242 pages
978-0-7923-4513-8 (ISBN)
Description
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.
More details
Series
Edition
1997 ed.
Language
English
Place of publication
Dordrecht
Netherlands
Target group
Professional and scholarly
Research
Illustrations
XI, 242 p.
Dimensions
Height: 241 mm
Width: 160 mm
Thickness: 19 mm
Weight
553 gr
ISBN-13
978-0-7923-4513-8 (9780792345138)
DOI
10.1007/978-94-011-5768-1
Schweitzer Classification
Other editions
Additional editions
Book
10/2012
Springer
€53.49
Shipment within 15-20 days
Content
1 Fundamentals of Linear Algebra.- 1.1 Integers, Rationals and Real Numbers.- 1.2 Real Numbers and Displacements.- 1.3 Groups.- 1.4 Rings and Fields.- 1.5 Linear Spaces.- 1.6 Inner Product Spaces.- 1.7 Algebras.- 1.8 Complex Numbers.- 2 Quaternions.- 2.1 Inventing Quaternions.- 2.2 Quaternion Algebra.- 2.3 The Exponential Form and Root Extraction.- 2.4 Frobenius' Theorem.- 2.5 Inner Product for Quaternions.- 2.6 Quaternions and Rotations in 3- and 4-Dimensions.- 2.7 Relation to the Rotation Matrix.- 2.8 Matrix Formulation of Quaternions.- 2.9 Applications to Spherical Trigonometry.- 2.10 Rotating Axes in Mechanics.- 3 Complexified Quaternions.- 3.1 Scalars, Pseudoscalars, Vectors and Pseudovectors.- 3.2 Complexified Quaternions: Euclidean Metric.- 3.3 Complexified Quaternions: Minkowski Metric.- 3.4 Application of Complexified Quaternions to Space-Time.- 3.5 Quaternions and Electromagnet ism.- 3.6 Quaternionic Representation of Bivectors.- 3.7 Null Tetrad for Space-time.- 3.8 Classification of Complex Bivectors and of the Weyl Tensor.- 4 Cayley Numbers.- 4.1 A Common Notation for Numbers.- 4.2 Cayley Numbers.- 4.3 Angles and Cayley Numbers.- 4.4 Cayley Number Identities.- 4.5 Normed Algebras and the Hurwitz Theorem.- 4.6 Rotations in 7-and 8-Dimensional Euclidean Space.- 4.7 Basis Elements for Cayley Numbers.- 4.8 Geometry of 8-Dimensional Rotations.- Appendix 1 Clifford Algebras.- Appendix 2 Computer Algebra and Cayley Numbers.- References.