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.

Klappentext

This monograph is an accessible account of the normed algebras over the real field, particularly the quaternions and the Cayley numbers. The application of quaternions to spherical geometry and to mechanics is considered and the relation between quaternions and rotations in 3- and 4-dimensional Euclidean space is fully developed. The algebra of complexified quaternions is described and applied to electromagnetism and to special relativity. By looking at a 3-dimensional complex space we explore the use of a quaternion formalism to the Lorentz transformation and we examine the classification of electromagnetic and Weyl tensors. In the final chapter, extensions of quaternion algebra to the alternative non-associative algebra of Cayley numbers are investigated. The standard Cayley number identities are derived and their use in the analysis of 7- and 8-dimensional rotations is studied. Appendices on Clifford algebras and on the use of dynamic computation in Cayley algebra are included. br/ emAudience:/em This volume has been written at a level suitable for final year and postgraduate students.

Inhalt
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.