Quaternions and Cayley Numbers

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.