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Group theory --- Quaternios. --- Álgebra. --- Quaternions. --- Cayley numbers. --- Quaternions --- 511.147 --- Complex numbers. Imaginaries --- 511.147 Complex numbers. Imaginaries --- Álgebra. --- Cayley numbers --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis --- Cayley numbers (Algebra) --- Octonions --- Nombres hypercomplexes --- Geometrie

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Ordered algebraic structures --- Lie algebras. --- Exceptional Lie algebras. --- Nombres hypercomplexes --- Octonions --- Cayley numbers (Algebra) --- Galois cohomology --- Cohomologie galoisienne.

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"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 familiar octonion algebra, concentrating on its remarkable "triality symmetry" after an appropriate study of Moufang loops. The authors also describe the arithmetics of the quaternions and octonions. The book concludes with a new theory of octonion factorization. Topics covered include the geometry of complex numbers, quaternions and 3-dimensional groups, quaternions and 4-dimensional groups, Hurwitz integral quaternions, composition algebras, Moufang loops, octonions and 8-dimensional geometry, integral octonions, and the octonion projective plane."--Provided by publisher.

Quaternions. --- Cayley numbers (Algebra) --- Cayley octave (Algebra) --- Cayley's numbers (Algebra) --- Cayley's octave (Algebra) --- Octonions --- Cayley algebras --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis

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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.

Algebra --- 511.6 --- Cayley numbers --- Quaternions --- Algebra, Universal --- Algebraic fields --- Curves --- Surfaces --- Numbers, Complex --- Vector analysis --- Algebraic number fields --- Quaternions. --- Cayley numbers. --- 511.6 Algebraic number fields --- Associative rings. --- Rings (Algebra). --- Nonassociative rings. --- Matrix theory. --- Algebra. --- Mathematical physics. --- Applied mathematics. --- Engineering mathematics. --- Associative Rings and Algebras. --- Non-associative Rings and Algebras. --- Linear and Multilinear Algebras, Matrix Theory. --- Theoretical, Mathematical and Computational Physics. --- Applications of Mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Physical mathematics --- Physics --- Mathematics --- Rings (Algebra) --- Algebraic rings --- Ring theory