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Hypercomplex numbers. An elementary introduction to algebras. Transl. by A. Shenitzer. (English) Zbl 0669.17001
New York etc.: Springer-Verlag. x, 169 p. DM 78.00 (1989).

[For a review of the German translation (Teubner 1978) see Zbl 0395.17001.]

The original Russian edition appeared in 1973 as a text-book, intended for students of science high schools. As a matter of fact one cannot expect that this book contains new material and further developments in the study of hypercomplex systems. Nevertheless it was worthwhile to translate it into English because of the lucid way in which the subject is presented by the Russian authors.

The book is subdivided into three chapters. The first part deals with the introduction of complex numbers, quaternions and Cayley numbers, presented as hypercomplex number systems and in the third chapter, titled ‘The exceptional position of four algebras’, it is shown that the number in the “sum of squares” can take on just the four values 1,2,4 and 8 to obtain all division algebras: the reals, the complex numbers, the quaternions and the Cayley algebras.

The second part, of an auxiliary nature, is an elementary exposition of the fundamental concepts of linear algebra.

Reviewer: A.H.Boers

17-01Textbooks (nonassociative rings and algebras)
15-01Textbooks (linear algebra)
17A35Division algebras
16KxxDivision rings and semi-simple Artin rings
17D05Alternative rings
16P10Finite associative rings and finite-dimensional algebras
30G35Functions of hypercomplex variables and generalized variables