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Structure of Hypercomplex Units and Exotic Numbers as Sections of Bi-Quaternions

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A survey of all families of hypercomplex (HC-) numbers is suggested with emphasis on exotic sets. Systematic description of variety of representations of HC-units is given, and interior structure of the units is studied. Elementary math objects constituting the structure are demonstrated to possess variously algebraic, geometric and physical properties, being eigenfuctions of HC-vector operators, ideals of idempotent matrices, dyads (Lame coefficients) linking two 2-dimensional surfaces, projectors of matrix-vectors onto given axis, and spinors. It is also shown that full set of bi-quaternion numbers comprises as special cases real, complex, quaternion numbers and as well exotic sets split-complex and dual numbers. In particular a HC-unit of double numbers is found to be represented by a Pauli-type matrix, and a simple formula for null-modulus HC-unit of dual numbers is indicated.

Document Type: Research Article


Publication date: December 1, 2010

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  • ADVANCED SCIENCE LETTERS is an international peer-reviewed journal with a very wide-ranging coverage, consolidates research activities in all areas of (1) Physical Sciences, (2) Biological Sciences, (3) Mathematical Sciences, (4) Engineering, (5) Computer and Information Sciences, and (6) Geosciences to publish original short communications, full research papers and timely brief (mini) reviews with authors photo and biography encompassing the basic and applied research and current developments in educational aspects of these scientific areas.
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