Structure of hypercomplex units and exotic numbers as sections of Bi-quaternions

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. © 2010 American Scientific Publishers.