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The generalizations of complex numbers considered here differ from well-known generalizations such as quaternions, octonions, bicomplex and multicomplex numbers and Clifford algebras, to name some of the well-known ones, in at least two fundamental respects.
First of all, products of elements of such traditional algebraic structures are explained by the fact that certain expressions in brackets are formally treated as when multiplying expressions in brackets of real numbers, with additional assumptions made for the multiplication of so-called basic elements. In the present work, on the one hand, suitable vector-valued vector products are introduced and used which are geometrically motivated as rotations and stretches.
Furthermore, traditional generalizations of complex numbers do not provide any information about which concrete mathematical objects fulfill the formulated wishes with regard to multiplication and whether the fulfillment of these wishes is unequivocal or ambiguous, while in concrete objects they are always specified in the present work and, in particular, no place is left for the mystification of imaginary numbers.
The new vector products allow the introduction of vector-valued vector division, vector powers and exponential functions. A corresponding generalization of the Euler formula applies to the theory of directional probability laws. New consequences also arise in the special case of classical complex numbers, as shown by the Fourier transformation of probability densities. The new general vector division opens new perspectives of differentiability and function theory.
