Algebra of Quadriform Numbers

Orchard-Hays, W. (1975). Algebra of Quadriform Numbers. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-75-160

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Abstract

In a separate paper by the same author, a set of hypernumbers, called quadriform, were defined in terms of 4 x 4 real matrices which represent a kind of generalization of 2 x 2 matrix representation of complex numbers. The object was to provide for the other two square roots of unity which are always ignored in the complex field and then to be able to take square roots of all numbers in terms of real matrices. Although it was found possible to do so, the resulting set of entities do not form a consistent and usable algebra. This effort and the negative results obtained are summarized in Appendix B of this paper.

It was found, however, in the author's earlier paper, that the complete set of 2 x 2 real matrices do lead to a consistent and usable algebra provided one is prepared to recognize certain restrictions on the range of allowable quantities. In this paper, it is this set which is referred to by the name "quadriform". It is not quite true that they form a field but they are not less than a field in the sense of a ring, but rather more than a field, in particular the complex field which is a proper subset of the quadriforms. The limitations on the range of allowable quadriforms are well defined and present no unusual difficulties with the following exception: addition (and subtraction) of two allowable numbers may give an unallowable result. This does not occur within the complex subset and all attributes of the complex field are retained within the subset.

In general, quadriforms are not commutative under multiplication. Although this does not prevent a consistent algebra, it does impose severe limitations on the generalization of complex functions. For example, there is no generalization of the exponential and natural log functions. More precisely, there are several possible generalizations of the exponential function, each of which is the same as e^z for the complex subset, but which do not retain the desired characteristics over the full quadriform set. This is discussed at some length in the last part of this paper.

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