Non-Associative Number Theory: Meditations Upon an Article by T. Evans

Abstract

The course of' learning tor a college math student is a search for unity. Throughout his four years of study, he is constantly trying to organize vectors, groups, differentials, integrals, and the like, into a meaningful. pattern. Such was the incentive for the choice of topic for this paper. The goal was a subject which would be self-contained, could be developed in a few axioms, and which would not require extensive study of other subjects. A paper on the axiomatic development· of non-associative numbers appeared to be the. answer. There would be only one primary source; everything could be developed from five postulates. Unfortunately, this was not the case. Ten years of work lay behind the writing of the source-paper, and, to understand it fully, other sources had to be studied - which led far away from the original idea. This paper is an attempt to bring order and simplicity back into non-associative number theory. It is aimed at the casual reader; and, in many cases, whole areas of the theory are left undiscussed if' they would not fit comfortably into this particular treatment of the subject. The paper is informal in its approach; consequently, the language should not be formal. It is not. For convenience, although contrary to established practice, the author will refer to himself as "I"; for he feels he .is not yet such a scholar that he can afford the humility of the "imperial We".