Hypernumbers—II. further concepts and computational applications

Abstract

In a previous paper [1] on hypernumbers it was suggested that there were significant ramifications to be explored and exemplified. In particular, the following important areas were only mentioned in passing, but are now more fully examined: the hypernumber use of idempotents, zero divisors, and nilpotents; the concept of bimatrices; and a survey, necessarily brief, of hypernumbers beyond ε and i, that is, beyond the proper square and fourth roots of unity respectively. The term hypernumber was introduced into mathematics by the present author in 1966 [2] to denote domains of number including or beyond the arithmetic of ordinary numbers, complex numbers, quaternions, matrices, or octaves (Cayley-Graves numbers); and the term has since been used by Kline [3] and by Spencer and Moon [4], but unnecessarily restricted to forms of i and ε, and too often only to matrix forms.