On hypercomplex number systems

Abstract

1. The usual theory relates to systems of numbers =,a,e, in which the co6rdinates ai range independently over all real nulmbers or else over all ordinary complex numbers; for example, the real quaternion system, or the complex quaternion system. As an obvious generalization,t the co6rdinates may range independently over all the marks of any field F; for example, the rational quaternion system. As a further generalization, the sets of co6rdinates al, *, a in the various numbers of a system may include only a part of the sets bl, -, b , each b ranging independently over F; for example, the integral quaternion system. The various coordinates a1, , a. need not have the same range; for example, the numbers (a + 2b V/2) el + (c + 4d 12 ) e2 (a, b, c, d arbitrary integers)