James Algebra

James algebra uses three types of spatial containers/boundaries to represent a wide variety of number types (integers, rationals, irrationals, reals, imaginaries), as well as to implement all common operations. The three top rows in the Image show the James algebra representation of addition, multiplication, and raising to a power. The bottom three rows show subtraction, division, and roots. James algebra is an entirely different approach to iconic arithmetic than those presented in the Arithmetic section. Several unique numerical concepts arise from this approach. Generalized cardinality applies to negative and fractional counts, as well as to integer counts. The generalized inverse unifies subtraction, division, roots, and logarithms into a single boundary structure differentiated by spatial nesting rather than by algebraic operation. The James imaginary, J is a new type of imaginary number, two of which compose into the conventional imaginary number, i. J is explored on the James Imaginary page. The content presented on this page and on the James Imaginary page is discussed in depth in Jeffrey James 1993 masters thesis from the University of Washington. The thesis, A Calculus of Number Based on Spatial Forms, is available online on the ancient Laws of Form website.

 

Note, October 2020: The James algebra content is being explored in a series of three volumes of Iconic Arithmetic described on this site:

Iconic Arithmetic Volume I

Iconic Arithmetic Volume II

Iconic Arithmetic Volume III which is due in early 2021.