Geometrization of the Real Number System

TL;DR

This paper discusses the extension of real numbers into geometric number systems, specifically Clifford algebras, which simplify higher mathematics and are consistent with established algebraic structures, reflecting a historical progression in mathematical development.

Contribution

It presents the geometrization of real numbers through Clifford algebras, emphasizing their consistency and role in advancing mathematical and scientific understanding.

Findings

Clifford algebras provide a consistent framework for geometric number systems.

Geometric algebras simplify tensor analysis and category theory.

These systems underpin modern scientific and technological progress.

Abstract

Geometric number systems, obtained by extending the real number system to include new anticommuting square roots of +1 and -1, provide a royal road to higher mathematics by largely sidestepping the tedious languages of tensor analysis and category theory. The well known consistency of real and complex matrix algebras, together with Cartan-Bott periodicity, firmly establishes the consistency of these geometric number systems, often referred to as Clifford algebras. The geometrization of the real number system is the culmination of the thousands of years of human effort at developing ever more sophisticated and encompassing number systems underlying scientific progress and advanced technology in the 21st Century. Complex geometric algebras are also considered.