Real numbers as infinite decimals -- theory and computation

TL;DR

This paper addresses the longstanding challenge of performing arithmetic directly on infinite decimal representations of real numbers, providing a comprehensive solution to a historical problem in the theory and computation of real numbers.

Contribution

It introduces a novel method for arithmetic on infinite decimals, resolving a problem considered 'fatal' by Hardy, and advances the understanding of real number representations.

Findings

Successfully defined arithmetic operations on infinite decimals

Resolved Hardy's 'fatal defect' in decimal representation

Bridged historical gap in real number theory

Abstract

In the 16th century, Simon Stevin initiated a modern approach to decimal representation of measuring numbers, marking a transition from the discrete arithmetic practised by the Greeks to the arithmetic of the continuum taken for granted today. However, how to perform arithmetic directly on infinite decimals remains a long-standing problem, which has seen the popular degeometrisation of real numbers since the first constructions were published in around 1872. Our article is devoted to solving this historical problem. An issue that Hardy called "a fatal defect" is also settled.