Defining arithmetical operations on infinite decimals

TL;DR

This paper develops a method to perform arithmetic directly on infinite decimal representations of real numbers, confirming the existence of algorithms for digit-wise computation of sums and products without relying on orderings.

Contribution

It introduces a novel approach to define arithmetic operations on infinite decimals directly, extending Hua's 1962 work and confirming algorithmic computability of real number operations.

Findings

Arithmetic operations can be defined directly on infinite decimals

Algorithms exist for digit-wise computation of sums and products

Supports Hua's approach to real numbers without orderings

Abstract

Completing Loo-Keng Hua's approach to the real number system pioneered in 1962, this paper defines arithmetical operations directly on infinite decimals without appealing to any ordering structure. Therefore, the widespread belief that there exists an algorithm for determining the digits of the product of two real numbers in terms of finite pieces of their decimal strings is essentially confirmed.