A new approach to the real numbers

TL;DR

This paper presents a comprehensive approach to defining real numbers using decimal representations, integrating classical constructions like Dedekind cuts and Cauchy sequences within this framework.

Contribution

It introduces a unified method for constructing and understanding real numbers through decimal representations, connecting various classical approaches.

Findings

Provides a complete decimal-based construction of real numbers

Explains Dedekind cuts and Cauchy sequences within the decimal framework

Characterizes real numbers as a complete ordered field

Abstract

In this paper we provide a complete approach to the real numbers via decimal representations. Construction of the real numbers by Dedekind cuts, Cauchy sequences of rational numbers, and the algebraic characterization of the real number system by the concept of complete ordered field are also well explained in the new setting.