# Construction and Categoricity of the Real Number System Using Decimals

## TL;DR

This paper defines real numbers as infinite decimals, establishes their properties as a complete ordered field, and proves their uniqueness up to isomorphism, providing a foundational perspective on real number construction.

## Contribution

It offers a detailed construction of real numbers via infinite decimals and proves their categoricity as a complete ordered field.

## Key findings

- Real numbers can be constructed as infinite decimals.
- The set of real numbers forms a complete ordered field.
- Any complete ordered field is isomorphic to this decimal-based construction.

## Abstract

In this expository article, the real numbers are defined as infinite decimals. After defining an ordering relation and the arithmetic operations, it is shown that the set of real numbers is a complete ordered field. It is further shown that any complete ordered field is isomorphic to the constructed set of real numbers.