A Canonical Bijection Between Finite-Decimal Real Numbers and Natural Numbers with Constant-Time Enumeration Formulas

TL;DR

This paper introduces a precise, constant-time method to bijectively map finite-decimal real numbers to natural numbers using a systematic parametrization, enabling efficient and accurate enumeration of all such real numbers.

Contribution

It provides the first explicit, constant-time bijection between finite-decimal real numbers and natural numbers with closed-form formulas and exact decimal arithmetic.

Findings

Constant-time enumeration formulas for real numbers

Complete indexing of finite-decimal real numbers

Exact decimal arithmetic ensures perfect accuracy

Abstract

We present an explicit bijection between finite-decimal real numbers and natural numbers ($\mathbb{N} = \{1, 2, 3, ...\}$) using a systematic 4-tuple parametrization with closed-form mathematical formulas for enumeration. Our enumeration system provides complete indexing of all real numbers with terminating decimal representations through the parametrization $(\text{sign}, N_1, N_2, N_3)$. Both forward and inverse mappings execute in O(1) constant time, achieved through closed-form lexicographic positioning formulas that eliminate enumeration loops. The system uses exact decimal arithmetic throughout, ensuring perfect accuracy across all representable numbers. This bijective correspondence demonstrates that finite-decimal real numbers can be systematically enumerated and indexed with optimal constant-time computational efficiency.