A Canonical Bijection Between Finite-Decimal Real Numbers and Natural Numbers with Constant-Time Enumeration Formulas
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915446748348416 |
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| author | Rithvik, S. K. |
| author_facet | Rithvik, S. K. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10750 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Canonical Bijection Between Finite-Decimal Real Numbers and Natural Numbers with Constant-Time Enumeration Formulas Rithvik, S. K. Number Theory 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. |
| title | A Canonical Bijection Between Finite-Decimal Real Numbers and Natural Numbers with Constant-Time Enumeration Formulas |
| topic | Number Theory |
| url | https://arxiv.org/abs/2508.10750 |