Square Root of Natural Numbers as an infinite Staircase of Continued Fractions
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2025
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| _version_ | 1866901520230907904 |
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| author | bhatnagar, anuj |
| author_facet | bhatnagar, anuj |
| contents | <div>When one lists the continued fractions of √ n across the natural numbers, a striking regularity appears: the coefficients rise in discrete steps, forming an arithmetical and visual staircase. This article reveals the algebraic and geometric underpinnings of that pattern. Each segment between consecutive perfect squares yields a constant pair of leading coefficients and a linearly increasing remainder, giving rise to a simple integer structure that unites the periodicity of quadratic irrationals with an illuminating global picture. We include a short proof of the fundamental relation and three complementary visualizations, along with a brief note on the golden ratio as a conceptual ”zeroth stair.</div> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17699285 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Square Root of Natural Numbers as an infinite Staircase of Continued Fractions bhatnagar, anuj continued fractions quadratic irrationals Lagrange theorem square roots irrational numbers <div>When one lists the continued fractions of √ n across the natural numbers, a striking regularity appears: the coefficients rise in discrete steps, forming an arithmetical and visual staircase. This article reveals the algebraic and geometric underpinnings of that pattern. Each segment between consecutive perfect squares yields a constant pair of leading coefficients and a linearly increasing remainder, giving rise to a simple integer structure that unites the periodicity of quadratic irrationals with an illuminating global picture. We include a short proof of the fundamental relation and three complementary visualizations, along with a brief note on the golden ratio as a conceptual ”zeroth stair.</div> |
| title | Square Root of Natural Numbers as an infinite Staircase of Continued Fractions |
| topic | continued fractions quadratic irrationals Lagrange theorem square roots irrational numbers |
| url | https://doi.org/10.5281/zenodo.17699285 |