$\pi$ and Arc-Length
| Autor: | Nathan, Joseph Amal |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Mathematics Newsletter, Vol.31 #1, March-June (2020) 26-30 |
| Druh dokumentu: | Working Paper |
| Popis: | We use the classical definitions (i) $\pi$ is the ratio of area to the square of the radius of a circle; (ii) $\pi$ is the ratio of circumference to the diameter of a circle, to prove $\pi$'s existence within the purview of Euclidean geometry. Next we show that the "arc-length" (Definition 1) is deducible from Euclidean geometry. Then we prove the Non-Euclidean-Axioms(NEA) of Archimedes (Corollary 4 and 5) and that the arc-length integral converges to the arc-length. We justify why `Euclidean Metric' (Definition 5) is a correct metric for arc-length; derive expressions for area, circumference of a circle and finally prove the equivalence of definitions (i) and (ii). Comment: 10 pages with 5figures |
| Databáze: | arXiv |
| Externí odkaz: |
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