A simple proof that e q is irrational

A simple proof that e q is irrational | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 2 September 2025 V1 Latest version Share on A simple proof that e q is irrational Author : Ali Chtatbi 0009-0005-0058-6835 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175683403.36542352/v1 383 views 121 downloads Contents Abstract Supplementary Material Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract In this paper we are gonna domonstrate a simple proof of the irrationality of e q in a more direct manner, using infinite series. 1 Background In 1737, Euler gave the first proof of irrationality of e by using the simple continued fraction expansion see [1] [2], Then Fourier introduce a really nice proof by contradiction [3][4], years after Liouville adapted Fourier's methods to prove that e 2 is also irrational, in fact, e can't be the root of a second degree equation with rational coefficients, and by using the same trick, he managed to prove the irrationality of e 4 [5], But e 3, e 5 seems to be difficult to obtain using the previous ideas, In 1891, Hurwitz explained how it is possible to prove that e is not root of a third degree [6]. However, in [7], Hermite uses a new idea and some calculus to prove the strong result that e q is irrational for any non-zero rational q. But the question remains Can Fourier's proof of the irrationality of e be modified to establish the irrationality of e q, where q is an integer? Supplementary Material File (a_simple_proof_that_e_q_is_irrational.pdf) Download 46.81 KB Information & Authors Information Version history V1 Version 1 02 September 2025 Copyright This work is licensed under a Creative Commons Attribution 4.0 International License Keywords irrational numbers mathematica mathematics number theory Authors Affiliations Ali Chtatbi 0009-0005-0058-6835 [email protected] View all articles by this author Metrics & Citations Metrics Article Usage 383 views 121 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Ali Chtatbi. A simple proof that e q is irrational. Authorea . 02 September 2025. 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