A simpler elementary proof that e is irrational

IF 0.7 Q3 EDUCATION & EDUCATIONAL RESEARCH

International Journal of Mathematical Education in Science and Technology Pub Date : 2023-09-20 DOI:10.1080/0020739x.2023.2255868

F. M. S. Lima

引用次数: 0

Abstract

AbstractIn this short note I present an elementary proof of irrationality for the number e, the base of the natural logarithm. It is simpler than other known proofs as it does not use comparisons with geometric series, nor Beukers' integrals, and it does not assume that e is a rational number from the beginning.Keywords: Irrationality proofEuler's numberMaclaurin seriesalternating seriesMathematic Subject classifications: 41-0197-0111J72 AcknowledgmentsThe author thanks M. R. Javier for some hints on how to simplify and reduce his initial proof without losing the mathematical rigour. Thanks are also due to the anonymous reviewers for their suggestions and hints on additional references.Disclosure statementNo potential conflict of interest was reported by the author.Notes1 The above conclusion that 1/e is not an integer makes unnecessary to check the case q=1.

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一个更简单的e是无理数的初等证明

在这篇短文中，我给出了自然对数底数e无理性的一个初等证明。它比其他已知的证明更简单，因为它不使用几何级数的比较，也不使用Beukers积分，而且它从一开始就不假设e是一个有理数。关键字:无理性证明uler数maclaurin系列交替系列数学学科分类:41-0197-0111J72致谢作者感谢mr . Javier在不失去数学严谨性的情况下如何简化和减少他的初始证明的一些提示。同时也要感谢匿名审稿人对其他参考文献的建议和提示。披露声明作者未报告潜在的利益冲突。注1由于以上1/e不是整数的结论，所以不需要检查q=1的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Mathematical Education in Science and Technology EDUCATION & EDUCATIONAL RESEARCH-

CiteScore

3.30

自引率

11.10%

发文量

123

期刊介绍： Mathematics is pervading every study and technique in our modern world, bringing ever more sharply into focus the responsibilities laid upon those whose task it is to teach it. Most prominent among these is the difficulty of presenting an interdisciplinary approach so that one professional group may benefit from the experience of others. The International Journal of Mathematical Education in Science and Technology provides a medium by which a wide range of experience in mathematical education can be presented, assimilated and eventually adapted to everyday needs in schools, colleges, polytechnics, universities, industry and commerce. Contributions will be welcomed from lecturers, teachers and users of mathematics at all levels on the contents of syllabuses and methods of presentation.