{"id":293,"date":"2025-05-21T22:52:01","date_gmt":"2025-05-21T22:52:01","guid":{"rendered":"https:\/\/qpr.ca\/blogs\/mathstuff\/?p=293"},"modified":"2025-05-21T22:57:18","modified_gmt":"2025-05-21T22:57:18","slug":"proof-that-e-is-irrational","status":"publish","type":"post","link":"https:\/\/qpr.ca\/blogs\/mathstuff\/2025\/05\/21\/proof-that-e-is-irrational\/","title":{"rendered":"Proof that e is Irrational\u00a0"},"content":{"rendered":"<p>Thanks to <a href=\"https:\/\/www.quora.com\/profile\/Alexander-Farrugia\">Alexander Farrugia<\/a> at Quora for passing on this simple proof that <em><span style=\"font-size: 16.95px; text-wrap-mode: nowrap;\">e<\/span><\/em> is irrational, that is understandable by a competent high schooler &#8211; apparently based on a version posted by Sridhar Ramesh several years ago.<\/p>\n<p>The basic idea is to use the alternating series representation of $#1\/e#$ to show that for all $#n#$, $#1\/e#$ is between $#S_n=\\frac{a}{n!}#$ and $#S_{n+1}=S_{n}\\pm\\frac{1}{(n+1)!}#$ and so between $#\\frac{a}{n!}#$ and $#\\frac{a\\pm 1}{n!}#$ for some integer $#a#$. This of course means that there is no $#n#$ for which $#1\/e#$ can be written as a fraction $#N\/D#$ with denominator $#D=n!#$, but if $#1\/e=N\/D#$ then it can also be written as $#\\frac{N(D-1)!}{D!}#$ and so <em>can<\/em> be written as a fraction with factorial denominator.<\/p>\n<p>Source: <em><a href=\"https:\/\/farrugiamaths.quora.com\/A-Simple-Proof-that-e-is-Irrational\">(1002) A Simple Proof that e is Irrational &#8211; Farrugia Maths &#8211; Quora<\/a><\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Thanks to Alexander Farrugia at Quora for passing on this simple proof that e is irrational, that is understandable by a competent high schooler &#8211; apparently based on a version posted by Sridhar Ramesh several years ago. The basic idea is to use the alternating series representation of $#1\/e#$ to show that for all $#n#$,&hellip;<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-293","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/posts\/293","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/comments?post=293"}],"version-history":[{"count":3,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/posts\/293\/revisions"}],"predecessor-version":[{"id":303,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/posts\/293\/revisions\/303"}],"wp:attachment":[{"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/media?parent=293"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/categories?post=293"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/qpr.ca\/blogs\/mathstuff\/wp-json\/wp\/v2\/tags?post=293"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}