{"id":26231,"date":"2020-10-30T12:44:29","date_gmt":"2020-10-30T16:44:29","guid":{"rendered":"https:\/\/blog.carlrobitaille.org\/?p=26231"},"modified":"2020-10-30T12:44:30","modified_gmt":"2020-10-30T16:44:30","slug":"%cf%80-is-irrational","status":"publish","type":"post","link":"https:\/\/blog.carlrobitaille.org\/?p=26231","title":{"rendered":"\u03c0 is irrational"},"content":{"rendered":"\n<figure class=\"wp-block-embed-twitter wp-block-embed is-type-rich is-provider-twitter\"><div class=\"wp-block-embed__wrapper\">\n<blockquote class=\"twitter-tweet\" data-width=\"550\" data-dnt=\"true\"><p lang=\"ca\" dir=\"ltr\">\u03c0 is irrational<br><br>Proof. If \u03c0=a\/b with a,b\u2208\u2115 then for any n\u2208\u2115 the integral<br>a\u207f\u03c0\u207f\u207a\u00b9 \u222b\u2080\u00b9 sin(\u03c0x) {x\u207f(1\u2013x)\u207f\/n!} dx<br>is an integer (apply integration by parts 2n-times &amp; use a\/\u03c0=b).<br>But<br>0 &lt; a\u207f\u03c0\u207f\u207a\u00b9 \u222b\u2080\u00b9 sin(\u03c0x) x\u207f(1\u2013x)\u207f\/n! dx &lt; a\u207f\u03c0\u207f\u207a\u00b9\/n! &lt; 1<br>for n large enough.<br>QED <a href=\"https:\/\/t.co\/MjD1if1cAz\">pic.twitter.com\/MjD1if1cAz<\/a><\/p>&mdash; Tam\u00e1s G\u00f6rbe (@TamasGorbe) <a href=\"https:\/\/twitter.com\/TamasGorbe\/status\/1114446647369588738?ref_src=twsrc%5Etfw\">April 6, 2019<\/a><\/blockquote><script async src=\"https:\/\/platform.twitter.com\/widgets.js\" charset=\"utf-8\"><\/script>\n<\/div><\/figure>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[19],"tags":[],"class_list":["post-26231","post","type-post","status-publish","format-standard","hentry","category-mathematiques"],"_links":{"self":[{"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=\/wp\/v2\/posts\/26231","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=26231"}],"version-history":[{"count":1,"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=\/wp\/v2\/posts\/26231\/revisions"}],"predecessor-version":[{"id":26232,"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=\/wp\/v2\/posts\/26231\/revisions\/26232"}],"wp:attachment":[{"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=26231"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=26231"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.carlrobitaille.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=26231"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}