{"id":138,"date":"2006-11-10T15:35:54","date_gmt":"2006-11-10T15:35:54","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=138"},"modified":"2006-11-10T15:35:54","modified_gmt":"2006-11-10T15:35:54","slug":"the-transcendence-of-texmboxhugepitex","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=138","title":{"rendered":"The Transcendence of <img src='\/maths\/latexrender\/pictures\/85a0d14370f5f0c6ab8988ca383c283c.gif' title='\\mbox{\\huge\\pi}' alt='\\mbox{\\huge\\pi}' align=absmiddle>"},"content":{"rendered":"<p>Many people have seen the proof that <img src='\/maths\/latexrender\/pictures\/e1671797c52e15f763380b45e841ec32.gif' title='e' alt='e' align=absmiddle> is irrational (eg <a href=\"http:\/\/planetmath.org\/encyclopedia\/EIsIrrationalProof.html\" target=\"_blank\">e is irrational<\/a>); fewer have read the proof that <img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> is irrational (eg <a href=\"http:\/\/pi314.at\/math\/irrational.html\" target=\"_blank\">Pi is irrational<\/a>) but how many have seen the proofs that <img src='\/maths\/latexrender\/pictures\/e1671797c52e15f763380b45e841ec32.gif' title='e' alt='e' align=absmiddle> and <img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> are <a href=\"http:\/\/en.wikipedia.org\/wiki\/Transcendental_number\" target=\"_blank\">transcendental<\/a> ie not the solution of any polynomial in integer (or rational) coefficients? It&#8217;s a shame as these results are fascinating and are easily accessible to degree level students.<\/p>\n<p>Of course, if you are willing to accept <a href=\"http:\/\/en.wikipedia.org\/wiki\/Lindemann-Weierstrass_theorem\" target=\"_blank\">Lindemann&#8217;s result<\/a> that <i>if <img src='\/maths\/latexrender\/pictures\/7b7f9dbfea05c83784f8b85149852f08.gif' title='\\alpha' alt='\\alpha' align=absmiddle> is algebraic then <img src='\/maths\/latexrender\/pictures\/7a60fda6f503962145990d761066e532.gif' title='e^{\\alpha}' alt='e^{\\alpha}' align=absmiddle> is transcendental<\/i>, then the fact that <img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> is transcendental follows immediately from the transcendence of <img src='\/maths\/latexrender\/pictures\/e1671797c52e15f763380b45e841ec32.gif' title='e' alt='e' align=absmiddle> (click on <i>read more<\/i> below if you can&#8217;t see why).<\/p>\n<p>But that&#8217;s silly as Lindemann&#8217;s theorem is hard to prove. However, I have some notes, typed out on green <a href=\"http:\/\/en.wikipedia.org\/wiki\/Spirit_duplicator\" target=\"_blank\">banda<\/a> sheets. They were a supplement to a course given by <a href=\"http:\/\/en.wikipedia.org\/wiki\/Ian_Stewart_%28mathematician%29\" target=\"_blank\">Ian Stewart<\/a>  back in the long forgotten past, well, 1970 to be precise. Very good they are too. I have LaTeXed them and you can read the proofs yourself here: <a href=\"http:\/\/sixthform.info\/maths\/files\/pitrans.pdf\" title=\"The Transcendence of Pi\" \/>The Transcendence of Pi<\/a>.<\/p>\n<p>Ian Stewart said (and I agree with him) that back in 1970, a student just needed to understand first-year undergraduate analysis to follow the proofs. I wonder if that is still true?<br \/>\n<!--more--><br \/>\nIf <img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> were algebraic then so would <img src='\/maths\/latexrender\/pictures\/d461c3e8dd824cc0d6211194d5c0052f.gif' title='i\\pi' alt='i\\pi' align=absmiddle> be. It would follow from Lindemann that <img src='\/maths\/latexrender\/pictures\/e66292953e4d21fb7ab2543f1cc919f7.gif' title='e^{i\\pi}' alt='e^{i\\pi}' align=absmiddle> is transcendental which is silly. So <img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> must be transcendental.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Many people have seen the proof that is irrational (eg e is irrational); fewer have read the proof that is irrational (eg Pi is irrational) but how many have seen the proofs that and are transcendental ie not the solution of any polynomial in integer (or rational) coefficients? It&#8217;s a shame as these results are [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-138","post","type-post","status-publish","format-standard","hentry","category-articles"],"_links":{"self":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/138","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=138"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/138\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}