{"id":39,"date":"2004-10-11T21:47:58","date_gmt":"2004-10-11T21:47:58","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=39"},"modified":"2004-10-11T21:47:58","modified_gmt":"2004-10-11T21:47:58","slug":"amazing-formula","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=39","title":{"rendered":"Amazing Formula"},"content":{"rendered":"<p>There are many interesting formulae in mathematics; <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/df37f7a54973d09fc88ba240bb219ec9.gif' title='\\displaystyle \\sum _{i=1}^\\infty \\frac{1}{n^2} = \\frac {\\pi^2}{6}' alt='\\displaystyle \\sum _{i=1}^\\infty \\frac{1}{n^2} = \\frac {\\pi^2}{6}' align=absmiddle><\/ul>\n<p>must be one of the most amazing of all.<\/p>\n<p>The first reaction is where did that <img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> come from? You can find 14 different proofs of this in a paper on <a href=\"http:\/\/www.maths.ex.ac.uk\/~rjc\/rjc.html\" target=\"_blank\">Robin Chapman&#8217;s Home Page<\/a> [<i>look for Evaluating zeta(2)]<\/i><\/p>\n<p>Given this result can you prove another amazing result?<br \/>\n<i>If you pick two positive integers at random, the probability of them having no common divisor is <\/i><img src='\/maths\/latexrender\/pictures\/73c7cd9f737d1ab67054523110eb62f9.gif' title='\\dfrac{6}{\\pi^2}' alt='\\dfrac{6}{\\pi^2}' align=absmiddle><\/p>\n<p><img src='\/maths\/latexrender\/pictures\/4f08e3dba63dc6d40b22952c7a9dac6d.gif' title='\\pi' alt='\\pi' align=absmiddle> gets everywhere! See <a href=\"http:\/\/en.wikipedia.org\/wiki\/Pi\" target=\"_blank\">Wikipedia<\/a> for more such as <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/f75ed91af7c2f6c8e1e4f83798a2cd2c.gif' title='\\displaystyle \\left(\\frac{1}{2}\\right)!=\\frac{\\sqrt{\\pi}}{2}' alt='\\displaystyle \\left(\\frac{1}{2}\\right)!=\\frac{\\sqrt{\\pi}}{2}' align=absmiddle> or <img src='\/maths\/latexrender\/pictures\/6a1a35fbca319e2638d92098359ac43e.gif' title='\\displaystyle \\frac{2}{\\pi}=\\frac{\\sqrt{2}}{2}\\frac{\\sqrt{2+\\sqrt{2}}}{2}\\frac{\\sqrt{2+\\sqrt{2+\\sqrt{2}}}}{2} \\ldots' alt='\\displaystyle \\frac{2}{\\pi}=\\frac{\\sqrt{2}}{2}\\frac{\\sqrt{2+\\sqrt{2}}}{2}\\frac{\\sqrt{2+\\sqrt{2+\\sqrt{2}}}}{2} \\ldots' align=absmiddle> or <img src='\/maths\/latexrender\/pictures\/0a619ea208c38398839dd79ab38064df.gif' title='\\displaystyle \\int_{-\\infty}^{\\infty}e^{-x^2}\\;dx=\\sqrt{\\pi}' alt='\\displaystyle \\int_{-\\infty}^{\\infty}e^{-x^2}\\;dx=\\sqrt{\\pi}' align=absmiddle><\/ul>\n","protected":false},"excerpt":{"rendered":"<p>There are many interesting formulae in mathematics; must be one of the most amazing of all. The first reaction is where did that come from? You can find 14 different proofs of this in a paper on Robin Chapman&#8217;s Home Page [look for Evaluating zeta(2)] Given this result can you prove another amazing result? If [&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-39","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\/39","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=39"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/39\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=39"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=39"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=39"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}