{"id":1340,"date":"2004-06-18T21:57:19","date_gmt":"2004-06-19T05:57:19","guid":{"rendered":"http:\/\/www.ogre.nu\/wp\/?p=1340"},"modified":"2024-11-11T15:42:39","modified_gmt":"2024-11-11T23:42:39","slug":"the-transcendental-dope","status":"publish","type":"post","link":"https:\/\/bendwavy.org\/wp\/?p=1340","title":{"rendered":"the transcendental dope"},"content":{"rendered":"<p>The Straight Dope currently has <a href=\"http:\/\/www.straightdope.com\/columns\/040618.html\">a column about the golden ratio<\/a> and <a href=\"http:\/\/www.straightdope.com\/mailbag\/mfibonacci.html\">a staff report about Fibonacci numbers<\/a>.  These have prompted <a href=\"http:\/\/boards.straightdope.com\/sdmb\/showthread.php?t=262245\">a thread on the Straight Dope Message Board<\/a> which raises a couple of mathematical questions to which I&#8217;d like to post my answers.  But the SDMB is now a pay site, and I&#8217;m not about to pay $14.95 to share my knowledge.<\/p>\n<p>A logical place to respond to the Straight Dope is alt.fan.cecil-adams, but I <a href=\".\/?p=999\">swore off<\/a> posting there.<\/p>\n<p>Well then, I&#8217;ll post here.<\/p>\n<p>One of the questions is the relation between transcendental and irrational numbers.  An <i>algebraic<\/i> number is a solution to a polynomial equation of the form <i>a<sub>0<\/sub>&nbsp;x<sup>0<\/sup> + .&nbsp;.&nbsp;. + a<sub>n<\/sub>&nbsp;x<sup>n<\/sup> = 0<\/i>, where the coefficients <i>a<sub>i<\/sub><\/i> are integers (whole numbers, not necessarily positive).  If <i>n=1<\/i> then <i>x = -a<sub>0<\/sub>\/a<sub>1<\/sub><\/i>, a <i>rational<\/i> number.  Real numbers other than algebraic numbers are called <i>transcendental<\/i>.  (Are there transcendentals which can be considered roots of integer polynomials of infinite degree?)  As the rationals are a special case of the algebraics, it follows that a transcendental number cannot be rational.<\/p>\n<p>The other interesting question is what it means to describe &phi; (phi) a.k.a. &tau; (tau) as the most irrational number.  Any real number can be expressed as a <a href=\"http:\/\/www.numericana.com\/answer\/fractions.htm\">continued fraction<\/a> &mdash; and I see that that page says all that I was about to say on the subject.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Straight Dope currently has a column about the golden ratio and a staff report about Fibonacci numbers. These have prompted a thread on the Straight Dope Message Board which raises a couple of mathematical questions to which I&#8217;d like &hellip; <a href=\"https:\/\/bendwavy.org\/wp\/?p=1340\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"class_list":["post-1340","post","type-post","status-publish","format-standard","hentry","category-mathematics"],"_links":{"self":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/1340","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1340"}],"version-history":[{"count":2,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/1340\/revisions"}],"predecessor-version":[{"id":4671,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=\/wp\/v2\/posts\/1340\/revisions\/4671"}],"wp:attachment":[{"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1340"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1340"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bendwavy.org\/wp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1340"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}