{"id":40,"date":"2004-10-18T15:43:00","date_gmt":"2004-10-18T15:43:00","guid":{"rendered":"http:\/\/www.sixthform.info\/maths\/?p=40"},"modified":"2004-10-18T15:43:00","modified_gmt":"2004-10-18T15:43:00","slug":"some-textbooks-misuse-infinity","status":"publish","type":"post","link":"https:\/\/www.sixthform.info\/maths\/?p=40","title":{"rendered":"Some textbooks misuse infinity"},"content":{"rendered":"<p>It&#8217;s happened again! <img src='\/maths\/latexrender\/pictures\/7ed9abff4dafd78d08e616c899412e92.gif' title='\\infty' alt='\\infty' align=absmiddle> used in a textbook (<i>unnamed to protect the guilty<\/i>) as if it were a real number instead of an idea. In a discussion of the formula for the acute angle <img src='\/maths\/latexrender\/pictures\/2554a2bb846cffd697389e5dc8912759.gif' title='\\theta' alt='\\theta' align=absmiddle> between two lines <\/p>\n<ul><img src='\/maths\/latexrender\/pictures\/2b2fad06b13cbed3ccca2af9f8622f2f.gif' title='\\theta=\\tan^{-1}\\left |\\dfrac{m_1-m_2}{1+m_1m_2}\\right |' alt='\\theta=\\tan^{-1}\\left |\\dfrac{m_1-m_2}{1+m_1m_2}\\right |' align=absmiddle><\/ul>\n<p> the following appears:<\/p>\n<ul><i>Putting <img src='\/maths\/latexrender\/pictures\/933ea89df3f14044b6dd67061b2c55e6.gif' title='m_1m_2=-1' alt='m_1m_2=-1' align=absmiddle> gives an angle <img src='\/maths\/latexrender\/pictures\/73f58ebbb5dfa6181cbf804fd05ad7fa.gif' title='\\tan^{-1}(\\infty)=90^{\\circ}' alt='\\tan^{-1}(\\infty)=90^{\\circ}' align=absmiddle>, confirming the condition for the lines to be perpendicular<\/i><\/ul>\n<p> This is of course complete nonsense. As I&#8217;ve said before <img src='\/maths\/latexrender\/pictures\/7f94aec329d482f316b62cbbac40f18b.gif' title='\\tan(90^{\\circ})' alt='\\tan(90^{\\circ})' align=absmiddle> doesn&#8217;t exist and <img src='\/maths\/latexrender\/pictures\/0e5308dd31476b3812978a7cffa2abd4.gif' title='\\tan^{-1}(x)' alt='\\tan^{-1}(x)' align=absmiddle> is only defined on <img src='\/maths\/latexrender\/pictures\/2369a2488f59aa39a3fca53e0eff9f88.gif' title='\\mathbb{R}' alt='\\mathbb{R}' align=absmiddle> ie for <img src='\/maths\/latexrender\/pictures\/f65a12f07093615348bb1ad051ecc1a0.gif' title='-\\infty&lt;x &lt;\\infty' alt='-\\infty&lt;x &lt;\\infty' align=absmiddle><br \/>\nThe textbook was written by the examiners (which is one reason why we use it); this worries me even more.<br \/>\nI suppose this is better than one well-known textbook back in the eighties which solved the equation <img src='\/maths\/latexrender\/pictures\/95be225911690b0f4f917cf6689003b2.gif' title='t(t-3)=t^2-4' alt='t(t-3)=t^2-4' align=absmiddle> by putting <img src='\/maths\/latexrender\/pictures\/bd51c49e8aeb3243980de367dc4cd201.gif' title='\\frac{1}{m}=t' alt='\\frac{1}{m}=t' align=absmiddle> then &#8216;showing&#8217; <img src='\/maths\/latexrender\/pictures\/b00c079c0af9d49ad641b915c7b7dcc9.gif' title='t=\\infty' alt='t=\\infty' align=absmiddle> or <img src='\/maths\/latexrender\/pictures\/85c8238ae8a8751b6a8634ed27636424.gif' title='t=\\frac{4}{3}' alt='t=\\frac{4}{3}' align=absmiddle>. This seems to show that all linear equations are quadratics in disguise; or cubics, quartics &#8211; who knows where this nonsense leads \ud83d\ude15<br \/>\nSee also <a href=\"http:\/\/www.sixthform.info\/maths\/index.php?m=200403#16\" target=\"_blank\">Division by zero shock!<\/x><\/p>\n","protected":false},"excerpt":{"rendered":"<p>It&#8217;s happened again! used in a textbook (unnamed to protect the guilty) as if it were a real number instead of an idea. In a discussion of the formula for the acute angle between two lines the following appears: Putting gives an angle , confirming the condition for the lines to be perpendicular This is [&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-40","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\/40","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=40"}],"version-history":[{"count":0,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=\/wp\/v2\/posts\/40\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=40"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=40"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.sixthform.info\/maths\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=40"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}