{"id":337,"date":"2019-06-26T23:39:10","date_gmt":"2019-06-26T23:39:10","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=337"},"modified":"2019-12-20T21:50:50","modified_gmt":"2019-12-20T21:50:50","slug":"quick-square-roots","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/quick-square-roots\/","title":{"rendered":"Quick Square Roots"},"content":{"rendered":"\n<p>Here&#8217;s a very quick way to generate the square root of N. Let A<sub>0<\/sub>=N. Then generate a\u00a0sequence\u00a0of numbers A<sub>1<\/sub>, A<sub>2<\/sub>, A<sub>3<\/sub>, etc. (on your calculator, for instance) by using the formula:<\/p>\n\n\n\n<p style=\"text-align:center\">A<sub>k+1<\/sub>\u00a0= 1\/2 ( A<sub>k<\/sub>\u00a0+ (N\/A<sub>k<\/sub>) ).<\/p>\n\n\n\n<p>This will give a sequence that converges very quickly to the square root of N. In fact, it converges so quickly, that it generally doubles the number of correct digits after each step!<\/p>\n\n\n\n<p>This formula arises as a result of using\u00a0Newton&#8217;s method. Can you figure out how?<\/p>\n\n\n\n<p><strong>Presentation&nbsp;Suggestions:<\/strong><br>Draw a picture, if it is helpful, of how Newton&#8217;s method works. Challenge them to explore what happens if you start off with different values of A<sub>0<\/sub>.<\/p>\n\n\n\n<p><strong>The\u00a0Math\u00a0Behind\u00a0the\u00a0Fact:<\/strong><br>Repeatedly applying a function over and over is called\u00a0<em>iteration<\/em>. Iterated functions are studied in\u00a0<em>dynamical systems<\/em>. Newton&#8217;s method is one example of how iteration can be very useful.<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Quick Square Roots.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n\n\n\n<p><strong>Fun Fact suggested by<\/strong>: <br>Francis Su&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here&#8217;s a very quick way to generate the square root of N. Let A0=N. Then generate a\u00a0sequence\u00a0of numbers A1, A2,&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[2,20,7,131,130,129,4,155,10],"class_list":["post-337","page","type-page","status-publish","hentry","tag-algebra","tag-analysis","tag-calculus","tag-dynamical-systems","tag-functions","tag-iteration","tag-medium","tag-newtons-method","tag-numtheory"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/337","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=337"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/337\/revisions"}],"predecessor-version":[{"id":1593,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/337\/revisions\/1593"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=337"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=337"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}