{"id":348,"date":"2019-03-21T11:23:11","date_gmt":"2019-03-21T16:23:11","guid":{"rendered":"http:\/\/wordpress.morningside.edu\/meyersh\/?p=348"},"modified":"2019-03-21T11:23:13","modified_gmt":"2019-03-21T16:23:13","slug":"newtons-method","status":"publish","type":"post","link":"https:\/\/wordpress.morningside.edu\/meyersh\/2019\/03\/21\/newtons-method\/","title":{"rendered":"Newtons&#8217; Method"},"content":{"rendered":"<p>Squaring is a lossy process. That is, when a number is squared the root is lost. The search for roots is further complicated\u00a0 because many whole numbers have irrational roots.<\/p>\n<p>Even intuition is misleading, take a guess that the square root of 4 might be half (and you&#8217;d be right), but geometric growth would indicate that this is not the case as the root rapidly becomes a smaller and smaller portion of the square.<\/p>\n<p>But Newton. He provides a method to make a guess and increase it by some small delta. His method, given a guess <em>z<\/em> looks like this:<\/p>\n<blockquote><img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=z_1+%3D+z_0+-+%5Cfrac%7Bz%5E2+-x%7D%7B2z%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"z_1 = z_0 - &#92;frac{z^2 -x}{2z}\" class=\"latex\" \/><\/blockquote>\n<p>Turning around and feeding z back into the expression iteratively yields an answer which tends closer and closer to the root.<\/p>\n<p>Mind boggling basic calculus.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Squaring is a lossy process. That is, when a number is squared the root is lost. The search for roots is further complicated\u00a0 because many whole numbers have irrational roots. Even intuition is misleading, take a guess that the square root of 4 might be half (and you&#8217;d be right), but geometric growth would indicate &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/wordpress.morningside.edu\/meyersh\/2019\/03\/21\/newtons-method\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Newtons&#8217; Method&#8221;<\/span><\/a><\/p>\n","protected":false},"author":11,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[4057],"tags":[],"class_list":["post-348","post","type-post","status-publish","format-standard","hentry","category-maths","entry"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p16T7G-5C","_links":{"self":[{"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/posts\/348","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/comments?post=348"}],"version-history":[{"count":8,"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/posts\/348\/revisions"}],"predecessor-version":[{"id":428,"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/posts\/348\/revisions\/428"}],"wp:attachment":[{"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/media?parent=348"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/categories?post=348"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wordpress.morningside.edu\/meyersh\/wp-json\/wp\/v2\/tags?post=348"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}