{"id":372,"date":"2007-04-04T21:54:05","date_gmt":"2007-04-04T21:54:05","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/04\/04\/surreal-division-a-weak-post\/"},"modified":"2007-04-04T21:54:05","modified_gmt":"2007-04-04T21:54:05","slug":"surreal-division-a-weak-post","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/04\/04\/surreal-division-a-weak-post\/","title":{"rendered":"Surreal Division (A weak post)"},"content":{"rendered":"<p> Coming back from games to numbers, I promised earlier that I would define<br \/>\ndivision. Division in surreal numbers is, unfortunately, <em>ugly<\/em>. We start with<br \/>\na simple, basic identity: if a=b&times;c, and a is not zero, then c=a&times;(1\/b). So if we can define how to take the reciprocal of a surreal number, then division falls out naturally from combining it the reciprocal with multiplication.<\/p>\n<p> This is definitely one of my weaker posts; I&#8217;ve debated whether or not to post it at all, but I promised that I&#8217;d show how surreal division is defined, and I don&#8217;t foresee my having time to do a better job of explaining it in a reasonable time frame.. So my apologies if this is harder to follow than my usual posts.<\/p>\n<p><!--more--><\/p>\n<p> We can define the reciprocal for surreals. It&#8217;s an iterative process, which will eventually converge on a solution, where eventually is defined in surreal terms&#8230; meaning<br \/>\nthat it might take forever to get the exactly answer. (After all, many simple reciprocals &#8211; e.g., 1\/3, aren&#8217;t born until generation &omega;, so starting from 3 (which is a third generation number), we need to follow the procedure until we can get to generation &omega; numbers.)<\/p>\n<p> To make notations easier, let&#8217;s say we have a number x. What we want to do is<br \/>\nfind a number y such that xy=1.<\/p>\n<p> We&#8217;ll define reciprocal only for <em>positive<\/em> numbers &#8211; we can derive it for negative numbers using multiplication. Our first step is to &#8220;normalize&#8221; x. What that means is, convert x into a form where &forall;n&isin;x<sub>L<\/sub>, 0&le;n. The normal form<br \/>\nof any positive number contains 0 and other positive numbers in its left set &#8211; and obviously, only positive number in its right set.<\/p>\n<p> So if we have x in normal form, then we can say y={0,L<sub>1<\/sub>,L<sub>2<\/sub>|R<sub>1<\/sub>,R<sub>2<\/sub>} where:<\/p>\n<ul>\n<li>L<sub>1<\/sub> = (1+(x<sub>R<\/sub>-x)y<sub>L<\/sub>)\/x<sub>R<\/sub><\/li>\n<li>L<sub>2<\/sub>= (1+(x<sub>L<\/sub>-x)y<sub>R<\/sub>)\/x<sub>L<\/sub><\/li>\n<li>R<sub>1<\/sub>=(1+(x<sub>L<\/sub>-x)y<sub>L<\/sub>)\/x<sub>L<\/sub><\/li>\n<li>R<sub>2<\/sub>=(1+(x<sub>R<\/sub>-x)y<sub>R<\/sub>)\/x<sub>R<\/sub><\/li>\n<\/ul>\n<p> Just looking at that, you can see the recursion in it &#8211; it involves divisions by x<sub>L<\/sub> and x<sub>R<\/sub>; and the definition of y itself involves the left or right sets of y. Basically, you keep computing values of Y &#8211; each step, you get closer. <\/p>\n<p> Conway uses 1\/3 as an example. Start with x=3={0,1,2|}. y={0,1\/2(1-y<sub>R<\/sub>)) | 1\/2(1-y<sub>L<\/sub>). If we start with y<sub>L<\/sub>=0, and then iterate, we&#8217;ll get y={0,1\/4,5\/16,&#8230;|1\/2,3\/8, &#8230;}. <\/p>\n<p> Unfortunately, I can&#8217;t give you much in the way of intuition for why this works. I can barely wrap my head around it. The only way to really get it is to work through some<br \/>\nexamples; but even doing example, the book-keeping is a nightmare. I tried to do an example on my own to show you how it works for a number where it will converge relatively quickly, but I keep losing track of where I am&#8230; <\/p>\n<p> Fortunately, in some sense, it <em>doesn&#8217;t matter<\/em>. The important thing to know is that you <em>can<\/em> define division in a valid way for the surreal numbers; you can prove that the reciprocal exists for any surreal number except zero; the reciprocal of zero is undefined using this definition; and that the result of surreal division using surreal reciprocals is provably exactly the same as the result of division using the traditional version of the real numbers.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Coming back from games to numbers, I promised earlier that I would define division. Division in surreal numbers is, unfortunately, ugly. We start with a simple, basic identity: if a=b&times;c, and a is not zero, then c=a&times;(1\/b). So if we can define how to take the reciprocal of a surreal number, then division falls out [&hellip;]<\/p>\n","protected":false},"author":1,"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":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[62],"tags":[],"class_list":["post-372","post","type-post","status-publish","format-standard","hentry","category-surreal-numbers"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-60","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/372","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=372"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/372\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=372"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=372"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=372"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}