{"id":3166,"date":"2012-03-02T11:35:33","date_gmt":"2012-03-02T16:35:33","guid":{"rendered":"http:\/\/www.decisionsciencenews.com\/?p=3166"},"modified":"2012-03-04T16:19:59","modified_gmt":"2012-03-04T21:19:59","slug":"how-to-square-numbers-in-your-head","status":"publish","type":"post","link":"https:\/\/www.decisionsciencenews.com\/?p=3166","title":{"rendered":"How to square numbers in your head"},"content":{"rendered":"<p>MENTALLY MULTIPLY NUMBERS BY THEMSELVES<\/p>\n<p style=\"text-align: center;\"><a href=\"http:\/\/www.decisionsciencenews.com\/wp-content\/uploads\/2012\/03\/diff.eng_.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3169\" title=\"diff.eng\" src=\"http:\/\/www.decisionsciencenews.com\/wp-content\/uploads\/2012\/03\/diff.eng_.jpg\" alt=\"\" width=\"480\" height=\"320\" \/><\/a><\/p>\n<p style=\"text-align: center;\"><em>Babbage&#8217;s Difference Engine is fueled by squares<\/em><\/p>\n<p>Even in the age of ubiquitous computing, its usually faster to do a simple operation like squaring a number in your head as opposed to doing it on paper or firing up <a href=\"http:\/\/www.r-project.org\/\">R<\/a>. Everyday decision making in science needs to happen in a fast and frugal manner. Assuming you know your multiplication tables up to 10&#215;10, here&#8217;s how to compute the squares of numbers up to 100 in your noggin (and beyond if you are willing to bootstrap).<\/p>\n<p>A. If the number ends in 0, chop off the 0s, square what is left, and put back two 0s for each one you knocked off. Examples:<\/p>\n<ul>\n<li>50: chopping off the zero gives 5, squaring that gives 25, replacing the 0 with two 0s gives 2500<\/li>\n<li>100: chopping off the zero gives 1, squaring that gives 1, replacing each chopped 0 with two gives 10,000<\/li>\n<\/ul>\n<p>B. If the number ends in 5, chop of the five, take what is left and multiply it by the next highest number, stick 25 on the end Examples:<\/p>\n<ul>\n<li>35: chopping off the 5 gives 3, multiplying what&#8217;s left (3) by the next highest number (4) gives 12, sticking 25 on the end gives 1225<\/li>\n<li>95: chopping off the 5 gives 9, multiplying what&#8217;s left (9) that by the next highest number (10) gives 90, sticking 25 on the end gives 9025<\/li>\n<\/ul>\n<p>C. If the number is 1 greater than a number that ends in 5 or 0, first square the number ending in 5 or 0, as above, then add to this the number ending in 5 or 0 and the number. Examples:<\/p>\n<ul>\n<li>51: 51 is one greater than 50, the square of which we know from above is 2500. To this we add the number ending in 5 or 0 (50) and the number (51). 2500 + 50 + 51 = 2601<\/li>\n<li>36: 36 is one greater than 35, the square of which we know from above is 1225. To this we add the number ending in 5 or 0 (35) and the number (36). 1225 + 35 + 36 = 1296<\/li>\n<\/ul>\n<p>If the number is 1 less than a number that ends in 5 or 0, first square the number ending in 5 or 0, as above, then subtract from this the number ending in 5 or 0 and the number. Examples:<\/p>\n<ul>\n<li>49: 49 is one less than 50, the square of which we know from above is 2500. From this we subtract the number ending in 5 or 0 (50) and the number (49). 2500 &#8211; 50 &#8211; 49 = 2401<\/li>\n<li>34: 34 is one less than 35, the square of which we know from above is 1225. From this we subtract the number ending in 5 or 0 (35) and the number (34). 1225 &#8211; 35 &#8211; 34 = 1156<\/li>\n<\/ul>\n<p>D. If the number is 2 greater than a number that ends in 5 or 0, first square the number ending in 5 or 0, as above, then add to this four times the number that is 1 greater than the number ending in 5 or 0. Examples:<\/p>\n<ul>\n<li>52: 52 is two greater than 50, the square of which we know from above is 2500. To this we add four times the number that is 1 greater than the number ending in 5 or 0 (51). 2500 + 4 * 51 = 2704<\/li>\n<li>37: 37 is two greater than 35, the square of which we know from above is 1225. To this we add four times the number that is 1 greater than the number ending in 5 or 0 (36). 1225 + 4 * 36 = 1369<\/li>\n<\/ul>\n<p>If the number is 2 less than a number that ends in 5 or 0, first square the number ending in 5 or 0, as above, then subtract from this four times the number that is 1 less than the number ending in 5 or 0. Examples:<\/p>\n<ul>\n<li>48: 48 is two less than 50, the square of which we know from above is 2500. From this we subtract four times the number that is 1 less than the number ending in 5 or 0 (49). 2500 &#8211; 4 * 49 = 2304<\/li>\n<li>33: 33 is two less than 35, the square of which we know from above is 1225. From this we subtract four times the number that is 1 less than the number ending in 5 or 0 (34). 1225 &#8211; 4 * 34 = 1089<\/li>\n<\/ul>\n<p>Since every number is either 1 or 2 greater or less than a number ending in 0 or 5, we are done.<\/p>\n<p>Happy squaring!<\/p>\n<p>Proof of C: First part: (N+1)^2=N^2+N+(N+1), Second part (N-1)^2=N^2-N-N+1=N^2-N-(N-1)<br \/>\nProof of D: First part: (N+2)^2=N^2+4N+4=N^2+4(N+1), Second part (N-2)^2=N^2-4N+4=N^2-4(N-1)<\/p>\n<p><span style=\"font-size: xx-small;\">Photo credit: http:\/\/www.flickr.com\/photos\/mrgiles\/325903759\/<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>MENTALLY MULTIPLY NUMBERS BY THEMSELVES<\/p>\n<p>Assume you know your multiplication tables up to 10&#215;10. Here&#8217;s how to compute the squares of numbers from 11 to 100.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","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,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[4,16,74],"tags":[20,345,341,342,343,1208,344],"class_list":["post-3166","post","type-post","status-publish","format-standard","hentry","category-encyclopedia","category-ideas","category-r","tag-decision-making","tag-howto","tag-math","tag-mental-math","tag-multiplication","tag-r","tag-squaring"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4LKj-P4","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=\/wp\/v2\/posts\/3166","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3166"}],"version-history":[{"count":13,"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=\/wp\/v2\/posts\/3166\/revisions"}],"predecessor-version":[{"id":3179,"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=\/wp\/v2\/posts\/3166\/revisions\/3179"}],"wp:attachment":[{"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3166"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3166"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.decisionsciencenews.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}