{"id":1342,"date":"2015-04-20T07:57:17","date_gmt":"2015-04-20T05:57:17","guid":{"rendered":"http:\/\/www.glc.us.es\/~jalonso\/exercitium\/?p=1342"},"modified":"2015-04-27T07:46:21","modified_gmt":"2015-04-27T05:46:21","slug":"con-minimo-comun-denominador","status":"publish","type":"post","link":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/con-minimo-comun-denominador\/","title":{"rendered":"Con m\u00ednimo com\u00fan denominador"},"content":{"rendered":"<p>Los n\u00fameros racionales se pueden representar como pares de enteros:<\/p>\n<pre lang=\"text\">\n   type Racional a = (a,a)\n<\/pre>\n<p>Definir la funci\u00f3n<\/p>\n<pre lang=\"text\">\n   reducida :: Integral a => [Racional a] -> [Racional a]\n<\/pre>\n<p>tal que (reducida xs) es la lista de los n\u00fameros racionales donde cada uno es igual al correspondiente elemento de xs y el denominador de todos los elementos de (reducida xs) es el menor n\u00famero que cumple dicha condici\u00f3n; es decir, si xs es la lista<\/p>\n<pre lang=\"text\">\n   [(x_1, y_1), ..., (x_n, y_n)]\n<\/pre>\n<p>entonces (reducida xs) es<\/p>\n<pre lang=\"text\">\n   [(z_1, d), ..., (z_n, d)]\n<\/pre>\n<p>tales que<\/p>\n<pre lang=\"text\">\n   z_1\/d = x_1\/y_1, ..., z_n\/d = x_n\/y_n\n<\/pre>\n<p>y d es el menor posible. Por ejemplo,<\/p>\n<pre lang=\"text\">\n   reducida [(1,2),(1,3),(1,4)]  ==  [(6,12),(4,12),(3,12)]\n   reducida [(1,2),(1,3),(6,4)]  ==  [(3,6),(2,6),(9,6)]\n   reducida [(-7,6),(-10,-8)]    ==  [(-14,12),(15,12)]\n   reducida [(8,12)]             ==  [(2,3)]\n<\/pre>\n<h4>Soluciones<\/h4>\n<pre lang=\"haskell\">\ntype Racional a = (a,a) \n\nreducida :: Integral a => [Racional a] -> [Racional a]\nreducida xs = \n    [(x * (m `div` y), m) | (x,y) <- ys]\n    where ys = map fraccionReducida xs\n          m  = mcm [y | (_,y) <- ys]\n\n-- (fraccionReducida r) es el n\u00famero racional igual a r con menor\n-- denominador positivo. Por ejemplo,\n--    fraccionReducida ( 6, 10)  ==  ( 3,5)\n--    fraccionReducida (-6, 10)  ==  (-3,5)\n--    fraccionReducida ( 6,-10)  ==  (-3,5)\n--    fraccionReducida (-6,-10)  ==  ( 3,5)\n--    fraccionReducida ( 3,  5)  ==  ( 3,5)\nfraccionReducida :: Integral a => (a,a) -> (a,a)\nfraccionReducida (x,y) =\n    (s * x1 `div` m, y1 `div` m)\n    where s  = signum x * signum y      \n          x1 = abs x\n          y1 = abs y\n          m  = gcd x1 y1\n          \n-- (mcm xs) es el m\u00ednimo com\u00fan m\u00faltiplo de xs. Por ejemplo,\n--    mcm [2,6,10]  ==  30\nmcm :: Integral a => [a] -> a\nmcm = foldl lcm 1\n<\/pre>\n","protected":false},"excerpt":{"rendered":"<p>Los n\u00fameros racionales se pueden representar como pares de enteros: type Racional a = (a,a) Definir la funci\u00f3n reducida :: Integral a => [Racional a] -> [Racional a] tal que (reducida xs) es la lista de los n\u00fameros racionales donde cada uno es igual al correspondiente elemento de xs y el denominador de todos los&#8230;<\/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,"_kad_post_transparent":"","_kad_post_title":"","_kad_post_layout":"","_kad_post_sidebar_id":"","_kad_post_content_style":"","_kad_post_vertical_padding":"","_kad_post_feature":"","_kad_post_feature_position":"","_kad_post_header":false,"_kad_post_footer":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"footnotes":"","_jetpack_memberships_contains_paid_content":false},"categories":[4],"tags":[130,8,30,185,288,10,287,11,151],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1342"}],"collection":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/comments?post=1342"}],"version-history":[{"count":3,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1342\/revisions"}],"predecessor-version":[{"id":1390,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/posts\/1342\/revisions\/1390"}],"wp:attachment":[{"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/media?parent=1342"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/categories?post=1342"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.glc.us.es\/~jalonso\/exercitium\/wp-json\/wp\/v2\/tags?post=1342"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}