{"id":22389,"date":"2022-10-18T13:24:46","date_gmt":"2022-10-18T12:24:46","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=22389"},"modified":"2022-10-19T01:55:57","modified_gmt":"2022-10-19T00:55:57","slug":"considera-os-numeros-racionais","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=22389","title":{"rendered":"Considera os n\u00fameros racionais"},"content":{"rendered":"\n<p><ul id='GTTabs_ul_22389' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_22389' class='GTTabs_curr'><a  id=\"22389_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_22389' ><a  id=\"22389_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_22389'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considera os n\u00fameros racionais seguintes:<br \/>\\[\\begin{array}{*{20}{c}}{\\frac{{12}}{{105}}}&amp;{\\rm{e}}&amp;{\\frac{{135}}{{300}}}\\end{array}\\]<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Indica qual destes n\u00fameros admite uma representa\u00e7\u00e3o em d\u00edzima finita.<\/li>\n<li>Representa-os na forma de d\u00edzima finita ou infinita peri\u00f3dica.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_22389' onClick='GTTabs_show(1,22389)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_22389'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>Comecemos por determinar as formas irredut\u00edveis das fra\u00e7\u00f5es dadas:<br \/>\\[\\begin{array}{*{20}{l}}{\\frac{{12}}{{105}} = \\frac{{2 \\times 2 \\times 3}}{{3 \\times 5 \\times 7}} = \\frac{{2 \\times 2}}{{5 \\times 7}} = \\frac{4}{{35}}}\\\\{}\\\\{\\frac{{135}}{{300}} = \\frac{{3 \\times 3 \\times 3 \\times 5}}{{2 \\times 2 \\times 3 \\times 5 \\times 5}} = \\frac{{3 \\times 3}}{{2 \\times 2 \\times 5}} = \\frac{9}{{20}}}\\end{array}\\]<br \/>Das fra\u00e7\u00f5es dadas, aquela que admite uma representa\u00e7\u00e3o em d\u00edzima finita \u00e9 \\(\\frac{{135}}{{300}}\\), pois \u00e9 a \u00fanica que \u00e9 equivalente a uma fra\u00e7\u00e3o decimal, visto que, na forma irredut\u00edvel, o seu denominador n\u00e3o tem na sua decomposi\u00e7\u00e3o fatores primos diferentes de \\(2\\) e de \\(5\\).<br \/><br \/><\/li>\n<li>Apresentam-se seguidamente os dois n\u00fameros racionais na forma de d\u00edzima infinita peri\u00f3dica e d\u00edzima finita, respetivamente:<br \/>\\[\\begin{array}{*{20}{c}}{\\frac{{12}}{{105}} = \\frac{4}{{35}} = 0,1\\left( {142857} \\right)}&amp;{\\rm{e}}&amp;{\\frac{{135}}{{300}} = \\frac{9}{{20}} = \\frac{{45}}{{100}} = 0,45}\\end{array}\\]<br \/><br \/>\\[\\begin{array}{*{20}{l}}{4,}&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;0&amp;{}&amp;{}&amp;3&amp;5&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}\\\\\\hline{}&amp;5&amp;0&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{0,}&amp;1&amp;1&amp;4&amp;2&amp;8&amp;5&amp;7&amp;1&amp;4\\\\{}&amp;1&amp;5&amp;0&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}\\\\{}&amp;{}&amp;1&amp;0&amp;0&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}\\\\{}&amp;{}&amp;{}&amp;3&amp;0&amp;0&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}\\\\{}&amp;{}&amp;{}&amp;{}&amp;2&amp;0&amp;0&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}\\\\{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;2&amp;5&amp;0&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}\\\\{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;0&amp;5&amp;0&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}\\\\{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;1&amp;5&amp;0&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}\\\\{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;1&amp;0&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_22389' onClick='GTTabs_show(0,22389)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considera os n\u00fameros racionais seguintes:\\[\\begin{array}{*{20}{c}}{\\frac{{12}}{{105}}}&amp;{\\rm{e}}&amp;{\\frac{{135}}{{300}}}\\end{array}\\] Indica qual destes n\u00fameros admite uma representa\u00e7\u00e3o em d\u00edzima finita. Representa-os na forma de d\u00edzima finita ou infinita peri\u00f3dica. Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19189,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[100,97,664],"tags":[424,261,665,666,667],"series":[],"class_list":["post-22389","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-8--ano","category-aplicando","category-numeros-reais","tag-8-o-ano","tag-dizima","tag-dizima-finita","tag-dizima-infinita-periodica","tag-fracao-decimal"],"views":255,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat75.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/22389","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=22389"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/22389\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19189"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=22389"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=22389"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=22389"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=22389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}