{"id":8947,"date":"2012-05-06T21:44:36","date_gmt":"2012-05-06T20:44:36","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8947"},"modified":"2021-12-29T00:53:43","modified_gmt":"2021-12-29T00:53:43","slug":"mostre-que-todo-o-numero-complexo-nao-nulo-tem-inverso-em-mathbbc","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8947","title":{"rendered":"Mostre que todo o n\u00famero complexo n\u00e3o nulo tem inverso em $\\mathbb{C}$"},"content":{"rendered":"<p><ul id='GTTabs_ul_8947' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8947' class='GTTabs_curr'><a  id=\"8947_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8947' ><a  id=\"8947_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_8947'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Mostre que todo o n\u00famero complexo n\u00e3o nulo tem inverso em $\\mathbb{C}$<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8947' onClick='GTTabs_show(1,8947)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8947'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p>Para\u00a0$z = a + bi$, n\u00e3o nulo, temos: $$\\begin{array}{*{20}{l}} \u00a0 {\\frac{1}{z}}&amp; = &amp;{\\frac{1}{{a + bi}}} \\\\ \u00a0 {}&amp; = &amp;{\\frac{1}{{a + bi}} \\times \\frac{{a &#8211; bi}}{{a &#8211; bi}}} \\\\ \u00a0 {}&amp; = &amp;{\\frac{{a &#8211; bi}}{{{a^2} + {b^2}}}} \\\\ \u00a0 {}&amp; = &amp;{\\frac{a}{{{a^2} + {b^2}}} &#8211; \\frac{b}{{{a^2} + {b^2}}}i} \\end{array}$$<\/p>\n<p>Portanto, todo o n\u00famero complexo n\u00e3o nulo tem inverso em $\\mathbb{C}$.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8947' onClick='GTTabs_show(0,8947)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Mostre que todo o n\u00famero complexo n\u00e3o nulo tem inverso em $\\mathbb{C}$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19178,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,18],"series":[],"class_list":["post-8947","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-12--ano","category-aplicando","category-numeros-complexos-12--ano","tag-12-o-ano","tag-numeros-complexos"],"views":1706,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat69.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8947","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=8947"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8947\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8947"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8947"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8947"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8947"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}