{"id":8934,"date":"2012-05-06T19:50:47","date_gmt":"2012-05-06T18:50:47","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=8934"},"modified":"2021-12-29T00:49:22","modified_gmt":"2021-12-29T00:49:22","slug":"mostre-que-4","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=8934","title":{"rendered":"Mostre que"},"content":{"rendered":"<p><ul id='GTTabs_ul_8934' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_8934' class='GTTabs_curr'><a  id=\"8934_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_8934' ><a  id=\"8934_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_8934'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Sendo ${z_1} = {a_1} + {b_1}i$ e ${z_2} = {a_2} + {b_2}i$, mostre que:<\/p>\n<ol>\n<li>$\\overline {{z_1} + {z_2}}\u00a0 = \\overline {{z_1}}\u00a0 + \\overline {{z_2}} $<\/li>\n<li>$\\overline {{z_1}.{z_2}}\u00a0 = \\overline {{z_1}} .\\overline {{z_2}} $<\/li>\n<li>$\\overline {{z_1} &#8211; {z_2}}\u00a0 = \\overline {{z_1}}\u00a0 &#8211; \\overline {{z_2}} $<\/li>\n<li>$\\overline {\\left( {\\frac{{{z_1}}}{{{z_2}}}} \\right)}\u00a0 = \\frac{{\\overline {{z_1}} }}{{\\overline {{z_2}} }}$, para ${z_2} \\ne 0$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_8934' onClick='GTTabs_show(1,8934)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_8934'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\overline {{z_1} + {z_2}} }&amp; = &amp;{\\overline {\\left( {{a_1} + {b_1}i} \\right) + \\left( {{a_2} + {b_2}i} \\right)} }&amp;{}&amp;{}&amp;{\\overline {{z_1}}\u00a0 + \\overline {{z_2}} }&amp; = &amp;{\\overline {\\left( {{a_1} + {b_1}i} \\right)}\u00a0 + \\overline {\\left( {{a_2} + {b_2}i} \\right)} } \\\\<br \/>\n{}&amp; = &amp;{\\overline {\\left( {{a_1} + {a_2}} \\right) + \\left( {{b_1} + {b_2}} \\right)i} }&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\left( {{a_1} &#8211; {b_1}i} \\right) + \\left( {{a_2} &#8211; {b_2}i} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\left( {{a_1} + {a_2}} \\right) &#8211; \\left( {{b_1} + {b_2}} \\right)i}&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\left( {{a_1} + {a_2}} \\right) &#8211; \\left( {{b_1} + {b_2}} \\right)i}<br \/>\n\\end{array}$$<br \/>\nPortanto, $\\overline {{z_1} + {z_2}}\u00a0 = \\overline {{z_1}}\u00a0 + \\overline {{z_2}} $.<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\overline {{z_1}.{z_2}} }&amp; = &amp;{\\overline {\\left( {{a_1} + {b_1}i} \\right).\\left( {{a_2} + {b_2}i} \\right)} }&amp;{}&amp;{}&amp;{\\overline {{z_1}} .\\overline {{z_2}} }&amp; = &amp;{\\overline {\\left( {{a_1} + {b_1}i} \\right)} .\\overline {\\left( {{a_2} + {b_2}i} \\right)} } \\\\<br \/>\n{}&amp; = &amp;{\\overline {{a_1}{a_2} + {a_1}{b_2}i + {a_2}{b_1}i &#8211; {b_1}{b_2}} }&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\left( {{a_1} &#8211; {b_1}i} \\right).\\left( {{a_2} &#8211; {b_2}i} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\overline {\\left( {{a_1}{a_2} &#8211; {b_1}{b_2}} \\right) + \\left( {{a_1}{b_2} + {a_2}{b_1}} \\right)i} }&amp;{}&amp;{}&amp;{}&amp; = &amp;{{a_1}{a_2} &#8211; {a_1}{b_2}i &#8211; {a_2}{b_1}i &#8211; {b_1}{b_2}} \\\\<br \/>\n{}&amp; = &amp;{\\left( {{a_1}{a_2} &#8211; {b_1}{b_2}} \\right) &#8211; \\left( {{a_1}{b_2} + {a_2}{b_1}} \\right)i}&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\left( {{a_1}{a_2} &#8211; {b_1}{b_2}} \\right) &#8211; \\left( {{a_1}{b_2} + {a_2}{b_1}} \\right)i}<br \/>\n\\end{array}$$<br \/>\nPortanto, $\\overline {{z_1}.{z_2}}\u00a0 = \\overline {{z_1}} .\\overline {{z_2}} $.<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\overline {{z_1} &#8211; {z_2}} }&amp; = &amp;{\\overline {\\left( {{a_1} + {b_1}i} \\right) &#8211; \\left( {{a_2} + {b_2}i} \\right)} }&amp;{}&amp;{}&amp;{\\overline {{z_1}}\u00a0 &#8211; \\overline {{z_2}} }&amp; = &amp;{\\overline {\\left( {{a_1} + {b_1}i} \\right)}\u00a0 &#8211; \\overline {\\left( {{a_2} + {b_2}i} \\right)} } \\\\<br \/>\n{}&amp; = &amp;{\\overline {\\left( {{a_1} &#8211; {a_2}} \\right) + \\left( {{b_1} &#8211; {b_2}} \\right)i} }&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\left( {{a_1} &#8211; {b_1}i} \\right) &#8211; \\left( {{a_2} &#8211; {b_2}i} \\right)} \\\\<br \/>\n{}&amp; = &amp;{\\left( {{a_1} &#8211; {a_2}} \\right) &#8211; \\left( {{b_1} &#8211; {b_2}} \\right)i}&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\left( {{a_1} &#8211; {a_2}} \\right) &#8211; \\left( {{b_1} &#8211; {b_2}} \\right)i}<br \/>\n\\end{array}$$<br \/>\nPortanto, $\\overline {{z_1} &#8211; {z_2}}\u00a0 = \\overline {{z_1}}\u00a0 &#8211; \\overline {{z_2}} $.<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n$$\\begin{array}{*{20}{l}}<br \/>\n{\\overline {\\left( {\\frac{{{z_1}}}{{{z_2}}}} \\right)} }&amp; = &amp;{\\overline {\\left( {\\frac{{{a_1} + {b_1}i}}{{{a_2} + {b_2}i}}} \\right)} }&amp;{}&amp;{}&amp;{\\frac{{\\overline {{z_1}} }}{{\\overline {{z_2}} }}}&amp; = &amp;{\\frac{{\\overline {{a_1} + {b_1}i} }}{{\\overline {{a_2} + {b_2}i} }}} \\\\<br \/>\n{}&amp; = &amp;{\\overline {\\left( {\\frac{{{a_1} + {b_1}i}}{{{a_2} + {b_2}i}} \\times \\frac{{{a_2} &#8211; {b_2}i}}{{{a_2} &#8211; {b_2}i}}} \\right)} }&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\frac{{{a_1} &#8211; {b_1}i}}{{{a_2} &#8211; {b_2}i}} \\times \\frac{{{a_2} + {b_2}i}}{{{a_2} + {b_2}i}}} \\\\<br \/>\n{}&amp; = &amp;{\\overline {\\left( {\\frac{{\\left( {{a_1}{a_2} + {b_1}{b_2}} \\right) + \\left( {{a_2}{b_1} &#8211; {a_1}{b_2}} \\right)i}}{{{{\\left( {{a_2}} \\right)}^2} + {{\\left( {{b_2}} \\right)}^2}}}} \\right)} }&amp;{}&amp;{}&amp;{}&amp; = &amp;{\\frac{{\\left( {{a_1}{a_2} + {b_1}{b_2}} \\right) &#8211; \\left( {{a_2}{b_1} &#8211; {a_1}{b_2}} \\right)i}}{{{{\\left( {{a_2}} \\right)}^2} + {{\\left( {{b_2}} \\right)}^2}}}} \\\\<br \/>\n{}&amp; = &amp;{\\frac{{\\left( {{a_1}{a_2} + {b_1}{b_2}} \\right) &#8211; \\left( {{a_2}{b_1} &#8211; {a_1}{b_2}} \\right)i}}{{{{\\left( {{a_2}} \\right)}^2} + {{\\left( {{b_2}} \\right)}^2}}}}&amp;{}&amp;{}&amp;{}&amp;{}&amp;{}<br \/>\n\\end{array}$$<br \/>\nPortanto, $\\overline {\\left( {\\frac{{{z_1}}}{{{z_2}}}} \\right)}\u00a0 = \\frac{{\\overline {{z_1}} }}{{\\overline {{z_2}} }}$, para ${z_2} \\ne 0$.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_8934' onClick='GTTabs_show(0,8934)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Sendo ${z_1} = {a_1} + {b_1}i$ e ${z_2} = {a_2} + {b_2}i$, mostre que: $\\overline {{z_1} + {z_2}}\u00a0 = \\overline {{z_1}}\u00a0 + \\overline {{z_2}} $ $\\overline {{z_1}.{z_2}}\u00a0 = \\overline {{z_1}} .\\overline&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19581,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[226,97,310],"tags":[427,312,18],"series":[],"class_list":["post-8934","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-conjugado","tag-numeros-complexos"],"views":2470,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat195.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8934","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=8934"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/8934\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19581"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8934"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8934"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8934"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=8934"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}