{"id":430,"date":"2018-12-24T10:21:33","date_gmt":"2018-12-24T09:21:33","guid":{"rendered":"https:\/\/blogs.ua.es\/matesfacil\/?page_id=430"},"modified":"2019-06-13T08:52:58","modified_gmt":"2019-06-13T07:52:58","slug":"modulo-y-argumento-de-un-numero-imaginario","status":"publish","type":"page","link":"https:\/\/blogs.ua.es\/matesfacil\/secundaria-numeros-operaciones\/numeros-imaginarios\/modulo-y-argumento-de-un-numero-imaginario\/","title":{"rendered":"M\u00f3dulo y argumento de un n\u00famero imaginario"},"content":{"rendered":"<p>Dado un n\u00famero complejo en su forma bin\u00f3mica\u00a0<span id=\"MathJax-Element-1-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-1\" class=\"mjx-math\"><span id=\"MJXc-Node-2\" class=\"mjx-mrow\"><span id=\"MJXc-Node-3\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><span id=\"MJXc-Node-4\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-5\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-6\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-7\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-8\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">i<\/span><\/span><\/span><\/span><\/span>,\u00a0 se define el <strong>m\u00f3dulo<\/strong> de\u00a0<span id=\"MathJax-Element-2-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-9\" class=\"mjx-math\"><span id=\"MJXc-Node-10\" class=\"mjx-mrow\"><span id=\"MJXc-Node-11\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><\/span><\/span><\/span>\u00a0como<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-modulo-argumento-angulo-propiedades.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"el m\u00f3dulo de un complejo z = a+bi es la ra\u00edz cuadrada de a^2 + b^2\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img3\/T0.png\" alt=\"Definici\u00f3n de m\u00f3dulo, argumento y conjugado de los n\u00fameros complejos, con interpretaci\u00f3n geom\u00e9trica y ejemplos. Enunciamos las propiedades b\u00e1sicas del conjugado y del m\u00f3dulo (de la suma, del producto, del cociente, etc.). Matem\u00e1ticas. N\u00fameros complejos. Secundaria. Bachillerato. Universidad. \" width=\"138\" height=\"27\" \/><\/a><\/p>\n<p>Se define el <strong>argumento<\/strong> de\u00a0<span id=\"MathJax-Element-3-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-12\" class=\"mjx-math\"><span id=\"MJXc-Node-13\" class=\"mjx-mrow\"><span id=\"MJXc-Node-14\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><\/span><\/span><\/span>\u00a0como<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-modulo-argumento-angulo-propiedades.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"el argumento de un complejo z = a+bi es la arcotangente de b\/a\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img3\/T1.png\" alt=\"Definici\u00f3n de m\u00f3dulo, argumento y conjugado de los n\u00fameros complejos, con interpretaci\u00f3n geom\u00e9trica y ejemplos. Enunciamos las propiedades b\u00e1sicas del conjugado y del m\u00f3dulo (de la suma, del producto, del cociente, etc.). Matem\u00e1ticas. N\u00fameros complejos. Secundaria. Bachillerato. Universidad. \" width=\"248\" height=\"48\" \/><\/a><\/p>\n<p><strong>Nota 1:<\/strong>\u00a0la funci\u00f3n arcotangente proporciona el \u00e1ngulo entre -45\u00ba y 45\u00ba.<\/p>\n<p><strong>Nota 2:<\/strong>\u00a0observad que, por ejemplo, la funci\u00f3n arcotangente proporciona el mismo \u00e1ngulo para\u00a0<span id=\"MathJax-Element-4-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-15\" class=\"mjx-math\"><span id=\"MJXc-Node-16\" class=\"mjx-mrow\"><span id=\"MJXc-Node-17\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><span id=\"MJXc-Node-18\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-19\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-20\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-21\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-22\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">i<\/span><\/span><\/span><\/span><\/span>\u00a0y para\u00a0<span id=\"MathJax-Element-5-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-23\" class=\"mjx-math\"><span id=\"MJXc-Node-24\" class=\"mjx-mrow\"><span id=\"MJXc-Node-25\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">w<\/span><\/span><span id=\"MJXc-Node-26\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-27\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-28\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-29\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-30\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-31\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">i<\/span><\/span><\/span><\/span><\/span>. Sin embargo,\u00a0<span id=\"MathJax-Element-6-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-32\" class=\"mjx-math\"><span id=\"MJXc-Node-33\" class=\"mjx-mrow\"><span id=\"MJXc-Node-34\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><\/span><\/span><\/span>\u00a0y\u00a0<span id=\"MathJax-Element-7-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-35\" class=\"mjx-math\"><span id=\"MJXc-Node-36\" class=\"mjx-mrow\"><span id=\"MJXc-Node-37\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">w<\/span><\/span><\/span><\/span><\/span>\u00a0est\u00e1n en cuadrantes distintos, as\u00ed que su argumento es distinto. Para solucionar esto:<\/p>\n<ul>\n<li>Si el complejo est\u00e1 en el segundo cuadrante (<span id=\"MathJax-Element-8-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-38\" class=\"mjx-math\"><span id=\"MJXc-Node-39\" class=\"mjx-mrow\"><span id=\"MJXc-Node-40\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-41\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-42\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>,\u00a0<span id=\"MathJax-Element-9-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-43\" class=\"mjx-math\"><span id=\"MJXc-Node-44\" class=\"mjx-mrow\"><span id=\"MJXc-Node-45\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-46\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-47\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>), hay que\u00a0<strong>sumar<\/strong>\u00a0180\u00ba al \u00e1ngulo obtenido.<\/li>\n<li>Si el complejo est\u00e1 en el tercer cuadrante (<span id=\"MathJax-Element-10-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-48\" class=\"mjx-math\"><span id=\"MJXc-Node-49\" class=\"mjx-mrow\"><span id=\"MJXc-Node-50\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-51\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-52\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>,\u00a0<span id=\"MathJax-Element-11-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-53\" class=\"mjx-math\"><span id=\"MJXc-Node-54\" class=\"mjx-mrow\"><span id=\"MJXc-Node-55\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-56\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-57\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>), hay que\u00a0<strong>restar<\/strong>\u00a0180\u00ba al \u00e1ngulo obtenido.<\/li>\n<\/ul>\n<p><strong>Nota 3:<\/strong>\u00a0si\u00a0<span id=\"MathJax-Element-12-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-58\" class=\"mjx-math\"><span id=\"MJXc-Node-59\" class=\"mjx-mrow\"><span id=\"MJXc-Node-60\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-61\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-62\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span>, el argumento es<\/p>\n<ul>\n<li>0\u00b0 (0 radianes) si\u00a0<span id=\"MathJax-Element-13-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-63\" class=\"mjx-math\"><span id=\"MJXc-Node-64\" class=\"mjx-mrow\"><span id=\"MJXc-Node-65\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-66\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-67\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/li>\n<li>90\u00b0 (<span id=\"MathJax-Element-14-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-68\" class=\"mjx-math\"><span id=\"MJXc-Node-69\" class=\"mjx-mrow\"><span id=\"MJXc-Node-70\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">\u03c0<\/span><\/span><span id=\"MJXc-Node-71\" class=\"mjx-texatom\"><span id=\"MJXc-Node-72\" class=\"mjx-mrow\"><span id=\"MJXc-Node-73\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\/<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-74\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><\/span><\/span><\/span>\u00a0radianes) si\u00a0<span id=\"MathJax-Element-15-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-75\" class=\"mjx-math\"><span id=\"MJXc-Node-76\" class=\"mjx-mrow\"><span id=\"MJXc-Node-77\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-78\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&gt;<\/span><\/span><span id=\"MJXc-Node-79\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/li>\n<li>270\u00b0 (<span id=\"MathJax-Element-16-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-80\" class=\"mjx-math\"><span id=\"MJXc-Node-81\" class=\"mjx-mrow\"><span id=\"MJXc-Node-82\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-83\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">\u03c0<\/span><\/span><span id=\"MJXc-Node-84\" class=\"mjx-texatom\"><span id=\"MJXc-Node-85\" class=\"mjx-mrow\"><span id=\"MJXc-Node-86\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\/<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-87\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">2<\/span><\/span><\/span><\/span><\/span>\u00a0radianes) si\u00a0<span id=\"MathJax-Element-17-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-88\" class=\"mjx-math\"><span id=\"MJXc-Node-89\" class=\"mjx-mrow\"><span id=\"MJXc-Node-90\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-91\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">&lt;<\/span><\/span><span id=\"MJXc-Node-92\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">0<\/span><\/span><\/span><\/span><\/span><\/li>\n<\/ul>\n<p>Adem\u00e1s, se denomina\u00a0<strong>argumento principal<\/strong>\u00a0de\u00a0<span id=\"MathJax-Element-18-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">z<\/span><\/span>,\u00a0<span id=\"MathJax-Element-19-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-96\" class=\"mjx-math\"><span id=\"MJXc-Node-97\" class=\"mjx-mrow\"><span id=\"MJXc-Node-98\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">A<\/span><\/span><span id=\"MJXc-Node-99\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">r<\/span><\/span><span id=\"MJXc-Node-100\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">g<\/span><\/span><span id=\"MJXc-Node-101\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">(<\/span><\/span><span id=\"MJXc-Node-102\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><span id=\"MJXc-Node-103\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">)<\/span><\/span><\/span><\/span><\/span>, al argumento de\u00a0<span id=\"MathJax-Element-20-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-104\" class=\"mjx-math\"><span id=\"MJXc-Node-105\" class=\"mjx-mrow\"><span id=\"MJXc-Node-106\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><\/span><\/span><\/span>\u00a0en el intervalo\u00a0<span id=\"MathJax-Element-21-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-107\" class=\"mjx-math\"><span id=\"MJXc-Node-108\" class=\"mjx-mrow\"><span id=\"MJXc-Node-109\" class=\"mjx-mrow\"><span id=\"MJXc-Node-110\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">]<\/span><\/span><span id=\"MJXc-Node-111\" class=\"mjx-mrow\"><span id=\"MJXc-Node-112\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-113\" class=\"mjx-msubsup\"><span class=\"mjx-base\"><span id=\"MJXc-Node-114\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">180<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-115\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2218<\/span><\/span><\/span><\/span><span id=\"MJXc-Node-116\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-117\" class=\"mjx-msubsup MJXc-space1\"><span class=\"mjx-base\"><span id=\"MJXc-Node-118\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">180<\/span><\/span><\/span><span class=\"mjx-sup\"><span id=\"MJXc-Node-119\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2218<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-120\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">]<\/span><\/span><\/span><\/span><\/span><\/span>\u00a0o, si es en radianes,\u00a0<span id=\"MathJax-Element-22-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-121\" class=\"mjx-math\"><span id=\"MJXc-Node-122\" class=\"mjx-mrow\"><span id=\"MJXc-Node-123\" class=\"mjx-mrow\"><span id=\"MJXc-Node-124\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">]<\/span><\/span><span id=\"MJXc-Node-125\" class=\"mjx-mrow\"><span id=\"MJXc-Node-126\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">\u2212<\/span><\/span><span id=\"MJXc-Node-127\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">\u03c0<\/span><\/span><span id=\"MJXc-Node-128\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">,<\/span><\/span><span id=\"MJXc-Node-129\" class=\"mjx-mi MJXc-space1\"><span class=\"mjx-char MJXc-TeX-math-I\">\u03c0<\/span><\/span><\/span><span id=\"MJXc-Node-130\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-main-R\">]<\/span><\/span><\/span><\/span><\/span><\/span>.<\/p>\n<p>Si representamos el complejo\u00a0<span id=\"MathJax-Element-23-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 21.424px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-131\" class=\"mjx-math\"><span id=\"MJXc-Node-132\" class=\"mjx-mrow\"><span id=\"MJXc-Node-133\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><span id=\"MJXc-Node-134\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-135\" class=\"mjx-mi MJXc-space3\"><span class=\"mjx-char MJXc-TeX-math-I\">a<\/span><\/span><span id=\"MJXc-Node-136\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-137\" class=\"mjx-mi MJXc-space2\"><span class=\"mjx-char MJXc-TeX-math-I\">b<\/span><\/span><span id=\"MJXc-Node-138\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">i\u00a0<\/span><\/span><\/span><\/span><\/span>en el plano complejo, su longitud es su\u00a0<strong>m\u00f3dulo<\/strong>\u00a0y el \u00e1ngulo que forma con la parte positiva del eje horizontal es su\u00a0<strong>argumento<\/strong>:<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-modulo-argumento-angulo-propiedades.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"representaci\u00f3n del n\u00famero complejo a+bi en el plano complejo\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img3\/G2.png\" alt=\"Definici\u00f3n de m\u00f3dulo, argumento y conjugado de los n\u00fameros complejos, con interpretaci\u00f3n geom\u00e9trica y ejemplos. Enunciamos las propiedades b\u00e1sicas del conjugado y del m\u00f3dulo (de la suma, del producto, del cociente, etc.). Matem\u00e1ticas. N\u00fameros complejos. Secundaria. Bachillerato. Universidad. \" width=\"213\" height=\"205\" \/><\/a><\/p>\n<p style=\"text-align: left\" align=\"center\">Ejemplo:<\/p>\n<p>Calculamos el m\u00f3dulo de\u00a0<span id=\"MathJax-Element-24-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 16.432px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span id=\"MJXc-Node-139\" class=\"mjx-math\"><span id=\"MJXc-Node-140\" class=\"mjx-mrow\"><span id=\"MJXc-Node-141\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">z<\/span><\/span><span id=\"MJXc-Node-142\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-143\" class=\"mjx-mn MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><span id=\"MJXc-Node-144\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-145\" class=\"mjx-mn MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">5<\/span><\/span><span id=\"MJXc-Node-146\" class=\"mjx-mi\"><span class=\"mjx-char MJXc-TeX-math-I\">i<\/span><\/span><\/span><\/span><\/span>:<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-modulo-argumento-angulo-propiedades.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"el m\u00f3dulo del complejo 3+5i es ra\u00edz(34)\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img3\/P1-1.png\" alt=\"Definici\u00f3n de m\u00f3dulo, argumento y conjugado de los n\u00fameros complejos, con interpretaci\u00f3n geom\u00e9trica y ejemplos. Enunciamos las propiedades b\u00e1sicas del conjugado y del m\u00f3dulo (de la suma, del producto, del cociente, etc.). Matem\u00e1ticas. N\u00fameros complejos. Secundaria. Bachillerato. Universidad. \" width=\"158\" height=\"86\" \/><\/a><\/p>\n<p>Calculamos el argumento de\u00a0<span id=\"MathJax-Element-25-Frame\" class=\"mjx-chtml MathJax_CHTML\" style=\"line-height: 0;text-indent: 0px;text-align: left;font-style: normal;font-weight: normal;font-size: 16.432px;letter-spacing: normal;float: none;direction: ltr;max-width: none;max-height: none;min-width: 0px;min-height: 0px;border: 0px;margin: 0px;padding: 1px 0px\" role=\"presentation\"><span class=\"MJX_Assistive_MathML\" role=\"presentation\">z<\/span><\/span>:<\/p>\n<p align=\"center\"><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-modulo-argumento-angulo-propiedades.html\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" title=\"el argumento de z = 3+5i es 59.04\u00ba\" src=\"https:\/\/www.problemasyecuaciones.com\/complejos\/img3\/P1-2.png\" alt=\"Definici\u00f3n de m\u00f3dulo, argumento y conjugado de los n\u00fameros complejos, con interpretaci\u00f3n geom\u00e9trica y ejemplos. Enunciamos las propiedades b\u00e1sicas del conjugado y del m\u00f3dulo (de la suma, del producto, del cociente, etc.). Matem\u00e1ticas. N\u00fameros complejos. Secundaria. Bachillerato. Universidad. \" width=\"194\" height=\"71\" \/><\/a><\/p>\n<p style=\"text-align: left\">M\u00e1s informaci\u00f3n:<\/p>\n<ul>\n<li><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-definicion-representacion-raiz-negativos-i.html\">Introducci\u00f3n a los n\u00fameros complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-imaginarios-forma-polar-trigonometrica-binomica-ejemplos-problemas.html\">Formas bin\u00f3mica y polar<\/a><\/li>\n<li><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/numeros-complejos-modulo-argumento-angulo-propiedades.html\">M\u00f3dulo y argumento de complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.problemasyecuaciones.com\/complejos\/sumar-restar-multiplicar-dividir-numeros-complejos-imaginarios-ejemplos-formulas.html\">Operaciones entre complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/ejercicios-resueltos-producto-complejos.html\">Producto y cociente de complejos en forma bin\u00f3mica<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/BAC\/complejos\/numeros-complejos-forma-polar-binomica-calculadora-producto-problemas-resueltos.html\">Producto y cociente de complejos en forma polar<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/ejercicios-resueltos-demostraciones-complejos.html\">Propiedades de los n\u00fameros complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/BAC\/complejos\/raices\/raices-n-esimas-numeros-complejos-imaginarios-poligono-regular-argumento-modulo-ejemplos.html\">Ra\u00edces de n\u00fameros complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/ejercicios-resueltos-producto-complejos.html\">Calculadora de operaciones entre complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/BAC\/complejos\/numeros-complejos-forma-polar-binomica-calculadora-producto-problemas-resueltos.html\">Calculadora de forma polar y bin\u00f3mica de complejos<\/a><\/li>\n<li><a href=\"https:\/\/www.matesfacil.com\/SegundoGrado\/ECUACIONES-SEGUNDO-GRADO-SOLUCIONES-COMPLEJAS.html\">Ecuaciones cuadr\u00e1ticas con soluciones complejas<\/a><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Dado un n\u00famero complejo en su forma bin\u00f3mica\u00a0z=a+bi,\u00a0 se define el m\u00f3dulo de\u00a0z\u00a0como Se define el argumento de\u00a0z\u00a0como Nota 1:\u00a0la funci\u00f3n arcotangente proporciona el \u00e1ngulo entre -45\u00ba y 45\u00ba. Nota 2:\u00a0observad que, por ejemplo, la funci\u00f3n arcotangente proporciona el mismo \u00e1ngulo para\u00a0z=a\u2212bi\u00a0y para\u00a0w=\u2212a+bi. Sin embargo,\u00a0z\u00a0y\u00a0w\u00a0est\u00e1n en cuadrantes distintos, as\u00ed que su argumento es distinto. Para &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/blogs.ua.es\/matesfacil\/secundaria-numeros-operaciones\/numeros-imaginarios\/modulo-y-argumento-de-un-numero-imaginario\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> \u00abM\u00f3dulo y argumento de un n\u00famero imaginario\u00bb<\/span><\/a><\/p>\n","protected":false},"author":4324,"featured_media":0,"parent":412,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-430","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages\/430","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/users\/4324"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/comments?post=430"}],"version-history":[{"count":2,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages\/430\/revisions"}],"predecessor-version":[{"id":613,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages\/430\/revisions\/613"}],"up":[{"embeddable":true,"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/pages\/412"}],"wp:attachment":[{"href":"https:\/\/blogs.ua.es\/matesfacil\/wp-json\/wp\/v2\/media?parent=430"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}