{"id":2712,"date":"2022-05-06T12:03:01","date_gmt":"2022-05-06T10:03:01","guid":{"rendered":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/imaginarni-jednotka\/"},"modified":"2022-05-14T18:06:15","modified_gmt":"2022-05-14T16:06:15","slug":"imaginarni-jednotka","status":"publish","type":"lesson","link":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/imaginarni-jednotka\/","title":{"rendered":"Imagin\u00e1rn\u00ed jednotka a s\u010d\u00edt\u00e1n\u00ed a sou\u010din komplexn\u00edch \u010d\u00edsel"},"content":{"rendered":"<ul>\n<li>Imagin\u00e1rn\u00ed jednotku zna\u010d\u00edme p\u00edsmenem <strong><em>i -1<\/em><\/strong><\/li>\n<li><em><strong>i=[0;1]\u00a0<\/strong><\/em><\/li>\n<li>Plat\u00ed n\u00e1sleduj\u00edc\u00ed rovnost: <strong><em>i\u00b2=i\u22c5i=[0;1]\u22c5[0;1]=[\u22121;0]=\u22121 \u21d2 i\u00b2=-1<\/em><\/strong><\/li>\n<li>Pokud bychom cht\u011bli komplexn\u00ed \u010d\u00edslo <em><strong>[a,b] <\/strong><\/em>zapsat v algebraick\u00e9m tvaru, vypadalo by takto:<\/li>\n<\/ul>\n<p style=\"text-align: center\"><em><strong>\u00a0a+bi; a,b\u2208R<\/strong><\/em><\/p>\n<ul>\n<li>abychom mohli s komplexn\u00edmi \u010d\u00edsly pracovat, mus\u00edme si zav\u00e9st z\u00e1kladn\u00ed matematick\u00e9 operace<\/li>\n<\/ul>\n<p><strong>S\u010c\u00cdT\u00c1N\u00cd KOMPLEXN\u00cdCH \u010c\u00cdSEL<\/strong><\/p>\n<p style=\"text-align: center\">Komplexn\u00ed \u010d\u00edsla mezi sebou m\u016f\u017eeme s\u010d\u00edtat. N\u00e1zorn\u011b si uk\u00e1\u017eeme jak:<\/p>\n<p>$$x=1+2i;y=1+4i$$<\/p>\n<p>$$x+y=(1+1)+(2+4)i=2+6i$$<\/p>\n<p style=\"text-align: center\">Jak je z toho p\u0159\u00edkladu patrn\u00e9, kdy\u017e mezi sebou s\u010d\u00edt\u00e1me komplexn\u00ed \u010d\u00edsla,<\/p>\n<p style=\"text-align: center\">zvl\u00e1\u0161\u0165 s\u010d\u00edt\u00e1me jejich re\u00e1lnou \u010d\u00e1st a zvl\u00e1\u0161\u0165 tu imagin\u00e1rn\u00ed<\/p>\n<p style=\"text-align: center\">Pro s\u010d\u00edt\u00e1n\u00ed komplexn\u00edch \u010d\u00edsel pou\u017e\u00edv\u00e1me n\u00e1sleduj\u00edc\u00ed vzorec:<\/p>\n<p>$$x=x_1+x_2i;y=y_1+y_2i$$<\/p>\n<p>$$x+y=(x_1+y_1)+(x_2+y_2)i$$<\/p>\n<p><strong>SOU\u010cIN KOMPLEXN\u00cdCH \u010c\u00cdSEL<\/strong><\/p>\n<p style=\"text-align: center\">Sou\u010din komplexn\u00edch \u010d\u00edsel je snadn\u00fd, pokud m\u00e1me neust\u00e1le na pam\u011bti, \u017ee <strong><em>i\u00b2=-1<\/em><\/strong><\/p>\n<p>$$x=1+2i; y=1+4i$$<\/p>\n<p>$$x\\cdot{y}=(1+2)\\cdot{(1+4i)}=1+4i+2i+{8i}^{2}=1+6i-8=-7+6i$$<\/p>\n","protected":false},"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-2712","lesson","type-lesson","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Imagin\u00e1rn\u00ed jednotka a s\u010d\u00edt\u00e1n\u00ed a sou\u010din komplexn\u00edch \u010d\u00edsel | Studijni-svet.cz<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/imaginarni-jednotka\/\" \/>\n<meta property=\"og:locale\" content=\"cs_CZ\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Imagin\u00e1rn\u00ed jednotka a s\u010d\u00edt\u00e1n\u00ed a sou\u010din komplexn\u00edch \u010d\u00edsel | Studijni-svet.cz\" \/>\n<meta property=\"og:description\" content=\"Imagin\u00e1rn\u00ed jednotku zna\u010d\u00edme p\u00edsmenem i -1 i=[0;1]\u00a0 Plat\u00ed n\u00e1sleduj\u00edc\u00ed rovnost: i\u00b2=i\u22c5i=[0;1]\u22c5[0;1]=[\u22121;0]=\u22121 \u21d2 i\u00b2=-1 Pokud bychom cht\u011bli komplexn\u00ed \u010d\u00edslo [a,b] zapsat v algebraick\u00e9m tvaru, vypadalo by takto: \u00a0a+bi; a,b\u2208R abychom mohli s komplexn\u00edmi \u010d\u00edsly pracovat, mus\u00edme si zav\u00e9st z\u00e1kladn\u00ed matematick\u00e9 operace S\u010c\u00cdT\u00c1N\u00cd KOMPLEXN\u00cdCH \u010c\u00cdSEL Komplexn\u00ed \u010d\u00edsla mezi sebou m\u016f\u017eeme s\u010d\u00edtat. 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a,b\u2208R abychom mohli s komplexn\u00edmi \u010d\u00edsly pracovat, mus\u00edme si zav\u00e9st z\u00e1kladn\u00ed matematick\u00e9 operace S\u010c\u00cdT\u00c1N\u00cd KOMPLEXN\u00cdCH \u010c\u00cdSEL Komplexn\u00ed \u010d\u00edsla mezi sebou m\u016f\u017eeme s\u010d\u00edtat. N\u00e1zorn\u011b si uk\u00e1\u017eeme jak: $$x=1+2i;y=1+4i$$ $$x+y=(1+1)+(2+4)i=2+6i$$ ... 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