{"id":36,"date":"2021-12-15T09:53:00","date_gmt":"2021-12-15T09:53:00","guid":{"rendered":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/die-komplexen-zahlen\/"},"modified":"2021-12-15T09:53:00","modified_gmt":"2021-12-15T09:53:00","slug":"die-komplexen-zahlen","status":"publish","type":"chapter","link":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/chapter\/die-komplexen-zahlen\/","title":{"raw":"Die komplexen Zahlen","rendered":"Die komplexen Zahlen"},"content":{"raw":"\n<style>.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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\n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z3ddc3635dd4a\" class=\"sectionHead\"><span class=\"titlemark\">2.3 <\/span> <a id=\"x1-560003\"><\/a>Die komplexen Zahlen<\/h3> <p class=\"noindent\">Unter Verwendung der reellen Zahlen k\u00f6nnen wir die Menge der komplexen Zahlen<a id=\"dx1-56001\"><\/a> als <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2102<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">definieren. Wir schreiben ein Element <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> viel h\u00e4ufiger in der Form <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span> wobei das Symbol <math display=\"inline\"><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/math> als die <span class=\"ecbx-1095\">imagin<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">re Einheit <\/span>bezeichnet wird. Man beachte, dass bei dieser Identifikation <math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo><\/math> vorerst als Ersatz f\u00fcr das Komma zu verstehen ist. Die Zahl <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> wird als der <span class=\"ecbx-1095\">Realteil<\/span> von <math display=\"inline\"><mi>z<\/mi><\/math> bezeichnet und man schreibt <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">;<\/span> die Zahl <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ist der <span class=\"ecbx-1095\">Imagin<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">rteil <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Elemente von <math display=\"inline\"><mi>\u2102<\/mi><\/math> mit Imagin\u00e4rteil <math display=\"inline\"><mn>0<\/mn><\/math> bezeichnet man auch als <span class=\"ecbx-1095\">reell <\/span>und die Elemente mit Realteil <math display=\"inline\"><mn>0<\/mn><\/math> als <span class=\"ecbx-1095\">rein imagin<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">r<\/span>. Via der injektiven Abbildung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> identifizieren wir <math display=\"inline\"><mi>\u211d<\/mi><\/math> mit der Teilmenge der reellen Elemente von <math display=\"inline\"><mi>\u2102<\/mi><\/math> (der \u201e <math display=\"inline\"><mi>x<\/mi><\/math>-Achse\u201c). <\/p> <div class=\"center\"> <p class=\"noindent\"> <\/p><p class=\"noindent\"><\/p><div class=\"mefigcentered\" id=\"wpsize=410&amp;url=Pictures\/Reelle_Zahlen\/komplexe_Ebene\/komplexe_ebene2.pdf\"><img id=\"z2c57f6142b78\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Reelle_Zahlen\/komplexe_Ebene\/komplexe_ebene2.svg\" width=\"410\"><\/div>  <\/div> <p class=\"noindent\">Die Menge <math display=\"inline\"><mi>\u2102<\/mi><\/math> (inklusive deren graphische Darstellung wie oben) wird ganz im Sinne der Identifikation <math display=\"inline\"><mi>\u2102<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> auch <span class=\"ecbx-1095\">komplexe Ebene <\/span>(alternativ <span class=\"ecbx-1095\">Gausssche Zahlenebene <\/span>oder auch <span class=\"ecbx-1095\">Argand-Ebene<\/span>) genannt. In der geometrischen Denkweise wird die Menge der reellen Punkte als die <span class=\"ecbx-1095\">reelle Achse <\/span>und die Menge der rein imagin\u00e4ren Punkte als die <span class=\"ecbx-1095\">imagin<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">re Achse <\/span>bezeichnet. <\/p><p class=\"indent\">Wie Sie vielleicht schon erwartet haben, soll <math display=\"inline\"><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> eine Wurzel von <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> sein. Formal ausgedr\u00fcckt, wollen wir, dass <math display=\"inline\"><mi>\u2102<\/mi><\/math> einen K\u00f6rper darstellt, in dem die Rechenoperationen von <math display=\"inline\"><mi>\u211d<\/mi><\/math> \u201everallgemeinert\u201c werden, und dass <math display=\"inline\"><msup><mrow><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> gilt. Die Addition auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> definieren wir \u201ekomponentenweise\u201c durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Multiplikation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> definieren wir hingegen durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere gilt <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> und die Addition und Multiplikation auf&nbsp;<math display=\"inline\"><mi>\u2102<\/mi><\/math> erweitern die entsprechenden Operationen auf&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zf6f351b7ef2d\"> <a id=\"x1-56002r33\"><\/a> <span class=\"ecbx-1095\">Proposition 2.33 <\/span>(Komplexe Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Mit den oben definierten Verkn<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">pfungen definiert <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">einen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper, den <\/span><span class=\"ecbi-1095\">K<\/span><span class=\"ecbi-1095\">\u00f6<\/span><span class=\"ecbi-1095\">rper der komplexen Zahlen<\/span><span class=\"ecti-1095\">. Hierbei ist die Null gleich <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">und die Eins gleich <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr die Geschichte der komplexen Zahlen verweisen wir auf den <a href=\"http:\/\/www.bbc.co.uk\/programmes\/p003hyd9\" target=\"_blank\" rel=\"noopener\">Podcast<\/a> der BBC (zum Beispiel ab der 14. oder 20. Minute). <\/p> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zb45fdc34423b\"> <a id=\"x1-56003r34\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.34.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">W<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">mit obiger Addition  und  mit  der  (komponentenweisen)  Multiplikation  definiert  durch<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mi>d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">auch ein K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper? Genauer: Welche K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rperaxiome gelten in diesem Fall?<\/span> <\/p> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis von Proposition <a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56002r33\">2.33<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Wir verifizieren die K\u00f6rperaxiome. Wie wir sehen werden, folgen die Eigenschaften der Addition auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> aus den Eigenschaften der Addition auf <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir beginnen mit der Kommutatitivit\u00e4t der Addition (da dies die \u00dcberpr\u00fcfung der anderen Axiome ein wenig vereinfacht): Seien <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Das Element <math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> ist ein (und schlussendlich also das) Nullelement der Addition, denn <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die additive Inverse eines Elements <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ist <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span>                                                                                                                                                                           denn <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Addition ist assoziativ: Seien <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">+<\/mo><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Eigenschaften der Multiplikation fordern etwas mehr Aufwand. Wir zeigen zuerst, dass die Multiplikation kommutativ ist. F\u00fcr <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> haben wir                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Das Element <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> ist ein Einselement, denn <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi>i<\/mi><\/math> und f\u00fcr <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir geben nun die multiplikative Inverse eines Elements <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi>i<\/mi><\/math> (das heisst <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> oder <math display=\"inline\"><mi>y<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math>), an. Wir bemerken zuerst, dass <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">:<\/span> Nehmen wir vorerst an, dass <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> dann ist <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und damit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>y<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> gilt ebenso <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und damit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Die multiplikative Inverse ist gegeben durch <span class=\"maperiod\"><math display=\"inline\"> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span>                                                                                                                                                                           denn <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die verbleibenden beiden Axiome (Assoziativit\u00e4t der Multiplikation und Distributivit\u00e4t) lassen sich durch abstraktere Argumente beweisen, die aber auch etwas mehr Wissen ben\u00f6tigen. Wir best\u00e4tigen diese Axiome deswegen durch zwei konkrete Rechnungen. <\/p><p class=\"indent\">Die Multiplikation ist assoziativ: Seien <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Nun berechnet man <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Es bleibt nur noch die Distributivit\u00e4t: Seien also <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">womit gezeigt w\u00e4re, dass <math display=\"inline\"><mi>\u2102<\/mi><\/math> zusammen mit der oben definierten Addition und der oben definierten Multiplikation ein K\u00f6rper ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"zdabf7f69b797\"> <a id=\"x1-56004r35\"><\/a> <span class=\"ecbx-1095\">Applet 2.35 <\/span>(Komplexe Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><\/p><div class=\"geoapplet\" style=\"width: 687px\"><iframe height=\"465px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/zhttt4bs\/width\/687\/height\/465\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten die K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rperoperationen (Addition, Multiplikation, multiplikatives Inverse)<\/span> <span class=\"ecti-1095\">auf den komplexen Zahlen. Die wahre geometrische Bedeutung der Multiplikation und des<\/span> <span class=\"ecti-1095\">multiplikativen Inversen l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich hier bereits erahnen, doch werden wir diese erst sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter<\/span> <span class=\"ecti-1095\">besprechen.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z03d99b3a1375\"> <span class=\"ecti-1095\">Bemerkung <\/span>(Andere Konstruktionen der komplexen Zahlen)<span class=\"ecti-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> (i)<\/dt><dd class=\"enumerate\">Wenn Sie die ersten Eigenschaften der Matrixmultiplikation kennen, l\u00e4sst sich obiger Beweis deutlich vereinfachen. In diesem Fall l\u00e4sst sich <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mi>x<\/mi><\/mtd><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mi>y<\/mi><\/mtd> <mtd class=\"array\" columnalign=\"center\"> <mi>x<\/mi><\/mtd> <\/mtr><\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo><msub><mrow><mi class=\"qopname\"> Mat<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u211d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">identfizieren. Die Multiplikation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> entspricht dann der Multiplikation der Matrizen in <math display=\"inline\"><msub><mrow><mi class=\"qopname\">Mat<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u211d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und insbesondere folgt beispielsweise die Assoziativit\u00e4t der Multiplikation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> aus der Assoziativit\u00e4t der Matrixmultiplikation. Gleiches gilt f\u00fcr die Distributivit\u00e4t. (Kommutativit\u00e4t der Multiplikation und die Existenz der multiplikativen Inversen m\u00fcssen aber nach wie vor direkt \u00fcberpr\u00fcft werden, da diese beiden Eigenschaften im Allgemeinen nicht f\u00fcr die Matrixmultiplikation gelten.) <\/p><\/dd><dt class=\"enumerate\"> (ii)<\/dt><dd class=\"enumerate\">Ebenso l\u00e4sst sich <math display=\"inline\"><mi>\u2102<\/mi><\/math> aus <math display=\"inline\"><mi>\u211d<\/mi><\/math> konstruieren, wenn man den Ring der Polynome mit reellen Koeffizienten kennt (siehe Abschnitt <a href=\"..\/..\/chapter\/polynome#x1-810002\">3.2<\/a>).<\/dd><\/dl> <\/div> <p class=\"indent\">Wie schon zuvor angedeutet, wollen wir <math display=\"inline\"><mi>\u211d<\/mi><\/math> als eine Teilmenge von <math display=\"inline\"><mi>\u2102<\/mi><\/math> auffassen. Vielmehr nennt man <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> auch einen Unterk\u00f6rper, da Addition und Multiplikation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> eingeschr\u00e4nkt auf                                                                                                                                                                           <math display=\"inline\"><mi>\u211d<\/mi><\/math> die Addition und Multiplikation auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> ergeben. Wir werden deswegen von nun an f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> k\u00fcrzer <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> und <math display=\"inline\"><mi>x<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <\/math> schreiben. Insbesondere wollen wir auch <span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <\/math> und <math display=\"inline\"><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <\/math> schreiben. Per Definition der Multiplikation gilt nun <math display=\"inline\"><msup><mrow><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> wie gew\u00fcnscht. <\/p><p class=\"indent\">Wir wollen ebenso bemerken, dass die komplexen Zahlen keinen angeordneten K\u00f6rper bilden \u2013 unabh\u00e4ngig davon, welche Ordnung man auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> w\u00e4hlt. Angenommen es g\u00e4be eine Ordnung&nbsp;<span class=\"maperiod\"><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2264<\/mo><\/mrow><mrow><mi>\u2102<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mi>\u2102<\/mi><\/math> mit <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2264<\/mo> <\/mrow><mrow><mi>\u2102<\/mi> <\/mrow> <\/msub> <\/math> ein angeordneter K\u00f6rper ist. In einem angeordneten K\u00f6rper sollte <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <msub><mrow><mo class=\"MathClass-rel\">&lt;<\/mo> <\/mrow><mrow><mi>\u2102<\/mi> <\/mrow> <\/msub> <mn>0<\/mn><\/math> gelten, was aber <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <msub><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><\/mrow><mrow><mi>\u2102<\/mi><\/mrow><\/msub><mn>0<\/mn><\/math> widerspricht (siehe Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-460002\">2.1.2<\/a>). <\/p><p class=\"indent\">An dieser Stelle m\u00f6chten wir uns kurz fragen, wieso die komplexen Zahlen \u00fcberhaupt von Interesse sind. W\u00e4hrend eine der sch\u00f6nen Eigenschaften der reellen Zahlen deren vollst\u00e4ndige Ordnung ist, so zeichnen sich die komplexen Zahlen unter anderem durch algebraische Sch\u00f6nheit aus. Auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> hat nicht nur die Gleichung <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> eine L\u00f6sung, sondern auch jede andere Gleichung der Form <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Diese Tatsache (\u201e <math display=\"inline\"><mi>\u2102<\/mi><\/math> ist algebraisch abgeschlossen\u201c) ist Inhalt des sogenannten Fundamentalsatzes der Algebra, den wir im zweiten Semester beweisen werden. Intuitiv sollte man in Analogie zu \u201e<math display=\"inline\"><mi>\u211d<\/mi><\/math> ist vollst\u00e4ndig, da <math display=\"inline\"><mi>\u211d<\/mi><\/math> keine L\u00fccken hat\u201c  den Fundamentalsatz der Algebra lesen als \u201e<math display=\"inline\"><mi>\u2102<\/mi><\/math> hat algebraisch keine L\u00fccken\u201c. <\/p><p class=\"indent\">Zum Abschluss dieses ersten Exkurses in das Reich der komplexen Zahlen wollen wir die komplexe Konjugation definieren. Diese ist im Wesentlichen nichts anderes als eine Spiegelung um die reelle Zahlengerade (und wird zum Beispiel in der Linearen Algebra in der Untersuchung von                                                                                                                                                                           komplexen inneren Produkten unentbehrlich sein). <\/p> <div class=\"me metheorem\"> <p class=\"indent\"><\/p><h4 id=\"zd10424957a02\"> <a id=\"x1-56007r36\"><\/a> <span class=\"ecbx-1095\">Definition 2.36 <\/span>(Konjugation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Die <span class=\"ecbx-1095\">komplexe Konjugation <\/span>ist die Abbildung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mover accent=\"true\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Im Beweis von Proposition <a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56002r33\">2.33<\/a> wurde die komplexe Konjugation indirekt schon verwendet: die multiplikative Inverse eines von Null verschiedenen Elements <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"zb7141f507a9e\"> <a id=\"x1-56008r37\"><\/a> <span class=\"ecbx-1095\">Lemma 2.37 <\/span>(Eigenschaften der Konjugation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die komplexe Konjugation erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt folgende Eigenschaften:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>z<\/mi><mover accent=\"true\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><mover accent=\"true\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Des Weiteren gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><mi>z<\/mi><mover accent=\"true\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">genau dann, wenn <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>w<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">\u22c5<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <p class=\"indent\"> <\/p> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir \u00fcberlassen der Leserin\/dem Leser Teil <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> als \u00dcbung. Seien <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/math> und <math display=\"inline\"><mi>w<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">\u22c5<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was zu zeigen war. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"z2d79712754f1\"> <a id=\"x1-56012r38\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.38.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie (i) in Lemma <\/span><a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56008r37\"><span class=\"ecti-1095\">2.37<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wie schon angemerkt wurde, gelten die Folgerungen (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450071\">a<\/a>)-(<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4502712\">l<\/a>) in Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450001\">2.1.1<\/a> f\u00fcr alle K\u00f6rper und insbesondere auch f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me melemma\"> <p class=\"indent\"><\/p><h4 id=\"ze7b23f1ecdf4\"> <a id=\"x1-56013r39\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.39.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die Rechenregeln<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">sowie<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z1d99779ffc21\"> <a id=\"x1-56014r40\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.40.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie die Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow> <mrow><mn>2<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><span class=\"ecti-1095\">. Schliessen<\/span> <span class=\"ecti-1095\">Sie insbesondere, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Was bedeutet diese Gleichheit geometrisch?<\/span> <\/p> <\/div> <div class=\"me meexample\"> <p class=\"indent\"><\/p><h4 id=\"z6886f96c9413\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Wie            wir            gesehen            haben,            l\u00e4sst            sich            auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> keine Ordnung  definieren,  die  zur  Addition  und  zur  Multiplikation  kompatibel  ist. Dennoch l\u00e4sst  sich  auf  den  komplexen  Zahlen  Analysis  betreiben,  was  zum  Teil in diesem Kurs  aber  vor  allem  im  Kurs  \u201eFunktionentheorie\u201c  im  zweiten  Studienjahr des Mathematik-   und   Physikstudiums   thematisiert   wird.   Grund   daf\u00fcr   ist,   dass <math display=\"inline\"><mi>\u2102<\/mi><\/math> eine Verallgemeinerung des Vollst\u00e4ndigkeitsaxiom erf\u00fcllt (welches wir erst nach etwas mehr Theorie besprechen k\u00f6nnen). <\/p> <\/div> <a id=\"x1-56015r55\"><\/a> <h4 id=\"ze41f18ccd5a1\" class=\"subsectionHead\"><span class=\"titlemark\">2.3.1 <\/span> <a id=\"x1-570001\"><\/a>Verwendung der komplexen Zahlen<\/h4> <p class=\"noindent\">Unsere Konstruktion von&nbsp;<math display=\"inline\"><mi>\u2102<\/mi><\/math> aus&nbsp;<math display=\"inline\"><mi>\u211d<\/mi><\/math> mag etwas formal gewesen sein, doch muss man sich eigentlich nur merken, dass&nbsp;<math display=\"inline\"><msup><mrow><mi class=\"qopname\">i<\/mi><mo>  <\/mo>  <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> und sonst alle gew\u00f6hnlichen Eigenschaften f\u00fcr die Addition und Multiplikation gelten. Sogar die Formel f\u00fcr das multiplikative Inverse von&nbsp;<math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> muss man nicht auswendig lernen wenn man sich stattdessen merkt, dass man den Bruch&nbsp;<math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>z<\/mi><\/mrow><\/mfrac><\/math> mit dem konjugierten Element&nbsp;<math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> erweitert. Sie werden der komplexen Konjugation noch \u00f6fter und insbesondere in der Linearen Algebra-Vorlesung in der Diskussion von \u201einneren Produkten auf Vektorr\u00e4umen \u00fcber&nbsp;<math display=\"inline\"><mi>\u2102<\/mi><\/math> \u201c begegnen. Wir bemerken noch, dass wir manchmal die Variable&nbsp;<math display=\"inline\"><mi>i<\/mi><\/math> (zum Beispiel als Indexvariable) verwenden werden. Man sollte dies allerdings vermeiden, wenn gleichzeitig komplexen Zahlen eine wesentliche Rolle in der Diskussion spielen. <\/p><p class=\"indent\">Wir verwenden h\u00e4ufig die Variablen&nbsp;<math display=\"inline\"><mi>z<\/mi><\/math>                                                                                                                                                                           und <math display=\"inline\"><mi>w<\/mi><\/math> f\u00fcr Elemente der komplexen Zahlen.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-57001r56\"><\/a> <\/p> \n","rendered":"\n<style scoped=\"scoped\">.cmr-5{font-size:50%;}\n.cmr-7{font-size:70%;}\n.cmmi-5{font-size:50%;font-style: italic;}\n.cmmi-7{font-size:70%;font-style: italic;}\n.cmmi-10{font-style: italic;}\n.cmsy-5{font-size:50%;}\n.cmsy-7{font-size:70%;}\n.cmbx-10{ font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-10{font-weight: bold;}\n.cmbsy-7{font-size:70%;font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-7{font-weight: bold;}\n.cmbsy-5{font-size:50%;font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmbsy-5{font-weight: bold;}\n.cmex-7{font-size:70%;}\n.cmex-7x-x-71{font-size:49%;}\n.msam-7{font-size:70%;}\n.msam-5{font-size:50%;}\n.msbm-7{font-size:70%;}\n.msbm-5{font-size:50%;}\n.cmr-17{font-size:170%;}\n.cmr-12{font-size:120%;}\n.cmti-10{ font-style: italic;}\np{margin-top:0;margin-bottom:0}\np.indent{text-indent:0;}\np + p{margin-top:1em;}\np + div, p + pre {margin-top:1em;}\ndiv + p, pre + p {margin-top:1em;}\n@media print {div.crosslinks {visibility:hidden;}}\na img { border-top: 0; 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margin-top:0.25em; margin-left:1em; }\n.abstract p {margin-left:5%; margin-right:5%;}\ndiv.abstract {width:100%;}\ndiv.tabular, div.center div.tabular {text-align: center; margin-top:0.5em; margin-bottom:0.5em; }\ntable.tabular td p{margin-top:0em;}\ntable.tabular {margin-left: auto; margin-right: auto;}\ntd p:first-child{ margin-top:0em; }\ntd p:last-child{ margin-bottom:0em; }\ndiv.td00{ margin-left:0pt; margin-right:0pt; }\ndiv.td01{ margin-left:0pt; margin-right:5pt; }\ndiv.td10{ margin-left:5pt; margin-right:0pt; }\ndiv.td11{ margin-left:5pt; margin-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\ntd.td00{ padding-left:0pt; padding-right:0pt; }\ntd.td01{ padding-left:0pt; padding-right:5pt; }\ntd.td10{ padding-left:5pt; padding-right:0pt; }\ntd.td11{ padding-left:5pt; padding-right:5pt; }\ntable[rules] {border-left:solid black 0.4pt; border-right:solid black 0.4pt; }\n.hline hr, .cline hr{ height : 0px; margin:0px; }\n.hline td, .cline td{ padding: 0; }\n.hline hr, .cline hr{border:none;border-top:1px solid black;}\n.equation-star td{text-align:center; vertical-align:middle; }\ntable.equation-star { width:100%; border-bottom-color: rgb(255,255,255); }\n#content table.equation-star, #content table.equation-star tbody tr td { border: 0px none rgb(255,255,255); }\nmtd.align-odd{margin-left:2em; text-align:right;}\nmtd.align-even{margin-right:2em; text-align:left;}\n.boxed{border: 1px solid black; padding-left:2px; padding-right:2px;}\n.rotatebox{display: inline-block;}\n.item-head{float:left;width:2em;clear:left;}\n.item-content{margin-left:2em;}\n .foreignobject {line-height:100%; font-size:120%; font-family:STIXgeneral,Times,Symbol,cmr10,CMSY10,CMEX10;padding:0; margin:0; text-align:center; }\nmath {vertical-align:baseline; line-height:100%; font-size:100%; font-family:STIXGeneral,Times,Symbol, cmr10,cmsy10,cmex10,cmmi10; font-style: normal; margin:0; padding:0; }\n\n.entry-title{display: none}\n\ndiv.newtheorem { margin-bottom: 2em; margin-top: 2em; border: 1px solid #333; background: #c7e4da; border-color: #4eb79e;}\ndiv.newtheorem h3 { background: #4eb79e; color: white; padding: 0px 15px 0px 15px; margin-top: 12px}\ndiv.newtheorem p { padding: 15px 15px 15px 15px; }\n\ndiv.newtheorem p span.head .ecbx-1095{font-weight: bold}\ndiv.newtheorem p .ecti-1095{font-style: italic}\ndiv.newtheorem div.custom-itemize{font-style: italic}\ndiv.quote{font-style: italic}\ndiv.newtheorem dl, dl.enumerate {display: grid; grid-template-columns: 5% auto; align-items: start; margin-top: 1em}\ndiv.newtheorem dl dd, dl.enumerate dd {margin-bottom: 0.5em}\ndiv.newtheorem dl dt, dl.enumerate dt {font-weight: normal; margin-top: 0px; text-align: right; margin-right: 15%}\ndiv.newtheorem dl dd {font-style: italic}\ndiv.newtheorem dl dt {font-style: italic}\ndiv.proof p span.ecti-1095 {font-style: italic}\ndiv.figure p img { margin-left: auto; margin-right: auto; display: block; }\ndiv.mefigcentered, div.figure { text-align: center }\n\ndl:after {content:\"\";display:table;clear:both;}\ndd {padding:.5em 0;}\ndl {width:100%;}\ndt, dd {display:inline-block; width:125%;}\ndt {text-align:right; font-weight:bold; clear:left; float:left;}\ndd {width:100%; padding-left:1em; padding-top: 0px; clear:right;}\ndd + dd {float:right; clear:both;}\ndd + dt {clear:both;}\ndt + dt {width: 100%; float: none; padding: 0 70% 0 0;}\ndt + dt + dd {margin-top: -2em;}\ndt + dt + dd + dt {margin-top: 2em;}\n<\/style>\n<style scoped=\"scoped\">\n\/* CSS Analysis-Skript D-Math ETHZ *\/\n\n\/* Uniform Font, also for headers *\/\nh3 {\n\tfont-family: \"Times New Roman\", serif;\n\tmargin-bottom: 35px;\n}\nh4 {\n\tfont-family: \"Times New Roman\", serif;\n}\nh5 {\n\tfont-family: \"Times New Roman\", serif;\n}\n\n\/* Bold font, e.g. for definitions *\/\n.ecbx-1095 {font-weight: 550 ;}\n\n\n\/* Uniform spacing, indent: larger, noindent, enumerate, itemize *\/\np.indent {\n\tmargin: 25px 0px 0px 0px;\n\ttext-indent: 0px; \n}\np.noindent {\n\tmargin: 15px 0px 0px 0px;\n\ttext-indent: 0px; \n}\ndl.enumerate {\n\tmargin: 0px 0px 0px 0px;\n}\ndl.enumerate dt, dl.enumerate dd {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\ndiv.custom-itemize {\n\tmargin: 0px 0px 0px 0px;\n}\ndiv.custom-itemize div.item-head {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n\ttext-align: center;\n}\ndiv.custom-itemize div.item-head:first-of-type {\n\tmargin-top: 0px;\n} \ndiv.custom-itemize div.item-content {\n\tmargin-top: 15px;\n\tmargin-bottom: 0px;\n}\n.MJXc-display {\n\tmargin: 15px 0px 0px 0px;\n}\n\n\n\n\/* green metheorem\/melemma CSS class for more\/medium important latex-theorem-environments *\/\n\/* metheorem box+header *\/\ndiv.metheorem {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #c7e4da;\n}\ndiv.metheorem h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* melemma box+header *\/\ndiv.melemma {\n    margin-bottom: 40px;\n    margin-top: 40px;\n\tpadding: 0px 15px 15px 15px;\n    border: 1px solid #333;\n    border-color: #4eb79e;\n    background: #F2F2F2;\n}\ndiv.melemma h4 {\n    background: #4eb79e;\n    color: white;\n\tmargin-top: 12px;\n\tmargin-left: -15px;\n\tmargin-right: -15px;\n\tpadding: 0px 15px 0px 15px;\n}\n\/* meexample box+header *\/\ndiv.meexample {\n    margin-bottom: 30px;\n    margin-top: 30px;\n\tpadding: 0px 15px 15px 15px;\n\tborder-color: gainsboro;\n\tborder-style: solid;\n\tborder-width: thin;\n}\ndiv.meexample h4 {\n\tfont-size: inherit;\n\tfont-weight: bold;\n    padding: 15px 0px 0px 0px;\n\tmargin-top: 0px;\n\tmargin-bottom: 5px;\n}\ndiv.meexample h4+p.noindent, div.meexample h4+p.indent {\n\tmargin-top: 5px;\n\ttext-indent: 0px;\n}\n\/* padding and margins for stuff inside these boxes, CSS-selector &gt; doesn't work in WP *\/\ndiv.me details {\n\tmargin: 10px 0px 0px 0px;\n}\ndiv.me dd {\n    width: calc(100% - 30px);\n}\t\n\n\n\/* fixing background of pictures *\/\nimg {\n\tbackground: white;\n}\n\n\/* div-container for centered geoapplet *\/\ndiv.geoapplet {\n\tmargin-left: auto;\n\tmargin-right: auto;\n\tmargin-top: 15px;\n\tmax-width: 100%;\n}\ndiv.geoapplet iframe {\n\tborder-style: none;\n\tmax-height: 110vw;\n}\n\n\/* div-container for centered squeezed tables *\/\ndiv.websqueeze {\n\tmargin-left: auto;\n\tmargin-right: auto;\n}\n\n\/* two containers for squeezing text sizes *\/\ndiv.mesmalltext, div.mesmalltext * {\n\tfont-size: 15px;\n}\nspan.metinytext, span.metinytext * {\n\tfont-size: 12px;\n}\n\n\n\/* removing grid lines in equations *\/\n#content table.equation tr td, #content table.equation tr th {\n    border: none;\n}\n#content table.equation {\n    border: none;\n}\n\n\/* hover\/click-solution for short inline explanations and footnotes *\/\n.hover-text {    \/* hidden part *\/\n    display: none;\n}\n.marginpar {     \/* style for footnote as marginpar *\/\n\ttext-decoration: none;\n\tborder: solid;\n\tborder-width: 1pt;\n\tpadding: 3pt;\t\n\twidth: 30%;\n\tbackground: white;\n}\n.hover-trigger { \/* style for hover\/click-trigger text\/symbol *\/\n\tbackground: none;\n\tborder: none;\n\tpadding: 0;\n\toutline: inherit;\t\n\ttext-transform: none;\n\tfont: inherit;\n\tposition: inherit;\n\tvertical-align: baseline;\n    color: #FF7F00;\n\tcursor: help;\n}\n.hover-trigger:hover +.hover-text{\n    display: inline;\n}\n.hover-trigger:active +.hover-text{\n    display: inline;\n}\n\n\/* simplifying style of details\/summary, removing triangle *\/\ndetails summary {\n  background: none;\n  list-style: none;\n  outline: none;\n  cursor: pointer;\n}\ndetails summary::-webkit-details-marker { \n  display: inline;\n  display: none;\n}\n\n\/* MC-True\/False as inline details\/summary *\/\ndetails.mcquest, div.me details.mcquest {\n\tdisplay: inline;\n\tmargin-top: 0px;\n}\nsummary.mcquest {\n\tdisplay: inline;\n\tcolor: #FF7F00;\n\tcursor: help;\n}\n\n\/* proof style: simple black box with gray background \n                little black square at the end on the right *\/\ndiv.proof {\n\tborder-color: black;\n\tborder-style: solid;\n\tborder-width: thin;\n\tbackground-color: #F2F2F2;\n\tpadding: 15px;\n\tmargin-top: 1em; \n}\ndiv.proof p:first-of-type {\n\tmargin: 0px;\n}\ndiv.qed {\n\tmargin-top: -25px;\n\tmargin-bottom: -7px;\n\ttext-align: right;\n}\ntable.equation+div.qed {\n\tmargin-top: -65px;\n}\n\n\/* The following is making also math-formulas inside the headers of Lemmas, etc., white. *\/\ndiv.melemma h4 span {\n    color: white;\n}\ndiv.metheorem h4 span {\n    color: white;\n}\n\n\/* The following are used to avoid fullstop, period, colon, semicolon, and endquote (broader) to move by itself to the next line after a formula.\n   The math-environment before needs to be wrapped in span.maperiod and the fullstop etc. in a span.period --- together they achieve what we want.  *\/\nspan.maperiod {\n       margin-right: 5px;\n}\nspan.period {\n       display: inline-block;\n       width: 0px;\n       margin-left: -5px;\n       margin-right: 4.9px;\n\t   text-indent: 0px;\n}\nspan.maendquote {\n       margin-right: 8px;\n}\nspan.endquote {\n       display: inline-block;\n       width: 0px;\n       margin-left: -8px;\n       margin-right: 7.9px;\n}\n\n\n\/* The following is removing an extra space left of the equation side in aligned equations *\/\nspan.mjx-mtd {\n    padding-left: 0em !important;\n}\n\n\/* The following fixes the weird problem that math appears smaller if it was rendered while the details tag was closed. *\/\ndetails span.mjx-chtml, details span.MathJax_CHTML {\n font-size: 100% !important;\n}\n\n\/* trying to fix line breaks in verbatim, new lines are missing *\/\npre.verbatim {\n\twhite-space: pre-wrap;\n\tfont-size: small;\n}\n<\/style><h3 id=\"z3ddc3635dd4a\" class=\"sectionHead\"><span class=\"titlemark\">2.3 <\/span> <a id=\"x1-560003\"><\/a>Die komplexen Zahlen<\/h3> <p class=\"noindent\">Unter Verwendung der reellen Zahlen k\u00f6nnen wir die Menge der komplexen Zahlen<a id=\"dx1-56001\"><\/a> als <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi>\u2102<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-rel\">\u2223<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">definieren. Wir schreiben ein Element <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> viel h\u00e4ufiger in der Form <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span> wobei das Symbol <math display=\"inline\"><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/math> als die <span class=\"ecbx-1095\">imagin<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">re Einheit <\/span>bezeichnet wird. Man beachte, dass bei dieser Identifikation <math display=\"inline\"><mo class=\"MathClass-bin\">+<\/mo><\/math> vorerst als Ersatz f\u00fcr das Komma zu verstehen ist. Die Zahl <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> wird als der <span class=\"ecbx-1095\">Realteil<\/span> von <math display=\"inline\"><mi>z<\/mi><\/math> bezeichnet und man schreibt <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math><\/span><span class=\"period\">;<\/span> die Zahl <math display=\"inline\"><mi>y<\/mi> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ist der <span class=\"ecbx-1095\">Imagin<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">rteil <\/span>von <span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Elemente von <math display=\"inline\"><mi>\u2102<\/mi><\/math> mit Imagin\u00e4rteil <math display=\"inline\"><mn>0<\/mn><\/math> bezeichnet man auch als <span class=\"ecbx-1095\">reell <\/span>und die Elemente mit Realteil <math display=\"inline\"><mn>0<\/mn><\/math> als <span class=\"ecbx-1095\">rein imagin<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">r<\/span>. Via der injektiven Abbildung <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><mo class=\"MathClass-rel\">\u21a6<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> identifizieren wir <math display=\"inline\"><mi>\u211d<\/mi><\/math> mit der Teilmenge der reellen Elemente von <math display=\"inline\"><mi>\u2102<\/mi><\/math> (der \u201e <math display=\"inline\"><mi>x<\/mi><\/math>-Achse\u201c). <\/p> <div class=\"center\"> <div class=\"wp-nocaption \"><\/div><div class=\"wp-nocaption \"><\/div><div class=\"mefigcentered\" id=\"wpsize=410&amp;url=Pictures\/Reelle_Zahlen\/komplexe_Ebene\/komplexe_ebene2.pdf\"><img decoding=\"async\" id=\"z2c57f6142b78\" alt=\"PIC\" src=\"https:\/\/people.math.ethz.ch\/~einsiedl\/Pictures\/Reelle_Zahlen\/komplexe_Ebene\/komplexe_ebene2.svg\" width=\"410\" \/><\/div>  <\/div> <p class=\"noindent\">Die Menge <math display=\"inline\"><mi>\u2102<\/mi><\/math> (inklusive deren graphische Darstellung wie oben) wird ganz im Sinne der Identifikation <math display=\"inline\"><mi>\u2102<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> auch <span class=\"ecbx-1095\">komplexe Ebene <\/span>(alternativ <span class=\"ecbx-1095\">Gausssche Zahlenebene <\/span>oder auch <span class=\"ecbx-1095\">Argand-Ebene<\/span>) genannt. In der geometrischen Denkweise wird die Menge der reellen Punkte als die <span class=\"ecbx-1095\">reelle Achse <\/span>und die Menge der rein imagin\u00e4ren Punkte als die <span class=\"ecbx-1095\">imagin<\/span><span class=\"ecbx-1095\">\u00e4<\/span><span class=\"ecbx-1095\">re Achse <\/span>bezeichnet. <\/p><p class=\"indent\">Wie Sie vielleicht schon erwartet haben, soll <math display=\"inline\"><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> eine Wurzel von <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn><\/math> sein. Formal ausgedr\u00fcckt, wollen wir, dass <math display=\"inline\"><mi>\u2102<\/mi><\/math> einen K\u00f6rper darstellt, in dem die Rechenoperationen von <math display=\"inline\"><mi>\u211d<\/mi><\/math> \u201everallgemeinert\u201c werden, und dass <math display=\"inline\"><msup><mrow><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-bin\">\u22c5<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> gilt. Die Addition auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> definieren wir \u201ekomponentenweise\u201c durch <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die Multiplikation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> definieren wir hingegen durch                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Insbesondere gilt <math display=\"inline\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> und die Addition und Multiplikation auf&nbsp;<math display=\"inline\"><mi>\u2102<\/mi><\/math> erweitern die entsprechenden Operationen auf&nbsp;<span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zf6f351b7ef2d\"> <a id=\"x1-56002r33\"><\/a> <span class=\"ecbx-1095\">Proposition 2.33 <\/span>(Komplexe Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Mit den oben definierten Verkn<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">pfungen definiert <\/span><math display=\"inline\"><mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">einen K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper, den <\/span><span class=\"ecbi-1095\">K<\/span><span class=\"ecbi-1095\">\u00f6<\/span><span class=\"ecbi-1095\">rper der komplexen Zahlen<\/span><span class=\"ecti-1095\">. Hierbei ist die Null gleich <\/span><math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> <span class=\"ecti-1095\">und die Eins gleich <\/span><span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">.<\/span> <\/p> <\/div> <p class=\"indent\">F\u00fcr die Geschichte der komplexen Zahlen verweisen wir auf den <a href=\"http:\/\/www.bbc.co.uk\/programmes\/p003hyd9\" target=\"_blank\" rel=\"noopener\">Podcast<\/a> der BBC (zum Beispiel ab der 14. oder 20. Minute). <\/p> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb45fdc34423b\"> <a id=\"x1-56003r34\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.34.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">W<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">re<\/span> <math display=\"inline\"><msup><mrow><mi>\u211d<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <\/math> <span class=\"ecti-1095\">mit obiger Addition  und  mit  der  (komponentenweisen)  Multiplikation  definiert  durch<\/span> <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u00d7<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>a<\/mi><mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>b<\/mi><mi>d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> <span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r<\/span> <math display=\"inline\"><mi>a<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>b<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>c<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>d<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">auch ein K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rper? Genauer: Welche K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rperaxiome gelten in diesem Fall?<\/span> <\/p> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis von Proposition <a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56002r33\">2.33<\/a>.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\"> Wir verifizieren die K\u00f6rperaxiome. Wie wir sehen werden, folgen die Eigenschaften der Addition auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> aus den Eigenschaften der Addition auf <span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Wir beginnen mit der Kommutatitivit\u00e4t der Addition (da dies die \u00dcberpr\u00fcfung der anderen Axiome ein wenig vereinfacht): Seien <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Das Element <math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> ist ein (und schlussendlich also das) Nullelement der Addition, denn <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">f\u00fcr alle <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Die additive Inverse eines Elements <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> f\u00fcr <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> ist <span class=\"maperiod\"><math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span>                                                                                                                                                                           denn <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>x<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>y<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Addition ist assoziativ: Seien <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-bin\">+<\/mo><\/mtd> <mtd class=\"align-even\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die Eigenschaften der Multiplikation fordern etwas mehr Aufwand. Wir zeigen zuerst, dass die Multiplikation kommutativ ist. F\u00fcr <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> haben wir                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Das Element <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> ist ein Einselement, denn <math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi>i<\/mi><\/math> und f\u00fcr <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>0<\/mn><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mn>1<\/mn><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Wir geben nun die multiplikative Inverse eines Elements <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> wobei <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-punc\">,<\/mo> <mi>y<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> und <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi>i<\/mi><\/math> (das heisst <math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> oder <math display=\"inline\"><mi>y<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math>), an. Wir bemerken zuerst, dass <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">:<\/span> Nehmen wir vorerst an, dass <span class=\"maperiod\"><math display=\"inline\"><mi>x<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo><mn>0<\/mn><\/math><\/span><span class=\"period\">,<\/span> dann ist <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und damit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> F\u00fcr <math display=\"inline\"><mi>y<\/mi><mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> gilt ebenso <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math> und <math display=\"inline\"><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math> und damit <span class=\"maperiod\"><math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Die multiplikative Inverse ist gegeben durch <span class=\"maperiod\"><math display=\"inline\"> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><mo class=\"MathClass-bin\">+<\/mo><msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span>                                                                                                                                                                           denn <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mfrac><mrow> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mi>y<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mfrac><mrow><mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Die verbleibenden beiden Axiome (Assoziativit\u00e4t der Multiplikation und Distributivit\u00e4t) lassen sich durch abstraktere Argumente beweisen, die aber auch etwas mehr Wissen ben\u00f6tigen. Wir best\u00e4tigen diese Axiome deswegen durch zwei konkrete Rechnungen. <\/p><p class=\"indent\">Die Multiplikation ist assoziativ: Seien <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>3<\/mn><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Nun berechnet man <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mrow><\/mrow><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"><mspace class=\"quad\" width=\"1em\" \/><mspace class=\"quad\" width=\"1em\" \/> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">Es bleibt nur noch die Distributivit\u00e4t: Seien also <math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mi>k<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>k<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mn>1<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo> <mn>3<\/mn> <\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><\/mtd> <mtd class=\"align-even\"> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow> <mo class=\"MathClass-bin\">+<\/mo><mrow><mo class=\"MathClass-open\" fence=\"true\" mathsize=\"1.19em\">(<\/mo><mrow><mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\">)<\/mo><\/mrow><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><mtr><mtd class=\"align-odd\" columnalign=\"right\" \/> <mtd class=\"align-even\"> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>3<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\" \/> <mtd class=\"align-label\"> <mspace width=\"2em\" \/><\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">womit gezeigt w\u00e4re, dass <math display=\"inline\"><mi>\u2102<\/mi><\/math> zusammen mit der oben definierten Addition und der oben definierten Multiplikation ein K\u00f6rper ist. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zdabf7f69b797\"> <a id=\"x1-56004r35\"><\/a> <span class=\"ecbx-1095\">Applet 2.35 <\/span>(Komplexe Zahlen)<span class=\"ecbx-1095\">.<\/span> <\/h4> <div class=\"wp-nocaption \"><\/div><div class=\"geoapplet\" style=\"width: 687px\"><iframe height=\"465px\" scrolling=\"no\" src=\"https:\/\/www.geogebra.org\/material\/iframe\/id\/zhttt4bs\/width\/687\/height\/465\/border\/888888\/rc\/false\/ai\/false\/sdz\/false\/smb\/false\/stb\/false\/stbh\/false\/ld\/false\/sri\/false\" style=\"border:0px\"><\/iframe><\/div><p class=\"indent\"><span class=\"ecti-1095\">Wir betrachten die K<\/span><span class=\"ecti-1095\">\u00f6<\/span><span class=\"ecti-1095\">rperoperationen (Addition, Multiplikation, multiplikatives Inverse)<\/span> <span class=\"ecti-1095\">auf den komplexen Zahlen. Die wahre geometrische Bedeutung der Multiplikation und des<\/span> <span class=\"ecti-1095\">multiplikativen Inversen l<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">sst sich hier bereits erahnen, doch werden wir diese erst sp<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ter<\/span> <span class=\"ecti-1095\">besprechen.<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z03d99b3a1375\"> <span class=\"ecti-1095\">Bemerkung <\/span>(Andere Konstruktionen der komplexen Zahlen)<span class=\"ecti-1095\">.<\/span> <\/h4> <dl class=\"enumerate\"><dt class=\"enumerate\"> (i)<\/dt><dd class=\"enumerate\">Wenn Sie die ersten Eigenschaften der Matrixmultiplikation kennen, l\u00e4sst sich obiger Beweis deutlich vereinfachen. In diesem Fall l\u00e4sst sich <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"> <mrow><mo fence=\"true\" form=\"prefix\"> (<\/mo><mrow><mtable align=\"axis\" class=\"array\" columnlines=\"none none none none none none none none none\" equalcolumns=\"false\" equalrows=\"false\"> <mtr><mtd class=\"array\" columnalign=\"center\"><mi>x<\/mi><\/mtd><mtd class=\"array\" columnalign=\"center\"><mo class=\"MathClass-bin\">\u2212<\/mo><mi>y<\/mi><\/mtd><\/mtr> <mtr><mtd class=\"array\" columnalign=\"center\"><mi>y<\/mi><\/mtd> <mtd class=\"array\" columnalign=\"center\"> <mi>x<\/mi><\/mtd> <\/mtr><\/mtable> <\/mrow><mo fence=\"true\" form=\"postfix\">)<\/mo><\/mrow> <mo class=\"MathClass-rel\">\u2208<\/mo><msub><mrow><mi class=\"qopname\"> Mat<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u211d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">identfizieren. Die Multiplikation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> entspricht dann der Multiplikation der Matrizen in <math display=\"inline\"><msub><mrow><mi class=\"qopname\">Mat<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><mo class=\"MathClass-punc\">,<\/mo><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-open\">(<\/mo><mi>\u211d<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> und insbesondere folgt beispielsweise die Assoziativit\u00e4t der Multiplikation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> aus der Assoziativit\u00e4t der Matrixmultiplikation. Gleiches gilt f\u00fcr die Distributivit\u00e4t. (Kommutativit\u00e4t der Multiplikation und die Existenz der multiplikativen Inversen m\u00fcssen aber nach wie vor direkt \u00fcberpr\u00fcft werden, da diese beiden Eigenschaften im Allgemeinen nicht f\u00fcr die Matrixmultiplikation gelten.) <\/p><\/dd><dt class=\"enumerate\"> (ii)<\/dt><dd class=\"enumerate\">Ebenso l\u00e4sst sich <math display=\"inline\"><mi>\u2102<\/mi><\/math> aus <math display=\"inline\"><mi>\u211d<\/mi><\/math> konstruieren, wenn man den Ring der Polynome mit reellen Koeffizienten kennt (siehe Abschnitt <a href=\"..\/..\/chapter\/polynome#x1-810002\">3.2<\/a>).<\/dd><\/dl> <\/div> <p class=\"indent\">Wie schon zuvor angedeutet, wollen wir <math display=\"inline\"><mi>\u211d<\/mi><\/math> als eine Teilmenge von <math display=\"inline\"><mi>\u2102<\/mi><\/math> auffassen. Vielmehr nennt man <math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-rel\">\u2286<\/mo> <mi>\u2102<\/mi><\/math> auch einen Unterk\u00f6rper, da Addition und Multiplikation auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> eingeschr\u00e4nkt auf                                                                                                                                                                           <math display=\"inline\"><mi>\u211d<\/mi><\/math> die Addition und Multiplikation auf <math display=\"inline\"><mi>\u211d<\/mi><\/math> ergeben. Wir werden deswegen von nun an f\u00fcr alle <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> k\u00fcrzer <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math> und <math display=\"inline\"><mi>x<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mi>x<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <\/math> schreiben. Insbesondere wollen wir auch <span class=\"maperiod\"><math display=\"inline\"><mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>1<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/math><\/span><span class=\"period\">,<\/span> <math display=\"inline\"><mn>0<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>0<\/mn><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <\/math> und <math display=\"inline\"><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <\/math> schreiben. Per Definition der Multiplikation gilt nun <math display=\"inline\"><msup><mrow><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> wie gew\u00fcnscht. <\/p><p class=\"indent\">Wir wollen ebenso bemerken, dass die komplexen Zahlen keinen angeordneten K\u00f6rper bilden \u2013 unabh\u00e4ngig davon, welche Ordnung man auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> w\u00e4hlt. Angenommen es g\u00e4be eine Ordnung&nbsp;<span class=\"maperiod\"><math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2264<\/mo><\/mrow><mrow><mi>\u2102<\/mi><\/mrow><\/msub><\/math><\/span><span class=\"period\">,<\/span> so dass <math display=\"inline\"><mi>\u2102<\/mi><\/math> mit <math display=\"inline\"> <msub><mrow><mo class=\"MathClass-rel\">\u2264<\/mo> <\/mrow><mrow><mi>\u2102<\/mi> <\/mrow> <\/msub> <\/math> ein angeordneter K\u00f6rper ist. In einem angeordneten K\u00f6rper sollte <math display=\"inline\"><mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <msub><mrow><mo class=\"MathClass-rel\">&lt;<\/mo> <\/mrow><mrow><mi>\u2102<\/mi> <\/mrow> <\/msub> <mn>0<\/mn><\/math> gelten, was aber <math display=\"inline\"> <mo class=\"MathClass-bin\">\u2212<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo><msup><mrow> <mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <msub><mrow><mo class=\"MathClass-rel\">\u2265<\/mo><\/mrow><mrow><mi>\u2102<\/mi><\/mrow><\/msub><mn>0<\/mn><\/math> widerspricht (siehe Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-460002\">2.1.2<\/a>). <\/p><p class=\"indent\">An dieser Stelle m\u00f6chten wir uns kurz fragen, wieso die komplexen Zahlen \u00fcberhaupt von Interesse sind. W\u00e4hrend eine der sch\u00f6nen Eigenschaften der reellen Zahlen deren vollst\u00e4ndige Ordnung ist, so zeichnen sich die komplexen Zahlen unter anderem durch algebraische Sch\u00f6nheit aus. Auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> hat nicht nur die Gleichung <math display=\"inline\"><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mn>1<\/mn> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> eine L\u00f6sung, sondern auch jede andere Gleichung der Form <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <msup><mrow><mi>x<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msup> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> f\u00fcr <math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2115<\/mi><\/math> und <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>0<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">.<\/mo><mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> mit <math display=\"inline\"><msub><mrow><mi>a<\/mi><\/mrow><mrow><mi>n<\/mi> <\/mrow> <\/msub> <mo class=\"MathClass-rel\">\u2260<\/mo> <mn>0<\/mn><\/math> und <span class=\"maperiod\"><math display=\"inline\"><mi>n<\/mi> <mo class=\"MathClass-rel\">&gt;<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> Diese Tatsache (\u201e <math display=\"inline\"><mi>\u2102<\/mi><\/math> ist algebraisch abgeschlossen\u201c) ist Inhalt des sogenannten Fundamentalsatzes der Algebra, den wir im zweiten Semester beweisen werden. Intuitiv sollte man in Analogie zu \u201e<math display=\"inline\"><mi>\u211d<\/mi><\/math> ist vollst\u00e4ndig, da <math display=\"inline\"><mi>\u211d<\/mi><\/math> keine L\u00fccken hat\u201c  den Fundamentalsatz der Algebra lesen als \u201e<math display=\"inline\"><mi>\u2102<\/mi><\/math> hat algebraisch keine L\u00fccken\u201c. <\/p><p class=\"indent\">Zum Abschluss dieses ersten Exkurses in das Reich der komplexen Zahlen wollen wir die komplexe Konjugation definieren. Diese ist im Wesentlichen nichts anderes als eine Spiegelung um die reelle Zahlengerade (und wird zum Beispiel in der Linearen Algebra in der Untersuchung von                                                                                                                                                                           komplexen inneren Produkten unentbehrlich sein). <\/p> <div class=\"me metheorem\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zd10424957a02\"> <a id=\"x1-56007r36\"><\/a> <span class=\"ecbx-1095\">Definition 2.36 <\/span>(Konjugation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\">Die <span class=\"ecbx-1095\">komplexe Konjugation <\/span>ist die Abbildung <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-punc\">:<\/mo> <mi>\u2102<\/mi> <mo class=\"MathClass-rel\">\u2192<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"nbsp\" width=\"0.33em\" \/><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-rel\">\u21a6<\/mo><mover accent=\"true\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <p class=\"indent\">Im Beweis von Proposition <a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56002r33\">2.33<\/a> wurde die komplexe Konjugation indirekt schon verwendet: die multiplikative Inverse eines von Null verschiedenen Elements <math display=\"inline\"><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\"> i<\/mi><mo>  <\/mo>  <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> ist <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><msup><mrow><mo class=\"MathClass-open\">(<\/mo><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo><\/mrow><mrow><mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/mrow><\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-bin\">\u2212<\/mo> <mfrac><mrow><mi>y<\/mi><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>x<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>y<\/mi><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow> <mrow><msup><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup> <mo class=\"MathClass-bin\">+<\/mo> <msup><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msup><\/mrow><\/mfrac><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"zb7141f507a9e\"> <a id=\"x1-56008r37\"><\/a> <span class=\"ecbx-1095\">Lemma 2.37 <\/span>(Eigenschaften der Konjugation)<span class=\"ecbx-1095\">.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Die komplexe Konjugation erf<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">llt folgende Eigenschaften:<\/span> <\/p><dl class=\"enumerate\"><dt class=\"enumerate\"> <span class=\"ecti-1095\">(i)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">ist <\/span><math display=\"inline\"><mi>z<\/mi><mover accent=\"true\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math> <span class=\"ecti-1095\">und <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi><mover accent=\"true\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">\u2265<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Des Weiteren gilt f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><\/span><span class=\"period\">,<\/span> <span class=\"ecti-1095\">dass <\/span><math display=\"inline\"><mi>z<\/mi><mover accent=\"true\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math> <span class=\"ecti-1095\">genau dann, wenn <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mn>0<\/mn><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(ii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">.<\/span> <\/dd><dt class=\"enumerate\"> <span class=\"ecti-1095\">(iii)<\/span><\/dt><dd class=\"enumerate\"><span class=\"ecti-1095\">F<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">gilt <\/span><span class=\"maperiod\"><math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>w<\/mi> <\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">\u22c5<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math><\/span><span class=\"period\">.<\/span><\/dd><\/dl> <\/div> <div class=\"wp-nocaption \"><\/div> <div class=\"proof\"> <p class=\"indent\"><span class=\"head\"><\/span><\/p><details open=\"open\"><summary><b>Beweis.<\/b><\/summary><p class=\"indent\" style=\"margin-top: 10\">Wir \u00fcberlassen der Leserin\/dem Leser Teil <math display=\"inline\"><mo class=\"MathClass-open\">(<\/mo><mi>i<\/mi><mo class=\"MathClass-close\">)<\/mo><\/math> als \u00dcbung. Seien <math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><\/math> und <math display=\"inline\"><mi>w<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn> <\/mrow> <\/msub> <mo class=\"MathClass-punc\">,<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-punc\">,<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u211d<\/mi><\/math><\/span><span class=\"period\">.<\/span> Dann gilt <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">und                                                                                                                                                                           <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">+<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">i<\/mi><mo>  <\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>1<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u22c5<\/mo> <mo class=\"MathClass-open\">(<\/mo><msub><mrow><mi>x<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub> <mo class=\"MathClass-bin\">\u2212<\/mo> <msub><mrow><mi>y<\/mi><\/mrow><mrow><mn>2<\/mn><\/mrow><\/msub><mi class=\"qopname\"> i<\/mi><mo>  <\/mo><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover> <mo class=\"MathClass-bin\">\u22c5<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>w<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><mo class=\"MathClass-punc\">,<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\">was zu zeigen war. <span>&nbsp;&nbsp;<\/span><\/p><div class=\"qed\">\u25a0<\/div><\/details><\/div> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z2d79712754f1\"> <a id=\"x1-56012r38\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.38.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie (i) in Lemma <\/span><a href=\"..\/..\/chapter\/die-komplexen-zahlen#x1-56008r37\"><span class=\"ecti-1095\">2.37<\/span><\/a><span class=\"ecti-1095\">.<\/span> <\/p> <\/div> <p class=\"indent\">Wie schon angemerkt wurde, gelten die Folgerungen (<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450071\">a<\/a>)-(<a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-4502712\">l<\/a>) in Abschnitt <a href=\"..\/..\/chapter\/die-axiome-der-reellen-zahlen#x1-450001\">2.1.1<\/a> f\u00fcr alle K\u00f6rper und insbesondere auch f\u00fcr <span class=\"maperiod\"><math display=\"inline\"><mi>\u2102<\/mi><\/math><\/span><span class=\"period\">.<\/span> <\/p> <div class=\"me melemma\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"ze7b23f1ecdf4\"> <a id=\"x1-56013r39\"><\/a> <span class=\"ecbx-1095\">Wichtige <\/span><span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.39.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi><mo class=\"MathClass-punc\">,<\/mo><mi>w<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> <span class=\"ecti-1095\">die Rechenregeln<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">sowie<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mi class=\"qopname\"> Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-bin\">+<\/mo><mi class=\"qopname\"> Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>w<\/mi><mo class=\"MathClass-close\">)<\/mo><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo><mo class=\"MathClass-punc\">.<\/mo><\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z1d99779ffc21\"> <a id=\"x1-56014r40\"><\/a> <span class=\"ecbx-1095\">\u00dc<\/span><span class=\"ecbx-1095\">bung 2.40.<\/span> <\/h4> <p class=\"indent\"><span class=\"ecti-1095\">Zeigen Sie die Identit<\/span><span class=\"ecti-1095\">\u00e4<\/span><span class=\"ecti-1095\">ten<\/span> <\/p><math display=\"block\"><mtable class=\"align-star\" columnalign=\"left\"> <mtr><mtd class=\"align-odd\" columnalign=\"right\"><mi class=\"qopname\">Re<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">+<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow> <mrow><mn>2<\/mn><\/mrow><\/mfrac> <mo class=\"MathClass-punc\">,<\/mo><mspace class=\"quad\" width=\"1em\" \/><mi class=\"qopname\">Im<\/mi><mo>  <\/mo><mo class=\"MathClass-open\">(<\/mo><mi>z<\/mi><mo class=\"MathClass-close\">)<\/mo> <mo class=\"MathClass-rel\">=<\/mo> <mfrac><mrow><mi>z<\/mi> <mo class=\"MathClass-bin\">\u2212<\/mo><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow> <mrow><mn>2<\/mn><mi class=\"qopname\">i<\/mi><mo>  <\/mo><\/mrow><\/mfrac> <\/mtd> <mtd class=\"align-even\"><mspace width=\"2em\" \/><\/mtd> <mtd class=\"align-label\" columnalign=\"right\"> <\/mtd><\/mtr><\/mtable><\/math> <p class=\"noindent\"><span class=\"ecti-1095\">f<\/span><span class=\"ecti-1095\">\u00fc<\/span><span class=\"ecti-1095\">r alle <\/span><math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math><span class=\"ecti-1095\">. Schliessen<\/span> <span class=\"ecti-1095\">Sie insbesondere, dass <\/span><span class=\"maperiod\"><math display=\"inline\"><mi>\u211d<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mrow><mo fence=\"true\" form=\"prefix\"> {<\/mo><mrow><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><mo class=\"MathClass-rel\">\u2223<\/mo><mi>z<\/mi> <mo class=\"MathClass-rel\">=<\/mo> <mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/mrow><mo fence=\"true\" form=\"postfix\">}<\/mo><\/mrow><\/math><\/span><span class=\"period\">.<\/span> <span class=\"ecti-1095\">Was bedeutet diese Gleichheit geometrisch?<\/span> <\/p> <\/div> <div class=\"me meexample\"> <div class=\"wp-nocaption \"><\/div><h4 id=\"z6886f96c9413\"> <span class=\"ecti-1095\">Bemerkung.<\/span><\/h4> <p class=\"indent\">Wie            wir            gesehen            haben,            l\u00e4sst            sich            auf <math display=\"inline\"><mi>\u2102<\/mi><\/math> keine Ordnung  definieren,  die  zur  Addition  und  zur  Multiplikation  kompatibel  ist. Dennoch l\u00e4sst  sich  auf  den  komplexen  Zahlen  Analysis  betreiben,  was  zum  Teil in diesem Kurs  aber  vor  allem  im  Kurs  \u201eFunktionentheorie\u201c  im  zweiten  Studienjahr des Mathematik-   und   Physikstudiums   thematisiert   wird.   Grund   daf\u00fcr   ist,   dass <math display=\"inline\"><mi>\u2102<\/mi><\/math> eine Verallgemeinerung des Vollst\u00e4ndigkeitsaxiom erf\u00fcllt (welches wir erst nach etwas mehr Theorie besprechen k\u00f6nnen). <\/p> <\/div> <a id=\"x1-56015r55\"><\/a> <h4 id=\"ze41f18ccd5a1\" class=\"subsectionHead\"><span class=\"titlemark\">2.3.1 <\/span> <a id=\"x1-570001\"><\/a>Verwendung der komplexen Zahlen<\/h4> <p class=\"noindent\">Unsere Konstruktion von&nbsp;<math display=\"inline\"><mi>\u2102<\/mi><\/math> aus&nbsp;<math display=\"inline\"><mi>\u211d<\/mi><\/math> mag etwas formal gewesen sein, doch muss man sich eigentlich nur merken, dass&nbsp;<math display=\"inline\"><msup><mrow><mi class=\"qopname\">i<\/mi><mo>  <\/mo>  <\/mrow><mrow><mn>2<\/mn> <\/mrow> <\/msup> <mo class=\"MathClass-rel\">=<\/mo> <mo class=\"MathClass-bin\">\u2212<\/mo><mn>1<\/mn><\/math> und sonst alle gew\u00f6hnlichen Eigenschaften f\u00fcr die Addition und Multiplikation gelten. Sogar die Formel f\u00fcr das multiplikative Inverse von&nbsp;<math display=\"inline\"><mi>z<\/mi> <mo class=\"MathClass-rel\">\u2208<\/mo> <mi>\u2102<\/mi><\/math> muss man nicht auswendig lernen wenn man sich stattdessen merkt, dass man den Bruch&nbsp;<math display=\"inline\"><mfrac><mrow><mn>1<\/mn><\/mrow> <mrow><mi>z<\/mi><\/mrow><\/mfrac><\/math> mit dem konjugierten Element&nbsp;<math display=\"inline\"><mover accent=\"false\" class=\"mml-overline\"><mrow><mi>z<\/mi><\/mrow><mo accent=\"true\">\u00af<\/mo><\/mover><\/math> erweitert. Sie werden der komplexen Konjugation noch \u00f6fter und insbesondere in der Linearen Algebra-Vorlesung in der Diskussion von \u201einneren Produkten auf Vektorr\u00e4umen \u00fcber&nbsp;<math display=\"inline\"><mi>\u2102<\/mi><\/math> \u201c begegnen. Wir bemerken noch, dass wir manchmal die Variable&nbsp;<math display=\"inline\"><mi>i<\/mi><\/math> (zum Beispiel als Indexvariable) verwenden werden. Man sollte dies allerdings vermeiden, wenn gleichzeitig komplexen Zahlen eine wesentliche Rolle in der Diskussion spielen. <\/p><p class=\"indent\">Wir verwenden h\u00e4ufig die Variablen&nbsp;<math display=\"inline\"><mi>z<\/mi><\/math>                                                                                                                                                                           und <math display=\"inline\"><mi>w<\/mi><\/math> f\u00fcr Elemente der komplexen Zahlen.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               <a id=\"x1-57001r56\"><\/a> <\/p> \n","protected":false},"author":1089,"menu_order":3,"template":"","meta":{"pb_show_title":"","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-36","chapter","type-chapter","status-publish","hentry"],"part":33,"_links":{"self":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/36","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/users\/1089"}],"version-history":[{"count":0,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/36\/revisions"}],"part":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/parts\/33"}],"metadata":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapters\/36\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/media?parent=36"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/pressbooks\/v2\/chapter-type?post=36"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/contributor?post=36"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/wp-prd.let.ethz.ch\/analysis19\/wp-json\/wp\/v2\/license?post=36"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}