From 2e9b5dc70e6aec67030fa8f90bd8a1fd81b04808 Mon Sep 17 00:00:00 2001
From: Sebastian Preisner <sebastian.preisner@gmail.com>
Date: Thu, 22 Jun 2017 08:12:52 +0000
Subject: [PATCH] Erzste Rechenoperationen

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 themen/KomplexeZahlen/rechenoperationen.md | 27 ++++++++++++++++++++++
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 create mode 100644 themen/KomplexeZahlen/rechenoperationen.md

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+## Rechenoperationen
+### Division
+$
+\frac { z_{ 1 } }{ z_{ 2 } } =\frac { a+bi }{ c+di } \quad =\frac { \left( a+bi \right)  }{ \left( c+di \right)  } \cdot \frac { \left( c-di \right)  }{ \left( c-di \right)  } =\frac { ac\quad -\quad adi\quad +\quad bci\quad -\quad bd{ i }^{ 2 } }{ { c }^{ 2 }-{ \left( di \right)  }^{ 2 } } =\frac { ac+bd+\left( bc-ad \right) i }{ { c }^{ 2 }+{ d }^{ 2 } } =\frac { ac+bd }{ { c }^{ 2 }+{ d }^{ 2 } } +\frac { \left( bc-ad \right)  }{ { c }^{ 2 }+{ d }^{ 2 } } 
+$
+
+### Multiplikation
+
+**Kadesische Form:**
+${ z }_{ 1 }\cdot { z }_{ 2 }=\left( a+bi \right) \cdot \left( c+di \right) =ac+adi+bci+bd{ i }^{ 2 }$
+
+**Trigonometrische Form:**
+$
+{ z }_{ 1 }\cdot { z }_{ 2 }=\left| { z }_{ 1 } \right| \left( \cos { \left( { \Phi  }_{ 1 } \right)  } +\sin { \left( { \varphi  }_{ 1 } \right)  } i \right) \cdot \left| { z }_{ 2 } \right| \left( \cos { \left( { \varphi  }_{ 2 } \right)  } \cdot \sin { \left( { \varphi  }_{ 2 } \right)  } i \right) =\left| { z }_{ 1 } \right| \cdot \left| { z }_{ 2 } \right| \left( \cos { \left( { \varphi  }_{ 1 } \right) \cdot \cos { \left( { \varphi  }_{ 2 } \right)  }  } +\sin { \left( { \varphi  }_{ 1 } \right)  } \cdot \sin { \left( { \varphi  }_{ 2 } \right)  } i \right) §
+
+### Addition
+${ z }_{ 1 }+{ z }_{ 2 }=\left( a+bi \right) +\left( c+di \right) =a+c+\left( b+d \right) i§
+
+### Subtraktion
+${ z }_{ 1 }-{ z }_{ 2 }=\left( a+bi \right) -\left( c+di \right) =a-c+\left( b-d \right) i$
+
+### Potenzierung
+$
+{ z }^{ n }={ \left( a+bi \right)  }^{ n }={ \left( \left| z \right| \cdot \left( \cos { \varphi  } +\sin { \varphi  } i \right)  \right)  }^{ n }={ \left| z \right|  }^{ n }\cdot \left( \cos { \left( n\cdot \varphi  \right)  } +\sin { \left( n\cdot \varphi  \right)  } i \right)
+$
+
+### Wurzel