Add Komplexe Zahlen and Images

8 years ago · f95e96a94b
parent c9c1bebc6f
commit f95e96a94b
8 changed files with 1195 additions and 3 deletions
--- a/Formelsammlung.pdf
+++ b/Formelsammlung.pdf
--- a/Formelsammlung.tex
+++ b/Formelsammlung.tex
@ -33,14 +33,227 @@
 \maketitle   % requires ./img/Logo.pdf


-% Section
+% Mengenlehre
 % ----------------------------------------------------------------------
 \section{Mengenlehre}

-\input{themen/Mengenlehre/definition.tex}
-\input{themen/Mengenlehre/operations.tex}
+\begin{sectionbox}

+\subsection{Definizion}
+	Ist $E$ eine Eigenschaft, die ein Element haben kann oder auch nicht, so beschreibt man die Menge der $E$ erfüllenden Elemente durch:
+	
+	A = $\lbrace x \vert x $ hat Eigenschaft $ E \rbrace$
+	
+\subsection{Teilmengen}
+	Sind A und B Mengen, so heißt A Teilmenge oder auch Untermenge von B, wenn jedes Element von A auch Element von B ist.
+	\begin{cookbox}{Merke zu Teilmengen}
+		\item Jede Menge A ist Teilmenge von sich selbst, das heißt $A \subset A$
+		\item Jede Menge A hat die leere Menge als Teilmenge, das heißt: $\emptyset \subset A$
+		\item Ist $A \subseteq B$ und $B \subseteq C$, so folgt $A \subseteq C$
+		\item Aus $A \subseteq B$ und $B \subseteq A$ folgt $A = B$
+	\end{cookbox}
+	
+\subsection{Operationen}
+	\begin{tablebox}{lll}
+		$A \subseteq B$ &  & A ist Teilmenge von B \\
+		$A \cup B$ & A vereinigt B & $A \cup B = \lbrace x \vert x \in A$ oder $x \in B \rbrace$ \\
+		$A \cap B$ & A geschnitten B & $A \cap B = \lbrace x \vert x \in A$ und $x \in B \rbrace$ \\
+		$A \setminus B$ & A ohne B & $A \cup B = \lbrace x \vert x \in A$ und $x \notin B \rbrace$ \\
+		$\mathcal{P}(A)$ & Potenzmenge A & Potenzmenge der Menge A\\
+		$A \in B$ & A Element von B & A ist ein Element von B\\
+		$A \notin B$ & A kein Element von B & A ist nicht in B enthalten \\
+	\end{tablebox}
+	
+\subsection{Potenzmenge}
+	Die Potenzmenge ist die Menge aller Teilmengen.
+	\begin{quote}
+	Es sei A eine Menge. Dann versteht man unter der Potenzmenge $\mathcal{P}(A)$ der Menge A die Menge aller Teilmengen von A. Auch die Menge $\emptyset$ hat eine Teilmenge es gilt: $\mathcal{P}(\emptyset) = \lbrace \emptyset \rbrace$.
+	\end{quote}
+	
+	Berechnet wird die Potenzmenge mit Hilfe von $2^{\vert A \vert}$ (Zwei hoch Kardinalität von A)

+\subsection{Kardinalität}
+	Beschreibt die Menge aller Elemente einer Menge.
+	
+	\begin{quote}
+	Es sei A eine endliche Menge. Dann versteht man unter der Kardinalität oder auch Mächtigkeit von A die Anzahl der Elemente von A und schreibt dafür $\vert A \vert$, manchmal auch $\#A$. Hat A unendlich viele Elemente, so sagt man, A hat die Kardinalität unendlich, und schreibt $\vert A \vert = \infty$
+	\end{quote}
+	
+	\begin{cookbox}{Beispiel}
+		$M = \lbrace 1, 2\rbrace$ \\
+		$P\left(M \right) = \lbrace \lbrace \rbrace, \lbrace 1 \rbrace, \lbrace 2 \rbrace, \lbrace 1, 2 \rbrace \rbrace $ \\
+		Nicht jedoch $\lbrace 2,1 \rbrace$! Es gilt $\lbrace 1,2 \rbrace = \lbrace 2,1 \rbrace$.
+
+	\end{cookbox}
+
+\subsection{Komplement}
+	Das Komplement ist die Differenz zwischen gegebener Menge und Grundmenge. 
+	
+	
+\end{sectionbox}
+
+\begin{sectionbox}
+\subsection{Lösungsalgorithmus}
+\begin{cookbox}{Arbeitsablauf}
+		\item $\setminus$ entfernen
+		\item De Morgen Gesetze anwenden
+		\item Assoziativ- und Distributiv- Gesetze im Wechsel mit dem Vereinfachen
+	\end{cookbox}
+	
+\begin{cookbox}{Arbeitsablauf}
+		\item $\setminus$ entfernen
+		\item De Morgen Gesetze anwenden
+		\item Assoziativ- und Distributiv- Gesetze im Wechsel mit dem Vereinfachen
+	\end{cookbox}
+	
+\subsection{Regeln}
+	\begin{tablebox}{ll}
+	Kommutativ & $A \cup B = B \cup A$\\
+	& $A \cap B = B \cap A$ \\
+	\ctrule
+	Assoziativ & $A \cap \left( B \cap C \right) = \left( A \cap B \right) \cap C$ \\
+	& $A \cup \left( B \cup C \right) = \left( A \cup B \right) \cup C$ \\
+	\ctrule
+	Distributiv & $A \cup \left( B \cap C \right) = \left( A \cup B \right) \cap \left(A \cup C \right)$ \\
+	& $A \cap \left( B \cup C \right) = \left( A \cap B \right) \cup \left(A \cap C \right)$ \\
+	\ctrule
+	Adjunktiv & $A \cup \left( A \cap B \right) = A $ \\
+	& $A \cap \left( A \cup B \right) = A$ \\
+	\ctrule
+	de Morganschen Regeln & $A \setminus \left( B \cap C \right) = \left( A \setminus B \right) \cup \left( A \setminus C \right)$ \\
+	& $A \setminus \left( B \cup C \right) = \left( A \setminus B \right) \cap \left( A \setminus C \right)$ \\
+	\ctrule
+	de Morganschen Gesetz & $A \setminus B = A \cap \overline{B}$ \\
+	\end{tablebox}
+	
+\subsection{Vereinfachen}
+	\begin{tablebox}{lll}
+		$A \cup A = A$ & $A \cap \emptyset = \emptyset $ & $\overline{\overline{A}} = A$ \\
+		$A \cap A = A$ & $A \cup \overline{A} = G $ & $\overline{\emptyset} = G  $ \\
+		$A \cup G = G$ & $A \cap \overline{A} = \emptyset $ & $\overline{G} = \emptyset $ \\
+		$A \cap G = A$ & $\overline{A \cup B} = \overline{A} \cap \overline{B} $ & $\emptyset \neq \lbrace \emptyset \rbrace $!!! \\
+		$A \cup \emptyset = A $ & $\overline{A \cap B} = \overline{A} \cup \overline{B}$ & $ $ \\
+	\end{tablebox} 
+\end{sectionbox}
+
+% Komplexe Zahlen
+% ----------------------------------------------------------------------
+\section{Komplexe Zahlen}
+
+\begin{sectionbox}
+\subsection{Potenzen von i}
+
+\begin{tablebox}{ll}
+		$i = \sqrt{-1} $ & ${ i }^{ 4 } = 1 $ \\
+		${ i }^{ 2 } = -1$ & ${ i }^{ 5 } = i$  \\
+		${ i }^{ 3 } = -i$ & ${ i }^{ 6 } = -1 $...  \\
+	\end{tablebox} 
+	
+\subsection{Visualisierung}
+\begin{minipage}{0.49\textwidth}
+	\includegraphics[width=\textwidth]{img/einheitskreis_komplexe_zahlen.png}
+\end{minipage}
+\begin{minipage}{0.49\textwidth}
+	\includegraphics[width=\textwidth]{img/visualisierung_komplexe_zahlen.png}
+\end{minipage}
+
+	\begin{tablebox}{lll}
+		$\vert z \vert = \sqrt{ {a}^{2} + {b}^{2} }$& $\varphi = \arctan \left( \cfrac{b}{a} \right) $ & siehe Tabelle xxx \\
+		$\tan\left(\varphi\right) = \cfrac{\vert b \vert}{ \vert a \vert}$ & $\cos\left(\varphi\right) = \cfrac{a}{\vert z \vert}$  & $\sin\left(\varphi\right) = \cfrac{b}{\vert z \vert}$ \\
+	\end{tablebox} 
+	
+\end{sectionbox}
+
+\begin{sectionbox}
+
+\begin{tablebox}{lll}
+\textbf{x,y} & \textbf{(in Grad)} & \textbf{(im Bogenmaß)} \\
+$x > 0, y \ge 0$    & $\varphi = \arctan \cfrac{y}{x} $ & $\varphi = arctan \cfrac{y}{x}$  \\
+$x < 0$   & $\varphi = \arctan \cfrac{y}{x} + 180^\circ $ & $\varphi = arctan \cfrac{y}{x} + \pi $ \\
+$x > 0, y \le 0 $ & $\varphi = \arctan \cfrac{y}{x} + 360^\circ $  & $\varphi = \arctan \cfrac{y}{x} + 2\pi $   \\
+$x = 0, y > 0 $ & $\varphi = 90^\circ $ &   $\varphi = \cfrac{\pi}{2} $           \\
+$x = 0, y < 0 $   & $\varphi = 270^\circ $     &      $\varphi = \cfrac{3}{2}\pi $        \\
+$x = 0, y = 0 $   & $\varphi = 0^\circ $    &       $\varphi = 0 $       \\ 
+\end{tablebox}
+
+\subsection{Formen}
+	\textbf{Kartesische Form:}
+	\begin{align*}
+	 { z }_{ 1 } \cdot { z }_{ 2 } & =  \left( a+bi \right) \cdot \left( c+di \right)  \\
+	  & =  ac+adi+bci+bd{ i }^{ 2 } \\
+	\end{align*}
+	
+	\textbf{Trigonometrische Form:}
+	\begin{align*}
+	{ z }_{ 1 }\cdot { z }_{ 2 } & =\left| { z }_{ 1 } \right| \left( \cos { \left( { \varphi  }_{ 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)
+	\end{align*}
+
+\subsection{Rechenoperationen}
+\begin{minipage}{0.49\textwidth}
+	\textbf{Addition}
+\begin{align*}
+	 { z }_{ 1 } + { z }_{ 2 } &= \left(a + bi\right) + \left(c + di \right)\\
+	 &= a + c + (b + d)i
+	\end{align*}
+\end{minipage}
+\begin{minipage}{0.49\textwidth}
+	\textbf{Subtraktion}
+	\begin{align*}
+	 { z }_{ 1 } - { z }_{ 2 } &= \left(a + bi \right) - \left(c + di \right)\\
+	 &= a - c + (b - d)i
+	\end{align*}
+\end{minipage}
+
+\textbf{Multiplikation}
+\begin{align*}
+	 { z }_{ 1 } \cdot  { z }_{ 2 } &= \left(a + bi \right) \cdot \left(c + di \right) \\
+	 &= ac + adi + bci + bd { i }^{ 2 }
+	\end{align*}
+	\begin{align*}
+	{ z }_{ 1 }\cdot { z }_{ 2 } & =\left| { z }_{ 1 } \right| \left( \cos { \left( { \varphi  }_{ 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)
+	\end{align*}
+	
+\textbf{Division}
+\begin{align*}
+\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 } } 
+\end{align*}
+
+\textbf{Potenzierung}
+\begin{align*}
+{ 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)
+\end{align*}
+
+\textbf{Wurzel} $\lbrace k \in \mathbb{N}  \vert k = 0 bis n-1 \rbrace$
+\begin{align*}
+\sqrt[n]{z} &= \sqrt[n]{ a+bi } \\
+{ z }_{ k } &= \sqrt[n]{\vert z \vert} \cdot \left( \cos{ \left( \cfrac{ \varphi + k \cdot 360}{n} \right) } +\sin{\left( \cfrac{\varphi + k \cdot 360}{n} \right)} i \right) 
+\end{align*}
+Es gibt immer $n$ Ergebnisse die in ${ z }_{ k } $ für $k= 0$ bis $k= n-1$ berechnet werden.
+
+\end{sectionbox}
+
+% Tipps und Tricks
+% ----------------------------------------------------------------------
+\section{Tipps}
+
+\begin{sectionbox}
+
+\subsection{Sinus \& Cosinus}
+\begin{tablebox}{ll}
+		$\cos 0^\circ = 1$ & $\sin 0^\circ = 0$ \\
+		$\cos 30^\circ = \cfrac{1}{2}\sqrt{3} \cong 0.8660254$ & $\sin 30^\circ = \cfrac{1}{2}$  \\
+		$\cos 45^\circ = \cfrac{1}{2}\sqrt{2} \cong 0.70710678$ & $\sin 45^\circ = \cfrac{1}{\sqrt{2}} \cong 0.70710678$  \\
+		$\cos 90^\circ = 0$ & $\sin 90^\circ = 1$  \\
+	\end{tablebox} 
+
+\end{sectionbox}
 % ======================================================================
 % End
 % ======================================================================
--- a/img/einheitskreis.svg
+++ b/img/einheitskreis.svg
@ -0,0 +1,180 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+   xmlns:dc="http://purl.org/dc/elements/1.1/"
+   xmlns:cc="http://creativecommons.org/ns#"
+   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+   xmlns:svg="http://www.w3.org/2000/svg"
+   xmlns="http://www.w3.org/2000/svg"
+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+   width="82.142616mm"
+   height="82.142609mm"
+   viewBox="0 0 82.142616 82.142609"
+   version="1.1"
+   id="svg8"
+   inkscape:version="0.92.1 r"
+   sodipodi:docname="einheitskreis.svg">
+  <defs
+     id="defs2">
+    <marker
+       inkscape:stockid="Arrow1Send"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="marker4865"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4863"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         transform="matrix(-0.2,0,0,-0.2,-1.2,0)"
+         inkscape:connector-curvature="0" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Send"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="Arrow1Send"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4532"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         transform="matrix(-0.2,0,0,-0.2,-1.2,0)"
+         inkscape:connector-curvature="0" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow2Mend"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="Arrow2Mend"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4544"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+         d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+         transform="scale(-0.6)"
+         inkscape:connector-curvature="0" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Lend"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="Arrow1Lend"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4520"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         transform="matrix(-0.8,0,0,-0.8,-10,0)"
+         inkscape:connector-curvature="0" />
+    </marker>
+  </defs>
+  <sodipodi:namedview
+     id="base"
+     pagecolor="#ffffff"
+     bordercolor="#666666"
+     borderopacity="1.0"
+     inkscape:pageopacity="0.0"
+     inkscape:pageshadow="2"
+     inkscape:zoom="2.8"
+     inkscape:cx="175.24515"
+     inkscape:cy="154.55776"
+     inkscape:document-units="mm"
+     inkscape:current-layer="layer1"
+     showgrid="false"
+     showguides="true"
+     inkscape:guide-bbox="true"
+     fit-margin-top="0"
+     fit-margin-left="0"
+     fit-margin-right="0"
+     fit-margin-bottom="0"
+     inkscape:window-width="1920"
+     inkscape:window-height="1172"
+     inkscape:window-x="1280"
+     inkscape:window-y="0"
+     inkscape:window-maximized="1" />
+  <metadata
+     id="metadata5">
+    <rdf:RDF>
+      <cc:Work
+         rdf:about="">
+        <dc:format>image/svg+xml</dc:format>
+        <dc:type
+           rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
+        <dc:title></dc:title>
+      </cc:Work>
+    </rdf:RDF>
+  </metadata>
+  <g
+     inkscape:label="Ebene 1"
+     inkscape:groupmode="layer"
+     id="layer1"
+     transform="translate(-49.712126,-50.753891)">
+    <circle
+       style="opacity:0.56699997;fill:none;fill-opacity:1;stroke:#000000;stroke-width:0.30250454;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
+       id="path4489"
+       cx="90.337097"
+       cy="92.27153"
+       r="17.382805" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#marker4865)"
+       d="M 49.712126,92.271525 H 130.96207"
+       id="path4493"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4495"
+       d="M 90.337096,132.8965 V 51.646561"
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow1Send)" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="M 107.72349,93.850091 V 90.692969"
+       id="path4499"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4501"
+       d="M 72.950703,93.850091 V 90.692969"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="m 88.758536,74.885122 h 3.157122"
+       id="path4503"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4505"
+       d="m 88.758536,109.65793 h 3.157122"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="M 55.61431,93.850091 V 90.692969"
+       id="path4507"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4509"
+       d="M 125.05989,93.850091 V 90.692969"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4511"
+       d="m 88.758536,57.548712 h 3.157122"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="m 88.758536,126.99433 h 3.157122"
+       id="path4513"
+       inkscape:connector-curvature="0" />
+  </g>
+</svg>
--- a/img/einheitskreis_komplexe_zahlen.png
+++ b/img/einheitskreis_komplexe_zahlen.png
--- a/img/einheitskreis_komplexe_zahlen.svg
+++ b/img/einheitskreis_komplexe_zahlen.svg
@ -0,0 +1,332 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+   xmlns:dc="http://purl.org/dc/elements/1.1/"
+   xmlns:cc="http://creativecommons.org/ns#"
+   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+   xmlns:svg="http://www.w3.org/2000/svg"
+   xmlns="http://www.w3.org/2000/svg"
+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+   width="82.142616mm"
+   height="82.142609mm"
+   viewBox="0 0 82.142616 82.142609"
+   version="1.1"
+   id="svg8"
+   inkscape:version="0.92.1 r"
+   sodipodi:docname="einheitskreis_komplexe_zahlen.svg">
+  <defs
+     id="defs2">
+    <marker
+       inkscape:stockid="Arrow1Mstart"
+       orient="auto"
+       refY="0.0"
+       refX="0.0"
+       id="marker5166"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path5164"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         style="fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1;fill:#000000;fill-opacity:1"
+         transform="scale(0.4) translate(10,0)" />
+    </marker>
+    <marker
+       inkscape:isstock="true"
+       style="overflow:visible"
+       id="marker5096"
+       refX="0"
+       refY="0"
+       orient="auto"
+       inkscape:stockid="Arrow1Send">
+      <path
+         inkscape:connector-curvature="0"
+         transform="matrix(-0.2,0,0,-0.2,-1.2,0)"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         id="path5094" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Mstart"
+       orient="auto"
+       refY="0.0"
+       refX="0.0"
+       id="Arrow1Mstart"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4523"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         style="fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1;fill:#000000;fill-opacity:1"
+         transform="scale(0.4) translate(10,0)" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Lend"
+       orient="auto"
+       refY="0.0"
+       refX="0.0"
+       id="marker4994"
+       style="overflow:visible;"
+       inkscape:isstock="true">
+      <path
+         id="path4992"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         style="fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1;fill:#000000;fill-opacity:1"
+         transform="scale(0.8) rotate(180) translate(12.5,0)" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Send"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="marker4865"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4863"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         transform="matrix(-0.2,0,0,-0.2,-1.2,0)"
+         inkscape:connector-curvature="0" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Send"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="Arrow1Send"
+       style="overflow:visible"
+       inkscape:isstock="true"
+       inkscape:collect="always">
+      <path
+         id="path4532"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         transform="matrix(-0.2,0,0,-0.2,-1.2,0)"
+         inkscape:connector-curvature="0" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow2Mend"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="Arrow2Mend"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4544"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+         d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+         transform="scale(-0.6)"
+         inkscape:connector-curvature="0" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Lend"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="Arrow1Lend"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4520"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         transform="matrix(-0.8,0,0,-0.8,-10,0)"
+         inkscape:connector-curvature="0" />
+    </marker>
+  </defs>
+  <sodipodi:namedview
+     id="base"
+     pagecolor="#ffffff"
+     bordercolor="#666666"
+     borderopacity="1.0"
+     inkscape:pageopacity="0.0"
+     inkscape:pageshadow="2"
+     inkscape:zoom="2.8"
+     inkscape:cx="121.00592"
+     inkscape:cy="144.55714"
+     inkscape:document-units="mm"
+     inkscape:current-layer="layer1"
+     showgrid="false"
+     showguides="true"
+     inkscape:guide-bbox="true"
+     fit-margin-top="0"
+     fit-margin-left="0"
+     fit-margin-right="0"
+     fit-margin-bottom="0"
+     inkscape:window-width="1920"
+     inkscape:window-height="1172"
+     inkscape:window-x="1280"
+     inkscape:window-y="0"
+     inkscape:window-maximized="1" />
+  <metadata
+     id="metadata5">
+    <rdf:RDF>
+      <cc:Work
+         rdf:about="">
+        <dc:format>image/svg+xml</dc:format>
+        <dc:type
+           rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
+        <dc:title></dc:title>
+      </cc:Work>
+    </rdf:RDF>
+  </metadata>
+  <g
+     inkscape:label="Ebene 1"
+     inkscape:groupmode="layer"
+     id="layer1"
+     transform="translate(-49.712126,-50.753891)">
+    <circle
+       style="opacity:0.761;fill:none;fill-opacity:1;stroke:#000000;stroke-width:0.30250454;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
+       id="path4489"
+       cx="90.337097"
+       cy="92.27153"
+       r="17.382805" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#marker4865)"
+       d="M 49.712126,92.271525 H 130.96207"
+       id="path4493"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4495"
+       d="M 90.337096,132.8965 V 51.646561"
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow1Send)" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="M 107.72349,93.850091 V 90.692969"
+       id="path4499"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4501"
+       d="M 72.950703,93.850091 V 90.692969"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="m 88.758536,74.885122 h 3.157122"
+       id="path4503"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4505"
+       d="m 88.758536,109.65793 h 3.157122"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="M 55.61431,93.850091 V 90.692969"
+       id="path4507"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4509"
+       d="M 125.05989,93.850091 V 90.692969"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4511"
+       d="m 88.758536,57.548712 h 3.157122"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="m 88.758536,126.99433 h 3.157122"
+       id="path4513"
+       inkscape:connector-curvature="0" />
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="89.047058"
+       y="73.686531"
+       id="text4924"><tspan
+         sodipodi:role="line"
+         id="tspan4922"
+         x="89.047058"
+         y="73.686531"
+         style="stroke-width:0.26458332">i</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="104.93449"
+       y="95.818787"
+       id="text4928"><tspan
+         sodipodi:role="line"
+         id="tspan4926"
+         x="104.93449"
+         y="95.818787"
+         style="stroke-width:0.26458332">1</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="87.633194"
+       y="113.41182"
+       id="text4932"><tspan
+         sodipodi:role="line"
+         id="tspan4930"
+         x="87.633194"
+         y="113.41182"
+         style="stroke-width:0.26458332">-i</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="69.310188"
+       y="95.818787"
+       id="text4936"><tspan
+         sodipodi:role="line"
+         id="tspan4934"
+         x="69.310188"
+         y="95.818787"
+         style="stroke-width:0.26458332">-1</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="86.690613"
+       y="56.923286"
+       id="text4940"><tspan
+         sodipodi:role="line"
+         id="tspan4938"
+         x="86.690613"
+         y="56.923286"
+         style="stroke-width:0.26458332">2i</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="85.276749"
+       y="126.09293"
+       id="text4944"><tspan
+         sodipodi:role="line"
+         id="tspan4942"
+         x="85.276749"
+         y="126.09293"
+         style="stroke-width:0.26458332">-2i</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="122.41405"
+       y="95.862198"
+       id="text4948"><tspan
+         sodipodi:role="line"
+         id="tspan4946"
+         x="122.41405"
+         y="95.862198"
+         style="stroke-width:0.26458332">2</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="51.22403"
+       y="95.862198"
+       id="text4952"><tspan
+         sodipodi:role="line"
+         id="tspan4950"
+         x="51.22403"
+         y="95.862198"
+         style="stroke-width:0.26458332">-2</tspan></text>
+    <circle
+       r="23.097219"
+       cy="92.27153"
+       cx="90.337097"
+       id="circle4978"
+       style="opacity:0.849;fill:none;fill-opacity:1;stroke:#000000;stroke-width:0.40194973;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;marker-end:url(#Arrow1Mstart);marker-mid:url(#marker5166)" />
+  </g>
+</svg>
--- a/img/einheitskreis_komplexe_zahlen2.svg
+++ b/img/einheitskreis_komplexe_zahlen2.svg
@ -0,0 +1,438 @@
+<?xml version="1.0" encoding="UTF-8" standalone="no"?>
+<!-- Created with Inkscape (http://www.inkscape.org/) -->
+
+<svg
+   xmlns:dc="http://purl.org/dc/elements/1.1/"
+   xmlns:cc="http://creativecommons.org/ns#"
+   xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#"
+   xmlns:svg="http://www.w3.org/2000/svg"
+   xmlns="http://www.w3.org/2000/svg"
+   xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd"
+   xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
+   width="82.142616mm"
+   height="82.142609mm"
+   viewBox="0 0 82.142616 82.142609"
+   version="1.1"
+   id="svg8"
+   inkscape:version="0.92.1 r"
+   sodipodi:docname="einheitskreis_komplexe_zahlen2.svg"
+   inkscape:export-filename="/home/sebastian/Dokumente/Privat/Studium/WBH/Mathe/Formelsammlung/img/einheitskreis_komplexe_zahlen.png"
+   inkscape:export-xdpi="92.769997"
+   inkscape:export-ydpi="92.769997">
+  <defs
+     id="defs2">
+    <marker
+       inkscape:stockid="Arrow1Mstart"
+       orient="auto"
+       refY="0.0"
+       refX="0.0"
+       id="marker5166"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path5164"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         style="fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1;fill:#000000;fill-opacity:1"
+         transform="scale(0.4) translate(10,0)" />
+    </marker>
+    <marker
+       inkscape:isstock="true"
+       style="overflow:visible"
+       id="marker5096"
+       refX="0"
+       refY="0"
+       orient="auto"
+       inkscape:stockid="Arrow1Send">
+      <path
+         inkscape:connector-curvature="0"
+         transform="matrix(-0.2,0,0,-0.2,-1.2,0)"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         id="path5094" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Mstart"
+       orient="auto"
+       refY="0.0"
+       refX="0.0"
+       id="Arrow1Mstart"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4523"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         style="fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1;fill:#000000;fill-opacity:1"
+         transform="scale(0.4) translate(10,0)" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Lend"
+       orient="auto"
+       refY="0.0"
+       refX="0.0"
+       id="marker4994"
+       style="overflow:visible;"
+       inkscape:isstock="true">
+      <path
+         id="path4992"
+         d="M 0.0,0.0 L 5.0,-5.0 L -12.5,0.0 L 5.0,5.0 L 0.0,0.0 z "
+         style="fill-rule:evenodd;stroke:#000000;stroke-width:1pt;stroke-opacity:1;fill:#000000;fill-opacity:1"
+         transform="scale(0.8) rotate(180) translate(12.5,0)" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Send"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="marker4865"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4863"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         transform="matrix(-0.2,0,0,-0.2,-1.2,0)"
+         inkscape:connector-curvature="0" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Send"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="Arrow1Send"
+       style="overflow:visible"
+       inkscape:isstock="true"
+       inkscape:collect="always">
+      <path
+         id="path4532"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         transform="matrix(-0.2,0,0,-0.2,-1.2,0)"
+         inkscape:connector-curvature="0" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow2Mend"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="Arrow2Mend"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4544"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.625;stroke-linejoin:round;stroke-opacity:1"
+         d="M 8.7185878,4.0337352 -2.2072895,0.01601326 8.7185884,-4.0017078 c -1.7454984,2.3720609 -1.7354408,5.6174519 -6e-7,8.035443 z"
+         transform="scale(-0.6)"
+         inkscape:connector-curvature="0" />
+    </marker>
+    <marker
+       inkscape:stockid="Arrow1Lend"
+       orient="auto"
+       refY="0"
+       refX="0"
+       id="Arrow1Lend"
+       style="overflow:visible"
+       inkscape:isstock="true">
+      <path
+         id="path4520"
+         d="M 0,0 5,-5 -12.5,0 5,5 Z"
+         style="fill:#000000;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1.00000003pt;stroke-opacity:1"
+         transform="matrix(-0.8,0,0,-0.8,-10,0)"
+         inkscape:connector-curvature="0" />
+    </marker>
+  </defs>
+  <sodipodi:namedview
+     id="base"
+     pagecolor="#ffffff"
+     bordercolor="#666666"
+     borderopacity="1.0"
+     inkscape:pageopacity="0.0"
+     inkscape:pageshadow="2"
+     inkscape:zoom="2.8"
+     inkscape:cx="129.41069"
+     inkscape:cy="169.16985"
+     inkscape:document-units="mm"
+     inkscape:current-layer="layer1"
+     showgrid="false"
+     showguides="true"
+     inkscape:guide-bbox="true"
+     fit-margin-top="0"
+     fit-margin-left="0"
+     fit-margin-right="0"
+     fit-margin-bottom="0"
+     inkscape:window-width="1920"
+     inkscape:window-height="1172"
+     inkscape:window-x="1280"
+     inkscape:window-y="0"
+     inkscape:window-maximized="1" />
+  <metadata
+     id="metadata5">
+    <rdf:RDF>
+      <cc:Work
+         rdf:about="">
+        <dc:format>image/svg+xml</dc:format>
+        <dc:type
+           rdf:resource="http://purl.org/dc/dcmitype/StillImage" />
+        <dc:title></dc:title>
+      </cc:Work>
+    </rdf:RDF>
+  </metadata>
+  <g
+     inkscape:label="Ebene 1"
+     inkscape:groupmode="layer"
+     id="layer1"
+     transform="translate(-49.712126,-50.753891)">
+    <circle
+       style="opacity:0.761;fill:none;fill-opacity:1;stroke:#000000;stroke-width:0.30250454;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1"
+       id="path4489"
+       cx="90.337097"
+       cy="92.27153"
+       r="17.382805" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#marker4865)"
+       d="M 49.712126,92.271525 H 130.96207"
+       id="path4493"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4495"
+       d="M 90.337096,132.8965 V 51.646561"
+       style="fill:none;stroke:#000000;stroke-width:0.5;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1;marker-end:url(#Arrow1Send)" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="M 107.72349,93.850091 V 90.692969"
+       id="path4499"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4501"
+       d="M 72.950703,93.850091 V 90.692969"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="m 88.758536,74.885122 h 3.157122"
+       id="path4503"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4505"
+       d="m 88.758536,109.65793 h 3.157122"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="M 55.61431,93.850091 V 90.692969"
+       id="path4507"
+       inkscape:connector-curvature="0" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4509"
+       d="M 125.05989,93.850091 V 90.692969"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       inkscape:connector-curvature="0"
+       id="path4511"
+       d="m 88.758536,57.548712 h 3.157122"
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.40000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="m 88.758536,126.99433 h 3.157122"
+       id="path4513"
+       inkscape:connector-curvature="0" />
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="89.047058"
+       y="73.686531"
+       id="text4924"><tspan
+         sodipodi:role="line"
+         id="tspan4922"
+         x="89.047058"
+         y="73.686531"
+         style="stroke-width:0.26458332">i</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="87.633194"
+       y="113.41182"
+       id="text4932"><tspan
+         sodipodi:role="line"
+         id="tspan4930"
+         x="87.633194"
+         y="113.41182"
+         style="stroke-width:0.26458332">-i</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="69.310188"
+       y="95.818787"
+       id="text4936"><tspan
+         sodipodi:role="line"
+         id="tspan4934"
+         x="69.310188"
+         y="95.818787"
+         style="stroke-width:0.26458332">-1</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="86.690613"
+       y="56.923286"
+       id="text4940"><tspan
+         sodipodi:role="line"
+         id="tspan4938"
+         x="86.690613"
+         y="56.923286"
+         style="stroke-width:0.26458332">2i</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="85.276749"
+       y="126.09293"
+       id="text4944"><tspan
+         sodipodi:role="line"
+         id="tspan4942"
+         x="85.276749"
+         y="126.09293"
+         style="stroke-width:0.26458332">-2i</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="122.41405"
+       y="95.862198"
+       id="text4948"><tspan
+         sodipodi:role="line"
+         id="tspan4946"
+         x="122.41405"
+         y="95.862198"
+         style="stroke-width:0.26458332">2</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="51.22403"
+       y="95.862198"
+       id="text4952"><tspan
+         sodipodi:role="line"
+         id="tspan4950"
+         x="51.22403"
+         y="95.862198"
+         style="stroke-width:0.26458332">-2</tspan></text>
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+       d="m 90.250073,92.169567 13.890627,-10.299851 0.0945,10.583335"
+       id="path5403"
+       inkscape:connector-curvature="0" />
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="96.557549"
+       y="95.217003"
+       id="text5407"><tspan
+         sodipodi:role="line"
+         id="tspan5405"
+         x="96.557549"
+         y="95.217003"
+         style="stroke-width:0.26458332">a</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="108.15668"
+       y="89.547356"
+       id="text5411"><tspan
+         sodipodi:role="line"
+         id="tspan5409"
+         x="108.15668"
+         y="89.547356"
+         style="stroke-width:0.26458332">b</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="21.824791"
+       y="126.8493"
+       id="text5415"
+       transform="rotate(-37.000703)"><tspan
+         sodipodi:role="line"
+         id="tspan5413"
+         x="21.824791"
+         y="126.8493"
+         style="stroke-width:0.26458332">|z|=1</tspan></text>
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+       d="m 104.21112,89.804629 c -0.0366,-0.0028 -0.0733,-0.0047 -0.11007,-0.0057 -0.0505,1.11e-4 -0.10102,0.002 -0.15141,0.0057 -0.071,0.0051 -0.14172,0.0137 -0.21187,0.02584 -0.0703,0.01201 -0.14,0.02753 -0.20877,0.04651 -0.0686,0.01918 -0.13622,0.04177 -0.20258,0.0677 -0.0665,0.02583 -0.13172,0.05498 -0.19533,0.08733 -0.0636,0.0322 -0.12548,0.06756 -0.18552,0.105939 -0.0599,0.03859 -0.11787,0.08016 -0.17363,0.124539 -0.0559,0.04432 -0.10953,0.0914 -0.16072,0.141076 -0.0512,0.04973 -0.0998,0.101988 -0.14572,0.15658 -0.0459,0.05448 -0.089,0.111228 -0.1292,0.170016 -0.0402,0.05874 -0.0775,0.119447 -0.11162,0.181901 -0.0342,0.06266 -0.0652,0.126989 -0.093,0.192752 -0.0278,0.06557 -0.0523,0.132489 -0.0734,0.200504 -0.021,0.06815 -0.0386,0.137314 -0.0527,0.207224 -0.0142,0.06987 -0.0249,0.140415 -0.032,0.211357 -0.007,0.07091 -0.0108,0.142145 -0.0109,0.213424 2.1e-4,0.05053 0.002,0.101027 0.006,0.15141 0.005,0.071 0.0133,0.141714 0.0253,0.211873 0.007,0.03603 -0.009,-0.03561 0,0 l 2.24713,-0.03064 z"
+       id="path5435"
+       inkscape:connector-curvature="0"
+       sodipodi:nodetypes="ccccccccccccccccccccccc" />
+    <path
+       style="fill:none;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:0.26499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-dasharray:none;stroke-opacity:1"
+       d="m 90.250073,92.169567 3.709841,0.0512 c 0.01981,-0.115841 0.03355,-0.232597 0.0418,-0.349823 0.0066,-0.08318 0.0099,-0.166568 0.01024,-0.249998 -1.66e-4,-0.117691 -0.0064,-0.235302 -0.01791,-0.352383 -0.01172,-0.117129 -0.02946,-0.233607 -0.0529,-0.348971 -0.02328,-0.115428 -0.05236,-0.229623 -0.08703,-0.342144 -0.03484,-0.112308 -0.07526,-0.222791 -0.121157,-0.331052 -0.0459,-0.108576 -0.09712,-0.214796 -0.153586,-0.318256 -0.02927,-0.05358 -0.06353,-0.104184 -0.09557,-0.156144 z"
+       id="path5443"
+       inkscape:connector-curvature="0" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+       d="m 90.250073,92.169567 23.606737,-17.471004 0.13364,17.506154 z"
+       id="path5445"
+       inkscape:connector-curvature="0" />
+    <path
+       style="fill:none;stroke:#000000;stroke-width:0.26458332px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1"
+       d="m 91.740259,93.741517 c 0.267269,1.603616 -0.200453,4.877668 1.937703,4.744035 2.138156,-0.133636 7.884448,-2.405428 8.753078,-0.267272 0.86863,2.13816 1.06908,2.47224 1.06908,2.47224 0,0 -0.46772,-2.739508 1.06908,-2.672691 1.5368,0.06682 8.55262,2.472241 9.28761,1.002261 0.73499,-1.469983 0.0668,-4.409948 0.0668,-4.409948"
+       id="path5447"
+       inkscape:connector-curvature="0" />
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;baseline-shift:baseline;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="99.491074"
+       y="103.09595"
+       id="text5451"><tspan
+         sodipodi:role="line"
+         id="tspan5449"
+         x="99.491074"
+         y="109.7123"
+         style="baseline-shift:baseline;stroke-width:0.26458332" /></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="102.43105"
+       y="104.16503"
+       id="text5461"><tspan
+         sodipodi:role="line"
+         id="tspan5459"
+         x="102.43105"
+         y="104.16503"
+         style="stroke-width:0.26458332">a</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.28959513px;line-height:5.13999271px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.2055997"
+       x="105.03276"
+       y="105.11384"
+       id="text5465"><tspan
+         sodipodi:role="line"
+         id="tspan5463"
+         x="105.03276"
+         y="105.11384"
+         style="stroke-width:0.2055997">n</tspan></text>
+    <text
+       xml:space="preserve"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:4.23333311px;line-height:6.61458302px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.26458332"
+       x="119.33583"
+       y="84.787987"
+       id="text5469"><tspan
+         sodipodi:role="line"
+         id="tspan5467"
+         x="119.33583"
+         y="84.787987"
+         style="stroke-width:0.26458332">b</tspan></text>
+    <text
+       id="text5477"
+       y="85.736794"
+       x="121.60347"
+       style="font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;font-size:3.28959513px;line-height:5.13999271px;font-family:'Liberation Sans';-inkscape-font-specification:'Liberation Sans';letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;stroke-width:0.2055997"
+       xml:space="preserve"><tspan
+         style="stroke-width:0.2055997"
+         y="85.736794"
+         x="121.60347"
+         id="tspan5475"
+         sodipodi:role="line">n</tspan></text>
+    <path
+       inkscape:connector-curvature="0"
+       id="path5479"
+       d="m 114.51962,91.146383 c 1.11649,-0.186082 3.39599,0.13956 3.30296,-1.349098 -0.093,-1.488652 -1.67475,-5.489413 -0.18609,-6.094182 1.48866,-0.604769 1.72126,-0.744329 1.72126,-0.744329 0,0 -1.90734,0.325642 -1.86082,-0.74433 0.0465,-1.069971 1.72126,-5.954619 0.69781,-6.466344 -1.02345,-0.511724 -3.07035,-0.04651 -3.07035,-0.04651"
+       style="fill:none;stroke:#000000;stroke-width:0.18421175px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1" />
+  </g>
+</svg>
--- a/img/visualisierung_komplexe_zahlen.png
+++ b/img/visualisierung_komplexe_zahlen.png
--- a/latex4ei/README.md
+++ b/latex4ei/README.md
@ -0,0 +1,29 @@
+# LaTeX4EI Template
+Write beautiful latex cheat sheets with minimal effort.
+
+## How to use
+put `latex4ei` folder in `./template-files/` into the same folder as
+your latex file. Specify \documentclass{latex4ei/latex4ei_sheet} and compile your code.
+See the documentation for further details.
+
+## Permanent installation
+Copy the `latex4ei` folder into your tex-distribution directory
+
+### Windows
+Copy files to `C:\texlive\XXXX\texmf-dist\tex\latex\latex4ei`
+
+### Linux
+```bash
+sudo ln -s ./pkg /usr/share/texlive/texmf-dist/tex/latex/latex4ei
+sudo mktexlsr
+```
+
+### Mac OS X
+```bash
+ln -s ./pkg /usr/local/texlive/texmf-local/tex/latex/latex4ei
+```
+
+If you want to share your documents please refer to the license.txt
+Read the changelog.txt and known_bugs.txt and check [www.latex4ei.de](http://latex4ei.de) for updates.
+
+© 2011-2016, LaTeX4EI