From 3bfb6d24009759a656602b1f84d3dec434e1cb7b Mon Sep 17 00:00:00 2001 From: Manuel Thalmann Date: Wed, 11 Jan 2023 23:35:06 +0100 Subject: [PATCH] Add chapter about complex numbers --- .../ComplexNumberMath.png | Bin 0 -> 21638 bytes .../ComplexNumbers.png | Bin 0 -> 8687 bytes .../ConjugatedComplexNumber.png | Bin 0 -> 15407 bytes .../RootComplexNumber.png | Bin 0 -> 26806 bytes .../TrigonometricComplexNumber.png | Bin 0 -> 15370 bytes .../Zusammenfassung.md | 177 ++++++++++++++++++ 6 files changed, 177 insertions(+) create mode 100644 Notes/Semester 3/HM1 - Höhere Mathematik/ComplexNumberMath.png create mode 100644 Notes/Semester 3/HM1 - Höhere Mathematik/ComplexNumbers.png create mode 100644 Notes/Semester 3/HM1 - Höhere Mathematik/ConjugatedComplexNumber.png create mode 100644 Notes/Semester 3/HM1 - Höhere Mathematik/RootComplexNumber.png create mode 100644 Notes/Semester 3/HM1 - Höhere Mathematik/TrigonometricComplexNumber.png diff --git a/Notes/Semester 3/HM1 - Höhere 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z4cO^?L*@CYlv;_Jw0fz^nW9*p*6GR2X@(2ld_omX;;J9+kM3+XE4C-91$=nxKh1eu z1S+bgCdcZIXAF&b%AT|BpJLKT=r(>RJuA-Ol%r~%;h2=$wpS%* z`KAryDMme-HyA`gGrQ&wo>j3G{_$u zz}kdq@|+lr&`WJEtL)6a#Y~Yo?D@t4^oO@fjr}v;z&rbM zLxoF)8mlhDNk*d-_=w~lylctx{JA4ye6(D z!EU_<0y MDNB_}y#Dlm0O#>_rvLx| literal 0 HcmV?d00001 diff --git a/Notes/Semester 3/HM1 - Höhere Mathematik/Zusammenfassung.md b/Notes/Semester 3/HM1 - Höhere Mathematik/Zusammenfassung.md index 277d1c9..1609aed 100644 --- a/Notes/Semester 3/HM1 - Höhere Mathematik/Zusammenfassung.md +++ b/Notes/Semester 3/HM1 - Höhere Mathematik/Zusammenfassung.md @@ -64,6 +64,8 @@ - [Jacobi-Verfahren](#jacobi-verfahren) - [Gauss-Seidel-Verfahren](#gauss-seidel-verfahren) - [Konvergenz](#konvergenz) + - [Komplexe Zahlen](#komplexe-zahlen) + - [Rechen-Regeln](#rechen-regeln) - [Formelbuchstaben](#formelbuchstaben) - [Glossar](#glossar) @@ -1213,6 +1215,179 @@ Für alle Matrizen, die diagonal-dominant sind gilt, dass sie für das Jacobi- u +## Komplexe Zahlen +Der Bereich der Komplexen Zahlen dient dazu, Werte abzubilden, die es eigentlich nicht geben kann. + +Beispiel einer komplexen Zahl: + +$$x^2 = -1$$ + +Es gibt keinen Wert, der $-1$ ergibt, wenn er quadriert wird. Es handelt sich also um eine komplexe Zahl. + +Dafür wird die imaginäre Einheit $i$ eingeführt mit folgender Eigenschaft: + +$$i^2 = -1$$ + +Für diese Definition wäre das Resultat von $x^2= -1$ also $x = \plusmn{i}$ + +In Python und in der Elektrotechnik wird der Buchstabe $j$ verwendet. + +Komplexe Zahlen $z$ mit $z = x + i \cdot y$ können nicht auf einem Zahlenstrahl dargestellt werden. + +Sie können in einem Koordinaten-System eingezeichnet werden, wobei $x$ der reale und $y$ der imaginäre Anteil sind: + +![](ComplexNumbers.png) + +Dieses Koordinaten-System nennt sich auch **Gaussche Zahlenebene**. + +
    + +***Komplexe Zahlen:*** + +Imaginäre Einheit $i$: + +$$i^2 = -1$$ + +Komplexe Zahlen $z$: + +$$z = x + i \cdot y$$ + +**Konjugierte** komplexe Zahl: + +$$z^* = x - i \cdot y$$ + +Betrag von $z$: + +$$|z| = \sqrt{x^2 + y^2}$$ + +Menge aller komplexen Zahlen $\mathbb{C}$: + +$$\mathbb{C} = \{ z | z = x + i \cdot y \text{ mit } x, y \in \mathbb{R}\}$$ + +
    + +Veranschaulichung einer konjugierten komplexen Zahl $z^*$: + +![](ConjugatedComplexNumber.png) + +
    + + - $\mathbb{C}$: Menge aller komplexen Zahlen + - $x$: Realteil einer komplexen Zahl + - $y$: Imaginärteil einer komplexen Zahl + - $z$: Komplexe Zahl + +
    + +
    + +***Darstellungsformen:*** + +Es gibt diverse Darstellungsformen für komplexe Zahlen: + + - Normalform (auch "algebraische" oder "kartesische" Form): + $$z = x + i \cdot y$$ + - Trigonometrische Form: + $$z = r \cdot (\cos(\varphi) + i \cdot \sin(\varphi))$$ + - Exponential-Form: + $$z = re^{i \cdot \varphi}$$ + +
    + +Beispiel einer komplexen Zahl $z$ in der Normalform und der Trigonometrischen Form: + +![](TrigonometricComplexNumber.png) + +
    + + - $r$: Die Länge des Vektors einer komplexen Zahl $z$ ($r = |z|$) + - $\varphi$: Der Winkel zwischen der x-Achse und dem Vektor der komplexen Zahl $z$ + +
    + +### Rechen-Regeln + +
    + +***Rechen-Regeln für komplexe Zahlen:*** + +Addition: + +$$z_1 + z_2 = (x_1 + x_2) + i \cdot (y_1 + y_2)$$ + +Subtraktion: + +$$z_1 - z_2 = (x_1 - x_2) + i \cdot (y_1 - y_2)$$ + +Multiplikation: + +$$z_1 \cdot z_2 = (x_1 \cdot x_2 - y_1 \cdot y_2) + + i \cdot(x_1 \cdot y_2 + x_2 \cdot y_2)$$ + +Division: + +$$\begin{aligned} + \frac{z_1}{z_2} &= + \frac{z_1 \cdot z_2^*}{z_2 \cdot z_2^*} = + \frac{(x_1 + i \cdot y_1) \cdot (x_2 - i \cdot y_2)}{(x_2 + i \cdot y_2) \cdot (x_2 - i \cdot y_2)} \\ + &= \frac{(x_1 \cdot x_2 + y_1 \cdot y_2) + i \cdot (y_1 \cdot x_2 - x_1 \cdot y_2)}{x_2^2 + y_2^2} \\ + &= \frac{(x_1 \cdot x_2 + y_1 \cdot y_2)}{x_2^2 + y_2^2} + i \cdot \frac{(y_1 \cdot x_2 - x_1 \cdot y_2)}{x_2^2 + y_2^2} + \end{aligned}$$ + +
    + +Visualisierung von Addition und Subtraktion zwei komplexer Zahlen $z_1$ und $z_2$: + +![](ComplexNumberMath.png) + +
    + +***Potenzieren in der Polarform:*** + +Für komplexe Zahlen in der Normalform gilt folgendes: + + - Sei $n \in \mathbb{N}$: + $$z^n = (r \cdot e^{i \cdot \varphi})^n = r^n \cdot e^{i \cdot n \cdot \varphi} = + r^n \cdot (\cos(n \cdot \varphi) + i \cdot \sin(n \cdot \varphi))$$ + +
    + +
    + +***Fundamentalsatz der Algebra:*** + +Eine algebraische Gleichung $n$-ten Grades mit komplexen Koeffizienten und Variablen $a_i, z \in \mathbb{C}$ + +$$a_n \cdot z^n + a_{n - 1} \cdot z^{n - 1} + \dots + a_1 \cdot z + a_0 = 0$$ + +besitzt in der Menge $\mathbb{C}$ der komplexen Zahlen genau $n$ Lösungen. + +
    + +
    + +***Ziehen der Wurzel einer komplexen Zahl:*** + +Die Gleichung für das Ziehen einer Wurzel $n$ der komplexen Zahl $a$ lautet: $z^n = a$. + +Für die Lösung dieser Gleichung existieren genau $n$ verschiedene Lösungen in der Menge $\mathbb{C}$: + +$$z_k = r \cdot (\cos(\varphi_k + i \cdot \sin(\varphi_k)) = r \dot e^{i \cdot \varphi_k}$$ + +für $k = 0, 1, 2, \dots, n - 1$: +mit + +$$r = \sqrt[n]{r_0}$$ +$$\varphi_k = \frac{\varphi + k \cdot 2 \cdot \pi}{n}$$ + +Die Bildpunkte der Ergebnisse liegen in der komplexen Zahlenebene auf einem Kreis um den Nullpunkt mit dem Radius $r = \sqrt[n]{r_0}$ und bilden die Ecken eines regelmässigen $n$-Ecks. + +
    + +Visualisierung des Ziehens der $6$-ten Wurzel einer komplexen Zahl $z$: + +![](RootComplexNumber.png) + ## Formelbuchstaben
    @@ -1226,6 +1401,8 @@ Für alle Matrizen, die diagonal-dominant sind gilt, dass sie für das Jacobi- u - $e$: Exponent der Maschinenzahl - $H$: Housholder-Matrix (siehe $QR$-Zerlegung) - $I$: Identitäts-Matrix (Matrix, überall den Wert $0$ und auf der Diagonalen den Wert $1$ hat) + - $i$: Imaginäre Einheit für die Darstellung komplexer Zahlen + - $j$: Alternative Schreibweise für $i$ in Python und in der Elektrotechnik - $K$: Konditionszahl - $L$: Untere Dreiecksmatrix/Normierte Matrix - $m$: Mantisse (Darstellbarer Bereich der Maschinenzahl)