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28 lines
1.7 KiB
Markdown
28 lines
1.7 KiB
Markdown
7 years ago
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## Rechenoperationen
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### Division
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$
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\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 } }
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$
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### Multiplikation
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**Kadesische Form:**
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${ z }_{ 1 }\cdot { z }_{ 2 }=\left( a+bi \right) \cdot \left( c+di \right) =ac+adi+bci+bd{ i }^{ 2 }$
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**Trigonometrische Form:**
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$
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{ 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) §
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### Addition
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${ z }_{ 1 }+{ z }_{ 2 }=\left( a+bi \right) +\left( c+di \right) =a+c+\left( b+d \right) i§
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### Subtraktion
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${ z }_{ 1 }-{ z }_{ 2 }=\left( a+bi \right) -\left( c+di \right) =a-c+\left( b-d \right) i$
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### Potenzierung
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$
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{ 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)
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$
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### Wurzel
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