MOD1_3.PPT - NOMBRES COMPLEXES a Définitions Le nombre...

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NOMBRES COMPLEXES a) Définitions Le nombre imaginaire i: i i i i i i i   1 1 1 2 3 4 5 ; ; ; ; ; ...
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NOMBRES COMPLEXES a) Définitions Le nombre imaginaire i: i i i i i i i   1 1 1 2 3 4 5 ; ; ; ; ; ... Le nombre complexe z: z a ib a : partie réelle de z b : partie imaginaire de z a z   Re b z   Im
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NOMBRES COMPLEXES a) Définitions Le nombre imaginaire i: i i i i i i i   1 1 1 2 3 4 5 ; ; ; ; ; ... Le nombre complexe z: z a ib a : partie réelle de z b : partie imaginaire de z a z   Re b z   Im Le complexe conjugué de z : z a ib *
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b) Représentations de z Im Re a b z z peut être considéré comme un "vecteur" dans le plan complexe
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Im Re a b z -b z * Représentations cartésiennes de z et z *
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Im Re a b z A Représentations cartésienne et "polaire" de z a A cos b A sin A a b 2 2 tan b a A : la norme de z
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Im Re a b z -b z *  A
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Identité d'Euler: e i i cos sin Im Re A=1 z e i
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Identité d'Euler: e i i cos( ) sin( ) e i i cos sin e i i cos sin  Im Re A=1 z e i z e i *
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Im Re a b z A Représentation exponentielle de z z a ib A i cos sin Ae i
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Représentations exponentielles de z et z * z a ib A i cos sin Ae i z a ib * Ae i A i cos sin  Im Re A z * z
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c) Opérations sur les nombres complexes - Somme z a ib 1 1 1 z a ib 2 2 2 z z z a a i b b a ib 3 1 2 1 2 1 2 3 3 a f a f
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c) Opérations sur les nombres complexes - Somme z a ib 1 1 1 z a ib 2 2 2 z z z a a i b b a ib 3 1 2 1 2 1 2 3 3 a f a f 1 2 Exemple: Sommer 3 2 et 1 3 z i z i     z i 3 4 5 Im Re
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-Multiplication z a ib A e i 1 1 1 1 1 z a ib A e i 2 2 2 2 2 z z a ib a ib z 3 1 2 1 1 2 2 a fa f z a a b b i a b a b a ib 3 1 2 1 2 1 2 2 1 3 3
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