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arithmetic_structure_of_complex_numbers [2014/01/29 19:22]
nikolaj
arithmetic_structure_of_complex_numbers [2014/03/21 11:11]
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-===== Arithmetic structure of complex numbers ===== 
-==== Set ==== 
-| @#FFBB00: $\langle \mathbb C,​+_\mathbb{C},​\cdot_\mathbb{C} \rangle$ | 
  
-| @#55EE55: $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ | 
-| @#55EE55: $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ | 
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-As defined in [[complex number]], the pattern with $x+iy$ denotes $\langle x,y\rangle$ with $x,y\in \mathbb R$. The operations $+_\mathbb{R}$ and $\cdot_\mathbb{R}$ on the right hand sides are these of [[arithmetic structure of real numbers]]. 
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-==== Discussion ==== 
-=== Theorems === 
-For $a,​b\in\mathbb R$ and $z,​u\in\mathbb C$ and $n,​k\in\mathbb N$, we have  
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-$\bullet\ \frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$,​ 
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-or  
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-$\bullet\ \frac{1}{z}=\frac{1}{|z|^2}\overline{z}$,​ 
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-and also 
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-$\bullet\ |z+u|^2=|z|^2+\mathrm{Re}(z\cdot\overline{u})+|u|^2$,​ 
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-$\bullet\ \mathrm{Re}(z\cdot\overline{u})\le |z\cdot\overline{u}|$,​ 
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-$\bullet\ |\sum_{k=1}^n z_k|\le \sum_k^n|z_k|$. 
-==== Parents ==== 
-=== Requirements === 
-[[Complex number]] 
-=== Element of === 
-[[Field]] 
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