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arithmetic_structure_of_complex_numbers [2014/01/29 19:12] nikolaj |
arithmetic_structure_of_complex_numbers [2014/03/21 11:11] (current) |
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| ===== Arithmetic structure of complex numbers ===== | ===== Arithmetic structure of complex numbers ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#FFBB00: $\langle \mathbb C,+_\mathbb{C},\cdot_\mathbb{C} \rangle$ | | + | | @#FFBB00: definiendum | @#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: postulate | @#55EE55: $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ | |
| - | | @#55EE55: $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ | | + | | @#55EE55: postulate | @#55EE55: $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ | |
| 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]]. | 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 ==== | ==== Discussion ==== | ||
| === Theorems === | === Theorems === | ||
| - | For $a,b\in\mathbb R$, we have | + | For $a,b\in\mathbb R$ and $z,u\in\mathbb C$ and $n,k\in\mathbb N$, we have |
| - | $\frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$ | + | ^ $ \frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$ ^ |
| + | ^ $\frac{1}{z}=\frac{1}{|z|^2}\overline{z}$ ^ | ||
| + | |||
| + | and | ||
| + | |||
| + | ^ $ |z+u|^2=|z|^2+\mathrm{Re}(z\cdot\overline{u})+|u|^2 $ ^ | ||
| + | ^ $ \mathrm{Re}(z\cdot\overline{u})\le |z\cdot\overline{u}| $ ^ | ||
| + | ^ $ |\sum_{k=1}^n z_k|\le \sum_k^n|z_k| $ ^ | ||
| ==== Parents ==== | ==== Parents ==== | ||
| - | === Requirements === | + | === Context === |
| [[Complex number]] | [[Complex number]] | ||
| === Element of === | === Element of === | ||
| [[Field]] | [[Field]] | ||
