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arithmetic_structure_of_complex_numbers [2014/01/29 19:11] nikolaj |
arithmetic_structure_of_complex_numbers [2014/01/29 19:35] nikolaj |
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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 |
| - | $\frac{1}{a+ib}=\frac{a-ib}{a^2+b^2}$ | + | ^ $ \frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$ ^ |
| - | === Reference === | + | ^ $\frac{1}{z}=\frac{1}{|z|^2}\overline{z}$ ^ |
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Complex_number|Complex number]] | + | |
| + | 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 === | === Requirements === | ||
