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arithmetic_structure_of_complex_numbers [2014/01/29 19:22] nikolaj |
arithmetic_structure_of_complex_numbers [2014/01/29 19:35] nikolaj |
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| For $a,b\in\mathbb R$ and $z,u\in\mathbb C$ and $n,k\in\mathbb N$, we have | For $a,b\in\mathbb R$ and $z,u\in\mathbb C$ and $n,k\in\mathbb N$, we have | ||
| - | $\bullet\ \frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$, | + | ^ $ \frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$ ^ |
| + | ^ $\frac{1}{z}=\frac{1}{|z|^2}\overline{z}$ ^ | ||
| - | or | + | and |
| - | $\bullet\ \frac{1}{z}=\frac{1}{|z|^2}\overline{z}$, | + | ^ $ |z+u|^2=|z|^2+\mathrm{Re}(z\cdot\overline{u})+|u|^2 $ ^ |
| - | + | ^ $ \mathrm{Re}(z\cdot\overline{u})\le |z\cdot\overline{u}| $ ^ | |
| - | and also | + | ^ $ |\sum_{k=1}^n z_k|\le \sum_k^n|z_k| $ ^ |
| - | + | ||
| - | $\bullet\ |z+u|^2=|z|^2+\mathrm{Re}(z\cdot\overline{u})+|u|^2$, | + | |
| - | + | ||
| - | $\bullet\ \mathrm{Re}(z\cdot\overline{u})\le |z\cdot\overline{u}|$, | + | |
| - | + | ||
| - | $\bullet\ |\sum_{k=1}^n z_k|\le \sum_k^n|z_k|$. | + | |
| ==== Parents ==== | ==== Parents ==== | ||
| === Requirements === | === Requirements === | ||
