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arithmetic_structure_of_complex_numbers [2014/01/29 19:12] nikolaj |
arithmetic_structure_of_complex_numbers [2014/01/29 19:22] 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 |
| - | $\frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$ | + | $\bullet\ \frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$, |
| + | |||
| + | or | ||
| + | |||
| + | $\bullet\ \frac{1}{z}=\frac{1}{|z|^2}\overline{z}$, | ||
| + | |||
| + | and also | ||
| + | |||
| + | $\bullet\ |z+u|^2=|z|^2+\mathrm{Re}(z\cdot\overline{u})+|u|^2$, | ||
| + | |||
| + | $\bullet\ \mathrm{Re}(z\cdot\overline{u})\le |z\cdot\overline{u}|$, | ||
| + | |||
| + | $\bullet\ |\sum_{k=1}^n z_k|\le \sum_k^n|z_k|$. | ||
| ==== Parents ==== | ==== Parents ==== | ||
| === Requirements === | === Requirements === | ||
