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