# Differences

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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 Both sides previous revision Previous revision 2014/01/29 19:35 nikolaj 2014/01/29 19:22 nikolaj 2014/01/29 19:12 nikolaj 2014/01/29 19:12 nikolaj 2014/01/29 19:11 nikolaj 2013/12/21 19:57 external edit2013/09/03 01:14 nikolaj created 2014/01/29 19:35 nikolaj 2014/01/29 19:22 nikolaj 2014/01/29 19:12 nikolaj 2014/01/29 19:12 nikolaj 2014/01/29 19:11 nikolaj 2013/12/21 19:57 external edit2013/09/03 01:14 nikolaj created Line 12: Line 12: 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 ===