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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
@@ Line 10: / Line 10: @@
 ==== Discussion ====
 === 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
-$\frac{1}{a+ib}=\frac{a-ib}{a^2+b^2}$
+$\bullet\ \frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$,
+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 ====
 === Requirements ===