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## Arithmetic structure of complex numbers

### Set

 $\langle \mathbb C,+_\mathbb{C},\cdot_\mathbb{C} \rangle$
 $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$

As defined in complex number, the pattern with $x+iy$ denotes $\langle x,y\rangle$ with $x,y\in \mathbb R$. The operations $+_\mathbb{R}$ and $\cdot_\mathbb{R}$ on the right hand sides are these of arithmetic structure of real numbers.

### Discussion

#### Theorems

For $a,b\in\mathbb R$ and $z,u\in\mathbb C$ and $n,k\in\mathbb N$, we have

$\bullet\ \frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$,

or

$\bullet\ \frac{1}{z}=\frac{1}{|z|^2}\overline{z}$,

and also

$\bullet\ |z+u|^2=|z|^2+\mathrm{Re}(z\cdot\overline{u})+|u|^2$,

$\bullet\ \mathrm{Re}(z\cdot\overline{u})\le |z\cdot\overline{u}|$,

$\bullet\ |\sum_{k=1}^n z_k|\le \sum_k^n|z_k|$.