Arithmetic structure of complex numbers

Set

definiendum

$\langle \mathbb C,+_\mathbb{C},\cdot_\mathbb{C} \rangle$

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

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

$\frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$
$\frac{1}{z}=\frac{1}{\|z\|^2}\overline{z}$

and

$\|z+u\|^2=\|z\|^2+\mathrm{Re}(z\cdot\overline{u})+\|u\|^2$
$\mathrm{Re}(z\cdot\overline{u})\le \|z\cdot\overline{u}\|$
$\|\sum_{k=1}^n z_k\|\le \sum_k^n\|z_k\|$

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Complex number

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Arithmetic structure of complex numbers

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Arithmetic structure of complex numbers

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Theorems

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