===== Arithmetic structure of complex numbers =====
==== Set ====
| @#FFBB00: definiendum | @#FFBB00: $\langle \mathbb C,+_\mathbb{C},\cdot_\mathbb{C} \rangle$ |

| @#55EE55: postulate   | @#55EE55: $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ |
| @#55EE55: postulate   | @#55EE55: $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ |

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| $ ^
==== Parents ====
=== Context ===
[[Complex number]]
=== Element of ===
[[Field]]