# Differences

This shows you the differences between two versions of the page.

 arithmetic_structure_of_complex_numbers [2014/01/29 19:12]nikolaj arithmetic_structure_of_complex_numbers [2014/03/21 11:11] (current) 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 Next 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 Line 1: Line 1: ===== Arithmetic structure of complex numbers ===== ===== Arithmetic structure of complex numbers ===== ==== Set ==== ==== Set ==== - | @#FFBB00: $\langle \mathbb C,​+_\mathbb{C},​\cdot_\mathbb{C} \rangle$ | + | @#FFBB00: definiendum ​| @#FFBB00: $\langle \mathbb C,​+_\mathbb{C},​\cdot_\mathbb{C} \rangle$ | - | @#55EE55: $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ | + | @#55EE55: postulate ​  | @#55EE55: $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ | - | @#55EE55: $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ | + | @#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]]. 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]]. Line 10: Line 10: ==== Discussion ==== ==== Discussion ==== === Theorems === === 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 - $\frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$ + ^ $\frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$ ​^ + ^ $\frac{1}{z}=\frac{1}{|z|^2}\overline{z}$ ^ + + 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 ==== ==== Parents ==== - === Requirements ​=== + === Context ​=== [[Complex number]] [[Complex number]] === Element of === === Element of === [[Field]] [[Field]]