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arithmetic_structure_of_complex_numbers [2014/01/29 19:12] nikolaj |
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| - | ===== Arithmetic structure of complex numbers ===== | ||
| - | ==== Set ==== | ||
| - | | @#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: $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ | | ||
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| - | 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 $a,b\in\mathbb R$, we have | ||
| - | |||
| - | $\frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$ | ||
| - | ==== Parents ==== | ||
| - | === Requirements === | ||
| - | [[Complex number]] | ||
| - | === Element of === | ||
| - | [[Field]] | ||
