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arithmetic_structure_of_complex_numbers [2014/01/29 19:22] nikolaj |
arithmetic_structure_of_complex_numbers [2014/03/21 11:11] |
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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]]. | ||
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- | ==== Discussion ==== | ||
- | === Theorems === | ||
- | For $a,b\in\mathbb R$ and $z,u\in\mathbb C$ and $n,k\in\mathbb N$, we have | ||
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- | $\bullet\ \frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$, | ||
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- | or | ||
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- | $\bullet\ \frac{1}{z}=\frac{1}{|z|^2}\overline{z}$, | ||
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- | and also | ||
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- | $\bullet\ |z+u|^2=|z|^2+\mathrm{Re}(z\cdot\overline{u})+|u|^2$, | ||
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- | $\bullet\ \mathrm{Re}(z\cdot\overline{u})\le |z\cdot\overline{u}|$, | ||
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- | $\bullet\ |\sum_{k=1}^n z_k|\le \sum_k^n|z_k|$. | ||
- | ==== Parents ==== | ||
- | === Requirements === | ||
- | [[Complex number]] | ||
- | === Element of === | ||
- | [[Field]] |