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Arithmetic structure of complex numbers
Set
| $\langle \mathbb C,+_\mathbb{C},\cdot_\mathbb{C} \rangle$ |
| $(a+ib)+_\mathbb{C}(c+id)=(a+_\mathbb{R}c)+i(b+_\mathbb{R}d)$ |
| $(a+ib)\cdot_\mathbb{C}(c+id)=(a\cdot_\mathbb{R} c-_\mathbb{R}b\cdot_\mathbb{R} d)+i(a\cdot_\mathbb{R} d +_\mathbb{R}b\cdot_\mathbb{R} c)$ |
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 $a,b\in\mathbb R$, we have
$\frac{1}{a+ib}=\frac{1}{a^2+b^2}(a-ib)$
Parents
Requirements
Element of
