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arithmetic_structure_of_real_numbers [2013/09/08 14:24] nikolaj |
arithmetic_structure_of_real_numbers [2013/09/08 14:40] nikolaj |
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| | @#FFBB00: $\langle \mathbb R,+_\mathbb{R},\cdot_\mathbb{R} \rangle$ | | | @#FFBB00: $\langle \mathbb R,+_\mathbb{R},\cdot_\mathbb{R} \rangle$ | | ||
| - | | @#55EE55: $ r +_\mathbb{R} s = \{q+_\mathbb{Q}p\ |\ q\in r\land s\in p\} $ | | + | | @#55EE55: $ r +_\mathbb{R} s = \{q+_\mathbb{Q}p\ |\ q\in r\land p\in s\} $ | |
| - | | @#55EE55: $ ... $ | | + | | @#55EE55: $ r -_\mathbb{R} s = \{q-_\mathbb{Q}p\ |\ q\in r\land p\in \mathbb Q\setminus s\} $ | |
| - | | @#55EE55: $ ... $ | | + | | @#55EE55: $ -_{\mathbb R}r = \{q-_\mathbb{Q}p\ |\ q<0\land p\in \mathbb Q\setminus r\} $ | |
| - | | @#55EE55: $ ... $ | | + | |
| - | The operations $+_\mathbb{Q}$ and $\cdot_\mathbb{Q}$ on the right hand sides are these of [[arithmetic structure of rational numbers]]. | + | | @#55EE55: $ r\ge 0\land s\ge 0\implies r\cdot_\mathbb{R}s = \{q\cdot_\mathbb{Q}p\ |\ (q\in r)\land (p\in s)\land (q,p\ge 0)\}\cup\{q\ |\ (q\in\mathbb Q)\land (q<0)\} $ | |
| + | | @#55EE55: $ r\ge 0\land s < 0\implies r\cdot_\mathbb{R}s = -(r\cdot_\mathbb{R}(-s)) $ | | ||
| + | | @#55EE55: $ r < 0\land s\ge 0\implies r\cdot_\mathbb{R}s = -((-r)\cdot_\mathbb{R}s) $ | | ||
| + | | @#55EE55: $ r < 0\land s < 0\implies r\cdot_\mathbb{R}s = (-r)\cdot_\mathbb{R}(-s) $ | | ||
| - | >todo: | + | | @#55EE55: $ r\ge 0\land s > 0\implies r/_\mathbb{R}s = \{q/_\mathbb{Q}p\ |\ (q\in r)\land (p\in \mathbb Q\setminus s)\} $ | |
| - | > | + | | @#55EE55: $ r\ge 0\land s < 0\implies r/_\mathbb{R}s = -(r/_\mathbb{R}(-s)) $ | |
| - | > http://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_by_Dedekind_cuts | + | | @#55EE55: $ r < 0\land s > 0\implies r/_\mathbb{R}s = -((-r)/_\mathbb{R}s) $ | |
| + | | @#55EE55: $ r < 0\land s < 0\implies r/_\mathbb{R}s = (-r)/_\mathbb{R}(-s) $ | | ||
| + | |||
| + | The operations $+_\mathbb{Q}$ and $\cdot_\mathbb{Q}$ on the right hand sides are these of [[arithmetic structure of rational numbers]]. | ||
| ==== Discussion ==== | ==== Discussion ==== | ||
