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arithmetic_structure_of_real_numbers [2013/09/08 14:40]
nikolaj
arithmetic_structure_of_real_numbers [2014/03/21 11:11] (current)
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 ===== Arithmetic structure of real numbers ===== ===== Arithmetic structure of real numbers =====
-==== Definition ​==== +==== Set ==== 
-| @#FFBB00: $\langle \mathbb R,​+_\mathbb{R},​\cdot_\mathbb{R} \rangle$ |+| @#FFBB00: definiendum ​| @#FFBB00: $\langle \mathbb R,​+_\mathbb{R},​\cdot_\mathbb{R} \rangle$ |
  
-| @#55EE55: $ r +_\mathbb{R} s = \{q+_\mathbb{Q}p\ |\ q\in r\land p\in s\} $ | +| @#55EE55: postulate ​  | @#55EE55: $ r +_\mathbb{R} s = \{q+_\mathbb{Q}p\ |\ (q\in r)\land (p\in s)\} $ | 
-| @#55EE55: $ r -_\mathbb{R} s = \{q-_\mathbb{Q}p\ |\ q\in r\land p\in \mathbb Q\setminus s\} $ | +| @#55EE55: postulate ​  | @#55EE55: $ r -_\mathbb{R} s = \{q-_\mathbb{Q}p\ |\ (q\in r)\land (p\in \mathbb Q\setminus s)\} $ | 
-| @#55EE55: $ -_{\mathbb R}r = \{q-_\mathbb{Q}p\ |\ q<​0\land ​p\in \mathbb Q\setminus r\} $ |+| @#55EE55: postulate ​  | @#55EE55: $ -_{\mathbb R}r = \{q-_\mathbb{Q}p\ |\ (p\in \mathbb Q\setminus r)\land (q<0)\} $ |
  
-| @#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: postulate ​  | @#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: postulate ​  | @#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: postulate ​  | @#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) ​ $ |+| @#55EE55: postulate ​  | @#55EE55: $ r  < 0\land s <  0\implies r\cdot_\mathbb{R}s = (-r)\cdot_\mathbb{R}(-s) ​ $ |
  
-| @#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: postulate ​  | @#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)) ​ $ | +| @#55EE55: postulate ​  | @#55EE55: $ r\ge 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) ​ $ | +| @#55EE55: postulate ​  | @#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) ​ $ |+| @#55EE55: postulate ​  | @#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]]. The operations $+_\mathbb{Q}$ and $\cdot_\mathbb{Q}$ on the right hand sides are these of [[arithmetic structure of rational numbers]].
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 Wikipedia: [[http://​en.wikipedia.org/​wiki/​Real_number|Real number]], [[http://​en.wikipedia.org/​wiki/​Construction_of_the_real_numbers|Construction of the real numbers]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Real_number|Real number]], [[http://​en.wikipedia.org/​wiki/​Construction_of_the_real_numbers|Construction of the real numbers]]
 ==== Parents ==== ==== Parents ====
-=== Requirements ​===+=== Context ​===
 [[Real number]] [[Real number]]
 === Element of === === Element of ===
 [[Field]] [[Field]]
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