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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) Both sides previous revision Previous revision 2013/09/08 14:41 nikolaj 2013/09/08 14:40 nikolaj 2013/09/08 14:40 nikolaj 2013/09/08 14:24 nikolaj 2013/09/08 14:24 nikolaj 2013/09/08 14:23 nikolaj 2013/09/08 14:23 nikolaj 2013/09/06 22:04 external edit2013/09/05 23:22 nikolaj 2013/09/03 01:15 nikolaj created Next revision Previous revision 2013/09/08 14:41 nikolaj 2013/09/08 14:40 nikolaj 2013/09/08 14:40 nikolaj 2013/09/08 14:24 nikolaj 2013/09/08 14:24 nikolaj 2013/09/08 14:23 nikolaj 2013/09/08 14:23 nikolaj 2013/09/06 22:04 external edit2013/09/05 23:22 nikolaj 2013/09/03 01:15 nikolaj created Line 1: Line 1: ===== 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]]. Line 23: Line 23: 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]]