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 arithmetic_structure_of_rational_numbers [2013/09/06 22:04]127.0.0.1 external edit arithmetic_structure_of_rational_numbers [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/09/08 14:11 nikolaj 2013/09/06 22:04 external edit2013/09/03 00:38 nikolaj 2013/09/03 00:38 nikolaj 2013/09/03 00:30 nikolaj created Next revision Previous revision 2013/09/08 14:11 nikolaj 2013/09/06 22:04 external edit2013/09/03 00:38 nikolaj 2013/09/03 00:38 nikolaj 2013/09/03 00:30 nikolaj created Line 1: Line 1: ===== Arithmetic structure of rational numbers ===== ===== Arithmetic structure of rational numbers ===== - ==== Definition ​==== + ==== Set ==== - | @#FFBB00: $\langle \mathbb Q,​+_\mathbb{Q},​\cdot_\mathbb{Q} \rangle$ | + | @#FFBB00: definiendum ​| @#FFBB00: $\langle \mathbb Q,​+_\mathbb{Q},​\cdot_\mathbb{Q} \rangle$ | - | @#55EE55: $[\langle a,​b\rangle]+_\mathbb{Q}[\langle m,​n\rangle]=[\langle a\ n+b\ m,b\ n\rangle]$ | + | @#55EE55: postulate ​  | @#55EE55: $[\langle a,​b\rangle]+_\mathbb{Q}[\langle m,​n\rangle]=[\langle a\ n+b\ m,b\ n\rangle]$ | - | @#55EE55: $[\langle a,​b\rangle]\cdot_\mathbb{Q}[\langle m,​n\rangle]=[\langle a\ m,b\ n\rangle]$ | + | @#55EE55: postulate ​  | @#55EE55: $[\langle a,​b\rangle]\cdot_\mathbb{Q}[\langle m,​n\rangle]=[\langle a\ m,b\ n\rangle]$ | The operations $+$ and $\cdot$ on the right hand sides are these of [[arithmetic structure of integers]]. The operations $+$ and $\cdot$ on the right hand sides are these of [[arithmetic structure of integers]]. Line 15: Line 15: We'll also often omit the multiplication sign. We'll also often omit the multiplication sign. - >We can also introduce numerator and denominator:​ + We can also introduce numerator and denominator:​ - > + - >$\mathrm{num}\frac{a}{b}\equiv a$ + ^ $\mathrm{num}\frac{a}{b}\equiv a$ ^ - >$\mathrm{den}\frac{a}{b}\equiv b$ + ^ $\mathrm{den}\frac{a}{b}\equiv b$ ^ === Theorems === === Theorems === Line 27: Line 27: Wikipedia: [[http://​en.wikipedia.org/​wiki/​Rational_number|Rational number]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Rational_number|Rational number]] ==== Parents ==== ==== Parents ==== - === Requirements ​=== + === Context ​=== [[Rational number]] [[Rational number]] === Element of === === Element of === [[Field]] [[Field]]