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arithmetic_structure_of_rational_numbers [2013/09/03 00:30] nikolaj created |
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| ===== 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]]. | ||
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| We'll also often omit the multiplication sign. | We'll also often omit the multiplication sign. | ||
| + | |||
| + | We can also introduce numerator and denominator: | ||
| + | |||
| + | ^ $ \mathrm{num}\frac{a}{b}\equiv a $ ^ | ||
| + | ^ $ \mathrm{den}\frac{a}{b}\equiv b $ ^ | ||
| + | |||
| + | === Theorems === | ||
| + | Division or rational numbers is given by | ||
| + | |||
| + | $\frac{[\langle a,b\rangle]}{[\langle m,n\rangle]}=[\langle a\ m,b\ m\rangle]$ | ||
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Rational_number|Rational number]] | Wikipedia: [[http://en.wikipedia.org/wiki/Rational_number|Rational number]] | ||
| - | ==== Context ==== | + | ==== Parents ==== |
| - | === Requirements === | + | === Context === |
| [[Rational number]] | [[Rational number]] | ||
| === Element of === | === Element of === | ||
| [[Field]] | [[Field]] | ||
