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arithmetic_structure_of_rational_numbers [2013/09/06 22:04] 127.0.0.1 external edit |
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- | ===== Arithmetic structure of rational numbers ===== | ||
- | ==== Definition ==== | ||
- | | @#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: $[\langle a,b\rangle]\cdot_\mathbb{Q}[\langle m,n\rangle]=[\langle a\ m,b\ n\rangle]$ | | ||
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- | The operations $+$ and $\cdot$ on the right hand sides are these of [[arithmetic structure of integers]]. | ||
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- | ==== Discussion ==== | ||
- | We'll generally use the notation introduced in [[integer]] as well as | ||
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- | $\frac{a}{b}\equiv\langle a,b\rangle$ | ||
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- | We'll also often omit the multiplication sign. | ||
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- | >We can also introduce numerator and denominator: | ||
- | > | ||
- | >$ \mathrm{num}\frac{a}{b}\equiv a $ | ||
- | >$ \mathrm{den}\frac{a}{b}\equiv b $ | ||
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- | === Theorems === | ||
- | Division or rational numbers is given by | ||
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- | $\frac{[\langle a,b\rangle]}{[\langle m,n\rangle]}=[\langle a\ m,b\ m\rangle]$ | ||
- | === Reference === | ||
- | Wikipedia: [[http://en.wikipedia.org/wiki/Rational_number|Rational number]] | ||
- | ==== Parents ==== | ||
- | === Requirements === | ||
- | [[Rational number]] | ||
- | === Element of === | ||
- | [[Field]] |