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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) |
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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: | + | 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 === | ||
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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]] |