===== Arithmetic structure of rational numbers =====
==== Set ====
| @#FFBB00: definiendum | @#FFBB00: $\langle \mathbb Q,+_\mathbb{Q},\cdot_\mathbb{Q} \rangle$ |

| @#55EE55: postulate   | @#55EE55: $[\langle a,b\rangle]+_\mathbb{Q}[\langle m,n\rangle]=[\langle a\ n+b\ m,b\ n\rangle]$ |
| @#55EE55: postulate   | @#55EE55: $[\langle a,b\rangle]\cdot_\mathbb{Q}[\langle m,n\rangle]=[\langle a\ m,b\ n\rangle]$ |

The operations $+$ and $\cdot$ on the right hand sides are these of [[arithmetic structure of integers]].

==== Discussion ====
We'll generally use the notation introduced in [[integer]] as well as

$\frac{a}{b}\equiv\langle a,b\rangle$

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 ===
Wikipedia: [[http://en.wikipedia.org/wiki/Rational_number|Rational number]]
==== Parents ====
=== Context ===
[[Rational number]]
=== Element of ===
[[Field]]