## Arithmetic structure of rational numbers

### Set

 definiendum $\langle \mathbb Q,+_\mathbb{Q},\cdot_\mathbb{Q} \rangle$
 postulate $[\langle a,b\rangle]+_\mathbb{Q}[\langle m,n\rangle]=[\langle a\ n+b\ m,b\ n\rangle]$ postulate $[\langle a,b\rangle]\cdot_\mathbb{Q}[\langle m,n\rangle]=[\langle a\ m,b\ n\rangle]$

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: Rational number