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Arithmetic structure of rational numbers

Definition

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

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