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
Parents
Context
Element of
