Fermat said:
Thanks. Out of interest, what is the problem with defining the rationals to be a subset of the integers.
It is because mathematicians are crazy about precision. Sometimes to the point of being ridiculous about it.
Start with the set $\mathbb{Z}$. Define the set $A = \{ (x,y) \in \mathbb{Z}\times \mathbb{Z} ~ | ~ y \not = 0 \}$. Define a relation $\sim$ on $A$ by saying $(a,b)\sim (c,d)$ iff $ad = bc$. Check that $\sim$ is an equivalence relation on $A$. Given any $(a,b) \in A$ we define,
$$ [(a,b)] = \{ (x,y) \in A ~ | ~ (x,y) \sim (a,b) \} $$
This is what we call the
equivalence class of $(a,b)$. Instead of using the complicated notation $[(a,b)]$ we just write $\frac{a}{b}$. Therefore, $ad = bc$ if and only if $(a,b)\sim (c,d)$ if and only if $\frac{a}{b} = \frac{c}{d}$.
Definition: $\mathbb{Q}$ is the set of all equivalence classes of $A$, i.e. $\mathbb{Q}$ is the set of all subsets of $A$ which are of the form $\frac{a}{b}$.
Thus, $\frac{2}{1} = \{ ...,(-6,-3),(-4,-2),(-2,-1),(2,1),(4,2),(6,3),... \}$ as you can see that is not the same thing as $2$.
This defines $\mathbb{Q}$ as a set. It is standard to immediately define binary operations on it that turn it into a ring. This is done by defining,
$$ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \text{ and }\frac{a}{b}\cdot \frac{c}{d} = \frac{ac}{bd} $$
Please note the sloppiness. The $+$ sign on the LHS is the addition operation defined on $\mathbb{Q}$ while $+$ sign on RHS is the standard addition operation on $\mathbb{Z}$. So it is better to call the one on LHS by $+'$ to distinguish the different operation. But this is just taking it too far with ridiculous precision. Same thing with multiplication, the one on LHS is the one we are defining on the one on RHS is the standard one for $\mathbb{Z}$.
It is also a standard exercise that prove that since we are choosing representatives for the classes that this type of definition for $\mathbb{Q}$ is well-defined. But I am not going to do that.