Here's what I thought:
Let's start with a fraction, it is just a number which represent a value by telling us how many pieces of a certain size we have.
Now, what about another way to represent this value, like considering it as a single piece of that size? Well, we can take a bigger number, and divide it into equal pieces of the size we need, to make one of that pieces equal to the value we want to represent.
So, what value, divided by another value, gives the piece we need? Exactly! The same values of a fraction, numerator and denominator, but why this relationship? Why does it work? Is there an axiom/theorem that makes it possible, or is it just a coincidence?
EDIT: e.g. What relationship makes the number 5/2 equals to what the division between the number 5 and the number 2 gives as result?
EDIT 2: What I'm looking for is, if possible, a more intuitive way to understand deeply this relationship without the use of algebraic expressions to prove it or rules that "make it true and stop".
EDIT 3: Ok so I thought this, let's say we have the fraction 1/5, it means we have one fifth of a whole and to get a fifth of a whole we can just divide the whole in 5 pieces and get the size of one.
Now, let's say we have 2/5, as a fraction, it still make sense, it still means we have two fifth of a whole, but as a division, it suddently stop working, because in a division, we're not working with a single whole anymore, we're working with two wholes.
Why does this works than? Well because division still takes only the size of a piece, while fractions can get the size of more pieces togheter, so how do we make it work? We can just double the starting integer so that when divided by the same size as before, the size of a single piece is equivalent to the size of two pieces of the previous value.
But now, why is the number (in this case the number 2), still valid for both the division and the fraction, why is it still appliable to both and so on with all the other numbers?