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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?

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    $\begingroup$ OP is asking why $p \div q = \dfrac pq = p \times \dfrac1q$. $\endgroup$ – Kenny Lau Sep 5 '17 at 15:57
  • $\begingroup$ A fraction is a number expressed as the ratio between two integers: $\dfrac 2 3$. The result of the division of $2$ by $3$ gives as result the number $\dfrac 2 3$. $\endgroup$ – Mauro ALLEGRANZA Sep 5 '17 at 15:57
  • $\begingroup$ I think the idea you may be looking for is reflected by the facts that you are working in a context in which every non-zero number has a unique multiplicative inverse (and perhaps also that multiplication is associative). Dividing by a number is then the same as multiplying by its inverse. $\endgroup$ – Mark Bennet Sep 5 '17 at 16:04
  • $\begingroup$ 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". $\endgroup$ – Johnathan Doe Sep 5 '17 at 16:10
  • $\begingroup$ The reason we resort to abstract definitions is because in order to ask something like "Why are $X$ and $Y$ the same?" you need to know exactly what $X$ and $Y$ mean in the first place. What does $p/q$ mean exactly? What does $p \div q$ mean? If you are just looking for intuition, it's hard to say more than you already have. If you are thinking of $1/6$ as the size of one piece of a whole object cut into six pieces then this is the exact same thing as taking the whole object, dividing it by $6$ and taking one of the pieces. $\endgroup$ – wgrenard Sep 5 '17 at 16:26

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