How to make sense of fractions?

Suppose that we are in a mindset where the word "number" means "positive integer" (let's leave negative numbers and zero out out the discussion, just to keep things simpler). Then, as you rightly point out, there is no answer to the problem of dividing $1$ by $2$, since there is no "number" that you can multiply by $2$ to obtain $1$.

So if there exists some object "one half" (whatever that would be) which solves this problem, it can't be a "number" in our old sense – it must be a new kind of number. Some would say that it's intuitively clear what "one half" is: it's just the mathematical object which corresponds to our idea of cutting a pizza in half. But this argument is not completely convincing; if it were, you wouldn't be asking this question, right? ;-)

So what mathematicians have done is that they have actually constructed a larger system of numbers which visibly includes this mysterious object. For the purposes of this answer, and just for fun, I will call these new numbers "lumbers", but they are nothing but the positive fractions (also called positive rational numbers).

By definition, a "lumber" is a set of equivalent pairs of numbers, where we use the pair $(a,b)$ to represent the idea of "dividing $a$ things into $b$ pieces", and the pair $(a,b)$ is considered equivalent to the pair $(c,d)$ if "dividing $c$ things into $d$ pieces" would give the same result (i.e., if $ad=bc$; this way of expressing things is a way of saying "$a/b=c/d$" without mentioning division, which is taboo at this point, since we haven't defined division yet).

For example, if $5$ people share $3$ pizzas equally, they each get the same amount of pizza as if $10$ people share $6$ pizzas, or if $15$ people share $9$ pizzas, etc. So the pairs of numbers $(3,5)$, $(6,10)$, $(9,15)$, etc., are equivalent, and the set of all these equivalent pairs is a "lumber" which we can give the name "three fifths": $$ \text{the lumber “three fifths”} = \{ (3,5), (6,10), (9,15), \dots \}. $$ Similarly, the mysterious object "one half" is defined to be the following lumber: $$ \text{the lumber “one half”} = \{ (1,2), (2,4), (3,6), \dots \}. $$

Now when it comes to the question of dividing the number $1$ by the number $2$, there is a problem, namely that a "number" is not a "lumber", but we can fix this by saying that to each number $n$ there is a unique corresponding lumber, namely the lumber containing the pair $(n,1)$. For example, $$ \text{the number $1$} $$ corresponds to $$ \text{the lumber “one”} = \{ (1,1), (2,2), (3,3), \dots \} $$ and the $$ \text{the number $2$} $$ corresponds to $$ \text{the lumber “two”} = \{ (2,1), (4,2), (6,3), \dots \}. $$ ("If $6$ pizzas are shared among $3$ people, they get $2$ pizzas each.")

Next, we define how to multiply two lumbers $X$ and $Y$: take any pair $(a,b)$ from the set of pairs $X$ and any pair $(c,d)$ from $Y$, and let $XY$ be the lumber containing the pair $(ac,bd)$. (This is our way of saying "$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$" without mentioning division.) Here I'm sweeping one detail under the rug; one needs to check that the definition makes sense, in that the result doesn't depend on the choice of pairs $(a,b)$ and $(c,d)$. Let's not go into that here.

Similarly, we let $X/Y$ be the lumber which contains the pair $(ad,bc)$ (which corresponds to the idea "$\frac{a/b}{c/d} = \frac{ad}{bc}$").

Now we can finally check that $$ \frac{\text{the lumber “one”}}{\text{the lumber “two”}} = \text{the lumber “one half”}, $$ which shows that in the "lumber system" we can indeed divide "one" by "two", something that we couldn't do in our old "number system". Here's how: the pair $(a,b)=(1,1)$ belongs to the lumber "one", and the pair $(c,d)=(2,1)$ belongs to the lumber "two", and the recipe for division was to form the pair $(ad,bc)=(1\cdot 1,1\cdot 2)=(1,2)$, which indeed belongs to the lumber "one half"!

Now, in practice it would be too tedious to keep making all these distinctions, so when one uses the symbol "$1$" it can mean either the number $1$ (a positive integer), or the lumber "one" (the set defined above), or many other things as well, all depending on the context. And instead of introducing some new special symbol for the lumber "one half", one simply says that it's the result of dividing $1$ (meaning the lumber "one") by $2$ (meaning the lumber "two"), and writes it as "$1/2$".

I like several of the other answers. I want to add that you have discovered something about what makes fractions hard, and hard for teachers to explain to kids. What we write as "1/2" has (at least) three separate (but related) meanings.

• It represents "cut something in two pieces and take one of them". That's the way fractions are first taught and understood.

• It's a "number" - the spot on the number line halfway between 0 and 1. (There's a prominent number line in almost every first and second grade classroom.)

• It's the answer to the question "what do you multiply 2 by to get 1?" That's the meaning kids have to grasp when they begin to grapple with algebra.

Mathematicians and other people who have somehow gotten used to these three meanings can easily and subconsciously switch back and forth among them in any particular situation. But explaining just how these three different ways to think about "1/2" are related is subtle - and more than I will attempt here.

I hope this helps you.

The symbol $1/n$ is just only a "symbol" that represents the number $x$ defined by $ n \cdot x=1$, i.e. the inverse of $n$ in the group $(\mathbb{Q}, \cdot)$ and the existence of this number is guaranteed by the axioms defining $\mathbb{Q}$.

Formally you can construct $\mathbb{Q}$ as the set of equivalence classes of ordered pairs of integers $(m,n)$ with $n \ne 0$, where the equivalence relation is $(m_1,n_1) \sim (m_2,n_2)$ iff $m_1n_2 - m_2n_1 = 0$ and with suitable definitions of the $+$ and $\cdot$ operations as you can see in the Wikipedia page.

So the symbol $m/n$ stay for a representative element of the equivalence class $(m,n)$ in such quotient set.