A rational number is one that can be written as $a/b$ where $a$ and $b$ are integers, $b\gt0$ ($a$ can take care of negative rationals), and I suppose $\gcd(a,b) = 1$.
Given some $n\in\mathbb{Q}$ where $n=a/b$, what is the next rational number?
At first, I naively thought that it was $(a+1)/b$ but of course that is absurd. Consider $n=1/2$. The next rational number is obviously not $1$.
I decided that one must make the "granularity" finer. For instance $n=1/2=2/4$, so by the above idea, the next one is $3/4$, which is better.
Extending that, is it correct that the next rational number is:
$$ \frac{1+\prod_{n\in\mathbb{Z^+},n\neq b}n}{\prod_{n\in\mathbb{Z^+}}n} $$
(We can factor in the sign of $a$ if we wanted to make this more correct)
The point of this exercise is to make you realize that there is no such thing as "the next rational number". You did well to realize $\frac{a+1}b$ does not work. But the problem is more general: suppose you do decide that some number $\frac c d$ is the "next" one after $\frac a b$. But ask yourself: what is the average of $\frac a b$ and $\frac c d$? And where is it located?