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)
$\endgroup$ 31 Answer
$\begingroup$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?
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