Is there any way to determine whether an integer lies between 2 rational numbers without knowing them?

Question

I want to know if there is a mathematical condition (not involving the floor function) for there to be an integer between 2 rational numbers $α$ & $β$. I know that $$β>[α]+1$$but I don't really know what to do with the Greatest Integer Function since I have no idea what the two numbers are.

Source of the problem:

Show that there is no fraction $\frac{e}{f}$ where $f<b+d$ that lies between 2 "neighbour fractions" $\frac{a}{b}$ & $\frac{c}{d}$ $(\frac{c}{d}-\frac{a}{b}=\frac{1}{bd})$

So far, I've determined that $e$ can be any number in the interval $[f\cdot(\frac{a}{b}):f\cdot(\frac{a}{b}+\frac{1}{bd})]$ and want to find the values of $f$ for which an integer lies in the interval

Answer 1

Multiplied woith all denominators, your conditions are as follows: $$ adf < ebd < cbf,\quad cb - ad = 1,\quad f < b+d. $$ In particular, $$ \alpha := (eb-af)d > 0\quad\text{and}\quad \beta := (cf-ed)b > 0. $$ But also $$ \alpha + \beta = (eb-af)d + (cf-ed)b = (cb-ad)f = f. $$ Hence, we have $d|\alpha$, $b|\beta$ (and thus $d\le\alpha$, $b\le\beta$), and $\alpha + \beta = f < b+d\le \alpha+\beta$, which is absurd.

Answer 2

Setting aside the possible application to "neighbour fractions", let's look at sufficient conditions on rational numbers $r,s$ that guarantee an integer $k$ strictly between them.

Obviously some knowledge about $r,s$ is needed because there are many pairs of rational numbers without an integer strictly between them. Without using the floor function one can derive the "integer exists in between" from information about how far apart $r,s$ are.

If $|r-s| \gt 1$, then there is an integer $k$ strictly between $r$ and $s$.

If at least one of $r,s$ is itself not an integer, this can be improved to require merely $|r-s| \ge 1$.