A few issues with this argument. First of all, you're clearly right that the RHS can yield an irrational/transcendental number. It would require insane mathematical coincidence for this not to be so. (It's really hard to have $e^x$ be algebraic, for instance.) And there's nothing in the formula about how many particles there are; we could imagine $N =2$, which would imply that the LHS can only be $0, .5, 1$!
First of all, let's recognize that this is a probabilistic formula like many in statistical mechanics. It holds on average, and yields the probability that an atom has velocity $v$. In general, you can interpret this as "fraction of atoms with that velocity" but clearly there are mathematical difficulties with that interpretation.
Next, many formulas in statistical mechanics hold only in the thermodynamic limit, which means that they assume $N \to \infty$. This then allows for any real number to appear on the LHS, since we have a continuum of values rather than just something that is $m/N$.
Finally, remember that physics is emphatically not math, and really never can be. Any real experiment inevitably brings in complications that we neglect so as to make the problem analytically tractable, so the physical systems we experiment with are never the mathematical systems we calculate with. We can aim for a close congruence between the two, but let's not forget that they really are different things.