Real life illustration of the fact that rationals have measure zero I wonder if there's any real world phenomenon that reflects the mathematical fact that $\Bbb Q^k$ has Lebesgue measure zero in $\Bbb R^k$, or put another way, the likelihood that we get a rational point by randomly picking points from $\Bbb R^k$ is zero. I'm not sure if such a phenomenon ever theoretically exists, since quantum physics seems to suggest the spacetime we live in is not a continuum. 
But, at least macroscopically, does there exists a system which depends on a parameter $p$ taken from, say, $\Bbb R$, and will behave perceptibly differently when $p$ is a "distinguishable" rational number (by which I mean fractions with simple, low terms like $1/2,4/7$ etc instead of something like $3.1415926535$) than when $p$ is irrational? Some mathematical dynamic systems seem to depend on parameter rationality, but I want to know if any perceptible real example exists. 
 A: This is perhaps not exactly what you had in mind (it is about real-world applications of sets of measure zero in general, as opposed to the rational numbers in particular), but I will float it out here anyway.
Considering the power iteration algorithm for finding eigenvalues: https://en.wikipedia.org/wiki/Power_iteration. If the matrix $A$ is singular (has eigenvalue 0), then there exists a vector $v$ such that $Av=0 \implies ||Av||=0$. Looking at the iteration formula, if we were unlucky enough to get $b_k=v$ for some $v$ such that $Av$, then the given expression for $b_{k+1}$ would be undefined and our algorithm would fail.
But of course the null space of any singular matrix (except for the zero matrix itself) has measure zero in the space as a whole (since any subspace which is not equal to the space itself has measure zero), so we have probability zero of choosing such a vector or any vector which would iterate to it, so in practice we can ignore this possibility entirely although in theory it is something we should have to worry about.
Another example illustrating a similar concept is determining whether or not a matrix is singular when calculating its inverse. Due to inevitable numerical errors perturbing the relevant vectors outside of the matrix's nullspace, even if we are starting with a singular matrix as an input, a naive algorithm would nevertheless calculate an "inverse" for the matrix (actually the inverse of a very similar but perturbed matrix which happens to not be singular), and oblivious users might not notice that there is any problem and use these bogus results in their subsequent calculations. Hence the algorithms which Matlab uses, for example, need to be more sophisticated in order to avoid letting numerical errors from preventing them from identifying matrices which are actually singular. Again, that such small perturbations arising from numerical error could do this with any non-negligible probability is only possible because the nullspace of any non-zero matrix has measure zero in the space as a whole.
