Given an arbitrarily chosen constant of nature (say, the speed of light c), we can confidently say that the fact that it is equal to 299 792 458 meters per second is a contingent fact about our universe (in other words, it is logically possible for c to equal some other value). In fact, it is logically possible for any constant to be equal to any positive real number other than the one it has in the actual world.
Let's also take into consideration the fact from mathematics that if you chose a random positive real number, the probability of choosing an algebraic number is exactly zero.
Given these two premises, does it follow that all of the nature's constants are not algebraic, i.e. transcendental (and hence irrational)?
EDIT: To make things more robust, let's make the following assumption: A Theory of Everything exists, and we are talking about it, it only and its constants.
NOTE: Constants varying with time don't change the essence of the question - if the constant is changing with time, then my question is about its value at a particular point t in time.