Practically there's uses in areas like cryptography and simulation where Pseudo-random number generators (PRNGs) are used. If the numbers aren't irrational then there may be flaws in the assumptions being used.
Beyond direct application, knowing a number is irrational can be a form of validation for theoretical modelling. If a number arising in a model turns out to be rational, it could mean an unexpected simplicity or symmetry, which is worth exploring further. Conversely, irrationality is often expected in complex systems and may confirm the soundness of a mathematical construct or physical model. I guess a good example of that is the relationship of light spectra and Planks constant.
How could it be relevant to cryptography/simulation? Short of a symbolic algebra system, all numbers on a computer are rational. Pi is irrational but M_PI is rational. How can a PRNG be based on an assumption of irrationality when it has no access to irrational numbers?
You can make PRNGs based on approximations to (disjunctive) irrational numbers, and the irrationality of the number being approximated is important to its quality.
I'm not aware of any widespread real-world PRNGs constructed this way because they're less efficient than traditional PRNGs. It's mostly a mathematical trick to be used in proofs and thought experiments.
I suspect they're referring to the more common practice of taking the first N digits of a well known number like Pi or e that happens to be irrational as a magic constant of known provenance. 1245678 is another common one though, which obviously isn't irrational.
Agreed re approximating and use of constants. Admittedly, I haven’t looked into PRNGs much since my numerical analysis college days! On simulation I did once do some work with Monte Carlo using quasi-random sequences (e.g Sobol), which can provide better coverage than pure randomness for certain problems.
Beyond direct application, knowing a number is irrational can be a form of validation for theoretical modelling. If a number arising in a model turns out to be rational, it could mean an unexpected simplicity or symmetry, which is worth exploring further. Conversely, irrationality is often expected in complex systems and may confirm the soundness of a mathematical construct or physical model. I guess a good example of that is the relationship of light spectra and Planks constant.