Real numbers mostly appear in calculus (e.g. the chain rule in gradient descent/backpropagation), but "discrete calculus" is then used as an approximation of infinitesimal calculus. It uses "finite differences" rather than derivatives, which doesn't require real numbers:

https://en.wikipedia.org/wiki/Finite_difference

I'm not sure about applications of real numbers outside of calculus, and how to replace them there.