Floats [if you ignore -0.0, infinities, and NaNs] are a subset of the rationals, themselves a subset of the real numbers.
It's generally accurate to consider floats an acceptable approximation of the [extended] reals, since it's possible to do operations on them that don't exist for rational numbers, like sqrt or exp.
> since it's possible to do operations on them that don't exist for rational numbers, like sqrt or exp
This kinda sent me on a spin, for a moment I thought my whole life was a lie and these functions don't take rationals as inputs somehow. Then I realized you mean rather that they typically produce non-rationals, so the outputs will be approximated.
It's generally accurate to consider floats an acceptable approximation of the [extended] reals, since it's possible to do operations on them that don't exist for rational numbers, like sqrt or exp.