You can count computable irrationals by numbering the algorithms that calculate (successive approximations to) them and lining them up in order. This is what Tu

You can count computable irrationals by numbering the algorithms that calculate (successive approximations to) them and lining them up in order. This is what Turing used his "Turing Machines" to do. So all of the irrationals that we use or could ever use in calculations are countable. The uncountability of the irrationals comes entirely from the uncomputable ones, which we will probably never see. The reals are deeply weird.