The infinities you're comparing (between two lines) aren't different in the sense that your BBC documentary was talking about. The reals (numbers whose decimal/binary/whatever expansions don't have to terminate) are a "bigger set" than the integers or the rationals (numbers that eventually terminate in some base n expansion) but not in the sense that there's some real number bigger than all the rational numbers. Have you ever tried reading through Cantor's diagonalization argument? (I might dump a run-through of it later... it's really not that bad).
If you want to see these differing infinities expressed within a number system, you're asking for the ordinals. And those are kind of crazy.
If you want to see these differing infinities expressed within a number system, you're asking for the ordinals. And those are kind of crazy.
