While looking at Cantors method of proof, which he used to show that the set of the rational numbers is countable and that it has got the same cardinality (Aleph-naught) as the set of the natural numbers, I recognized that if there were fractions that used transfinite numbers as their numerators and denominators, then those infinitely precisely defined fractions could be used within Cantors zizag-counting-grid to address not only all the rational numbers but all the real numbers (of course only in theory because transfinite numbers usually cannot be written down or spoken out very easily).
So my question is as stated above: Is there a mathematical concept of fractions using transfinite numbers as numerators and denominators? If yes, what is the name for these kind of fractions? Or is there a reason why one shouldn't use something like this.
A simple example of such a fraction would be a fraction where the numerator is an infinte sequence of 1s and the denominator is an infinte sequence of 2s.
A more complex example would be a fraction where the numerator would consist of the decimal places of Pi and the denominator would consist of the decimal places of 2^0.5.