I don't believe this is true. Consider the following number, whose name
I forget:
.123456789101112131415161718192021222324...979899100101102103104105...
This is created by writing the series 1,2,3,4,5,6,7,8,9,10,11,12,13,...
and concatenating all the digits. It is known to be an irrational number.
Clearly the number of digits in it is countably infinite, as with any
decimal. Yet you will not be able to convert it into a fraction.
Hal