To convert an infinitely recurring decimal to a fraction looks reasonably easy, e.g.:
let n = 0.222... (i)
then 10n = 2.222... (ii)
therefore 9n = (ii) - (i) = 2 ; so n = 2/9
Thats why they say: if it's a single digit recurring, just put it over 9. But this means that 0.999... = 9/9 = 1!
I can see that, for all practical purposes, this is so, but surely if we're being rigorous 0.999... is not 1; it never quite gets there.
This make suspect that the whole system of conversion is suspect. I even think that a recurring decimal didn't ought to be classed as a "rational" number because it can't properly be expressed as a fraction. My thinking must be flawed. Please help me before I go mad.