The reason you don't see two zeroes is as follows: you have
.123456789
then add 10 on the end, as the tenth digit after the decimal point, to get
.123456789(10)
where the parentheses denote a "digit" that's 10 or larger, which we'll have to deal with by carrying to get a well-formed decimal. Then carry twice to get
.12345678(10)0
.1234567900
So for a moment we have two zeroes, but now we need to add 11 to the 11th digit after the decimal point to get
1/81 is 0.012345679012345679....
no 8 in sight