Prove that rational numbers repeat the same finite sequence of digits endlessly, and that irrational numbers do not.
When you divide a by b there are only b possible remainders.
This means that after at most b steps in your long division the remainder must repeat itself.
Since a and b are integers, once your division gets past the decimal point, you always bring down a 0 in your division procedure.
If you always bring down a 0, the result of your division beyon any point depends only on the remainder at that point.
Thus once a remainder is repeated past the decimal point, the sequence of remainders between the repeat will continue repeating endlessly.
1/11 is .909090 ... or .(09)*.
1/7 is .142857142857142857 ... or .(142857)*.
If the decimal expansion of a number x repeats endlessly, as .(z)* for some sequence z that is k digits long, we can write
which is a rational number. If you have problems following this manipulation do not worry. We will talk about it later.