(a)

For any non-negative integer observe that

Thus, if we have

There are infinitely many non-negative integers and hence infinitely many such rationals

(b)

Note that for any Hence, in order to have we must have If is a positive non-integer rational, would have more decimal places than This would mean that their fractional parts cannot add up to 1. Hence there are no positive rationals such that

Remark: There do however exist positiveirrationalnumbers with To see this, note that is continuous on the interval and while the result follows by the intermediate-value theorem.