How many decimal representations are possible for the number 1

Question

I know that there at least two

$0.\overline{9}$ and 1

Is there a neat and more concrete way to understand this problem.

Answer 1

Assuming that you don't allow leading zeros, and that we identify all the representations with trailing zeros (e.g. $1.0, 1.00, 1.\overline0$, etc.), then these are the only two possible.

Certainly if the digit before the decimal point is $1$, then there can not be any non-zero numbers in the expansion; for example, if we had the digit $a$ appear at some point (with $a \in \{1, ..., 9\}$), then

$$1.0000...0a... \ge 1.0000...0a > 1$$

Likewise, if the representation starts with $0.9$, then no non-$9$ digits may appear; the reasoning is essentially the same: the existence of a digit smaller than $9$ implies that the number is too small, since

$$0.\underbrace{999...9}_{n}a... \le \underbrace{999...9}_{n}9 < 1$$