TITLE: Are there any other numbers like $0.999\ldots$?
QUESTION [4 upvotes]: In a manner similar to how the value $1$ can be represented as $0.(9)$ too, are there any other values that exhibit this property when represented in base 10?

REPLY [2 votes]: I seem to recall reading somewhere (therefore it's true!! (?)) that Johannes Kepler proposed a base-3 numeral system with three digits: $0$, $1$, and $-1$.  In that system, the number $1/2$ can be represented in two different ways:
$$
1.,\ -1,\ -1,\ -1,\ \ldots,
$$
and
$$
0.,\ 1,\ 1,\ 1, \ \ldots\ .
$$
And similarly for every binary rational number (i.e. rational number whose denominator is a power of $2$).
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