TITLE: Which real numbers have $2$ possible decimal representations?
QUESTION [5 upvotes]: I know that all positive and negative whole numbers have $2$ possible decimal representations. For example, $1+1+1+1$ could be represented as $4$ or as $3.99999...$ (I believe $4.000..1$ isn't a thing, right?). 
All terminating decimal numbers have $2$ representations as well, for example $1.5$ is the same as $1.4999..$ However, I can't really see how you would make a second representation for non-terminating decimals (especially irrational numbers) and zero.

REPLY [3 votes]: A decimal representation represents the number that the corresponding series converges to. For a number $x$ whose standard representation doesn't terminate, using any digit other than the one in its standard representation at any point in the series would make it impossible for the series to converge to $x$: If a greater digit is used, the partial sum is greater than $x$, and the remaining terms are all positive and hence can't make the series converge to $x$; and if a smaller digit is used in the $n$-th digit after the decimal point, the result is less than $x$ by more than $10^{-n}$, and the remaining terms can add up to at most $10^{-n}$, so again the series can no longer converge to $x$. Thus each digit is uniquely determined by $x$ unless the standard representation terminates.