Does 1/3 have a unique decimal representation?

Question

I think it does, but I’m not sure. And also there are rationals which have unique decimal representation besides irrational numbers. Am i right?

Answer 1

Yes, you are right. The only real numbers with more than one decimal representations are those that can be written as $\frac k{10^n}$, with $k\in\mathbb Z\setminus\{0\}$ and $n\in\{0,1,2,\ldots\}$. Those that I described have exactly two decimal representations. For instance, $\frac{31}5\left(=\frac{62}{10}\right)$ has two decimal representations:$$6.2\text{ and }6.1999999\ldots$$

Answer 2

Any rational that repeats digits that are not all $9$ and not all $0$ has a unique decimal representation. The issue is when the decimal expansion terminates, in which case you can decrease the last digit by $1$ and append an infinite string of $9$s.