Which real numbers have two representations?

Question

Are they only numbers that end with 9999... and 0000... after the dot or some other too? If so, can you give an example?

Answer 1

Any rational number whose decimal expansion terminates, i.e., numbers of the form $$\dfrac{p}{2^m 5^n}$$ where $p,m,n \in \mathbb{Z}$ can have two different representations. For instance, $$\dfrac75 = 1.4 = 1.3\bar{9}$$

Answer 2

The only real numbers with two decimal representations have their decimal representations agree up to some point, then one continues with $a999\ldots$ while the other with $b000\ldots$, where the digit $b$ is one more than the digit $a$. For example, $$5.5679999\ldots = 5.568000\ldots$$ Another example: $$34199.9999\ldots = 34200.0000\ldots$$