Can a real number $x \in (0,1)$ have more than different 2 binary expansions?

Question

I know that a real number $x \in (0,1)$ can have 2 different binary expansion. e.g 0.1 = 0.011111.... But is there any real number for which there are more than 2 different binary expansions?

Answer 1

No. Every terminating binary expansion has exactly one companion with an infinite string of 1's. Every other number has a unique binary expansion. In fact, you can define the real numbers by these two properties. A similar approach in the context of decimals was already developed in the 16th century by Simon Stevin. The first record we have of a mathematician aware of the non-uniqueness of decimal representation seems to be in Euler.