What points in $[0,1)$ will have two binary expansions?

Question

What points in $[0,1)$ will have two binary expansions?

I know that $\frac{1}{2}$ has the two expansions $0.1\bar{0}$ and $0.0\bar{1}$

$0.1\bar{0}$ is found by starting with $\frac{1}{2}$ and finding the binary expansion

$0.0\bar{1}$ is found by working backwards from the expansion to find the fraction.

But how do I go about finding other fractions that have two decimal expansions?

Answer 1

Partial answer: Any nonzero rational in $[0,1)$ with a terminating representation has two binary expansions.

Suppose $x\in (0,1)$ is rational, $x\ne 0$, and $$ x = 0.d_1 \dotsm d_n $$ for binary digits $d_i$. As $x\ne 0$, we have $d_n = 1$: $$ x = 0.d_1 \dotsm d_{n-1} 1 $$ But now it's easy to see and show that $$ x = 0.d_1 \dotsm d_{n-1} 0 \overline{1} $$ is another representation of $x$.

Answer 2

As BrianO explains in his answer, any rational number that can be written with a terminating binary expansion has two distinct binary representations: one in which the binary representation terminates in a $1$, and one in which the binary representation terminates in $0\overline{1}$.

So the question in the OP reduces to: which rational numbers in $[0,1)$ can be written as a terminating binary expansion?

The answer is: Any rational number which, when written in lowest terms, has a denominator that is a pure power of $2$.