Is it possible to prove that there exists a rational number with repeating decimal digits in base-10 representation that isn't repeating in binary?
For example, $0.\overline{0011}_2$ is a binary representation of $0.2_{10}$ which contains repeating digits to the right of the decimal point.
I'm wondering if there exists some $A \in \mathbb{Q}$ in which $A_{10}$ contains repeating decimal digits, but $A_2$ doesn't.
I'm asking this out of curiosity, this is not homework.
Best Answer
If "repeating" refers to "non-terminating", such numbers do not exist. Any non-repeating or terminating number in binary has the form $a/2^n$ for some integral $a,n$. Multiplying top and bottom by $5^n$ yields $5^na/10^n$, showing that the number is terminating in base 10 as well. Hence, by contraposition, any number repeating in decimal is also repeating in binary.