According to this question, the binary expansion of any rational number is always either terminate or repeated. And there are some irrational numbers in decimal with an irrational binary representation. For instance let us consider $\sqrt {2}$.
$$\sqrt {2}=(1.41421356237309504880\dots)_{10}\\=(1.01101010000010011110…)_2$$
My question: Is there any irrational number in decimal with a ratinal binary expansion?
Best Answer
No.
If a number is rational, then the expansion in any numeral system will repeat, and if it is irrational, then it will not repeat in any system.
The proof given in the question you linked is independent of the numeral system.