I know that any rational number has a repeating decimal and therefore the number 0.1011011101111... cannot be rational, however, I don't know the proof of that claim--and besides, I'm curious if there is some particularly easy proof in the case of this particularly simple-looking number. But nothing comes to mind. I've thought about doing slightly familiar tricks like multiplying by 10 and subtracting something but that doesn't seem to lead anywhere.
[Edit: One thought that occurs to me is that this number looks in some sense self-similar. I wonder if we can split it into self-similar pieces...]
[Further thought: If this number were in binary it might be easier to make a proof and then generalize it to other radixes.]