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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.]

Addem
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    You proved it when you said "I know that any rational number has a repeating decimal and therefore the number 0.1011011101111... cannot be rational" – Kaynex Mar 25 '17 at 05:24
  • You are saying you do not know the proof of the fact that rational numbers must have repeating decimals? Is that the question, or are you looking for alternate proofs? – Sarvesh Ravichandran Iyer Mar 25 '17 at 05:24
  • @астонвіллаолофмэллбэрг Yeah, I'm hoping for an alternate proof, if there's one that's simpler than the proof that all rational numbers have repeating decimals. – Addem Mar 25 '17 at 05:25
  • @Addem Proving that all rational numbers must have repeating decimals is very simple and I doubt you're going to get any simpler. – Kaynex Mar 25 '17 at 05:30
  • do you want to know how to prove that the number you are quoting is not periodic(eventually)? – clark Mar 25 '17 at 05:44
  • It's a non-repeating decimal, so it is irrational. It doesn't get much simpler than this... – suomynonA Mar 25 '17 at 05:54
  • @suomynonA the complication comes from proving that non-repeating decimals are irrational... – Addem Mar 25 '17 at 06:03
  • Still, there is no easier way. – Ivan Neretin Mar 25 '17 at 13:34

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