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Suppose that we travel along the digits of $\pi$ and when we arrive at $2$ or $3$ or $4$ or $5$ or $6$ or $7$ or $8$ or $9$ we replace those digits with the number $1$ and when we arrive at $1$ or $0$ we leave them unchanged. When we do this with all the digits will resulting number be irrational?

Since this looks quite hard you can, if you want, assume that $\pi$ is normal in order to prove this. But I do not know is normality enough?

(The statement that we can suppose that $\pi$ is normal was not in the original version of the question.)

Living Hell
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    A random sequence of $0, 1$ will make an irrational number. It's unlikely that $\pi$ has some property that avoids this. – Kaynex May 03 '17 at 21:51
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    I've never heard of anyone investigating this property personally, however it is widely expected (but not proven) that $\pi$ is a normal number, which should imply things about lengths of sequences of zeroes in its digits which should in turn imply the irrationality of the number you describe. – JMoravitz May 03 '17 at 21:52
  • Not much is known about the decimals of pi, and I suspect this would be an open problem as well. – TMM May 03 '17 at 21:52
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    It's unknown if the digits of $\pi$ are uniformly distributed: https://math.stackexchange.com/questions/51829/distribution-of-the-digits-of-pi – D Wiggles May 03 '17 at 21:53
  • I would describe it as something of a miracle that we are able to prove that $\pi$ is irrational, because even that is very difficult for most numbers. – Arthur May 03 '17 at 21:57
  • @Arthur $\pi$ was proven to be irrational long ago. – Eff May 03 '17 at 21:59
  • @Eff I'm not saying it's difficult in the case of $\pi$. I'm saying it's impossible for most real numbers, and we were lucky with $\pi$. – Arthur May 03 '17 at 22:02
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    @Eff ... which doesn't contradict Arhur. Miracles have the general tendency to have happened long ago ... – Hagen von Eitzen May 03 '17 at 22:02
  • @Arthur Ah, sorry. I read it as "[...] of a miracle if we are able to prove that $\pi$ is irrational". – Eff May 03 '17 at 22:03
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    So the question is, whether the zeros of $\pi$ appear periodically ... – user251257 May 03 '17 at 22:06
  • @projectilemotion I doubt we will ever know whether $\pi^e$ is irrational, for instance. Or, at the very least, it is a very long time into the future. – Arthur May 03 '17 at 22:06
  • @HagenvonEitzen comparing to the irrationality of $\sqrt 2$, it was pretty recent ... – user251257 May 03 '17 at 22:07
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    I believe it is not even known if there are infinitely many digits that are $0$. Clearly, if there were only finitely many digits $0$, it would be rational. – Thomas Andrews May 03 '17 at 22:15
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    Possibly related: https://mathoverflow.net/questions/265310/if-i-exchange-infinitely-many-digits-of-pi-and-e-are-the-two-resulting-num – Alp Uzman May 04 '17 at 13:19

1 Answers1

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So here is a question.

In a normal number, can the positions of the $0$s in the decimal be eventually periodic?

Answer: NO. In a normal number there are arbitrarily long strings of $7$s. (And also infinitely many $0$s.)

GEdgar
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