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This question is related with Does Pi contain all possible number combinations?. More specifically, I want to know if $\pi$ contains 1234567890. I checked this link https://www.facebook.com/notes/astronomy-and-astrophysics/what-is-the-exact-value-of-pi-%CF%80/176922585687811 and did not see it there. I think that $\pi$ does not contain 1234567890. It is true or not. If it is true, how to prove it?

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closed as too broad by Andres Caicedo, Danny Cheuk, Davide Giraudo, Brian Rushton, TMM Jun 28 '13 at 15:57

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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There is no good reason whatsoever to think $\pi$ does not contain that sequence. –  Gerry Myerson Jun 28 '13 at 12:29
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@Arjang, most real numbers contain it. –  Gerry Myerson Jun 28 '13 at 12:36
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You've checked 0% of the digits of $\pi$, and from that you want to draw a conclusion? Infinity is big... –  Thomas Andrews Jun 28 '13 at 12:40
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Did you read the answers to the question you linked? It is not known if $\pi$ is normal or disjunctive, so I'd say with near certainty that nobody here will be able to prove that $1234567890$ does not appear in the decimal expansion of $\pi$. I'm even more certain that nobody here will want to prove that it does, as this will involve searching through a ridiculous number of digits (if the string does appear). –  Cameron Buie Jun 28 '13 at 12:41
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Who knows. You can search the first 200 million digits here and see that those do not contain that particular string. "12345678" does, though. The decimals of pi have been calculated to ridiculous precision, so if you want, you could load up one of those pieces of software and go looking. (From that page: "If the digits were stored in an uncompressed ascii text file, the combined size of the decimal and hexadecimal digits would be 16.6 TB.") –  fuglede Jun 28 '13 at 12:47

1 Answer 1

The nature of most real numbers is that, in any base, you can find any sequence of digits infinitely many times. The definition of "most" is technical, but rigorous.

We don't know if $\pi$ has this property, but we don't know it doesn't. It appears to have this property in base $10$, but we can't prove it, yet, and "appears" is always a bit of nonsense when we are saying, "We've checked the first $N$ examples out of infinity."

So, as Cameron commented, you are not going to find anybody here who is going to be able to prove that it doesn't occur, since, if we could, we'd have answered a long unresolved question.

If $\pi$ acted like a string of random digits, then you'd expect to have to check on the order of $10^{10}$ or $10$ billion digits before you found $1234567890$. If you tested $1$ trillion digits and still didn't find this sequence, I'd be shocked. But I don't know where you can download $1$ trillion digits of $\pi$...

In the first 1 billion digits of $\pi$, I found two instances of $123456789$, but no instances of $1234567890$.

Here's a simple example. In the first billion digits, there were $10049$ instances of $12345.$ There were $969$ instances of $123456$. There were $97$ instances of $1234567$. There were $9$ instances of $12345678$. And there were two instances of $123456789.$ If the digits of $\pi$ were random, we expect that approximately one tenth of the instances of $123456789$ in any sample would have next digit $0$.

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+1, side note, if each digit is 1 byte (?), then one would need a terrabyte drive to download 1 trillion digits(?) ( (?) stands for signifying a shot in the dark try) –  Arjang Jun 28 '13 at 13:08
    
There is a highly up voted question on SO about computing pi, with an answer by Mystical who reports on computing 10 trillion digits of pi. More info can be found at Numberworld.org (posting from my phone so linking is inconvenient). –  hardmath Jun 28 '13 at 13:17
    
I found a site that used to host the previous record of $5$ trillion digits, but they had to take them down due to bandwidth issues. That appears to be the same site you mentioned with the $10$ trillion digits, @hardmath –  Thomas Andrews Jun 28 '13 at 13:32
    
The link to SO that @hardmath mentioned: stackoverflow.com/a/14283481/7061 It covers how to compute $\pi$, rather than properties of $\pi$. –  Thomas Andrews Jun 28 '13 at 13:38
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Well, I was being very conservative suggesting a trillion digits - we'd actually expect to see his sequence $100$ times in a trillion digits. Just wanted to give an extreme cutoff - if he hasn't found the string in that period, he might actually have something :) –  Thomas Andrews Jun 28 '13 at 13:51

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