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### N digit sequence within π

Posted: Thu Aug 11, 2016 2:59 pm UTC
I have asked this question to a few of my mathematics professors at* uni and have gotten different answers from all of them. Please excuse my lack of rigor, I have not gone beyond calculus 2.

Take an n digit long sequence of numbers within π.

For a 2 digit sequence, intuitively, given the properties of π every possible permutation exists within π. But is there a limit? Or does every n digit long sequence of permutations reside within π?

One professor told me that she thinks yes because π is infinite.
Another professor told me he thinks no because π is chaotic.
I asked a friend of mine and he said that I should read up on the probability distribution of the digits within π and I did, but the jury is still out on that one.

edit: mispelling

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 3:04 pm UTC
I'm pretty sure this is an open problem? It smells like one.

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 3:17 pm UTC
Also, this smells a lot like "if there's an infinite number of universes surely there's one where anything can happen", which is not how probability works. Take, for instance, the following number:
0.101001000100001...
(basically a single 1 and then more and more 0s). This number has infinite digits, it never repeats, I wouldn't be surprised if it's irrational. But you'll never find a "2" in there. Granted, this is not the same as pi, and I suppose this rant isn't directly related to the original post, but it's something that really bothers me with how some people think about infinity.

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 3:30 pm UTC
https://en.wikipedia.org/wiki/Normal_number
Pi is conjectured (but not proven) to be Normal - that all finite sequence of digits are equally likely to occur (in any base expansion).

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 3:37 pm UTC
Xenomortis wrote:https://en.wikipedia.org/wiki/Normal_number
Pi is conjectured (but not proven) to be Normal - that all finite sequence of digits are equally likely to occur (in any base expansion).

I was about to post that too. Pi is conjectured to be normal, because almost all real numbers are normal. That means that pi would have to be exceptionally special to not be normal, and there is no reason to think it is special in that way.

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 3:39 pm UTC
Well I don't know if that's true. There's one very specific special thing about pi, and about any other number ever used by humans in math - it's a number humans have used in math.

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 3:42 pm UTC
doogly wrote:I'm pretty sure this is an open problem? It smells like one.

At the very least the normality of π is unknown. If true, it would entail that any finite sequence of n digits in base b has a probability b^-n of occurring at every place in the number, which would entail that a chosen sequence occurs at least once almost surely somewhere in π.

But until there's an algorithm to generate digits of π from previous digits, there's no way to prove that every finite sequence of digits (or, I guess, any infinite subset) will occur in π.
You can do a proof by example for any sequence, though.

As explained below, if true, it would entail that every finite sequence would occur.[/edit]

pseudoedit: dammit, that's a lot of ninja'ing

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 3:59 pm UTC
Zohar wrote:Also, this smells a lot like "if there's an infinite number of universes surely there's one where anything can happen", which is not how probability works. Take, for instance, the following number:
0.101001000100001...
(basically a single 1 and then more and more 0s). This number has infinite digits, it never repeats, I wouldn't be surprised if it's irrational. But you'll never find a "2" in there. Granted, this is not the same as pi, and I suppose this rant isn't directly related to the original post, but it's something that really bothers me with how some people think about infinity.

Yeah, learning about convergent and divergent series got me out of that mind set of "infinite" means everything.

jaap wrote:
Xenomortis wrote:https://en.wikipedia.org/wiki/Normal_number
Pi is conjectured (but not proven) to be Normal - that all finite sequence of digits are equally likely to occur (in any base expansion).

I was about to post that too. Pi is conjectured to be normal, because almost all real numbers are normal. That means that pi would have to be exceptionally special to not be normal, and there is no reason to think it is special in that way.

That's the problem with conjectures, they're nothing more than an educated guess. Ramanujan had a lot of conjectures and I think most of them ended up being proven, but philosophically, you cant just build off of something you only think is true. But is math constructed or discovered?

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 4:23 pm UTC
Zohar wrote:Also, this smells a lot like "if there's an infinite number of universes surely there's one where anything can happen", which is not how probability works. Take, for instance, the following number:
0.101001000100001...
(basically a single 1 and then more and more 0s). This number has infinite digits, it never repeats, I wouldn't be surprised if it's irrational. But you'll never find a "2" in there. Granted, this is not the same as pi, and I suppose this rant isn't directly related to the original post, but it's something that really bothers me with how some people think about infinity.

A better thing to say is that you'll never find a "11" sequence in it, or a "1<some number of zeros>1<same number of zeros>1", or a decreasing number, or any increase in number other than only one additional zero.

That you'll never find a "2" in that sequence is a further abstraction to the definition (although less abstractions than saying you'll never find a "¶" in pi).

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 6:40 pm UTC
Flumble wrote:At the very least the normality of π is unknown. If true, it would entail that any finite sequence of n digits in base b has a probability b^-n of occurring at every place in the number, which would entail that a chosen sequence occurs at least once almost surely somewhere in π.

I'm not sure in what sense we can use probabilities here?

From your link, a normal number is always rich: its expansion contains every finite sequence.

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 7:26 pm UTC
Zohar wrote:This number has infinite digits, it never repeats, I wouldn't be surprised if it's irrational.
Such a number is most definitely irrational, as all rationals repeat. (Determining whether it's transcendental or algebraic is harder.)

Pi has been calculated to a bit past 10^13 consecutive digits, so I imagine every 11-digit sequence appears multiple times (or else it would be evidence against normality), but beyond that it's hard to say, since as mentioned it's still an open problem.

(And yes, of course pi is special in the sense that we use it for math, but that type of specialness is unrelated to the specialness of not being normal, which is what jaap was talking about.)

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 7:31 pm UTC
Yes, pretty much immediately after posting this I realized that of course it's irrational.

As for the specialness, yes, I wouldn't be surprised if it's true and pi is a normal number. It was more of a joke or silly philosophical thought on how humans choose numbers that are of interest to them. It's kind of like the "all natural numbers are interesting" proof by induction.

### Re: N digit sequence within π

Posted: Thu Aug 11, 2016 7:44 pm UTC
Meteoric wrote:
Flumble wrote:At the very least the normality of π is unknown. If true, it would entail that any finite sequence of n digits in base b has a probability b^-n of occurring at every place in the number, which would entail that a chosen sequence occurs at least once almost surely somewhere in π.

I'm not sure in what sense we can use probabilities here?

From your link, a normal number is always rich: its expansion contains every finite sequence.

Oops, you're right, there's no probability involved. (I've never read the part about disjunctiveness/richness)

π is disjunctive in base b → every finite sequence of digits appears in π in base b