About decimal expansion of Pi

Guilherme · #1 $Old$ July 10th, 2009, 08:27 PM

Hi all,

I'd like to know if all natural numbers can be found in some part of the decimal expansion of pi.
For example in 3.141592 we can find 14,15,92,1592.

The question is a little weaker than the concept of normal numbers, which I heard it's not known if pi is such. So If anyone ever heard about it, please share with us.

Regards,

arbolis · #2 $Old$ July 10th, 2009, 08:47 PM

Quote:

I'd like to know if all natural numbers can be found in some part of the decimal expansion of pi.

I believe so. It is not periodic while it is transcendental. Maybe that's enough to conclude.

AlephZero · #3 $Old$ July 10th, 2009, 08:58 PM

I believe it is equivalent to asking whether or not pi is a "normal" number, which is still unproven....

halbard · #4 $Old$ July 10th, 2009, 09:09 PM

Quote:

Originally Posted by arbolis:

I believe so. It is not periodic while it is transcendental. Maybe that's enough to conclude.

It is not enough. The number $\sum_{n=1}^\infty\frac1{10^{n!}}=0.110001000000000000000001000.....$ is known to be trancendental. Do you thing its decimal expansion contains the number $42$ ? And yet it is bound to contain the integer $10^{100}$ somewhere.

How confident are you that the decimal expansion of $\pi$ contains the integer $10^{100}$ somewhere? Very confident? Then I have a used car that you might be interested in ...

Guilherme · #5 $Old$ July 10th, 2009, 09:10 PM

Quote:

I believe it is equivalent to asking whether or not pi is a "normal" number, which is still unproven....

But what if I create a rational number like:

0. + 1 + 2 + 2 + 3 + 3 + 3 + ... + $\sum_{i=1}^k k$ + ...

where sum operation denotes concatenation.

It contains all natural numbers, but is that a "normal" number?

arbolis · #6 $Old$ July 10th, 2009, 10:14 PM

Quote:

Originally Posted by halbard $View Post$

It is not enough. The number $\sum_{n=1}^\infty\frac1{10^{n!}}=0.110001000000000000000001000.....$ is known to be trancendental. Do you thing its decimal expansion contains the number $42$ ? And yet it is bound to contain the integer $10^{100}$ somewhere.

How confident are you that the decimal expansion of $\pi$ contains the integer $10^{100}$ somewhere? Very confident? Then I have a used car that you might be interested in ...

Ok, good to know and good point. We cannot conclude.
However I'm confident (by intuition now, I'm not affirming) that the number $10^{100}$ appears somewhere in the decimals... if the digits are really "randomly" distributed (I'm not sure I making sense here) then $10^{100}$ should appear.
I find this article to be interesting : Infinite monkey theorem - Wikipedia, the free encyclopedia.

CaptainBlack · #7 $Old$ July 11th, 2009, 02:13 AM

Quote:

Originally Posted by halbard $View Post$

It is not enough. The number $\sum_{n=1}^\infty\frac1{10^{n!}}=0.110001000000000000000001000.....$ is known to be trancendental. Do you thing its decimal expansion contains the number $42$ ? And yet it is bound to contain the integer $10^{100}$ somewhere.

How confident are you that the decimal expansion of $\pi$ contains the integer $10^{100}$ somewhere? Very confident? Then I have a used car that you might be interested in ...

Quote:

Originally Posted by AlephZero $View Post$

I believe it is equivalent to asking whether or not pi is a "normal" number, which is still unproven....

When replying to a post, please quote that post. As it is the reader has to work at deducing what you are referring to. Remember a forum such as MHF is not a conversation between you and the previous poster.

CB