# About decimal expansion of Pi

• Jul 10th 2009, 06:27 PM
Guilherme
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,
• Jul 10th 2009, 06:47 PM
arbolis
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.
• Jul 10th 2009, 06:58 PM
AlephZero
I believe it is equivalent to asking whether or not pi is a "normal" number, which is still unproven....
• Jul 10th 2009, 07:09 PM
halbard
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 $\displaystyle \sum_{n=1}^\infty\frac1{10^{n!}}=0.110001000000000 000000001000.....$ is known to be trancendental. Do you thing its decimal expansion contains the number $\displaystyle 42$? And yet it is bound to contain the integer $\displaystyle 10^{100}$ somewhere.

How confident are you that the decimal expansion of $\displaystyle \pi$ contains the integer $\displaystyle 10^{100}$ somewhere? Very confident? Then I have a used car that you might be interested in ...
• Jul 10th 2009, 07:10 PM
Guilherme
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 + ... + $\displaystyle \sum_{i=1}^k k$ + ...

where sum operation denotes concatenation.

It contains all natural numbers, but is that a "normal" number?
• Jul 10th 2009, 08:14 PM
arbolis
Quote:

Originally Posted by halbard
It is not enough. The number $\displaystyle \sum_{n=1}^\infty\frac1{10^{n!}}=0.110001000000000 000000001000.....$ is known to be trancendental. Do you thing its decimal expansion contains the number $\displaystyle 42$? And yet it is bound to contain the integer $\displaystyle 10^{100}$ somewhere.

How confident are you that the decimal expansion of $\displaystyle \pi$ contains the integer $\displaystyle 10^{100}$ 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 $\displaystyle 10^{100}$ appears somewhere in the decimals... if the digits are really "randomly" distributed (I'm not sure I making sense here) then $\displaystyle 10^{100}$ should appear.
I find this article to be interesting : Infinite monkey theorem - Wikipedia, the free encyclopedia.
• Jul 11th 2009, 12:13 AM
CaptainBlack
Quote:

Originally Posted by halbard
It is not enough. The number $\displaystyle \sum_{n=1}^\infty\frac1{10^{n!}}=0.110001000000000 000000001000.....$ is known to be trancendental. Do you thing its decimal expansion contains the number $\displaystyle 42$? And yet it is bound to contain the integer $\displaystyle 10^{100}$ somewhere.

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

Quote:

Originally Posted by AlephZero
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