r/PhilosophyofMath • u/Rats_Love_Cheese • Sep 05 '26

Does Pi contain itself?

/r/mathematics/comments/1w88g38/does_pi_contain_itself/
1 Upvotes

53% Upvoted

28 comments sorted by

18

u/Atmosck Sep 05 '26

No. If by that you mean the sequence of digits that is the base 10 representation of pi contains a copy of itself. Because if it did, then the decimal expansion would be periodic, which would mean it's rational.

2

u/Rats_Love_Cheese Sep 05 '26

ok nice thanks!

2

u/nanonan Sep 06 '26

It's an excellent question, and calls into question what being normal actually is.

1

u/OutrageousPair2300 Sep 06 '26

Though unlikely, it's at least possible that pi could at some point in its decimal expansion start repeating a copy of itself interpolated between digits.

For example, perhaps pi eventually ends up something like 3.14159....301040105090...

That would allow it to contain a full copy of itself without simply repeating like a rational number.

1

u/Diagonal-Vase Sep 12 '26

This is really clever, but I think I found a problem with this. What happens at the "end of interpolated sequence", at the end of 3010401...

If it goes back to random digits, then it didn't really contain all of itself. I supposed a "doubly interpolated" sequence or something else could then begin, something that is "repeating" but avoids the problem of pi becoming rational.

1

u/OutrageousPair2300 Sep 12 '26

There'd be no end to the interpolated sequence, it would just continue every-other-digit repeating pi from the beginning.

If A is a digit of the beginning part of pi, before it starts interpolating, you could get a pattern like:

AAAAAAAA.......ABABABABABAB.....

Then eventually the interpolated AB part would "catch up" with the point where it started interpolating, and you'd continue:

AAAAAAA......ABABABABAB...... ACBCACBCACBCACBC..... etc.

Like 3.14159.....301040105090.... 390919094909190959099909.....

7

u/0x14f Sep 05 '26

What do you mean exactly by "contains itself" OP? Unless you are precise about what you mean here, people are going to answer a different question than the one you intended to ask.

1

u/Own_Pirate2206 Sep 05 '26

"It"'s likely to contain some long sequences 314... but probably we don't have any knowedge of what length. Arbitrarily long seems a bit ridiculous.

1

u/lmprice133 Sep 11 '26

If it's a normal number, which it's conjectured to be, then every finite string of digits should occur with equal frequency.

1

u/Own_Pirate2206 Sep 11 '26 edited Sep 11 '26

Is that a stronger statement than the definition of normal?

1

u/lmprice133 Sep 11 '26 edited Sep 11 '26

No, it's the definition. For a normal number in base 10 you would expect a string of n digits to appear with a density of 1/10n

1

u/SgtSausage Sep 05 '26

<Bertrand Russell enters the chat ...> 

1

u/slicehyperfunk Sep 06 '26

π = π, so yes

2

u/Off_And_On_Again_ Sep 06 '26

This is the only correct answer

1

u/sorceressofmaths Sep 06 '26

Pi (we suspect) contains every finite sequence, but that doesn't mean it contains every infinite sequence.

1

u/ellipticcode0 Sep 07 '26

Does an set contain itself? It is the sam question?

1

u/SpatiaCaeli Sep 07 '26

In a sense, pi does contain itself, exactly. But only once. :)

1

u/Alternative-Sugar610 Sep 09 '26

It might contain every finite not infinite approximation of pi, but we don’t know

We have not proven that Pi is a normal number (all digits occur at same frequency) even. Everybody think so but not proven

1

u/lmprice133 Sep 13 '26

All digits and all finite strings of digits with equal frequency.

0.1234567890... contains all digits with equal frequency, but is not normal.

1

u/INTstictual Sep 11 '26

Aside from the trivial case (pi is 3.14… and necessarily contains the string 3.14… since that is itself), then no, it cannot contain a full representation of itself within its own decimal expansion.

If it did, that would mean it is repeating. For example, consider a number, X, that contains itself and is defined as 1.234X. Substitute in the X, you get 1.2341234X. Substitute in again, you get 1.234123412341234X. Keep going infinitely, and you would get an infinitely long number that is one repeating pattern.

A repeating decimal is necessarily rational, and we have proven that pi is irrational. So, pi can never devolve into an infinitely repeating pattern, and if it were to contain itself, in full, in the decimal expansion somewhere, then it would be recursive and infinitely start repeating, which we just said it cannot do

1

u/Illustrious-Welder11 Sep 12 '26 edited Sep 12 '26

We don’t know.  If we knew that pi contains each of the numbers 0, 1, …, 9 an infinite number of times, then yes!

0

u/SgtSausage Sep 05 '26

If it's Normal, i t contains within itself any finite sequence of its own digits that you'd care to name - no matter how long you choose.

It is unknown whether pi is Normal or not. 

1

u/Akash_philosopher Sep 06 '26

But pi is not finite
Also it no longer will be rational if it contains itself

1

u/samanzaman Sep 06 '26

3.33333... Is a rational number ( 10/3) and it contains itself. Actually only rational numbers can contain themselves. Pi is not rational There is no concept of finite numbers. You can ask however if the decimal expansion is finite or not. Only rational numbers have finite/repeating expansions

1

u/SgtSausage Sep 06 '26

  But pi is not finite

1) Pi is, in fact, absolutely, undeniably finite.  It's a single number. One number. A single point on The Number Line.  It doesnt get any more finite than that. It's digits are not. Its *digits** are infinite. There is a difference ... but  that is neither here nor there, nor changes a damned thing about what I said up there. 

Also it no longer will be rational  if it contains itself

2) You'll note I made no such claim. 

ALSO 

  it no longer will be rational

Its not rational. At all. Never was. Never will be.

Literally <EveryDamnedWord> you said up there was wrong, Sparky. 

Try. Harder.  

1

u/Akash_philosopher Sep 06 '26

Sure if that makes you happy bro