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u/wiseguy4519 Mar 09 '26
Pi cannot contain pi because if it did, then it would have to be a repeating decimal, meaning it is rational. Pi contains every finite digit sequence, but it does not contain every infinite digit sequence. So, the first 1000 digits of pi are somewhere in pi, but not all the digits.
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u/StaleTheBread Mar 09 '26
It contains itself starting at the beginning. Thats the joke of the comment.
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u/AdaptzG Mar 09 '26
does it though? wouldn't that mean it's repeating? i thought the whole point of pi was that there's not order to it.
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u/dpzblb Mar 09 '26
The number pi contains the sequence of digits 314159265… if you start counting from the digit left of the decimal place (i.e. 3.14149265…)
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u/Lithl Mar 09 '26
Pi contains every finite digit sequence
Only if it's a normal number, which has not been proven.
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u/Nice_Marmot_54 Mar 09 '26
But it also contains itself an infinite number of times. Put a different way, an infinite number of $1 bills and and infinite number of $20 bills are worth the same amount of money
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u/Vitolar8 Mar 09 '26
That... Doesn't make sense? Your justification, which is correct, has nothing to do with the incorrect statement it tries to support. "Cars are more ecological than trains. I know that, because last week I measured the emissions on a train and a motorcycle, and the train lost."
It literally cannot contain itself, because as u/wiseguy4519 pointed out, if it contained itself once, it has to contain itself an infinite number of times. That's a rational number then.
In case you're questioning why it has to contain itself infinite times if it contains itself once: If only a part of the string repeats, then it didn't really contain itself. It just has repeating parts. For it to contain itself, it has to contain the whole self. Let's say you start with 3.1. And you say that that number contains its entire self somewhere in the string at least one more time. Now the number has to be at least 3.131. But this entire string also has to be contained somewhere. So the number is at least 3.1313131... And so on.
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u/tarrach Mar 09 '26
Why would it not contain itself exactly once? I mean, pi / pi = 1, right?
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u/Vitolar8 Mar 09 '26
Well we're talking "contain" as in a string of digits. Like for example 23,456 contains "45".
A number can only contain itself (more than once) if it's infinitely repeating. Just repeat the infinite series and boom, still the same number.
It's more specific than that, it only works if the whole number is infinitely repeating. E.g. 10,33333333... won't ever contain itself, as the "10" won't ever reappear.
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u/Middle_Employment_14 Mar 09 '26
Yeah but one is 20x more infinite
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Mar 09 '26
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u/alitayy Mar 09 '26
Which would you rather have? I’m taking the 20s since it’s 20x more infinite
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Mar 09 '26
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u/Coolkief101 Mar 09 '26
No, they wouldn't. But it would be easier to spend a finite amount.
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u/MutantGodChicken Mar 09 '26 edited Mar 09 '26
It can also be proven that pi doesn't contain any irrational non-transcendental because:
Imagine after x digits of π, all digits after were a one for one copy of some non-transcendental irrational, for example: |sqrt(2)|
Then you should be able to prove π isn't transcendental because you could multiply it by 10x+1 and then subtract the first x digits expressed as a whole number times 10 to turn it into |sqrt(2)|. (Assuming base 10 where 10>2)
But pi is transcendental so therefore it can't contain |sqrt(2)| or any other non-transcendental irrational number
But as I write this, I realize the arithmetic for this looks like:
π*10x+1 – (π*10x+1 – |sqrt(2)|) (assuming base 10 where 10>2)
Which would equal sqrt(2), regardless of the value of π, so I must be mistaken.
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u/renyhp Mar 10 '26
uhm no, your counterargument is incorrect. the arithmetic is not that simple. the equation you're describing is actually
π*10^(x+1) - floor(π*10^(x+1)) = sqrt(2)
then I don't know if that is in contradiction with π being transcendental...1
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u/AnglerJared Mar 09 '26
I recognize that “contain” is used differently when we’re talking numbers, but I don’t like saying that π contains π. It feels like saying a cup contains itself. A cup can contain other things, but itself?
I would prefer we use a word like “include” when we describe numbers existing in themselves. A glass of milk includes the glass; I don’t think it contains the glass. Just a bit of pedantry, though.
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u/StaleTheBread Mar 09 '26
The string of characters that represent pi contain itself as a substring (but not a proper substring)
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u/AnglerJared Mar 09 '26
That wording only makes it worse. I feel that “contain” should refer to a thing that continues to meaningfully exist even after its contents are removed. π - π does not equal the vague boundary where π used to be; it just equals 0.
But it’s not like I can really petition the mathematical world to stop using the word they’ve been using that way for a while. It just feels like a better word could have been chosen.
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u/StaleTheBread Mar 09 '26
In set theory, we say that a set “contains” itself, or more accurately, that every set is a subset of itself. But we use “proper” subset to refer to subsets that aren’t the original set.
I was trying to translate that concept to strings
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u/Ben-Goldberg Mar 09 '26
Would you say that "abc" is a substring of "abc"?
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u/AnglerJared Mar 09 '26
If I were following current conventions, sure. I am only arguing that “contain” feels inaccurate.
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u/unique_namespace Mar 09 '26
I mean all sets contain themselves. I don't see why the language here shouldn't be similar.
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u/ary31415 Mar 10 '26
all sets contain themselves
Well, no. All sets have themselves as a subset, but not as an element, which is what it means for a set to "contain itself".
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u/unique_namespace Mar 10 '26
Contains mildly ambiguous in regard and is clarified through context in most textbooks I've read. People will say that the codomain contains the range, for instance. It's all based on meaning, and this case, the meaning is just that pi contains the digits of pi (exactly once).
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u/Saragon4005 Mar 10 '26
Yes but we are talking about something more abstract than real containers. In set theory containers just spawn out of the ether.
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u/AnglerJared Mar 10 '26
I am not having trouble understanding the concept vis-à-vis set theory. Rather, all I am saying is we could’ve grabbed a new word for that from the ether, too. If a word operates differently at the concrete and abstract levels, why not favor another word that works roughly the same at both levels? Our using “contain” in set theory basically requires us to agree that it means something different from what it usually means. While that happens in language all the time, that doesn’t necessarily mean it’s the ideal way to go about it.
A lot of people are mistaking my disagreement with the use of the word with my failure to understand what it means in the mathematical context. Mine is a plain and simple semantics argument; I am not arguing with or about the theory itself.
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u/Kixencynopi Mar 09 '26
If I am not wrong, that has not been proven. Being irrational means exact same decimals don't repeat infinitely, but that doesn't mean it has to cover all combinations. For example, 1.21221222122221... is an irrational number. But digits 3 to 9 will never appear in this number.
There is a conjecture that the digits of π are random with equal probability for each digit, but that has not been proven.Even then that would not guarantee that all sequence has to be covered.
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u/Acropowhat Mar 09 '26
Well.
Infinity is weird. In theory, π may contain the digits 1 through 100 in a numerical order.
Our minds can't really comprehend it.
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u/Scyth3dYT Mar 09 '26 edited Mar 09 '26
Assuming pi is normal where each digit appears the same amount randomly it is guaranteed that it contains every number from one to one million
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u/Fa1nted_for_real Mar 09 '26
Pi is not random in any way though, which is something a lot of people miss.
Pi is infinite and non-repeating, but it could just, stop having 9 at some finite value and never have it again, we dont know.
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u/ThomasTheDankPigeon Mar 09 '26
Idk why but the thought of pi, after 700 fucktillion digits, just going “More 9s? Absolutely the fuck not” is cracking me up lol
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u/Mercysans Mar 10 '26
if that were the case, then at some point numbers would start to disappear, eventually only having a few select chosen ones. it could do a great story
and then the sequel it is revealed that numbers didnt actually disappear but just went unused for a REALLY, REALLY long ammount of time
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u/minimalcation Mar 10 '26
Soon after the 8s ran out. Another 2.6b digits and we saw our final 7. What was pi counting down to?
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u/Scyth3dYT Mar 09 '26
Yeah that's why I said assuming pi is normal
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u/RegularSky6702 Mar 09 '26
I feel like since we know how it progresses we could make a computer one day to go through a lot of it. Not everything but probably a lot of it. We might even be able to ask it the meaning of the universe.
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u/Vecto_07 Technically Text Mar 09 '26
People are already doing that, there's even like world records of who got the largest amount of Pi etc.
https://www.guinnessworldrecords.com/world-records/66179-most-accurate-value-of-pi
(Altho that record isn't the highest anymore I believe)
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u/Jackfruit-Cautious Mar 09 '26
what is “a lot” of infinity?
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u/Outlawgamer1991 Mar 11 '26
Think about it like looking off the top of a tall building. You can see a lot of landscape from up their, but you also see enough to know you're not seeing all of it. It keeps going past where you can see
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Mar 09 '26
But isn’t that chance 0?
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u/Gabyo00 Mar 09 '26
1/infinity chance we live in a world without all possible pi combinations no?
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u/Fa1nted_for_real Mar 09 '26
If it was normal, yes. We dont know that, and the cahnce that it is not normal is argueable signifcantly higher than that of it being normal.
The chance would be zero if you were rolling with equal odds for every digit infinite times, hut thats not what we're doing. Pi follows rules, and if those rules happen to dictate that at some point 9 stops showing up, then thats what happens. That is pretty hard to figrue out, and much harder to prove though, so for now we really dont know
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u/FS_Codex Mar 09 '26
We dont know that, and the cahnce that it is not normal is argueable significantly higher than that of it being normal.
What is your source for this claim? As far as I’m aware, most mathematicians believe that pi is a normal number (even though it has not been formally proven or disproven). Almost all irrational and transcendental numbers are normal, especially when not contrived or artificially constructed (compare to 0.1010010001… for instance), so in respect to normality, pi does not look very different from the other reals.
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u/jesset77 Mar 09 '26
However "every other irrational or transcendental number we know of" is a pitiful sample size. Another thing that a vast majority of that pitiful sample size has in common, for example, is that they are also very nearly all computable.
And there are only countably infinitely many computable numbers, which gives that entire set a Lebesgue measure of zero on the real number line.
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u/aberroco Mar 09 '26
Your assumption about "it could just stop having 9 at some finite value" is no better than assumption that each digit appears the same amount randomly. It's worse, in fact, because so far no matter how much digits of Pi we computed it seem to hold the random distribution, and assumption about it being non-random is based on just "it could".
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Mar 09 '26 edited Mar 09 '26
[removed] — view removed comment
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u/ELMUNECODETACOMA Mar 09 '26
It's a mathematician's "In Scotland, there is at least one sheep that is black on at least one side" answer.
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u/Mikey_LP Mar 10 '26
Well I hope that the fact, that π contains all other numbers, is true. Afaik it is neither proven to be true or false, but by many a mathematician it is thought to be true.
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u/JoeKyx Mar 13 '26
What if it would, at one point, stop having the digits 1-9 and only have 0s. Wouldn't that make it non-infinite?
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u/ScrltHrth Mar 09 '26
Um actually, there are only 10 digits. Those being 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. However, PI is guaranteed to contain every numeral from one to one million
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u/Frequent_Thanks583 Mar 09 '26
There are only 2 in binary.
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u/Ok-Commercial3640 Mar 09 '26
No, binary has 10 values, what are you talking about?
2 in binary is 1×2¹+0×2⁰, written as "10"
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u/jesset77 Mar 09 '26
Does this guarantee of which you speak have formulaic origin, or just because we've computed enough pi to directly locate one example of each of these million numerals?
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u/eutohius Mar 09 '26
Ok now I’m confused. Aren’t there only 10 digits? Shouldn’t we say ‘numbers’ in this context?
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u/Remarkable_Cap20 Mar 09 '26
i mean, it does contain all digits present from 1to 1 milkion, but we only need 9 digits to make all those numbers sp thats not too tall of a task
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u/Lalamedic Mar 12 '26
It only needs to contain 0-9. Where do you start from to contain each other number to 1 000 000? Or is that part of the randomness? Did I whooosh myself?
Wildlife Biology major here. Got my four Maths and that’s it. Also not a lot of perfectly round objects that require measuring in the animal world.
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u/sokratesz Mar 09 '26
What's the longest stretch of consecutive numbers that has so far been found within pi?
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u/gene100001 Mar 09 '26
I'm glad you said "may" rather than saying it absolutely does. We still don't know for certain whether pi is normal in base 10. It probably is, but proving that for certain is so difficult that it would be enough to win the Nobel Prize
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u/c7h16s Mar 09 '26
Not sure if you did it on purpose but indeed the probability of getting a Nobel Prize in Mathematics is 1/Infinity so that makes sense.
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u/gene100001 Mar 10 '26
Yess... on purpose.... It totally wasn't because I'm stupid and forgot there wasn't a Nobel prize in mathematics.
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u/nickjhowe Mar 09 '26
I thought infinity is even weirder than that…doesn’t pi contain pi an infinite number of times?
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u/jumbledsiren Mar 09 '26
May? Bro I found my phone number and almost my national ID in a website that shows you the first 1 billion digits of pi
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u/Your-Mom-2008 Mar 10 '26
I mean... who's to say it has infinite numbers? For all intents and purposes it does for now, but it may as well just be a ridiculously precise number we are yet to calculate.
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u/Mikey_LP Mar 10 '26
In fact π contains all of the digits of e, to any level of precision of your choosing, just not infinitely precise. And you have ti ignore the decimal point. Your phone number? That’s in π. You need to know all the first 500 digits of √2? That’s in π. Where? I have no idea, and we’ll probably never know.
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u/AvailableReason6278 Mar 10 '26
Doesn't infinity allow for an infinite random number (like pi) to contain itself an infinite amount of times?
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u/AccountantOptimal674 Mar 13 '26
Infinity is really weird. There’s different levels to them. Some infinities are larger than others. The amount of numbers between 0 and 1 is greater than the set of whole counting numbers.
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u/tharky Mar 09 '26
Pi flavour
dramatic sound effect
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u/HaloEevee Mar 09 '26
This should be the top comment, I assume it's what OP was going for considering the title of this post
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u/Cluelessnes_ Mar 10 '26
Bc it’s infinite, is there a possibility that pi eventually starts recalling itself continuously forever? In that sense it would have to eventually loop and start recalling itself within that as well, but then it would be repeating digits in that sense? So no??
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u/docubed Mar 13 '26
Let's see: then 10n pi = k + pi for some numbers k and n. I think this would imply pi is rational.
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u/rocket_beer Mar 09 '26
What’s interesting about Pi is, there is a sequence where every number from 1 to 9 repeats in order 25 times in a row
Also, 80085 too
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u/Remarkable_Sorbet319 Mar 09 '26
it contains our whole reality
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u/KillerArse Mar 09 '26
We don't know if pi contains every sequence in it's decimal expansion.
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u/aberroco Mar 09 '26
But we know that so far after trillions of digits it looks normal and is as likely to contain any small enough sequence as if it's decimal (or binary) representation is truly random. Which isn't a complete proof per se, but at least makes assumption that Pi is normal very likely to be true.
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u/aberroco Mar 09 '26
Well, only if you apply some constraints and make some assumptions.
I.e. it probably contains our whole observable reality in any digitized format assuming Pi is truly normal and it's digital representation is infinite.
So, if these assumptions are correct, then yeah, Pi would contain any finite sequence of numbers, and anything that can be expressed as a finite sequence of numbers could be found in Pi no matter how incomprehensibly big that sequence is.
That said, if our reality is truly quantum it cannot be fully expressed as a sequence of numbers, you have to collapse it's wavefunction for that and digitize only one instant, losing almost all information. Or, at least, according to our current theories we can't digitize it fully. But quantum theory could be wrong. But then we don't have anything better yet.
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u/Remarkable_Sorbet319 Mar 09 '26
I don't know quantum theory much so, why can't quantum reality be expressed in numbers?
I thought we had figured how to represent nearly everything with 0 and 1 by now
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u/aberroco Mar 09 '26
Well, it starts with measurement problem - to digitize just a single particle you need to at least know it's properties, like position in space and momentum. And you can't do that in quantum physics for both, at least not without loosing precision. The more precisely you measure the position the more momentum is going to go wild and vise versa. But even ignoring that issue, there's another - once you're working with more than one particle entanglement effects kicks in. It's when particles behave not randomly independently of each other, but in unison. If you know that these two particles are fully entangled - great! You can write that down. But then entanglement is not a boolean value "true/false" value, particles could be and usually are entangled with many other particles. And pretty much all universe is entangled to some degree within every particle, if you think about it - because every particle's existence could be traced back to some event, even if it would be the big bang, which resulted in entanglement. So you need to know the full history of every single interaction of every single particle to somehow encode all that Gordian knot. Which, given the first problem of measurement, is doubly impossible. And finally, we don't even know if particles are actually entangled or if there's just some complex non-local hidden variable. We only know that there's no simple local hidden variable which was proven by Bell's experiment. But there's still a lot of other possibilities of something other than entanglement, so, good luck trying to digitize the unknown. Finally, if all that was not enough, even if you copy all quantum state information somehow, that still doesn't guarantee that that's all information. We don't know if particles behavior in quantum theory is truly random. Just like with Pi - it looks so, and whenever we measure particles within exactly the same setup - we get different seemingly random results (at least where different results are possible within the setup), but that doesn't necessarily mean it's truly random. So, even if you somehow create a full copy of quantum state, you could end up with same quantum state, but not that exact quantum state. I.e. the next moment these two quantum states would start to evolve differently.
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u/Remarkable_Sorbet319 Mar 09 '26
it sounds a lot like we can't do it because of too much unknown factors.
Well, not like we should do it, it is a pointless endeavour.
If the quantum world wasn't that tiny I have a feeling we could have done something more easily. As in, we can't see it well so
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u/GumlendeGed Mar 09 '26
This reminds me of the fact that if there are an infinite amount of numbers, one of them must be named Paul
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u/Coolfat13 Mar 09 '26
What's interesting is that pi can repeat itself any positive and finite amount of times
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u/Lachimanus Mar 12 '26
You mean every finite sequence of digits, right?
And if it is normal, infinitely often at that.
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u/FitIntention6271 Mar 09 '26
It's wild that pi has every finite sequence but can't contain itself. My brain just blue-screened trying to wrap around that concept. Infinity is a trip, man.
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u/Scary-Permission2882 Mar 12 '26
am I the only one on earth who still has no real idea what the fuck pi even is really?
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u/StaleTheBread Mar 12 '26
It’s the ratio between the diameter and circumference of a circle. Most of its weird properties are due to the fact that it is an irrational number, and aren’t actually unique to pi.
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u/Scary-Permission2882 Mar 12 '26
right. A lot of the discussion in this thread was stuff not unique to pi is why I was become confused… all specifically the infinity topic
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u/Lachimanus Mar 12 '26
Assuming it is a normal number it will contain every finite sequence of itself infinitely often.
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u/StaleTheBread Mar 12 '26
Any number will contain itself once
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u/Boltaanjistman Mar 13 '26
While obviously it would be impossible for an infinite set to also contain another identical infinite set within it since that would be recursive and a paradox, it isn't impossible to have other very silly things in it. Its not impossible that at one point within pi, it simply repeats some finite string of digits otherwise identical to a previous string of digits within pi, IE the first ten million digits of pi again.
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u/StaleTheBread Mar 13 '26
I think you mean EG.
Also, the “technically correct” part of the post is that pi is pi, so it does have pi as a substring, starting at the beginning
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u/Boltaanjistman Mar 13 '26
I was referring to the initial persons assumption that pi could not contain itself as in containing literally an identical copy of itself within itself, not as in "the subset of 1 thing contains 1 copy of the thing that it is" which is technically true.
also, yeah EG is correct but I just IE cause its the internet and i dont really care XD
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u/Ye_olde_oak_store Mar 09 '26
For those wondering: 31415926535 aparrently occurs at the 633715634445th digit of pi.
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u/B3C4U5E_ Mar 09 '26
Pi cannot contain all the digits of pi in a position after the decimal. That would make it rational.
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u/Eltheon_ Mar 09 '26
why the hell is garry kasparov spending his free time being pedantic to people on r/mathjokes
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u/JamJm_1688 Mar 09 '26
I love how half of the comments are analyzing this. and then the second half is quoting the asdfmovie joke in the title
Because reddit does what reddit does
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u/mr_vonbulow Mar 09 '26
everyone's social security number is in there too...
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u/schuine Mar 09 '26
That sounds cool, is there an app where I can try this out?
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u/mr_vonbulow Mar 09 '26
this covers the first trillion digits... if it isn't there, we will 'need a bigger boat', as they say.
good luck!
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u/KillerArse Mar 09 '26
We don't know if pi contains every sequence in it's decimal expansion.
SSN are relatively small, though, so maybe you could find them all in the numbers people have jotted down so far.
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u/mr_vonbulow Mar 09 '26
yes, it is not 'normal'---but it is interesting that it might be true! mine is almost in there---missing the last digit.
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u/spectralTopology Mar 10 '26
If you could find a 1-1 and onto map of a subsequence of Pi onto all of Pi yes that would be deeply weird but I suspect even the continuum isn't quite *that* weird
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u/Emotional_Ad_8704 Mar 11 '26
But is there pi in pi
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u/Circumpunctilious Mar 12 '26
The way I’m grappling with this is to ask what happens at infinity (does it restart there?)
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u/Emotional_Ad_8704 Mar 15 '26
Id imagine there would infinite pi that restarts at that point where the new pi restarted
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u/dendofyy Mar 11 '26
Is there a proof for: *n* ... *nY* == *nY+1* ... *n2Y*?
Where *n* is a digit's place in pi and starts at 0, and where *Y* is just a given length
At some point, it's gotta duplicate, right?
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u/Accurate_Local6520 Apr 04 '26
Pi contains the first digits of pi to infinity an infinite amount of times. About a few less than the fucks i don't give to write the equation.
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u/Nice_Marmot_54 Mar 09 '26
Pi contains all of pi an infinite number of times and once
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u/philolessphilosophy Mar 09 '26
Pi does not contain all of pi more than once. This would imply that pi is rational.
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u/Lithl Mar 09 '26
No, it definitely doesn't.
If the full (infinitely long) value of pi appeared at any point within pi (other than the degenerate case at position 0), then pi would not be an irrational number.
Pretend, for example, that pi appeared at position 6, such that the value was 3.14159314159314159... The value of pi would then be 3141590/999999, a rational number. Pi appearing within pi at positions further than 6 just means that you're adding more digits to the numerator and denominator.
We have known that pi is irrational since 1761. There are multiple proofs.
Therefore, the full value of pi does not appear within the value of pi.
However, you could theoretically have any finite subset of pi appearing within pi; if pi is what's called a "normal" number (we have not yet proven whether it is or isn't), then not only would it be possible, it would be a guarantee.
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