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Does Pi necessarily contain any sequence of numbers?

PIAt work today, concerned about whether our new product release will actually work, we discussed the above question (totally unrelated to our product).

More formally, is there any property of Pi (assuming an infinite level of precision) that means it must contain any (possibly infinite) sequence of digits. For example, does it (by it's nature) contain the sequence 123.

Pi in fact DOES contain the sequence 123 (see 10000 Digits of Pi) but that's not the point! Can we prove that a non-random number must contain it.

After much arguing about the interaction between probability theory and infinity, trancendental numbers entropy and other related concepts, we seem to have agreed (between a Maths major, a Computer Science major and an Archaeoloy major ;) that it can't be concluded from the obvious properties of Pi, but we didn't rule out that a rigerous attack of the various Pi algorithms might in fact be able to prove otherwise.

Google and Wikipedia haven't shed any further light on the subject - can anyone comment further? (Note to self: get Dad to read this blog entry).

Clarification: We are of course dealing with Pi in decimal form. And yes, it is a stupid question that is of interest purely because of our human interest in patterns and order!

Image courtesy of xkcdA webcomic of romance, sarcasm, math, and language.

01:37 AM, 28 Jun 2005 by Mark Aufflick Permalink

Pi contains all finite sequences

I read somewhere long ago that Pi contains all finite sequences. So the probability of finding your own phone number in pi is 1. pi is sufficiently random and non-repeating that all finite sequences occur there. I'm pretty confident that there wasn't a proof in the book I saw this in :-) Anonymous Stuart-

by Unregistered Visitor on 07/01/05

Not sure, but...

I suspect you could use a variant of the triangular proof to show that it doesn't. The number of finite sequences within the decimal representation of pi is, surely ℵ-null (??), i.e. an enumerable sequence of them exists. Using the triangular proof you might create a sequence at any given size that is a counter example to the proposition.

I've been wrong before and I'm already doubting my suggestion. I do believe, however, that the result depends on the cardinality of the set of finite digit sequences. But then you also said "possibly infinite"; that takes me even further out of my depth than I was before, from ℵ-null metres to ℝ metres in the drink.

Semi from TramTown

by Unregistered Visitor on 07/02/05

Base Pi

That was along the lines of my thinking (although vastly more mathematically informed!), that the sequence of Pi can be enumerate, given sufficient time, and is therefore not random. Has anyone proven that Pi (in decimal format) has infinate precision, or might it be possible to enumerate the whole thing? As stated previously, this is all based on a decimal representation of Pi in base 10. If we were calculating in base Pi, the answer would be quite different ;)

by Mark Aufflick on 07/02/05

Fun essay covering base pi (and others)

by Mark Aufflick on 08/22/05

Normalcy of pi

It is believed, but not proven that pi contains every finite sequence of digits (in base 10). Pi does not contain every infinite sequence of digits.

by Unregistered Visitor on 07/26/06

That's interesting

I'd love to read more on that if you happen to check back here my anonymous friend!

by Mark Aufflick on 07/28/06

Pi contains everything

If you're able to convert all atomic data of yourself into digits then those digits should be located in pi somewhere. So that includes all of your memories, feelings, everything that makes you who you are. Not only are you located in pi somewhere, you are located in pi in an infinite number of places. Expanding further if you convert all data of the universe into digits that also should be located in pi somewhere in an infinite number of places. The same holds true for every possible alternate universe and every possible person that doesn't have to exist. Food for thought :-)

by Unregistered Visitor on 01/06/08

Must only be finite

Of course it's trivial to prove that Pi doesn't contain Pi, so thus it doesn't contain all infinite sequences. That still leaves the matter of all finite sequences. Time to implement a lazy Pi sequence generator in Haskell...

by Mark Aufflick on 03/04/08


Slightly rambling but sometimes on point discussion of Pi and the nature of 'countable' infinitive numbers:

We are in Digits of Pi and Live Forever

by Mark Aufflick on 03/04/08

Short proof that transcendental numbers don't necessarily contain all sequences

Fact: pi is transcendental, meaning that it is infinite and nonrepeating.
Fact: We cannot know all of pi, because it is infinite.

An infinite, nonrepeating sequence does not necessarily contain all possible sub-sequences. Imagine, if you will, that somewhere beyond the number of digits we know for pi, the number 9 unexpectedly stops occurring. Instead of continuing as an infinite, nonrepeating sequence of the digits 0 through 9, it becomes an infinite, nonrepeating sequence of the digits 0 through 8.

Such a sequence is obviously possible, because we can express pi in base 9, and it's still transcendental.

After the disappearance of 9, then, any sequence containing 9 that didn't appear before its disappearance WILL NEVER APPEAR.

Now, note that this is a proof that pi doesn't NECESSARILY contain all sequences, not a proof that there is some sequence that it absolutely DOESN'T contain. Because pi is infinite, we can never know the actual truth, but it is provably possible that it might not.

by Unregistered Visitor on 03/14/08

Does that really matter?

Hmmmm, I do not think that matters (the number 9 being dropped at some point). I'm not saying that pi contains every finite sequence, just that this counter-argument is false.


1) 9 is dropped after half of pi (or a third, or a fourth, ..., or some ratio). Then there are still an infinite number of finite sequences before the 9 is dropped.

2) The 9 is dropped after a finite number of digits. Then just convert the message into base-9.

More generally I would claim that:

Since it is proven that pi is non-repeating, it can not just end with an infinite number of the same digit. There has to be variation (and variation without a pattern), and even if that variation uses a subset of the digits 0-9 the possible subsets are finite. If we code our message in all possible, finite, subsets, at least one of them has to match an infinite region of numbers in pi.

In short: the information is in the variation, not in the numbers per se.

by Unregistered Visitor on 10/24/08

elegant maths (as opposed to elegant rabbit)

"In short: the information is in the variation, not in the numbers per se." That's a different angle - and beautifully put. If only my maths lecturers had been so succinct ;)

by Mark Aufflick on 10/25/08

it wouldnt drop the 9

it wouldnt drop the 9 for the same reason 0,999.. is the same as 1 :P think about it: The last 9 won't come ever ( a better compare is 0,00..001 being same as 0 )

by Unregistered Visitor on 12/20/08

Re: Does that really matter?

Well, if you put it THAT way, you're saying that every NUMBER appears in the digits of pi. The question is if every SEQUENCE appears in pi. If you reinterpret the sequence into a different base, you're changing the sequence. My phone number (say, "555-0987") isn't a number that can be reinterpreted in base X without changing its meaning. It's a sequence of digits that won't mean the same thing if you do anything to change it.

I'm not certain if the statement that "every number has a representation that can be found in some representation of pi" is going to be true or not. Certainly the counterargument isn't going to be as elegant as mine, if there is one. But it's definitely not the same statement.

And by the way, "it won't drop the 9" isn't a statement you can make based on such simple reasoning. I can invent an infinite, non-repeating number that has the property of not having any 9's past a certain point with a fairly simple construction: Start with 0.0, then start appending the natural numbers. (i.e. 0.0, 0.01, 0.012, blah blah blah, 0.0123456789101112) When you reach 1000000, skip any numbers that contain a 9. This number will contain every sequence of digits that doesn't contain a 9, and it will contain every sequence of digits of length 6 or less -- but any 7-digit sequence containing a 9 (for instance, the phone number I described earlier) won't appear in it.

by Unregistered Visitor on 03/25/09

The nine dropping

proof is invalid. It proves that pi doesn't contain all infinite sequences, but says nothing about finite sequences, which is what we're looking for. A sequence that contains no nines from a certain point on is an infinite sequence, and if it is not infinite, then it's definition does not prevent there being more nines in the part of pi that it does not cover.

And the proof is useless for infinite sequences as well, since there are much easier ways to prove the same thing. (pi does not contain the sequence [all digits in pi except one of them])

by Unregistered Visitor on 07/01/09

Handle infinity with care

The discussion about the possibility of 9 disappearing from some point of the decimal representation of Pi has been mathematically blurred.

1. It is not a proof considering Pi itself - just at proof that there exists irrational numbers that do not contain every finite sequence. It has not been proven here whether Pi is one of those or not!

2. It doesn't make sense to talk about some ratio of the decimal representation of Pi. This has to do with the concept of infinity itself because it is impossible to point out at which point we have reached one third of the digits of Pi!

3. It is not advisible to convert between base 10 and base 9 representation of a number without recalculating ALL of the digits. So when this shift occurs the original number in focus (here Pi) is left behind.

It is true, though, that the infinite sequence of digits of Pi cannot be found again anywhere in the digits of Pi. Let's say that Pi is written in base 10 as


Now assume that a_(n+1) equals a_1, a_(n+2) equals a_2 and so on - unlimited throughout the rest of the digits of Pi. From this we get that a_(i*n+j)=a_j and hence that the digits of Pi is a repeating sequence of the first n digits which contradicts Pi being irrational. So actually this argument goes for any irrational number.

by Unregistered Visitor on 07/26/09

very late post

The OP questions if pi can "contain any (possibly infinite) sequence of digits". It certainly can't contain any infinite sequence. Or at least, it can't contain more than one. Just ask, "which sequence starts first?" The paradox should be obvious. For instance it can't contain an infinite sequence of 1 (as defined by 1/9) and also an infinite sequence of 2 (as defined by 2/9). Which ever one occurs first in the decimal places of pi, it leaves no place for the other. The other can not be earlier in the pi sequence because we already defined this one to occur first. It can not be after because the digits of the series do not match and both repeat infinitely.

by Unregistered Visitor on 02/11/10

Oh Infinity (to the tune of O Canada)

Yes of course, that is obvious once you put it like that. Thanks!

by Mark Aufflick on 02/13/10


Pie is my everything. i love Pie. Cherry, and pumpkin, and apple, and ice cream, and even pig. PIE!!!!! YAY! ME =D PIE! Who here likes pie? yummy yummy yummy yummy yummy!

by Unregistered Visitor on 03/02/10

This reminds me of the infinite monkeys

If an infinite number of monkeys banged randomly on typewriters would they eventually bang out a duplicate of every classic book? Everybody says yes to that. The key is random, I guess. There is certainly no guarantee that the digits or pi are random. A coin toss is determined by billions of interactions that are determinable by no real observer. A random number from a computer is not truly random, but rather a good approximation of random for ordinary uses (but would it eventually produce every classic novel?)

Delving into the subject would actually blur the definition of random, because pi is a completely determined number. Pi is exactly the same no matter who calculates it. You can get any next digit of pi by continuously applying the following sequence (I know, there are many others, but this one is the most simple for humans to understand):

4/1 - 4/3 + 4/5 - 4/7 + 4/9 - ...

I have had a lot of people say it does not really equal pi, but it does. The way you can know is to look at the difference between the sum of n terms in the sequence and pi. EVERY SINGLE ONE gets closer to pi. (To get the digits out of this sequence, you wait until the digits stabilize).

Anyway, a number whos digits can be determined from such a simple sequence may not be random and may not contain every item of classic literature codified (or every finite sequence of digits).

by Unregistered Visitor on 05/07/10


I was memorizing digits of pi (I know I need to get a life) when I came upon the middle three number of my phone number. I then wondered if my entire phone number could be found in pi, and I decided since pi is infinite, it would have to be found somewhere. Then of course I wanted to look more into this so I googled it. I found this blog, which I read, along with most of the comments. I'm still a little undecided. If pi is infinite and never repeats, then eventually you would have to find every sequence of numbers, right? But although pi is infinite, it is not random and is completely pre-calculated. Something as simple as a phone number or a three digit sequence such as one two three would surely be found in it. But could I find the sequence that consists of one trillion ones, one trillion twos, one trillion threes, and so forth up until nine, all in a row? Because pi is infinite, obviously it goes on forever, so even a finite sequence nine trillion numbers long would have to be found somewhere in it since nine trillion is shorter than infinity. But, there is an infinite number of finite sequences, and there is because pi is predetermined, the odds of one finite sequence are greater than the other.

by Unregistered Visitor on 05/22/10

Base 256

If one were to convert all of the decimals in pie to ASCII (base 256). There would surely be a lot of gibberish; however, If pie really is infinite and non repeating then MAYBE it would contain all of the secrets of the universe in plain text. But then again it would also contain every possible scientific contradiction as well. How would you strip the correct answers out of pi and trash all of the nonsense and false info? So really the idea of using a transcendental number to look for information is no more useful than your imagination is.

by Unregistered Visitor on 06/29/10


I'm leaning towards "..." on this one. I don't think pi will include any given finite set of numbers, I think there are more bounds than that by virtue of its construction or initial conditions. Whatever imperative (axiomatic?) prevents it from repeating also I think would prevent a great many of the possible finite sequences from occurring. Of the known digits of pie what is the greatest number of any digit being repeated? Is it even possible to infer future structure from known values? My gut tells me no and yes simultaneously. Example: it seems clear to me that whatever stops it from being infinite ones, also stops it from including the maximum possible repetitions of one while still being finite. I'm thinking many features of pi might fall under the true but unprovable category described (demanded?) by this post's label. ~Innomen

by Unregistered Visitor on 10/25/10


Pi is infinite!!!! end of question. Don´t ever bother your mind with the naive hope that everything can be "solved" giving a certain result using strong enough computers. Pi was unsolvable the day that mathematicians accepted that a point is infinite small. look at it this way if the diameter in a circle is 1 pi equals its circumference, so pi in this case is the length around the circle we just wanna find out how long precise. here the problem arises because we accepted that a point is infinite small and what we really wanna do is finding the last point to end the circle so from were we might stand on the line on the circle and towards the end there is an infinite amount of points. So every time a computer adds another decimal it only gets a fraction closer to the end but hence there is an infinite amount of points there is an infinite amount of fractions between any given to numbers ergo we will never reach the end but only get a fraction closer.

by Unregistered Visitor on 11/30/10

Just for fun...

The fact that pi continues infinitely without repeating proves nothing in this argument. So does 0.101001000100001..., and it clearly does not contain every sequence of numbers. Why should pi contain every finite series of numbers? I've not yet seen a convincing argument.

by Unregistered Visitor on 03/14/11

Doesn't answer the question, but...

My roommate, my boyfriend and I were eating a late, lazy Sunday morning breakfast and debating this very question. We Googled it, and this blog entry came up first. Although it doesn't answer the question, we did find it worthy of comment that we are, respectively, an Archaeology major, a Computer Science major and a Maths major. Cheers to an infinite universe containing infinite repetitions of every given occurrence!

by Unregistered Visitor on 04/17/11

I like those odds

Good to see others are pondering the real issues of life also. And there should be more archaeologists in the world.

by Mark Aufflick on 04/19/11

According to Wikipedia: "Maybe."

by Unregistered Visitor on 05/09/11

sequences of PI

50 years or so ago, I read in a little book (from Emile Borel ?): 1/ PI does contain any sequence as long as you want (but not infinite) 2/ It does contain this sequence infinite (true infinite) times.

by Unregistered Visitor on 10/27/12


Many of the comments were really strange, I mean like if people know something specific about randomness, signals or wathever but they don't have a clue of how wrong their claims are for what concerns the purely logical axpect of the argument. Many people here have probably no clue of what cardinality is and what the fact that the real numbers are uncountable implies, nor they know what series and convergence mean it seems, but even though these really are all logical arguments other than simply mathematical ones they keep trying to use everiday logic to explain wrong claims.

I indeed have to thank the author of the last comment since he quoted Borel and with a simple wikipedia read about natural numbers I finally focused the problem, one way of solving this is "probably" all reduced to determine if pi is a natural number or not.

I say probably since I'm not sure if the properties of uniformity of natural numbers prove that number to contain every single finite succession to be contained in it but from what's written on wikipedia: "In mathematics, a normal number is a real number whose infinite sequence of digits in every base b[1] is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2, all b3 triplets of digits equally likely with density b−3, etc."

This meaning that all possible finite n-sequencies of numbers are equally likely, I'm not sure if this implies they are eventually found since i've just read for the first time what natural numbers are, but I'd say it indeed means just that, since the probability is more than zero and the decimals are infinite.

by Unregistered Visitor on 03/12/13


there was a miswrite, i obviously meant normal numbers, not natural.

by Unregistered Visitor on 03/12/13

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