patterns in e and pi

[RossGr]

The probability argrment given above leads to the conclusion that any randomly selected sequence of finite length MUST exist somewhere in PI. Since PI has been shown to be a random sequence of INFINITE length, you can find a infinite number of finite length sequences embedded in PI. Therefore if you randomly select a FINITE length sequence of digits you have an INFINITE number of subsquences of PI to compare to, irregardless on the length of your subsequence you MUST find it somewhere as everypossible combintation of sequences of that length must exist in PI. (OR e for that matter!)

 

[nortexoid]

Rossgr, that isn't true if i'm interpreting that correctly...

u're saying that for any possible finite number sequence (and thus every finite number sequence?), there's a one-to-one or one-to-many correspondence with a finite number sequence within pi or e simply because they contain infinitely many finite number sequences?

meaning, randomness + infinite set logically entails every finite number sequence??? - i don 't think so.

that is, for any infinite set, you could always use the "diagonal/slash" (cantor) method of deriving a number NOT within the set (proof of why some infinite sets are larger than others).

 

[VTHodge]

<< The Taylor polynomial of inverse tan is:

x - (x^3)/3 + (x^5)/5 - ... + ((-1)^k)[(x^(2k+1))/(2k+1)] >>



Thanks, thats the one I was thinking of.

I think what some people are getting confused is the difference between the logical nature of infinity and the statistical nature of infinity. Logically every number sequence must be present, because if you haven't found it yet, just keep looking. This is the argument that RossGr is making.

I am arguing for the statistical nature which esentially states that while the probability of finding the string increases as the length extends to infinity, you will never reach the point of gauranteed 100% probability, even at an infinite length series of numbers.

These two theories seem to contradict each other, but I think they are equally valid. They are both human interpretations of the true nature of mathematics. The human mind will never be able to completely understand the way infinity works. (But that won't stop us from discussing it on AT!)

 

[nortexoid]

VTHodge, I think u have that semi-backwards...or simply wrong.

u could have an infinitely long number string (and thus number sequences), while every number is simply a finite sequence repeated while not consisting of every finite number sequence - i.e. 1/3 as a decimal...

that number sequence does not logically entail the "instantiation" of any and every finite number sequence possible (since only 3 is infinitely repeated), no matter how hard u look.

EDIT: oh wait, that is obvious - we are talking about a random set....bah!

 

[sash1]

I don't know if you could prove it, or even if you could, it would definitely be hard. Pi, as you probably know since your asking, is a prededermined number: the ratio of the diameter over the circumference, and e, well that just escaped my mind, but IIRC its 2 point something. I can get my notes out from last years math and see what it is and some interesting things about it, but I'm too lazy. I don't think just random numbers will appear, since Pi isn't a random number itself. . .

I have no clue if it is provable or not, I'm a freshman in High School taking sophomore AP math (as well as sophomore AP Biology, if you cared; I like to brag about my nerdyness ... not). We have so many theorems, lemmes, postulates, definitions, and corolarries I can't even keep track of them, but I doubt any of them will help me prove this. Even with all thge stuff I have, I doubt I'll be able to justify what I'm doing past the first step.

I just got a 99 on my Geometry test, if you care! I'm happy about it! But then, I get A's on all my tests, so what else is new!

~Aunix

 

[flashbacck]

I think you can enter whatever string of numbers you want here, and it'll look for it, using some formula that determines the nth digit of pi w/o calculating any digits before it.

 

[RossGr]

I am not sure I understand the different view points

. You will note that I very carefully stated FINITE length sequences. It is the nature of infinity that PI MUST contain an infinite number of finite sequences of any specified length. I would think that a formal proof of this would depend on the fact that the digits of PI are random. Guess we need to think about what that statement means. My interpertation of it would be that the distribution of digits is evenly divided between the 10 digits, 0..9, so statistically speaking there are no more 1s then 2s etc. Ok. Now consider a randomly selected sequence of length N. Given that PI contains an infinite number of sequences of that length, the probability of that string NOT existing in that infinte collection of sequences is 0 (1/infinity).

OK, you volnteered to find a sequence how about the counting numbers in order 1-100. 12345678910111213141516...100

Actually I doubt that there is sufficient compting time in the world to actually find that sequence, so save your system for something useful like your fav game!

 

[jbond04]

This is the coolest damn thread I have ever seen in my life. I just started to learn about "e" last week in Calculus, (and how to integrate and differentiate it). Anyway, I can't wait until I learn about this stuff (even though it may be a few years down the road). Rock on, math guys!

Now, many of the posts above are WAY over my head, but, I will bring up this:

If a number is truly infinite (like pi or e), and it is truly random, then every conceivable number sequence will appear, because there will be an infinite opportunity for it to occur, therefore, it must happen sometime.

Please note that I could be completely wrong, since this is only my first year of Calculus.

 

[m0ti]

Infinite + random DOESN'T mean that it contains all possible sequences.

Random doesn't mean that the probability of any sequence is non-zero. Random means that there is a certain distribution which describes (statistically) the results. If pi were defined as a series of random digits, then yes, I would agree, that, given any sequence of numbers, it MUST appear somewhere in pi (but it is not guaranteed that you could find where the sequence is located in finite time).

A comment on the statement that the Nth binary digit of pi can be calculated without the need to calculate the N-1 preceding it. This effectively proves that the probability of finding a specific digit at a specific location is 1 for one specific value and 0 for the rest. What is interesting to consider here, is that there is a good deal of underlying knowledge that we have about pi that allows us to apparently escape the nescessity of Markov Chains, of an arbitrary order, IF we assume that the digits of pi, are indeed random (according to an unknown distribution). On the other hand, for answering this specific question, we would want to use Markov Chains. Using this method would, however, only provide us with an approximate statistical guess (of varying accuracy depending on the number of digits of pi used to calculate the statistic, and the order of the Markov Chain).

In any case, even if it might not provide a wholly accurate method, it could provide interesting statistics, and perhaps some insight (especially on the finite known sequences of pi).

 

[Agent004]

However, we all agreed that pi definitely contains numbers 0 - 9, albeit in random order and infinitely long. Since any finite sequence is consist of of those numbers in different arrangements, it must exists in an infinite sequence.



<< Infinite + random DOESN'T mean that it contains all possible sequences. >>



But the point is that pi contains an infinite number of sequence, solely because it's infinitely long. Plus we are taking about non-repeating sequences, which means it does contain all the possbilities, given an (random) infinite sequence consist of infintely many (random) sequence.

Simply put, in order to have an infinite long number/sequence, you must put all the number/sequence into it. However, how we do that? We put an infinitely many numbers/sequences into it, meaning all the possible sequence/numbers, therefore all the possbilities are contain within.

 

[m0ti]

That isn't correct.

First off, they aren't truly random, just pseudo-random.

Second, just saying it's infinite doesn't guarantee that a desired sequence is present. About the closest thing you can say is that it is present "on the far side" of infinity. In other words, you can't guarantee that the sequence is present upto a certain point n (finite) in pi. Saying, vaguely that it must be there because it's infinite, and SEEMINGLY non-repeating (as far as we know), does not constitute a formal proof.

I think that in order to prove or disprove it, one would have to find out something new and very encompassing about pi.

 

[RossGr]

moti:
I think that you are a bit confused. PI can be used to generate Pseudo-random numbers but according to this article linked to earlier in this thread it is beging to look very good for PI having a normal distribution of digits. IF this is indeed the case then I will stand by my earlier posts. The probability of finding any given finite sequence of digits embedded in PI is very nearly 1. This really seems incredible. When thinking of sequences like the one I mentioned in my last post it seems impossible that it could and by my own argument MUST exist some where in PI. The ability to accept this means truly coming to grips with the concept of infinity.

Could you actually find this sequence in PI, perhaps not, you may need to look at more digits then your (or anyone elses) computer is capable of dealing with. What is the largest floating point number IEEE format can deal with (10^3xx, on that order, I think) I am sure someone out there knows. But we really cannot use floating point to describe the digits of PI because we need to count the digits which implies we need to stick to the set of counting numbers this means the best we can do is a long integer. What ever the number of digits is it may not be enough to find ANY finite length sequence.

 

[Roguetech]

If you look at randomnessas infinite diversity in the finite data range (i.e. the numbers of 0-9), then EVERY combination MUST appear somewhere, ala Douglas Adams' Improbabilty Drive, or Raymond Feist's portal city in space. Sounds dumb to think that if the cosmos is infinite, and is random (which is most definity is NOT), then there must not only be another New York City, down to the last rat, but there actually must be an infinite number of New York Cities. Or that there would not only be the complete works of Shakespeare encoded in ASCII in Pi (assuming randomness and infitiness), but that the complete works would be in order of when he wrote it, but also in every language known to man (well, every language that can be encoded in ASCII .

That is the definition of infinite. If there is a probability of, say 1 to 1 quidrillion (rouphly speaking) of, say Windows ME staying up for more than 2 weeks, if it were to be continually rebooted for infinity, it would not only do just that, but it would do it an infinite number of times (infinity divided by 1 quidrillion = infinity; there's your proof, BTW).

Two problems with the idea, as everyone else has expressed... One, is it random? (almost certainly NOT - I'd hazard the guess that nothing is), and is it infinite? (perhaps, but you can't prove it. The OTHER paradox of Choas Theory.)

 

[OUdejavu]

Sorry, but y'all are wrong (at least those who say that every finite sequence MUST be contained in pi based on the assumption that pi is a pseudo-random number)

Here's why: Let's form a new psuedo-random number by removing every occurence of 0123456789 from pi, and lets call it phi. Surely phi is still an infinite sequence. Also phi is still psuedo-random as for every 0 we remove from pi, we also remove a 1,2,3.....9. The distribution of the numbers 0-9 is kept in tact. We now have an infinite, pseudo-random sequence that does not contain the sequence 0123456789 nor any finite sequence that contains 0123456789. So phi is an infinite, pseudo-random sequence that does not contain the infinite set of finite sequences containing 0123456789. QED.

P.S This is not to say that pi can't or won't contain every finite sequence, it very well may. But it is wrong to conclude that it will based on the fact that it is pseudo-random.

BOOYAH ~deja


edit - should known that wouldnt come out right the first time

 

[RossGr]

OUdejavu,
I like that argument, that is a very good for showing that a normal distribution of digits is not sufficient for the proof in general put what does it say about PI. E'gads these infinities running round in my head. Seems pretty incrediable that you can find an infinite number of 1234567890 sequences to delete from PI. What else do you suppose you would need, in addition to normal distribution, to prove the original question for PI?

 

[Barnaby W. Füi]

you guys are obiously omitting the expanse of just what infinity is. infinity NEVER ends, it's not really, really, long, it just keeps going FOREVER. hard to perceive as humans never really experience "infinity" in our lives, but it since pi is a random number, and infinity is infinitely expansive, then it you could take ANY sequence of numbers, yes even 1234567891111122222333333444445555566666777778888899999 and you would find that sequence in pi an infinite number of times.

am i making sense?

 

[m0ti]

RossGR,

Interesting article. It would tend to make sense. In any case, I agree intuitively with the argument, and it does make sense to me that pi would include all suggested sequences, but, unfortunately it's not a mathematic proof. It's currently been proven empirically, but not mathematically.

 

[Agent004]

To prove it mathematically is almost impossible, as how are we going to write down an infinite string. Here is where intuitive comes in. For example, take the hamonic series at limit of infinity. Even 1/ infinity has a value, that's infinitely small( gets closer and closer to 0, and it does exist in the real line). But surely this value has to be greater than 0, since it never attains(word?) the bound of 0. But fact we put it, it's equal to 0



<< What else do you suppose you would need, in addition to normal distribution, to prove the original question for PI? >>



Just create an axiom saying it's true and make sure all mathematicians to agreed on it.

Perhaps the foundamental problem is that the language itself is not rich enough for us solve this. A good example is ' the universe is infinitely big and yet, it has a definite size' ( a programme made by the BBC, couldn't remember the name of it).

 

[Hanpan]

Afaik it has not been proven that pi's decimal digits are truley random. We believe it to be so but as I mentioned earlier it does not follow from the fact that a number is irrational that it must have random decimal digits...

 

[OUdejavu]

Ross Gr,

All it proves is that people can't claim pi contains any finite subset, based on pi being pseudo-random which i believe that article said some smart guys were close to proving. As for being random, I don't think pi can be considered random because every time you calculate it you get 3.14..... which puts it more in the predictable than random category. You'd probably need some other information about pi beside pseudo randomness, but your guess is as good as mine.

BingBong,

If infinity is so long, think about the number of subsets that would have to be covered by a string that never stopped. Thats the nature of the beast.

pseudorandom is just not enough

~deja

 

[nortexoid]

is it just me or doesn't the the DIAGONAL SLASH procedure prove that not all finite number sequences occur even in an infinitely long number sequence (random or not).

it was initially used to show that there's no one-to-one correspondence between real numbers and natural numbers, but u could use it likewise to generate a finite number sequence not found in pi (or any number sequence for that matter).

 

[RossGr]

nortexoid, I don't think the diagonal array argument will work with PI because it relys on the ordered propertiy of the the real numbers. Since the digits of PI appear to be random they are certinaly not orderd so you cannot find a new value for PI by any method other then generate the next digit in the sequence.

 

[Kitros]

Ever read Hitchhikker's Guide??

It's highly improbable, so it is undefinitely so.

 

[Agent004]

Hitchhiker is cool, too bad the author has died

 

[nortexoid]

but ross, pi is NOT random...

it's a calculated ratio.

therefore the number sequence is determined, and the diagonal slash procedure should theoretically work to find a sequence not found in pi.