patterns in e and pi

[Turkey]

I think this all depends on whether or not the digits of pi are truly random (and if the number of digits of pi are infinite). Since no one knows, then we can't conclude anything.

On the other hand, if the digits of pi appear randomly (given any sequence of numbers, any other sequence of numbers is equally likely to follow) and are infinite in number, then it's an easy proof to prove that any string of numbers would appear in pi.

 

[OUdejavu]

I dont think the diagonal slash would work for sequences. For any n length group of finite sequences you would run out of digits way before you ran out of possible sequences and your new number created would be redundant........

deja

 

[nortexoid]

OU, i was thinking similarly about that for a bit...

however, it indeed should work given the length fo the number sequence you're looking for...

if you're attempting to find an infinitely long number sequence within pi, i believe the diagonal slash procedure could produce an infinitely long number sequence not contained within pi. Meaning, the number sequence generated would not be contained in pi if and only if it is infinite. Why?

if you take all possible number sequences within pi and perform a diagonal slash procedure on them, you'd end up w/ infinitely many number sequences, with at least one not contained within in pi...try it out.

 

[RossGr]

nortexoid, we are not talking about an INFINITE subsequece of PI. I am disussing only FINITE length subsequences. To start talking about an infinite subsequence is a whole different ball of wax, lets not go there.

No the number PI is not random it is indeed a fixed number. However, we do not know to infinite precision its exact value, we have numeric methods which successively generate higer and higher percision digits. These digits do not really change the value of PI but add to the precision with which we know it. I guess this is why people are refering to PI as psudeo random, it is very much predetermined it is just that we do not know what the next digit will be until it has been computed. That is where the subsequences which we are talking about occur, the order in which high precision (put low significance) digits appear.
I simply do not understand how you will create your table and why doing a diagonal on it generates a sequence which will not show up somewhere in the infinite tail of PI. It seems (not proven by any means) that if I wish to find a given subsequence all it need to do is continue generating more digits of PI until it shows up. Someone mentioned having computed and saved 32 million digits of PI with that many digits we can generate approximately 32 million different finite subsquences of length N (N<100) by defining each digit as the start of a new sequence consisting of the next N-1 digits. We will be less then 32million because the last digits N digits of the generated value of PI could not be used as starting points. Of course you can always just generate more digits. That is why I do not believe that the diagonal slash method will prove anything about subsequences of PI any array that you create will not contain or generate knowledge concerning the uncomputed digits.


Sorry if I am restating the obvious.

 

[Becks2k]

I havn't read all the replies, I skimed through them sorta but heres my 2 cents (which means sh*t).

You can't find any set of numbers in pi because pi isn't "random"

Random is you flip a coin and its heads or tails, theres no way to know whats going to happen. If you flipped the coin an infinite amount of times, eventually you would get a million heads in a row. You would have to, because if you didn't, head/tails wouldn't be random anymore. After 1million-1 times of flipping and getting heads you would be 100% sure of the outcome of the next one is going to be tails.

Pi isn't random, theres an equation for it. Theres an equation to find out the next number. Ahh I dunno how to say what I'm thinking. Okay with the coin flip eventually you WOULD get a million heads in a row, why? Because each flip is random, theres nothing stopping it from coming up a million times.

Pi isn't random, you can't say you'll find 1million 0s in a row becuase the numbers arn't random. They have to fit an equation. They can't be anything.

I dunno if that even makes sense.. basically the only way to prove eventually that a string would come up would be to prove the numbers were 100% random. But 100% random numbers don't fit an equation. You can't calculate 100% random numbers.

 

[nortexoid]

ross, i said you'd have to run the diagonal slash procedure on EVERY subsequence within pi to necessarily derive a subsequence itself not found within pi.

does it make intuitive sense? - to me anyway...

i'm not even sure the sequence has to infinite, merely exceptionally long..."near" infinite (rather poor relativization of infinity)...

anyway, either way, randomness or not does not logically entail that every number sequence exists in pi...

it is not even certain that pi itself is an infinite sequence.

 

[RossGr]

nortexoid, Ok now I think I am begining to understand our differences.

First of all, if PI is not an inifinte sequence then all of this is out the window. If PI turns out to have a repeating sequence then several things would change, it would cease be be irrational and would be classified as a rational number, it would contain a finite number of subsequences and certianly could not possibly contain all possible finite length subsequences.

But it is looking very much like it is an infinte sequence and random in the distribution of digits. IF it is infinte, then you cannot write ALL of the subsequences so the diagonal argument will not work. In addition it leaves the window open for the possibility of finding ANY finite length subsequence imbeded somewhere in the digits of PI

Next question, is PI indeed constant?

Consider this, draw a circle on the surface of a sphere, measure its diameter and curcumfrence ON THE SURFACE of the sphere, now calculate PI as circumfrence/Diameter, you will get a way different value then 3.14... The value will vary with the radius of the sphere. So this implies that PI depends on the curvature of the space in which it is calculated. If over time the curvature of our space time continum changes will PI change with it?

Becks2k,
At least read the post just prior to yours and attempt to understand what is meant when we refer to the "randomness" of PI.

 

[Becks2k]

I'm stupid tell me whats wrong with it :/

Wouldn't the only way to prove any number would come up in pi would be to prove pi is 100% random and infinite?

A 100% random number can't fit an equation?

Pi fits an equation

Puesdo randomness != 100% random

so isn't that the end

 

[nortexoid]

ok ross, but what u're really saying is that the probability of finding every fininite number sequence as a subsequence in pi is approaching/near 100% (since pi is, as interpreted from your comments, potentially infinite)...the fact that you're construal of infinity in the case of pi is potential undermines the possibility of finding all finite number sequences within pi

to find all finite number sequences in pi, pi itself would have to be actually infinite, otherwise it's only highly probable (granted the randomness of pi) that they be found in pi.

as for the spherical example, i don't see how u can draw a circle on a sphere while maintaining the 2-dimensionality of the circle...if u accomodate it to fit on the sphere, it becomes non-circular...it's bent and entirely a different 2-dimensional shape [than a circle].

however, that proposal seems similar to the convergence of parallel lines on curved (3d) surfaces...

 

[OUdejavu]

ok nortexoid, let me see if i can explain myself better, it always comes out diff than i think it

i am assuming the diag-slash method you are referring to is the one used to say that the set of infinite binary sequences is not countable (bin seq being 01101001.... like that) if i am wrong please tell me what it is.....

next it is necessary to compare only sets of the same length, we wont get anywhere comparing 1 to 41 to 5927, that is why i said to only compare sets of n length

so, consider this set: n equals 2
now we have a 00,01,02,...10,....99
we cant use diag-slash here because we have only 2 digits with 100 possible sets.. all we can prove is that the set of strings of lenth 2 cant be limited to 2 members....

as far as infinite sequences in pi, diag-slash will work to show that the set of infinite sequences in pi cant have both the properties of being countable and contain every possible subsequence.

deja

ps beck i think that is a good logical explanation of the concepts here

 

[RossGr]

Ok, concerning the randomness of PI please make and effort to read this Paper. If you do you find several interesting facts.

In the 1767 Lamber and Legendre proved Pi to be irrational therefore it is indeed infinite and non repeating.

Can we lay that issue to rest?

In 1882 Linderman proved it to be transcendental.

So therefore inspite of what you believe becks2k it is NOT the solution to a simple algebraic equation.

I learned several other interesting facts seems that the algorithm genertate digits base 16, not sure what this says about base 10 digits! The forumula used to genererate the digits is closely related to those found in chaotic systems so in some sense it could be said that the digits of PI are chaotic in nature. Currently something like 206 Billion digits have been computed these are essentially normally distributed. It has not yet been proven that PI is unconditionally normal (normal in all number bases) but the tools are now in place for someone to prove it.

Can I prove that PI contains every finite subsequence, no, If I could I quareentee you that I would not working as an HP tech!

Can You prove that it does NOT contain every finit subsequence, no, If you could then you would be not be doing whatever it is that you are currently doing.

So that leaves us with an interesting disscussion to which there can be no resolution.

If you wish to dispute the validity of old proofs I cited above there is a thread currently dealing with the validiy of math running in the poll section of Tomshardware's fourms. It is a poll on (dare I say it ) Does .9r=1 . NO, DO NOT START THAT AGAIN!.... Please

'bout sums it up
Ross

 

[nortexoid]

essentially the diag/slash procedure showed that there's no one-to-one correspondence between the set of irrational numbers and the real numbers. In other words, the set of irrational numbers is larger than the set of real numbers (since there's a member of the set of irrational numbers not coextensive w/ a real number)...

all the procedure does is generate a number using a simple substitution algorithm that doesn't map onto real numbers...in regards to pi, I was suggesting that if pi were to be completely expressed (as actually infinite - viz., as an infinitely long definite/determinate number sequence), u could take every number sequence contained in it and generate a number sequence not corresponding to any of the number sequence of which the procedure was carried out on...

i can see it in my head...and that's all that counts...joking - but honestly, it seems plausible that it can be carried out.

i) one side = all finite number sequences contained in pi
ii) other side = all finite number sequences generated by diag slash
iii) outcome = no one-to-one correspondence between i) and ii)

maybe i'm dillusional.

 

[OUdejavu]

ok nortexoid, i see what you are saying....

but you are assuming that the set of sequences contained in pi is countable
i.e. you could name them s1,s2,...,sn,sn+1,... and put them in a list and apply the diag-slash to them.
this may or may not be true

we already know that the set of ALL sequences is uncountable, something SO infinitely big you cant even describe them in a list....

so, if we assume the set of sequences contained in pi is countable(countably infinite is a less confusing description i guess) then we are done, pi doesnt contain every possible infinite subsequence, it doesnt even scratch the surface....

but, if we assume the set of sequences contained in pi is uncountable, then we cant put them in a list, and therefore we cant ues the diag-slash

hope i was clear in my explanation

- deja

 

[OUdejavu]

rossgr,

HAHAHA there was an argument about .9r=1? that is so funny of course it does!

i dont know what me finding it funny says about me though.............guess ill always be a math dork

 

[Becks2k]

"So therefore inspite of what you believe becks2k it is NOT the solution to a simple algebraic equation. "

It doesn't have to be a simple algebraic equation, the numbers still have to conform to some "equation"

How bout to P = 2pi r thats an equation. What if atfer the 300billionth digit of pi all of a sudden there was no more 9s. There is nothing stopping that from happening, a certain number doesn't have to come up. How can you prove that will never happen? Stuff like that COULD happen, just because we havn't seen it doesn't mean it doesn't happen.

I'm not trying to argue with you I just don't get how you can prove something is 100% random. The numbers have to conform to some type of equation, so then they're not random.

Infinite, nonrepeating and transendental doesn't prove 100% random.

what about a string that never has 9s in it, doesn't repeat and is infinite long... thats not 100% random

I don't see how you can prove 100% random.

(btw I don't know a sh*tload of math stuff, I don't think I'm right, I'm just asking why )

 

[RossGr]

Let's make sure that we are all talking about the same things here. Starting with what a mathematician calls the different sets of numbers and the size of the various sets.

1. Natural numbers or Integers are the basic counting numbers 1,2,3....
The size of the set of integers is said to be countable infinite.
2. Rational numbers are set of all fractions, 1/1, 1/2, 1/3, 1/4...
By creating a grid numbered with the Integers along the sides and top

/..1...|.2..|.3..|.4..|....
1 1/1,2/1,3/1,4/1...
2 1/2,2/2,3/2.4/2...
3 1/3.2/3.3/3,4/3
4 1/4,2/4,3/4,4/4
.
.
.

The intersection of each column & row forms a fraction of the row, column numbers, you can now start a line through the table in a diagonal fashion which hits every cell, for the above table the order would be 1/1,1/2,2/1,3/1,2/2,1/3,1/4,2/3,3/2,4/1,as you pass through a cell it can be assigned a Integer, thus the set of Rational numbers can be placed in a one to one correspondence with the counting numbers, this means it is a countable infinite set.

3. The Real numbers is the set formed by creating all possible finite and infinite length subsequences of integers (and a single '.' per number). Using a slight variation on the diagonal argument, as described in other posts of this thread, this can be shown to be a larger infinite then the countable set of Natural numbers, it is said to be UNCOUNTABLE infinite.

4. The Irrational numbers. The Real numbers contain as a subset the decimal representation of the rational numbers , delete this subset from the set of Reals and what is left is the Irrationals. The Irrationals also form an uncountable infinite set. The easiest way to see this is that by removing a countable infinite set from an uncountable infinite set, what is left is still uncountable. (Clear as mud, huh.~^)

Ok now lets look at PI, There is a countable infinite set of digits in PI, each digit has a place value defined by 10^-N, Since N is an integer each digit can be placed into a one to one correspondence with the Natural numbers. Now can we count the number of subsequences in PI? I have been very careful to specify FINITE length subsequences so that should mean that we could consider each subsequence a real number, now since I have specified that they be finite length subsequences they are all TERMINATING numbers therefore they are all RATIONAL, so the set of finite length subsequences in PI cannot be bigger then the set of Rational numbers, i.e. a countable infinite set.

Can I prove that every rational number can be found embedded in PI, absolutely NOT, still seems possible though.

Becks2k the definition of a transcendental number is that it is not Algebraic, that is why we cannot make the claim you are trying to make. Remember that by definition PI is the ratio of 2 MEASURED numbers. Ancients noted that their value of PI was only as good as their measurments.
'bout sums it up.
Ross

 

[OUdejavu]

I agree with you Ross, the set of finite subsequences is countably infinite, so the diag-slash isn't applicable. pi doesnt include all possible infinite subsequences, that should be obvious. That is also not the question .

So we are back to where we started. pi is apparently psuedo-random according to that article you linked, I would still say it is not random, guess I don't know if there is a formal definition of random or whether its just a concept. So, jumping the gun and assuming those dudes were right and pi is pseudorandom, we still can't use that fact to prove it contains all finite sequences, nor can we use the diagonal slash.

So were just stuck, we dont have enough information or ingenuity to do anything!

I have a gut feeling it doesnt contain all finite subsequences and you have a feeling it does, but no way to know. This sucks.

 

[OUdejavu]

Ya know, the more I think about it, the more diag-slash doesnt even apply here, no matter what sets are countable or uncountable.....it is just to differentiate between the 2.

My brain hurts....