patterns in e and pi

[MustPost]

is it possible to prove that in a transendental number such as e and pi, any non repeating sequence of numbers can be found.
For example, somewhere in pi will the digits go "389312032580345" which is a random sequesnce of numbers I typed.
pi has an infinate quantity of digits to the right of the decimal point with no repeating pattern, will this allow any sequence of numbers to show up somewhere in it?

 

[alpha88]

I dont know how to prove it, but it makes logical sense to me.

 

[silverpig]

I don't think you can say that any sequence of numbers will show up.
Take for example a pattern of one hundred million 1s in a row...

What may be even more interesting is to try and prove what is the longest series of given numbers that you can find. ie, there is definitely a 4 in pi... 3.14... and a 57 as well... 3.14157... (I think that's right anyways, doing this from memory)...

But what about 356? or 2349? What is the longest series of numbers that you can find for sure in pi? A series of 1 number is a certainty, as pi contains all of the digits (or does it?)... a series of 2 is probable... but what is the longest?

 

[Carceri]

No, that is not possible I believe. I don't have a proof, but my guess is that it can be shown from the definition of pi that without calculating the decimals, you have no knowledge of what will happen later in the sequence. Might look into it later if I get the time...

 

[Agent004]

<< is it possible to prove that in a transendental number such as e and pi, any non repeating sequence of numbers can be found >>



It would be possible, given the non repeating sequence of numbers, themselves and the neighbouring numbers (1 before and 1 after) are non repeating.

 

[MustPost]

<<Take for example a pattern of one hundred million 1s in a row...>>

can we prove that no where in pi there is 100 million consecutive 1s?

 

[RossGr]

Seems the due to the nature of infinite strings, you could argue that any given sequence of numbers MUST appear at some point in the sequence. Though I cannot prove it, it certianly feels correct. though the repetition of a single digit any significant number of times seems questionable. Again I cannot prove it either way. I believe this lies in the realm of number theory, I lean more to the applied side of math.

 

[alpha88]

I could probably work up a semi proof, but when dealing with something that is infinitely long and does not repeat, it isn't possible to prove that something WONT exist inside it, so it "must" exist.

 

[LordMagnusKain]

now if something MUST exist inside of it, then ther must also be a second repition of it; for example, in an infinit string of all numbers there must be a repition of the number 1; and of 14, and of 14159;

ok with that acepted it must mean that the first million numbers in Pi repeat someware else in pi; and that thewhole string ofnumbers starting from .14159... 1/2 ofinfinity must repeat someware, that sumware must be the other half of infinity; and being the other half of infinity the numbers must be repeated reflected in the center of infinity;

thus pi proves, as many have before, that infinity ends in 1;

 

[Shalmanese]

Umm, Infinity ends in 1 only in Base 10 or in other bases as well, What about in base pi?

 

[LordMagnusKain]

all intager bases have the number 1.

and to be a proper base of any type it must be able to reference all real numbers, and thus 1 could be referenced, and thus would still be the last number, becase it's the first number in pi.

 

[flood]

linkage!! [Are The Digits of Pi Random?] - slashdot



<< Steve Hamlin writes "A researcher at Lawrence Berkeley National Laboratory, and his colleague at the Center for Advanced Computation at Reed College, have taken a major step toward answering the age-old question of whether the digits of pi and other math constants are "random." In addition, a simple formula discovered makes it possible to calculate the Nth binary digit of Pi without computing any of the first N-1 digits, and do the computation with very little computing power. " >>

 

[Hanpan]

Just wanted to point out htat not all irrational numbers are random. For example the number 0.12112111211112111112... willnever contain a 6 or 4 or 3 or any number other than 2 or 1.

 

[VTHodge]

Assuming the digits do follow the definition of random (yes I have read the above posts, but work with me here) . . .

The probabilitiy of an x number sequence in a y digit string is finite and discernable.
As the length of the string y increases, the probability increases to certainty (100% - a differential).
As the length of the sequence x increases, the probability decreases to impossible (0% + a differential).

By increasing both x and y we determine that it is both certain and impossible for the string to occur.

THEREFORE . . . there is a finite but indiscernable PROBABILITY that any string will occur.
It CANNOT be guaranteed that it will or will not occur.

Just my two cents (where's my change?).

 

[Agent004]

<< It CANNOT be guaranteed that it will or will not occur >>



How do I intepret this sentence? Do I read it as ' it cannot be gurantee that it will occur' nor 'it can be gurantee to not occur' ? I apologized for pick bones out of this, but the definition of probability is very important.

You see ' it cannot be gurantee that it will occur' nor 'it can be gurantee to not occur' has very different meaning to It can't be guaranteed that " it will or will not occur"

The first sentence means the probibility is there, for it to happen(which is I think you meant). The latter is self-explainatory, but you can indeed gurantee that 'it will or will not occur'.

Anyway, what the sequence of numbers, does a longer sequence have a lower probability than a shorter sequence? For example, 97 may take many computations to spot in the infinite digits of pi, however a longer sequence may be obtain within the first few digits.

 

[Ice69]

well, Pi and irrational numbers seem to have peaked my interest the last few days


389312, occurs several times in pi, unfortunatly the way my data is set up(10 digits then a space), i can't search perfectly(i have no doubt that your "random" sequence is there)

i've got 32 million digits calculated and stored

purhaps it will allow any sequence, but the only problem is that no one has really tried, chances of finding a string of specific numbers decrease as the number gets bigger, ie take 1/3, its .3 repeating, 3s forever. but take 3/10, that's just .3 and 33/100's .33. the numbers of Pi stay near the 3.1415 range - smaller and smaller fractions - ie an easy if unaccurate way to calculate pi is take smaller and smaller fractions for each digit( 1/7, 1/9, 1/11 if i remember but you get the idea)

but Pi is just 1 irrational number- the ratio of the circumference of a unit circle to its diameter. If you were looking for something more specific you could look at your string and using some fancy math(that is way beyond me) then predict what root would give you that string.

Sorry i don't have any proofs, just sorta logic that yes you can find them- but depending on the ammount of time you want to hack through Pi- now up to the quadrillinth or so digits, you could find anything

 

[Hanpan]

n/m.

I should not make statements where I cannot remember the proof...

 

[Mday]

well, if they repeat, this would mean they are rational numbers. therefore making them algebraic numbers, which mean they are solutions to some polynomial with rational coefficients. HOWEVER, since it has been established through the use of the calculus that both pi and e (the natural number, 2.7....) are transcendental, and therefore are not solutions of any polynomials with rational coefficients.

it is not simply enough to say that e and pi are irrational, you must realize that there are irrational numbers, say the square root of 2, which are solutions to some polynomials with rational coefficients, and therefore algebraic. though the original question did deal with "rational" numbers.

the thing about random numbers is that, they are random. while is is possible for that string of digits to be part of pi, it is foolish to say that the string IS part of pi. for your specific case, just look at the huge number of digits of pi which have been generated so far.

 

[VTHodge]

<<

<< It CANNOT be guaranteed that it will or will not occur >>



How do I intepret this sentence? Do I read it as ' it cannot be gurantee that it will occur' nor 'it can be gurantee to not occur' ? I apologized for pick bones out of this, but the definition of probability is very important. >>



Sorry bout not being clear. What I mean is . . .
1) It is not 100% certain that any string, no matter how short, will appear (without calculating to prove it).
2) It is not 100% certain that any string, no matter how long, will not appear (even with calculations).

Hope that clears up my point.

BTW . . . can anybody post the Taylor Expansion series used to determine Pi. I used to know it, but haven't used it for a while.

 

[Buzzman151]

Does anyone know if Pi has been translated into Base 12 or any other base for that fact? I just read that they did it in binary. Just b/c we can't find any repitition in the Base 10 number doesn't mean that it there won't be any patterns/repetitions in other bases.


Drew

 

[Hanpan]

<< well, if they repeat, this would mean they are rational numbers. therefore making them algebraic numbers, which mean they are solutions to some polynomial with rational coefficients. HOWEVER, since it has been established through the use of the calculus that both pi and e (the natural number, 2.7....) are transcendental, and therefore are not solutions of any polynomials with rational coefficients.

it is not simply enough to say that e and pi are irrational, you must realize that there are irrational numbers, say the square root of 2, which are solutions to some polynomials with rational coefficients, and therefore algebraic. though the original question did deal with "rational" numbers.

the thing about random numbers is that, they are random. while is is possible for that string of digits to be part of pi, it is foolish to say that the string IS part of pi. for your specific case, just look at the huge number of digits of pi which have been generated so far. >>



That is what i was trying to say but alas I though I would get called on a proof and that is difficult.

 

[Agent004]

<< BTW . . . can anybody post the Taylor Expansion series used to determine Pi. I used to know it, but haven't used it for a while >>



Taylor's expansion requires a function, what kind of function will out put a pi? For Taylor's thorem to work, you need

f(b) = f(a) + f'(a)(b-a)+ (f^-2 (a)/2!)(b-a)^2 + ...

 

[nortexoid]

i haven't read all the posts prior to this one, but it if we speak in terms of logical necessity, it isn't logically necessary that any sequence of numbers be realized within pi.

why?

because randomness could generate an infinite set where each member is identical...in terms of random sets - which is probably different than the set of numbers after the decimal place specified by pi.

analogy: u flip a coin an infinite number of times - it is NOT logically necessary that given that set of flips, u flip a tails..u could flip heads infinitely many times (of course, remove the idea of successive synthesis from this notion since it runs counter-intuitive with infinity)

 

[Shalmanese]

well, using the relationship e^(i*pi) = 1, we can expand out e^ix at pi and get a relationship.

 

[Oda]

One of the ways to calculate pi is to use a Taylor expansion of a trigonometric function, for example the inverse tan of one is pi/4. The Taylor polynomial of inverse tan is:

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

To make the algorithm more efficient (since the above series converges quite slowly) we use the identity (inverse tan I will denote as "taninv"):

taninv(A) + taninv(B) = taninv[(A + B)/(1 - A*B)]

and choosing A and B so the RHS of the above expression gives pi/4.

Hope this helps