How is PI calculated?
DevNull
How is PI calculated? All of the information I can find shows several
very different and complex formulas for the calculation, none of which
give quite the same result after a certain precision.
But and I guess this is the real question, how does one know which
formula is the right one? It all seems very arbitrary to me, as though
you just pick which ever one is easiest to calculate out to the
precision needed. But is there such a thing as a true PI?
Anyways thanks.
Cloud Burst
There is no number that you can write down since it is a number
with a decimal fractional part that never repeats. Same goes
for most numbers, though. The closest we can get is to write
down a formula or notation that indicates the number, such as
"the square root of 2"
>Anyways thanks.
mensa...@aol.com
DevNull wrote:
> I've lately become enamoured with this question.
> How is PI calculated? All of the information I can find shows several
> very different and complex formulas for the calculation, none of which
> give quite the same result after a certain precision.
That's incorrect. They all give the same result to the same
precision. What you're seeing is convergence. One method may
display 150 digits, but it has only converged to 100. The digits
past the convergence point will be incorrect no matter which
method you use. When you see two different results, you must
ask if they have converged to the same precision, not how
total digits are being shown.
> But and I guess this is the real question, how does one know which
> formula is the right one?
They are all the right one.
> It all seems very arbitrary to me, as though
> you just pick which ever one is easiest to calculate out to the
> precision needed.
You also want to consider how fast the method converges.
If you have a math library that does unlimited precision
rationals, then any method that uses a sequence of rationals
will be "easy" to compute. But one method may converge to
100 digits of precision in fewer iterations than another,
so you pick that one.
> But is there such a thing as a true PI?
Of course, it's just that any finite decimal attempt at
representing it will be incorrect.
> Anyways thanks.
alexand...@ulbsibiu.ro
METHOD I. Let
a_n= 4( 2*n^2 +(4n+1)(n+2))/((4n+1)(4n+5))
b_n= 4*n^2 (2n+1)^2/( (4n-1)(4n+3)(4n+1)^2 )
(1) A_{n+1} = a_n*A_n - b_n*A_{n-1} , n=1,2,... ,
A_0= 4 , A_1= 76/15
(2) B_{n+1} = a_n*B_n - b_n*B_{n-1} , n=1,2,... ,
B_0 = 1 , B_1=8/5 .
Consider the sequence (Q_n) having the general term
==============================
A_n
(3) Q_n= ----- .
B_n
==============================
It's possible to prove that
0 < Q_{667}- Pi < 10^{-1001} .
Using (1)-(3), the numbers Q_0,Q_1,...,Q_{667}
can be computed.
=====================================================
METHOD II. (as a sum of a certain series) Determine the first
partial sums of
pi=4-Sum{k=1 to k=infty} (-1)^{k-1}a(k)
where
a(k)={40k^2+16k+1}*C(2k,k){16}^{k}/( 2k(4k+1)^2*C(4k,2k)^2 )
and C(n,k)=n!/(k!(n-k)!).
Regarding such questions: search on Internet the so-called WZ-method
(authors: H.Wilf, Doron Zeilberger) as well as
(in your library) the book
"A=B" by M.Petkovsek,D.Zeilberger and H.Wilf (?) .
However any book about accelerating the convergence of series is
useful (e.g. Markoff method, Norlund-method,...).
==================================================
METHOD III. (BBP-formula=(Bailey,Borwein Plouffe see Internet))
Try to solve following question :
Let T(k,z,a) =(1+z)^{8k+ a}/{8k+a} ,
P(k,z)= 2*T(k,z;1) +2*T(k,z;2)+ T(k,z;3)- T(k,z;5)*0.5 -
- T(k,z;6)*0.5 -T(k,z;7)*0.25 .
If |z+1| < sqrt(2) , then
SUM_{k=0 to k=infty}P(k,z)/16^{k} = ??
Suppose that you find the sum of the series.
Further select a convenable value for z (z=0 ?)
Here is the remarkable BBP formula (attributed to David Bailey ,
Peter Borwein and Simon Plouffe) for Pi ; that is
==================================
Pi =SUM_{k=0 to k=infty}c(k)/{16}^k
with
c(k):= 4/(8k+1) -2/(8k+4) -1/(8k+5) -1/(8k+6) .
==================================
This formula enables us to find the hexa-digits of pi .
=======================================
METHOD IV. In literature there are many so-called
"ALMOST IDENTITIES " which furnish us "good" approximations for pi.
For instance:
(1) pi =approx.= 2510613731736*sqrt(2)/1130173253125
If K:= 1/{10}^{10} then
(2) Pi =approx.=K*(SUM_{n=-infty to n=infty}e^{-K*n^2} )^{2}
This is not an identity but is correct to over 42 billion digits !!
It was discovered by Jonathan Borwein and Peter Borwein (1985 ??)
==========
METHOD V .
CONTINUED FRACTIONS approximation of Pi . See Internet...,
See also Method I.
alexand...@ulbsibiu.ro
Which definition you propose ??
Grover Hughes
by definition, being the ratio of the circumference of a circle to its
diameter. There are, as you imply, several formulas for approximating this
true value, and I'll offer a couple here in case you might not be aware of
them. The first one is
(pi^2)/8 = 1 + 1/3^2 + 1/5^2 + 1/7^2 + .... where the denominators are the
successive odd integers. The convergence is not very fast.
The second one is a continued fraction and is not easy to write on this
keyboard. If you wish, email me your snailmail address and I'll send you a
photocopy of the book page which is here before me. It looks sorta like
this:
pi/2 = 1 - ______________1____________________
3 - _________2x3_____________________
1 - _______1x2____________________
3 - ______4x5__________________
1 - _____3x4________________
3 - ____6x7______________
1 - _____5x6__________
3 - and so on,
where you can begin to see the rule for the pattern of the successive terms.
I haven't tried this one, so I don't know how fast or slow the convergence
is. By the way, the small x between two integers means "multiply"--- this
keyboard has no way that I know of to get the "raised dot" for implied
multiplication.
Hope this is of interest. With regards,
Grover Hughes
"DevNull" <smo...@gmail.com> wrote in message
news:1130691370....@o13g2000cwo.googlegroups.com...
John Bailey
This is a very profound question, if interpreted correctly. It gets
right down to Godel type provability. Having read the answers thus
far, they all pompously march right past the fundamental question--
Assuming any one of several definitions of Pi, which of the several
processes for generating a number resembling Pi are PROVABLY correct,
in the limit? Nonconstructive proofs disallowed.
reference:
http://www.math.vanderbilt.edu/~schectex/papers/difficult.html
John Bailey
http://home.rochester.rr.com/jbxroads/mailto.html
mensa...@aol.com
John Bailey wrote:
> On 30 Oct 2005 08:56:10 -0800, "DevNull" <smo...@gmail.com> wrote:
>
> >I've lately become enamoured with this question.
> >How is PI calculated? All of the information I can find shows several
> >very different and complex formulas for the calculation, none of which
> >give quite the same result after a certain precision.
> >But and I guess this is the real question, how does one know which
> >formula is the right one? It all seems very arbitrary to me, as though
> >you just pick which ever one is easiest to calculate out to the
> >precision needed. But is there such a thing as a true PI?
> >Anyways thanks.
>
> This is a very profound question, if interpreted correctly. It gets
> right down to Godel type provability. Having read the answers thus
> far, they all pompously march right past the fundamental question--
>
> Assuming any one of several definitions of Pi, which of the several
> processes for generating a number resembling Pi are PROVABLY correct,
> in the limit?
I thought all of them were.
> Nonconstructive proofs disallowed.
Why?
>
> reference:
> http://www.math.vanderbilt.edu/~schectex/papers/difficult.html
So? What has that to do with anything?
>
> John Bailey
> http://home.rochester.rr.com/jbxroads/mailto.html
eric.sc...@vanderbilt.edu
http://www.math.vanderbilt.edu/~schectex/papers/difficult.html
has been mentioned. So I thought I should offer my semi-expert
opinion: I agree with Mensanator. I don't see what constructivism
has to do with the calculation of pi. In fact, I would think that
any method of calculating pi would necessarily be constructive. Am
I overlooking something here?
Eric Schechter
(buy my new book)
John Bailey
<mensa...@aol.com> wrote:
>
>John Bailey wrote:
>> On 30 Oct 2005 08:56:10 -0800, "DevNull" <smo...@gmail.com> wrote:
>>
>> >I've lately become enamoured with this question.
>> >How is PI calculated? All of the information I can find shows several
>> >very different and complex formulas for the calculation, none of which
>> >give quite the same result after a certain precision.
>> >But and I guess this is the real question, how does one know which
>> >formula is the right one?
>>
>> right down to Godel type provability. Having read the answers thus
>> far, they all pompously march right past the fundamental question--
>>
>> Assuming any one of several definitions of Pi, which of the several
>> processes for generating a number resembling Pi are PROVABLY correct,
>> in the limit?
>
>I thought all of them were.
Well, the first problem is understanding what the problem is. Indeed,
even the compendium [L. Berggren, J. Borwein, P. Borwein, Pi: A Source
Book, Springer Verlag, New York 1997] fails to provide a proof of pi's
existence! Basically, you need to figure out what the exact definition
of pi is, and then rigorously prove that this defines a unique real
number. In effect, this problem is an exercise in mathematical rigor.
>> Nonconstructive proofs disallowed.
>
>Why?
http://www.rzuser.uni-heidelberg.de/~tvogt2/999.pdf.
The fact that you can’t compute the decimal expansion of a sum from
the decimal expansions of its addends is a well known phenomenon
that was noticed by Turing. In a fully constructive treatment of the
real numbers, this is often stated by saying (informally) that not
every positive real number has a decimal expansion. More precisely,
there is no constructive proof that every positive real number has a
decimal expansion (or at least we don’t know of one).
http://www.rbjones.com/rbjpub/cs/cs006.htm
Computable reals are defined in Turing's first classic paper
[Turing36] where he uses (what we now call) universal Turing machines
to show that there exist unsolvable problems in elementary number
theory. In this paper Turing defines computable reals as those whose
decimal expansion is computable by Turing machine. Pretty soon he
spotted that this is not so good (not what we would now call an
admissible representation) and comes out with a correction in
[Turing37], where he goes for convergent sequences of nested intervals
with rational end-points (which is an "admissible" representation).
>
>>
>> reference:
>> http://www.math.vanderbilt.edu/~schectex/papers/difficult.html
>
>So? What has that to do with anything?
I suppose I could say GET REAL, MAN; but that would be cheap.
A proof is nonconstructive if it asserts the existence of some object
without actually constructing or finding that object. Such proofs are
used freely in mainstream ("classical") mathematics. Constructivism
is the practice of avoiding such proofs or at least pointing them out
explicitly. Once a mathematician sees the distinction between
constructive and nonconstructive mathematics, he or she will choose
the former. (quoted from the questioned link, above)
Its a way a separating the adolescents from the adults. Hopefully the
OP won't back away now from asking questions about fundamentals.
John Bailey
http://home.rochester.rr.com/jbxroads/mailto.html
mensa...@aol.com
Well, I suppose if pi did NOT exist, then the proofs would
certainly be suspect. But you didn't read past the part you
quoted, did you? Didn't realize this article PROVES the
existence of pi? Is that why you think this quote somehow
supports your argument.
>
> >> Nonconstructive proofs disallowed.
> >
> >Why?
> http://www.rzuser.uni-heidelberg.de/~tvogt2/999.pdf.
> The fact that you can't compute the decimal expansion of a sum from
> the decimal expansions of its addends is a well known phenomenon
> that was noticed by Turing. In a fully constructive treatment of the
> real numbers, this is often stated by saying (informally) that not
> every positive real number has a decimal expansion. More precisely,
> there is no constructive proof that every positive real number has a
> decimal expansion (or at least we don't know of one).
Quoted from a paper that proves 0.999... < 1.
You really don't understand how to make a case for yourself,
do you?
And this does not answer my question: why are nonconstrutive
proofs disalowed?
>
<further bullshit snipped>
> >
> >>
> >> reference:
> >> http://www.math.vanderbilt.edu/~schectex/papers/difficult.html
> >
> >So? What has that to do with anything?
> I suppose I could say GET REAL, MAN; but that would be cheap.
>
> A proof is nonconstructive if it asserts the existence of some object
> without actually constructing or finding that object. Such proofs are
> used freely in mainstream ("classical") mathematics. Constructivism
> is the practice of avoiding such proofs or at least pointing them out
> explicitly. Once a mathematician sees the distinction between
> constructive and nonconstructive mathematics, he or she will choose
> the former. (quoted from the questioned link, above)
I can quote Eric Schecter too:
<quote>
</quote>
Now look who needs to GET REAL.
>
> Its a way a separating the adolescents from the adults. Hopefully the
> OP won't back away now from asking questions about fundamentals.
Hopefully, the OP won't get his answers from crackpots.
>
>
> John Bailey
> http://home.rochester.rr.com/jbxroads/mailto.html