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How can we express the value of pi in mathematical terms? (ie. how can we calculate the EXACT value of pi using a calculator?)

2006-06-26 04:46:36 · 33 answers · asked by Anonymous in Science & Mathematics Mathematics

33 answers

π = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...)

unfortunately, you need 100000 terms to calculate π to 5 decimal places.

Also,

π = 2√3 * (1 - 1/(3*3) + 1/(5*3^2) + ... + (-1)^k/[(2k+1)(3^k)] + ...]

or, a method used by computers to calculate π in the past by Stormer (1896) is:

π = 24arctan(1/8) + 8arctan(1/57) + 4arctan(1/239)

where arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...

So while it is true that you will never get the "exact" value of π to be stored in a calculator, you can determine it to an arbitrarily small error by adding in more terms from the above infinite series.

NOTE TO "THE SHADOW" Pi is not an exact value? Of course it is! It is the exact ratio of a circle's circumference to its diameter. Just because its transcendental doesnt mean it isnt exact. Its just that no numerical representation of pi will be its exact value.

NOTE TO "iandanielx" Of course there are equations that lead to pi. I just gave a few. (Only the follow up question implied the use of algebraic numbers due to a calculator being involved.) And accurate enough? 30 digits of pi is enough to calculate the circumference of the universe given its diameter to within the diameter of an atom.

NOTE TO: "countryboy_giddy_up" (IF IT GOES TO A VOTE, THE 4TH ANSWER!)
12 hours ago

Your plagerism is laughable. You obviously copied your answer from another website, and in pasting it into this thread, many of the text (formulae, etc.) that you stole to represent as your own was not pastable into the text editor of yahoo.

It is incredibly evident.

If this goes to a vote and anyone (besides him) votes 'best answer' I will have no further faith in this blog or at least its voters.

2006-06-26 04:54:37 · answer #1 · answered by Scott R 6 · 7 3

PI =
3.141592653589793238462643383279502884197169399375105820974944
592307816406286208998628034825342117067982148086513282306647
093844609550582231725359408128481117450284102701938521105559
644622948954930381964428810975665933446128475648233786783165
271201909145648566923460348610454326648213393607260249141273
724587006606315588174881520920962829254091715364367892590360
011330530548820466521384146951941511609433057270365759591953
092186117381932611793105118548074462379962749567351885752724
891227938183011949129833673362440656643086021394946395224737
190702179860943702770539217176293176752384674818467669405132
000568127145263560827785771342757789609173637178721468440901
224953430146549585371050792279689258923542019956112129021960
864034418159813629774771309960518707211349999998372978049951
059731732816096318595024459455346908302642522308253344685035
261931188171010003137838752886587533208381420617177669147303
598253490428755468731159562863882353787593751957781857780532
171226806613001927876611195909216420198938095257201065485863
278865936153381827968230301952035301852968995773622599413891
249721775283479131515574857242454150695950829533116861727855
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2006-06-27 08:49:54 · answer #2 · answered by Devil Hater 1 · 0 0

PI is an irrational number meaning it cannot be written in a closed fraction form.

as such, in decimal form it will never repeat or terminate...there will ALWAYS be more numbers so 3, 3.1, 3.14, 3.141, 3.1415

...3.1415926535897932384626433832795, and so forth.

It is impossible to compute the exact number PI...in more sophistication Pi has a finite amount of value and an infinite amount of error. (every digit about pi give you less and less amount of information about the 'exact' value of Pi, but there is still a infinite amount of information to be gained).

it like adding up the series 1/2, 1/4, 1/8, 1/16, ...

everybody knows that the total is 1 (which you can 'see' after a few numbers), but there is an infinate amout of information to be gained and you will never actually get to 1 but will get closer and closer.

2006-06-26 06:47:49 · answer #3 · answered by kmclean48 3 · 0 0

Pi is not an exact value at all. It is a continuing number that never repeats as far as we know and we have gotten as far as several hundred decimal places. Most calculators have a Pi button, so use it if you have it. Otherwise just use the 3.14. It is close but not exact.

2006-06-26 04:55:51 · answer #4 · answered by The Shadow 4 · 0 0

The exact value of pi is incalculable. The highest calculation ever had to be stopped at around 26,000,000 digits to avoid destroying the computer. It is equal to aproximately 3.14 or 3 and 1/7.

2006-06-26 06:28:02 · answer #5 · answered by Anonymous · 0 0

Nobody needs the absolute value of Pi with infinite digits; however, you can use Fourier series for a specific periodic function tn order to get the infinite number of digits to Pi.

and please do remember in float number format there is no EXACT value, and the Exact value for such a number is depend on the accuracy you want.

2006-06-26 05:37:48 · answer #6 · answered by ws 2 · 0 0

There is a book called "The History of Pi". In the back of the book Pi is shown to about 3000 decimal places. It never repeats and never terminates.

2006-06-26 04:57:40 · answer #7 · answered by Anonymous · 0 0

You Cannot. That is the definition of a transcendental number - there is no algebraic equation that can ever have Pi as an answer, let alone an equation that can give an accurate value.

2006-06-26 04:54:32 · answer #8 · answered by iandanielx 3 · 0 1

You can't express the exact value of pi using a calculator.

It is a transendental number -- meaning that it cannot be expressed in algebraic terms (it is not rational and cannot be expressed in terms of roots of any rational numbers).

2006-06-26 05:55:18 · answer #9 · answered by Ranto 7 · 0 0

Symbolic Computation software such as Maple or Mathematica can compute 10,000 digits of in a blink, and another 20,000-1,000,000 digits overnight (range depends on hardware platform).

It is possible to retrieve 1.25+ million digits of via anonymous ftp from the site wuarchive.wustl.edu, in the files pi.doc.Z and pi.dat.Z which reside in subdirectory doc/misc/pi. New York's Chudnovsky brothers have computed 2 billion digits of on a homebrew computer.

The current record is held by Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo with 51 billion digits of (51,539,600,000 decimal digits to be precise).

Nick Johnson-Hill has an interesting page of  trivia at: http://www.users.globalnet.co.uk/ nickjh/Pi.htm

The new record for the number of digits of is 4.29496 billion decimal digits of pi were calculated and verified by 28th August '95.

Related documents are available with anonymous ftp to www.cc.u-tokyo.ac.jp.

ftp://www.cc.u-tokyo.ac.jp/
This computations were made by Yasumasa Kanada, at the University of Tokyo.

There are essentially 3 different methods to calculate to many decimals.


One of the oldest is to use the power series expansion of together with formulas like . This gives about 1.4 decimals per term.
A second is to use formulas coming from Arithmetic-Geometric mean computations. A beautiful compendium of such formulas is given in the book and the AGM, (see references). They have the advantage of converging quadratically, i.e. you double the number of decimals per iteration. For instance, to obtain 1 000 000 decimals, around 20 iterations are sufficient. The disadvantage is that you need FFT type multiplication to get a reasonable speed, and this is not so easy to program.
A third one comes from the theory of complex multiplication of elliptic curves, and was discovered by S. Ramanujan. This gives a number of beautiful formulas, but the most useful was missed by Ramanujan and discovered by the Chudnovsky's. It is the following (slightly modified for ease of programming):
Set

Then , where





The great advantages of this formula are that

1) It converges linearly, but very fast (more than 14 decimal digits per term).

2) The way it is written, all operations to compute S can be programmed very simply. This is why the constant appearing in the denominator has been written this way instead of 262537412640768000. This is how the Chudnovsky's have computed several billion decimals.


An interesting new method was recently proposed by David Bailey, Peter Borwein and Simon Plouffe. It can compute the Nth hexadecimal digit of Pi efficiently without the previous N-1 digits. The method is based on the formula:





in O(N) time and O(log N) space. (See references.)

The following 160 character C program, written by Dik T. Winter at CWI, computes to 800 decimal digits.


int a=10000,b,c=2800,d,e,f[2801],g;main(){for(;b-c;)f[b++]=a/5;
for(;d=0,g=c*2;c-=14,printf("%.4d",e+d/a),e=d%a)for(b=c;d+=f[b]*a,
f[b]=d%--g,d/=g--,--b;d*=b);}

2006-06-26 04:50:15 · answer #10 · answered by Anonymous · 0 1

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