How is pi calculated?

My friend gave me the challenge of calculating pi, but I keep getting stuck. I keep getting pi defined in terms of itself. What am I doing wrong?

Code:

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x^2 + y^2 = 1  - Equation for a circle when r = 1
y = sqrt(1 - x^2)  - Move y to other side of equation
A/4 =  ∫(0 to 1) (sqrt(1 - x^2))  - solving area under the curve for the positive x and y
A/4 = (sqrt(1 − 1^2) * 1 + asin(1))/2 - (sqrt(1 − 0^2) * 0 + asin(0))/2 - definite integral
A/4 = (1 + asin(1))/2 - FUCK asin(1) = pi/4

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I know I've done this before. Grrr.

You can write Pi in terms of a taylor series. The more terms you tag on, the more precise your value for pi. If you can write Pi in terms of a trigonometric function (I think arctan is pretty standard for this), then all you need to do is replace "arctan(whatever)" with its taylor expansion, and you've got a way to numerically get at it.

I am not sure how the 1000s of digits are calculated most efficiently, but it's definitely some iterative process similar to expanding a taylor series. Wikipedia article should have that answer.

Ah here we go, I googled it and it came up with basically what I was saying:

Math Forum - Ask Dr. Math

EDIT: Here's some more fun: 1 Million Digits of Pi

Code:

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              $|=3,141592;sub _
          {print@_}sub o{_++$O[0
        ];_ 0for 1..$#O}sub O{$;=int
      $=/10,'0/^          ^';if($;<9)
    {_$_ for                @O;;@O=()
    ;0}push                    @O,$;;0
  ,;push@                      O,'.'if
  $^==1;                        0;if($;
  ==10){          print          ,o,@O=(
  )}}$?=        1000000        ;$-=10*
  (q/@O=      digits of pi      =10/,1)
  *int($?                      /3)+1;$
    _=2for@                    ,[0..$-]
    ,;for$^                  (1..$?){
    $"=$-;$O                =0;until
      ($"<0){$/=          2*$"+1;$/=
        10if$/==1;$==10*$,[$"]+$O;$,
          [$"]=$=%$/;$O=int($=/$/
              )*$"--,10}O}o

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I tested it on my box. It seems to work, though artificially slowly. :P

I did it once with a geometric figure, The number of itterations got one closer and closer, I do not remember if I kept it in the current version of The Delian Quest or not. I used Mathcad.

As pi can never be expressed precisely in decimal form, the best you can do is an approximation.

There are many ways. You could do it by integration as you currently are; what you should be doing is trying to approximate the numerical value of the integral in some way, for example by approximating the area underneath the curve as a series of rectangles, trapezia, or quadratic curves (the last one is called Simpson's Rule and is very efficient).

You could approximate a circle as a polygon of some sort and calculate its perimeter with the Cartesian length formula (it would probably be easiest to write the equation in parametric form, (cos(t), sin(t))).

The problem with the above is that they are single calculations with a preordained degree of accuracy which don't allow for any further refinement.

So you could as Replicon says use some infinite series. The classic way is the arctan integral, from 0 to 1 of (1 + x^2)^-1, which you do in two ways: the standard way gives you arctan(1) - arctan(0) = pi/4; for the second way you use the binomial theorem to express the integrand as an infinite polynomial, which you then integrate and end up with 1 - 1/3 + 1/5 - 1/7 ...

Hence pi = 4 (1 - 1/3 + 1/5 - 1/7...) which is rather beautiful.

However I don't think it converges very quickly. If the point of what you're doing is to understand what's going on then the above is the easiest option, but if you're after a fast iteration for some reason, I believe Wikipedia has a whole collection of them.

Quote:

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Originally Posted by Xei

As pi can never be expressed precisely in decimal form, the best you can do is an approximation..

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That is one of the differences between a tautologic, like common grammar, arithmetic, etc, versus a relatiologic, which is a geometric figure.

It also tells one the meaning of irrational, which simply means the inability to acquire a name for something, by the conventions of that grammar, in a given logic system.

What is irrational in arithmetic, is perfectly rational in geometry.

One can see the effect on a lot of geometric figures.

I think Taylor expansion is the best way

Deified is spelled the same way backwards as it is forwards. I'm not sure if that has anything to do with pi, but somehow it feels relevant to this thread as I have often compared its lack of conviction to that of Xei's inability to ban anyone thus far.
:cowbell:

Quote:

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Originally Posted by greenhavoc

Deified is spelled the same way backwards as it is forwards. I'm not sure if that has anything to do with pi, but somehow it feels relevant to this thread as I have often compared its lack of conviction to that of Xei's inability to ban anyone thus far.
:cowbell:

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Xei's "inability" to ban anyone could well be taken as just being generally awesome. I can't really see how palindromes are relevant to pi (although palindromic patterns are relevant to primes) but I could be wrong. There is certainly something circular about them. Maybe socially conservative politics is related to pi as well.

You're right, I am pretty clever. Absolute[ly] in fact.:P

Quote:

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Originally Posted by greenhavoc

You're right, I am pretty clever. Absolute[ly] in fact.:P

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xkcd: Words that End in GRY

The fifth panel also applies to you.

^[Ha] but whatever,the genius of my ignorance hasn't failed me yet, PhilosopherStoned.