Using the Bailey-Borwein-Plouffe algorithm, it is possible to calculate any digit in the hexadecimal expansion of pi without calculating any of the preceeding digits.

 

:pi: = :lsum: (4/(8n+1) - 2/(8n+4) - 1/(8n+5) - 1/(8n+6))*(1/16)ⁿ for n=0 to :inf:

 

Is there ever a hope of formulating an equivalent in decimal form? What does this say about whether pi might have an ending?

Who wants my elite VB program that utilizes this forumla.

 

Its rather slow though.

That and it's probably in base-10 rather than 6

Nah its in base-6. I've tested it out to 10,000 digits, matches up with what the formula should produce.

Originally posted by fafalone

What does this say about whether pi might have an ending?

 

It doesn't. It's formed from a Taylor series.

Originally posted by MrL_JaKiri

It doesn't. It's formed from a Taylor series.

 

which is...

Originally posted by NSX

 

which is...

 

The sum of an infinite number of ever expanding powers.

Originally posted by MrL_JaKiri

 

The sum of an infinite number of ever expanding powers.

 

Can you give me an example?

 

Is there ever a hope of formulating an equivalent in decimal form? What does this say about whether pi might have an ending? [/b]

 

Pi is exactly 3! -Professor J. Frink

Phi is (1+sqrt(5))/2

 

The Golden Ratio, :lcphi:

Originally posted by NSX

Can you give me an example?

 

he explained it in rather complex terminology

 

basically, if you have a function like cos(x), e^x or any function that has continuous derivatives, you can expand it in terms of x. a really crude way to think of it is kind of like a more complicated binomial expansion.

 

you start of by presuming that some function of x,

f(x) = a + bx + cx^2 + dx^3 + ...

 

where a, b, c, d etc are constants. for example, if you take f(x) = e^x, then you can say

 

when x = 0, a = 1.

 

then by differentiating it, you can see that b = 1, then again to see that c = 1/2 and d = 1/6, e = 1/24, etc

 

so therefore e^x = 1 + x + x^2/2! + x^3/3! + ... + x^n/n!

 

you can test it in your calculator if you want, but it works. btw, this is a bit overly simplified, i've not really studied these things in detail, but this is the basic principle behind it.

Isn't a series like that a sum tho; this allowed for calculating any digit of pi without calculating any of the preceeding digits.

Originally posted by fafalone

Isn't a series like that a sum tho; this allowed for calculating any digit of pi without calculating any of the preceeding digits.

 

If you calculate any digit of pi without having to know its surroundings, you would have to calculate an infinite number of 0's before you could be sure that it terminated.

So far there has been something around 6.5 billion digits in pi. And it's still not exact.

Originally posted by the GardenGnome

So far there has been something around 6.5 billion digits in pi. And it's still not exact.

 

Of course it isn't

 

It's a transcendetal number.

 

It has been proven to be infinite. (but between the values of 3.1 and 3.2)

More importantly, why does pi come up in all areas of science?

Nobody knows really - it's an extremely important constant, but it still surprises me when it turns up in a really obscure part of mathematics (like the infinite limits of sums going to things like pi^2/6). I suppose its because circles, angles, and more importantly radians are used in so many parts of science.

Originally posted by dave

 

he explained it in rather complex terminology

...

you can test it in your calculator if you want, but it works. btw, this is a bit overly simplified, i've not really studied these things in detail, but this is the basic principle behind it.

 

Cool. Thanks. It took a few minutes, but it sunk in