How is the value of Pi calculated with extreme accuracy?

[jobyts]

When we say we know the value of Pi upto - say 1 billion position accuracy -, how exactly they calculate it? Is it as simple as Circumference / diameter and the whole accuracy of the value of Pi is completely dependent on the accuracy to measure circumference and diameter?

 

[BobG]

I have no idea why anyone would calculate pi to that many places (or to a trillion decimal places, as has been done) since there is no instrument that could make measurements that precise.

Pi calculated to 39 decimal places is enough precision to determine the circumference of the observable universe to a precision equal to the radius of a hydrogen atom - provided of course that you could measure the diameter of the observable universe with a margin of error less than the radius of a hydrogen atom.

 

[gb7nash]

Found from wiki:

http://en.wikipedia.org/wiki/Pi

$$\pi = 4 \sum^\infty_{k=0}\frac{(-1)^k}{2k+1}$$

Have fun.

 

[Office_Shredder]

Bob, the reason for doing so is to show off how fast your computer is

 

[mathman]

Found from wiki:

http://en.wikipedia.org/wiki/Pi

$$\pi = 4 \sum^\infty_{k=0}\frac{(-1)^k}{2k+1}$$

This converges quite slowly. There are better ones.
http://en.wikipedia.org/wiki/Computing_π

 

[jobyts]

Found from wiki:

http://en.wikipedia.org/wiki/Pi

$$\pi = 4 \sum^\infty_{k=0}\frac{(-1)^k}{2k+1}$$

Have fun.

This converges quite slowly. There are better ones.
http://en.wikipedia.org/wiki/Computing_π

These are estimations. My question is when they say the value of Pi is accurate to so and so number of digits with respect to the reference value of Pi, how did they come up with the reference value of Pi.

 

[Office_Shredder]

These are estimations. My question is when they say the value of Pi is accurate to so and so number of digits with respect to the reference value of Pi, how did they come up with the reference value of Pi.

They're estimates of pi that are accurate to within 10^{-big number}. For example, the infinite series converges exactly to pi, and because it's alternating you can get a bound on the error if you truncate it at only finitely many terms

 

[jobyts]

They're estimates of pi that are accurate to within 10^{-big number}. For example, the infinite series converges exactly to pi, and because it's alternating you can get a bound on the error if you truncate it at only finitely many terms

Ah, ok...got it. Thanks all.

The mentioned wikipage also states "While that series is easy to write and calculate, it is not immediately obvious why it yields π.".
Does that mean there no proof for the Pi series equation?

 

[BobG]

Bob, the reason for doing so is to show off how fast your computer is

I probably shouldn't make fun of calculating pi to an enormous number of digits, considering I once figured out that I could count up to 1,099,511,627,775 using my fingers and toes.

Seemed kind of impressive until I figured out a rough estimate of how long it would take to count that high. Then I decided I'd save that task until someone invented immortality.

 

[brocks]

Ah, ok...got it. Thanks all.
The mentioned wikipage also states "While that series is easy to write and calculate, it is not immediately obvious why it yields π.".
Does that mean there no proof for the Pi series equation?

No, it means that some proofs are not immediately obvious.

In my own case, hardly any proofs are immediately obvious.

 

[DaveC426913]

I probably shouldn't make fun of calculating pi to an enormous number of digits, considering I once figured out that I could count up to 1,099,511,627,775 using my fingers and toes.

Speaking of an enormous number of digits, it would appear that you have 40 fingers and toes, twice the usual number of the average Homo Sapiens.

 

[DaveC426913]

In my own case, hardly any proofs are immediately obvious.