Is Pi the Most Precise Number in Existence?

[Christopher Small]

How do we know pi to so many decimal places? How can that be determined so precisely?

http://wikisource.org/wiki/Pi_to_1%2C250%2C000_places

 

[DeadWolfe]

http://mathworld.wolfram.com/PiFormulas.html

 

[Data]

There are integrals and series that converge exactly to $$\pi$$. In the case of integrals, we can approximate their values numerically to arbitrary precision, and in the case of series, we can evaluate enough terms so that the sum is as close as we like to the real decimal representation. Some examples are

$$\pi = 2\int_{-1}^1 \sqrt{1-x^2} dx$$

and

$$\pi = 2\sum_{k=0}^\infty \frac{n!}{(2n+1)!}$$

The integral above is just the area of a circle of radius 1, and the series is one derived due to a theorem by Euler. These and many more can be viewed at http://mathworld.wolfram.com/PiFormulas.html.

There are much more computationally efficient algorithms used to calculate $$\pi$$ quickly by programs like SuperPi.

 

[saltydog]

How do we know pi to so many decimal places? How can that be determined so precisely?

http://wikisource.org/wiki/Pi_to_1%2C250%2C000_places

The technique of "multi-precision" and other methods are used. Essentially, you store individual digits in arrays and do the arithmetic on the elements of the array. Check out "PIFAST" on the web (it's free) and is the fastest program to calculate PI and other numbers to lots of digits. I used it to calcuate e to 500 million places. Why? Uhhhhh ... for fun?

Oh yea, how do I know it's correct? Well, if you work with multi-precision, even write programs for doing it, you gain confidence that the algorithms are sound. I was very skeptical at first but after writing software for it, I'm 100% confident the results are correct.

 

[The Rev]

Kinda OT, but did anyone know that today is National Pi Day (because the date is 3.14).

Happy Pi Day to all!

:D

The Rev

 

[arildno]

Gee, is it Pi Day today?
Thanks for informing me..