Estimating $\pi$ very accurately

Question

Possible Duplicate:
Do We Need the Digits of $\pi$?

I often hear about people who compute a lot of digits of $\pi$.Does estimating $\pi$ to a large degree of precision have any importance (or potential use) in mathematics ?

Thank you

Answer 1

In my opinion, the value in itself is of as much importance to mathematics as the value of any number, say $\sqrt{2}$.

On the other hand, estimating the value of $\pi$ is of great importance due to the vast amount of techniques it generates.

Think of all the cute power series and inverse trig relations of $\pi$.

1) Machin's formula:

$$\frac{\pi}{4} = 4 \arctan\frac{1}{5} - \arctan\frac{1}{239}$$

2) Leibniz formula for $\pi$:

$$\frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \cdots = \arctan{1} = \frac{\pi}{4}$$

3) Euler Formula for $\pi$ $$ \frac{\pi}{4} = \frac{3}{4} \times \frac{5}{4} \times \frac{7}{8} \times \frac{11}{12} \times \cdots$$

4) Bailey-Borwein-Plouffe formula

$$\pi = \sum_{i = 0}^{\infty}\left[ \frac{1}{16^i} \left( \frac{4}{8i + 1} - \frac{2}{8i + 4} - \frac{1}{8i + 5} - \frac{1}{8i + 6} \right) \right]$$

Apparently, the above formula can be used to extract the digits of $\pi$ from an arbitrary location!! I am reading it now (see spigot algorithms) due to a suggestion from Potato and I am curious about its behavior.

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Consider the memorable sharp integral bounds you generate:

$$\frac{1}{1260} = \int_0^1\frac{x^4 (1-x)^4}{2}\,dx < \int_0^1\frac{x^4 (1-x)^4}{1+x^2}\,dx = \frac{22}{7} - \pi < \int_0^1\frac{x^4 (1-x)^4}{1}\,dx = {1 \over 630}$$

See Lucas for interesting extensions. Especially the error margin of $\frac{355}{113}$ approximation.

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Think of all the sneaky ways $\pi$ can crop up surprisingly, like in the Buffon's needle problem. We can use this experiment to empirically estimate the value of $\pi$.

Euler apparently proved that if you pick two integers at random, the probability that they are co-prime is $\frac{6}{\pi^2}$. The first thing I asked myself the first time I saw it was 'how did $\pi$ appear?'

With normal distribution containing $\pi$ in it's p.d.f and due to central limit theorem, I wont be surprised if there are so many other ways of estimating $\pi$ empirically.

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Estimating $\pi$ seems to be an interesting hobby that has given rise to some beautiful methods ranging from the Gauss-Legendre algorithm, continued fractions, empirical probabilistic techniques, complex numbers, geometry, integrals and infinite series.

So I think the spirit of estimating a number is very important to mathematics, while the value of the number in itself may not have much importance.