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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

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    I think figuring out these things can lead to unknown findings, for example, look at [**Pi Digits**](http://mathworld.wolfram.com/PiDigits.html) and [**Trillions of Digits using BBP (Bailey-Borwein-Plouffe )**](http://www.numberworld.org/misc_runs/pi-5t/details.html), and the [**Beautiful Formulas on this Wiki page**](http://en.wikipedia.org/wiki/Approximations_of_%CF%80)2012-12-15

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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.


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.


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.


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.

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    @Michael, loosely speaking, the probability that two randomly-chosen integers are relatively prime is $\frac{6}{\pi^2}$.2012-12-16