# Estimating $\pi$ very accurately [duplicate]

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

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No.${}{}{}{}{}{}$ – Graphth Dec 15 '12 at 23:14
Do you have any idea why its done ? – Amr Dec 15 '12 at 23:14

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Because we're nerds. – vermiculus Dec 15 '12 at 23:16

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It is of great importance, but the reasoning involved is circular. – copper.hat Dec 15 '12 at 23:18

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Knowing many digits of pi has no practical significance, but developing the methods needed to compute many digits of pi in a reasonable amount of time is of use, as those methods have applications in other places. Also, while attempting to compute digits of pi, we have created some beautiful mathematics, for example the Bailey–Borwein–Plouffe formula. – Potato Dec 15 '12 at 23:25
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## marked as duplicate by Henning Makholm, sdcvvc, Argon, Martin Argerami, Asaf KaragilaDec 16 '12 at 0:03

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## 1 Answer

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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Great answer. One small suggestion: Add the Bailey-Borwein-Plouffe formula! – Potato Dec 15 '12 at 23:48
Nice post, are there any other examples similar to Buffon's needle problem where $\pi$ shows up unexpectedly? – Michael Boratko Dec 15 '12 at 23:48
@Potato: Woah!! The idea of Bailey-Borwein-Plouffe formula is wonderful.Let me add it. – Isomorphism Dec 15 '12 at 23:55
@Michael, loosely speaking, the probability that two randomly-chosen integers are relatively prime is $\frac{6}{\pi^2}$. – Dan Brumleve Dec 16 '12 at 0:03
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