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This question already has an answer here:

What is the approximation of pi in a fraction form. I am very curious to know what it is. I have been seeing pi almost everywhere.

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marked as duplicate by user147263, Stahl, JimmyK4542, colormegone, Bruno Joyal Jan 20 '15 at 5:03

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ 314159265/100000000 $\endgroup$ – ASKASK Jan 20 '15 at 4:16
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    $\begingroup$ May I recommend Wikipedia as a first place to look up questions like this? They have a very comprehensive article on $\pi$. $\endgroup$ – Johanna Jan 20 '15 at 4:17
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    $\begingroup$ @ASKASK You should post that as an answer. It made me laugh. $\endgroup$ – Axoren Jan 20 '15 at 4:19
  • $\begingroup$ mathworld.wolfram.com/PiContinuedFraction.html $\endgroup$ – WillO Jan 20 '15 at 4:19
  • $\begingroup$ 22/7 and 355/113 $\endgroup$ – zed111 Jan 20 '15 at 4:26
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The idea here is that you can compute the continued fraction of $\pi$ and then truncate it somewhere to achieve an approximation of $\pi = 3.141592653589\dots$ $$ \pi = 3+ \cfrac{1}{7+ \cfrac{1}{15+ \cfrac{1}{1+ \cfrac{1}{292+ \cfrac{1}{1+ \cfrac{1}{1+ \cfrac{1}{1+ \cfrac{1}{2+\dotsb }}}}}}}} $$

The further along you truncate it, the more accurate your approximation. If we cut it off pretty early (at $7$), we get the classic $\frac{22}{7}$ approximation: $$ \pi \approx 3 + \frac{1}{7} = \frac{22}{7} = 3.14\color{red}{28571428571\dots} $$ If we cut it off later (say at $292$), we get a better approximation: $$\begin{align} \pi &\approx 3 + \cfrac{1}{7+\cfrac{1}{15+\cfrac{1}{1+\frac{1}{292}}}} \\ &\approx 3 + \cfrac{1}{7+\cfrac{1}{15+\frac{292}{293}}} \\ &\approx 3 + \cfrac{1}{7+\frac{293}{4687}} \\ &\approx 3 + \cfrac{4687}{33102} \\ \pi &\approx \frac{103993}{33102} = 3.141592653\color{red}{0119\dots} \\ \end{align}$$

You can use this method to get a rational number that is as close to $\pi$ as you need.

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I would say $\frac{314159265}{100000000}$

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There are many fraction forms of $\pi$, like $$\frac{\pi}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}...$$ which is based on the simple fact $$\int_{0}^{1} \frac{1}{1+x^2} dx=\arctan 1=\frac{\pi}{4}$$ And another famous form is the Wallis product $$\frac{\pi}{2}=\frac{2\centerdot2\centerdot4\centerdot4\centerdot6\centerdot6...}{1\centerdot3\centerdot3\centerdot5\centerdot5\centerdot7...}=\displaystyle\lim_{n\to\infty}\frac{2^{4n} (n!)^4}{[(2n)!]^4(2n+1)}$$ which is derived from the evaluation of $\int_{0}^{\frac{\pi}{2}} \sin^m xdx$.
With these, you can approach $\pi$ as close as you want, using a fraction. And as they converge quite fast, esp. the first series, it won't take much trouble to achieve a highly-accurate fraction approximation.

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    $\begingroup$ Actually the convergence of first series is very slow. $\endgroup$ – user207868 Jan 20 '15 at 5:19
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One approximation of $\pi$ is $\frac {22} 7$. It's approximate to 2 decimal places (and the third decimal place isn't that far off).

How approximate do you want it? We can always find a fraction that is approximately pi to so many decimal places.

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