What does Pi equal to 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.
 A: 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.
A: I would say $\frac{314159265}{100000000}$
A: 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.
A: 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.
