# What does Pi equal to [duplicate]

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.

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

• May I recommend Wikipedia as a first place to look up questions like this? They have a very comprehensive article on $\pi$. – Johanna Jan 20 '15 at 4:17
• mathworld.wolfram.com/PiContinuedFraction.html – WillO Jan 20 '15 at 4:19
• 22/7 and 355/113 – zed111 Jan 20 '15 at 4:26

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.

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

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.

• Actually the convergence of first series is very slow. – user207868 Jan 20 '15 at 5:19

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.