Palmosigx

2022-07-02

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.

Nicolas Calhoun

Expert

The idea here is that you can compute the continued fraction of $\pi$ and then truncate it somewhere to achieve an approximation of $\pi =3.141592653589\dots$
$\pi =3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{2+\cdots }}}}}}}}$
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:
$\pi \approx 3+\frac{1}{7}=\frac{22}{7}=3.1428571428571\dots$
If we cut it off later (say at 292), we get a better approximation:
$\begin{array}{rl}\pi & \approx 3+\frac{1}{7+\frac{1}{15+\frac{1}{1+\frac{1}{292}}}}\\ & \approx 3+\frac{1}{7+\frac{1}{15+\frac{292}{293}}}\\ & \approx 3+\frac{1}{7+\frac{293}{4687}}\\ & \approx 3+\frac{4687}{33102}\\ \pi & \approx \frac{103993}{33102}=3.1415926530119\dots \end{array}$
You can use this method to get a rational number that is as close to π as you need.

Augustus Acevedo

Expert

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