How are first digits of $\pi$ found? Since Pi or $\pi$ is an irrational number, its digits do not repeat. And there is no way to actually find out the digits of $\pi$ ($\frac{22}{7}$ is just a rough estimate but it's not accurate). I am talking about accurate digits by either multiplication or division or any other operation on numbers.
Then how are the first digits of $\pi$ found -

3.1415926535897932384626433832795028841971693993...

In fact, more than 100,000 digits of $\pi$ are found (sources - 100,000 digits of $\pi$)
How is that possible? If these digits of $\pi$ are found, then it must be possible to compute $\pi$ with some operations. (I am aware of breaking of circle into infinite pieces method but that doesn't give accurate results.)
How are these digits of $\pi$ found accurately? Can it be possible for a square root of some number to be equal to $\pi$?

One interesting property of $\pi$ is this (though it is very hard to prove):
\begin{align}
4>&\pi\\
&\pi>4-\frac43\\
4-\frac43+\frac45>&\pi\\
&\pi>4-\frac43+\frac45-\frac47\\
\end{align}
etc.
Theoretically, you can use this to get $\pi$ to any number of digits. However, you need to add up hundreds of terms just to prove that $3.14<\pi<3.15$ (or, what is the same, that the first three digits of pi are $3.14$).
