I have always been confused regarding the accuracy of $\pi$.
In the books which are written on this subject $\pi$ , there are references of people and their methods for finding the value of $\pi$. Person A found the value of $\pi$ and then it says that person B found more accurate value of $\pi$ (may be up to $100$ decimal places) and then person C found much more accurate value of $\pi$ and this goes on.
My question is: How does the mathematician know that the value of $\pi$ which he has calculated based on whatever algorithm he applied is more accurate then the value calculated by the previous mathematician?
Imagine a scenario, where the world has just started and there are only $2$ mathematicians (A & B) in it. Mathematician A has calculated the value of of $\pi$ as $3.1547$ (this value is in ancient Chinese text), now the other mathematician says he has calculated a more accurate value, i.e. $3.1416$.
My question is How is the Mathematician B so sure that $3.1416$ is a more accurate value?
I mean there is no standard with which to compare.
Wikipedia says: The Indian astronomer Aryabhata used a value of $3.1416$. Fibonacci in c. 1220 computed $3.1418$. Italian author Dante apparently employed the value $3.14142$.
When there is no standard to compare, how will I know which is the correct value of $\pi$?