How do you know one approximation to $\pi$ is more accurate than another? 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 &amp; 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 $\pi$?
We can prove that $$\pi=4\arctan 1 =4\sum_{i=1}^\infty \frac{-(-1)^i}{2i-1}=4\left(1-\frac 13+\frac 15-\frac 17+\dots\right)$$  There are series that converge much more quickly, but let's pretend that this is the only series so far known.  The alternating series theorem says that the error truncating a series like this is of the sign of and smaller than the first neglected term.  A lazy mathematician might compute the first hundred terms, getting $3.13159\frac {+0.02}{-0}$.  Another might sum the first thousand terms, which would give a maximum error of $0.002$ and would clearly be better.  Other formulas also come with a bound on the error, so we know how bad it can be.
This claims π is rational, which is known to be false.

Proof No.1

( π-3)×3^3= 4 
A circle of diameter 1 is inscribed within a square where the side of the square and the diameter of the circle is one and the same.

Now the circumference of the square 4 is divided by 3 diameters and the quotient of 1.33333333…is further divided by 3^2 (3 diameters squared) the result is 0.148148148… Accordingly, the rational pi is 3.148148148…

Applying this quantum to the equation ( π-3)×3^3= 4

  (3.148148148…- 3)×3^3= 4


We can arrive back to 4 which is the perimeter of the square. But if we substitute the irrational pi, then the result is

(3.141592653…- 3)×3^3= 3.823001631…

which is lesser than 4. But as the perimeter of the square is 4, therefore, the side of the square and the diameter of the circle inscribed within that square ought to be 1 each but it is only(3.823001631/4)0.9557504078.

Proof No.2. (Pythagorean)
( π-3)×3^3= 4 = 2^2

We can carve out an equilateral triangle from the Archimedean method of hemming a circle between two regular hexagons to approximate the circumference of the circle. Now we can divide the equilateral triangle into two right angles. It becomes a right angle of 30, 60 and 90 degrees.

It is in the proportion of hypotenuse being 2, adjacent being √3 and the opposite being 1 unit wise, the quantum rational pi could be arrived at.

( π-3)={(hypotenuse^2)/(adjacent^2 )}÷(hypotenuse +opposite)^2

( π-3)= {2^2/√(3^2 )}÷(2+1)^2 = 0.148148148........

Applying this quantum to the equation ( π-3)×3^3=4 = 2^2

(3.148148148…-3)×3^3= 2^2

We can return back to the unit of the hypotenuse. But it is impossible with the pi of 3.141592653…..which is irrational.