I’m not sure if I worded this correctly but I’ll try to elaborate more. I know that there are formulas that can calculate pi to the Nth digit. But my question is how can we be certain that the formulas accurately calculate the Nth digit of pi when we have nothing to compare it to since pi is an irrational number that keeps on going?
For example if I discovered a new formula than can calculate pi to 1 higher digit than what we currently know and the value I got was 4. How do I confirm that it is indeed 4 and not any other number? I have nothing else to compare it to?
I hope this makes sense
In: Mathematics
We have formulas, more specifically usually some kind of infinite sum that we can prove will converge towards pi. Often we can also say something about how quickly it converges. The fact that they converge to pi can be proven without having to know the digits of pi, with the same tools we use to prove other things in mathematics. For examples you can check out [https://en.wikipedia.org/wiki/Pi#Rapidly_convergent_series](https://en.wikipedia.org/wiki/Pi#Rapidly_convergent_series)
If you want to see what a proof might look like, you can look at the proofs for the [Leibnitz Formula](https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80#Proof_1)
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