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
There are lots of infinite sums that we know converge to multiples of pi for a variety of geometric reasons. For each of them, you can determine how close you are to pi based on how many terms you’ve added. So if the terms you have left to add, which keep getting smaller, have some upper bound, then you know how many decimal places you can count on.
[Matt Parker](https://youtube.com/@standupmaths) uses one of them each Pi Day to calculate pi in some goofy manner.
If all of those series produced different values for pi, then we’d have a problem. But they don’t.
If you came up with a new series for calculating pi, then first you’d want to prove it actually does work out to pi. After that, whatever new decimal place you got would be correct, assuming you did the sum right.
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