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
The formulas for pi come with a mathematical **proof**. An irrefutable argument from basic logics and our agreed-upon meaning of arithmetic. Everyone can check and verify it, and many indeed did already; nowadays computers can verify such proofs, too. Altogether a _lot_ of very good people would need to have overlooked an error if there is any.
And that’s not even where it stops: we can use another formula derived in a completely independent way. If it spits out the very same digits then we are even more certain that we didn’t screw up.
If your own formula disagrees you have to show not only it but also the proof of correctness to experts; and they very likely will spot the error very quickly.
Actually the by far largest chances for errors are in the hard- and software. There can be a coding mistake, which however usually gets spotted quickly when it goes off the rails; but nonetheless such codes often get verified by several programmers and even specialized programs. Hardware usually is fine, but there was a case (early 90s I think) where a rare and previously unknown fault in some intel CPU was found when people calculated pi on it and something like the million-th digit was wrong.
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