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
Two things.
First, because we can prove analytically that some formulae *must* be equal to π, or some multiple of π, even though we don’t know what all the digits of π are. As long as we precisely carry out those formulae, then there’s no need to “check” that it actually is π. We already know that analytically (meaning, based on the definitions of the things used, not on any *observations* about anything.)
Second, there are multiple distinct formulas that all get us π. The sum of 1/n² from 1 to infinity is ~~π/6~~ *π²/6*. The integral from -1 to 1 of √(1-x²) is π/2. Etc. By using different formulae, even if there were any uncertainty, we could check with a different calculation.
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