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What would be the theoretical mathematical implications/repercussions if computers kept calculating Pi, and then after some trillions of digits got all zeroes, when Pi stopped recording additional digits? Showing that Pi is actually the ratio of two whole numbers.
Are mathematicians sure that that could not happen, or are they sure that it eventually must happen? Or would it just show that computation had reached the limit of how it could process the entered data?
It would change the notion that it is not a ratio of whole numbers to the notion that it is. That would change nothing?
It would change that statement you made, that's it. Beyond that, what use does it have? Applying the number to any scale with 5 trillion decimal places would be even less than insignificant. It would not have an impact, anywhere, on anything.
Unlike physics, where the laws are fixed by the universe and one could reasonable ask what would happen if they were slightly different, any contradiction in a mathematical system means the contradiction of the entire system. I don't think one could create a theory of real numbers where Pi is rational.
What would be the theoretical mathematical implications/repercussions if computers kept calculating Pi, and then after some trillions of digits got all zeroes, when Pi stopped recording additional digits? Showing that Pi is actually the ratio of two whole numbers.
Are mathematicians sure that that could not happen, or are they sure that it eventually must happen? Or would it just show that computation had reached the limit of how it could process the entered data?
It can't stop. Once upon a when, I took a class on non-linear differential equations, while not exactly relevant to computing Pi, it did help with my understanding of infinite power series. Using the Taylor Series, it's possible to determine Pi is an irrational number which will never converge on a finite value.
Exactly how to prove this, would require communication with my younger and smarter self.
What would be the theoretical mathematical implications/repercussions if computers kept calculating Pi, and then after some trillions of digits got all zeroes, when Pi stopped recording additional digits? Showing that Pi is actually the ratio of two whole numbers.
Are mathematicians sure that that could not happen, or are they sure that it eventually must happen? Or would it just show that computation had reached the limit of how it could process the entered data?
If you mean adding a trail of zeroes instead of nonzero numbers, then that would mean that pi is a rational number, as opposed to an irrational number. Mathematicians would just categorize differently.
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