Not a mathematician here at all so perhaps my question is phrased incorrectly.
Let’s say through thorough testing in reality, we can prove with certainty Pi is correct up until 5 decimal places,
3.14159
The computers that are calculating Pi to an endless degree, how do they validate new values that are calculated as correct and cannot be otherwise?
In: 434
Ok. So in order to calculate pi you need an algorithm that converges to pi. So some series that’s limit is pi.
One algorithm would be its definition, the circumference of a circle devided by its diameter. Using a circle with radius ½ its circumference is pi.
So you approximate a circle with polygons with more and more sides and add the side lenghts. This approches pi.
Now there are other definitions, and way more efficient algorithms but the key is convergence:
In certain spaces, like what we are using now we can say that every series converges if its a Cauchy series. A Cauchy series is a series where the distance between the n -th and the n+1 -th element is smaller than between n and n-1. So lets say we have some distance function d so d(n,n+1)<d(n,n-1) for every n. If any series with this property converges we say our space is complete in this sense. If our space is a healthy vector space, we call this kind of space a Banach space.
We are going to find pi in our healthy Banach space so all we need to show is that a series we use has the Cauchy property. There are many series that converge to pi or something proportional to it. And we can call any such series the definition.
So how do we know that a Cauchy series approches pi and not something that really close to it? Well because the limit of many such series can be found analytically. So you don’t need to represent pi in its decimal form, you can find the limit analytically and pi would appear as a symbol. But if so this means that for any finite step this will be a better and better approximation as long as you actually care about the decimal representation.
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