If a number like Pi is infinite, how do we know each decimal that is newly calculated is valid?

Question

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

Answer 1

This is not how mathematics works. Mathematics is all about rigorous proofs. You cannot proof that a decimal representation of π is correct up to say 5 places by testing it a bunch of times. That doesn’t proof anything, maybe you just never found a counterexample even though one exists.

Instead you would show that a way of computing π computes a number that fulfills all defining characteristics of π.

Mathematics is not really about doing computations yourself. Instead it is about building “chains” of logical arguments. True statements lead to true statements which then lead to true statements and so on.

This is what computers do while looking for new decimals of π. Mathematicians have developed computational steps which can be formally (not by experiments!) proven to result in the correct decimal representation of π.