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

No-one seems to be really answering the question asked.

To verify “newly calculated digits of pi” you need to either:

1. Calculate it twice using two different methods, or
2. Randomly check some of the new digits using a method that only calculates one digit “anywhere” in the sequence.

In the first case it takes at least twice as long because if you’re calculating with the fastest algorithm you’ll have to check with only the second fastest algorithm.

In the second case you need to use an algorithm like Bailey-Borwein-Plouffe that can calculate one digit (actually one hexadecimal digit) anywhere in the sequence. So you might be calculating 1 million new digits but after you can just check 1000 of the new ones with BBP and if they’re all correct then you have confidence that all 1 million are correct.