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

There are various known algorithms that have been proven to spit out digits of pi. As long as these are implemented correctly, there isn’t really any doubt that they work. Some of these algorithms are “spigot algorithms” that allow you to calculate a given decimal place without having to calculate all the previous ones, which makes it especially easy to perform checks: if someone claims they have calculated a billion digits of pi, you can just pick a random sample and check them instead of repeating their entire calculation.

In practice, calculating new digits of pi is a pretty niche activity and nobody really uses the results for anything. So they may not be checked particularly carefully. It’s always possible that a bug or hardware fault will cause mistakes.