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

First of all, pi is not infinite.

Pi < 4 < infinity and pi > 3 > -infinity.

Pi has an infinite decimal expansion, but so do plenty of other numbers. You’ve been given examples of other irrational numbers like sqrt(2), but plenty of rational numbers have infinite decimal expansions too. 1/3, for example, is 0.333… that goes on infinitely too (the digit just happens to be conveniently repeating, which pi does not).

Another thing to consider is that the fact that we express numbers in a base-10 system is pretty much solely due to the fact that we happen to have 10 fingers. For example, in base pi, pi itself is just 10.