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

The decimal expansion for pi is not found by “testing in reality”. Using properties of circles and polygons inscribing and circumscribing a circle, mathematicians before the late 1600s found estimates on pi from below and above. For instance, if they found pi is between 3.1414 and 3.1416 then they knew pi starts out as 3.141… I am using decimals for their familiarity to you, but ancient mathematicians did not have the decimal system. But the idea of lower and upper estimates with fractions still made sense to them (a finite decimal is a special type of fraction.)

Once calculus was created, completely new methods were developed to rapidly compute pi to hundreds of digits. See the Veritasium video “The Discovery that Transformed Pi”.