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

Let’s say that you wanted to measure something, like the height of a room. You might start with a simple guess: it looks 10 feet high. You might then base it on a size you know: you’re 6 feet tall and it looks like 2 of you could fit the space, so it seems 12 feet high. You decide to be more accurate, so you grab your hockey stick and reach up to the ceiling, estimating that the space is 11 feet high. You then stand on top of a 5 foot shelving unit, on your tip toes, and put your head against the ceiling, getting a measurement of 11 feet 5 inches. You can continue this method over and over with more and more accurate measuring devices to get better measurements.

This is similar to many of the ways pi is calculated. You’re doing an infinite number of measurements to get better and better results. The first measurement might give the result 3. The next, 3.2. The next, 3.17. The next, 3.145. And so on. Each new measurement confirms a certain amount of numbers that came in the earlier calculations and gets you closer to cementing the next number in the sequence.