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

Pi is indeed an infinite number with an endless string of decimal places. It’s true that we cannot calculate or write down all of its digits because they go on forever. However, mathematicians have developed clever ways to make sure that each new decimal they calculate for Pi is valid.
They use something called algorithms, which are like special step-by-step instructions, to calculate Pi. These algorithms are based on mathematical formulas and rules that have been proven to work correctly. Mathematicians have checked and double-checked these algorithms many times to make sure they are accurate.
When a computer calculates Pi, it follows these algorithms and performs lots of calculations. The calculations become more and more precise with each step. While it cannot calculate all the digits of Pi at once, it can keep going and generate more and more accurate digits as it performs the calculations.
To make sure the new digits are correct, mathematicians use something called mathematical proofs. These proofs are like logical explanations that show why something is true. For Pi, mathematicians have proven that the algorithms used to calculate it will always produce the correct digits.
So, even though we can’t calculate all the digits of Pi at once, we trust that each new digit calculated using these algorithms is valid because they have been tested, verified, and proven to work correctly. This is how mathematicians and computers keep generating more and more accurate decimal places of Pi.