How do we know the digits of pi is infinite

Question

I know that some computers have calculated up to trillions of digits, but how do we know that it just doesn’t end at sone point?

In: Mathematics

Answer 1

1. We’ve proven that pi is irrational. An irrational number is one that can’t be represented as the ratio between two whole numbers.

https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational

The proof itself is unfortunately too complicated for ELI5.

2. We’ve proven that if a number’s decimal expansion is repeating, then it is rational. For example, if x=0.123123123… Then 1000x=123.123123123…, so we can calculate 1000x-x = 123, therefore x=123/999, i.e. x is rational. Therefore, if a number is irrational, its decimal expansion must be infinite and non repeating.