so the proof of the irrationality of pi is a bit more than i want to get into here, but there’s a very simple proof that there are infinite prime numbers which i will share here.
suppose that there is a finite number of prime numbers. write out a list of all of these prime numbers, and multiply them together. add one to this product, and you now have a number that is not divisible by any of our list of prime numbers, and thus should be another prime. this contradicts our initial assumption of finite primes, and therefore there are infinite primes.