Digits of irrationals

Question

I've been studying floating point arithmetic and I've read somewhere that numbers with infinitely many decimal digits without recursion are irrational.

But since we can't know all the digits of such a number then how did we come to the conclusion that its digits have no recursion? Does it have anything to do with formulae used to compute the $n$-th digit of a number?

(This is a question simply out of curiosity.)

Answer 1

The simpler (and ancient) way to know if a number $a$ is irrational is to explicitly show that it cannot be expressed as a quotient $\frac{n}{m}$ of two integers $n,m$.

But there are numbers, as the number $\pi+e$, for which we don't know if they are rational or irrational.

Answer 2

Floating-point (as well as fixed-point) arithmetic is just unable to represent irrationals and most rationals.

Actually, the floating-point numbers are essentially integers in a finite range, with a movable point, and can't have more than 16 (significant) decimal digits.