is there any way to represent irrational numbers with a finite amount of integers? I know that rational numbers can be represented with two integers $\frac{a}{b}$.
But is there any way to represent irrational numbers with an finite amount of integers?
My best guess is $\frac{a}{b} ^ \frac{c}{d}$.
it can represent any root of any number, but I don't know if it can represent things like $\sqrt{2}^\sqrt{2}$.

Suppose you have an encoding scheme allowing you to express a subset $C\subseteq \mathbb{R}$ of real numbers using, for each, a finite number of integers. Then you have a bijection between $C$ and
$$
\cup_{k\in \mathbb{N}} \mathbb{N}^k
$$
and since this is countable, then so is $C$. But the irrational numbers are uncountable.
