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 $\sqrt{2}^\sqrt{2}$.

If ${\frac{a}{b}}^{\frac{c}{d}}=\sqrt{2}$ then $(\log{\frac{a}{b}})*{\frac{c}{d}}=\frac{log{2}}{2}$, which implies
$\frac{a}{b}=2, \frac{c}{d}=\frac{1}{2}$.
But if ${\frac{a}{b}}^{\frac{c}{d}}=\sqrt{2}^\sqrt{2}$ then $\frac{c}{d}=\sqrt{2}$, and we would have found a rational number equal to $\sqrt{2}$. This is impossible, consequently there is no solution for ${\frac{a}{b}}^{\frac{c}{d}}=\sqrt{2}^\sqrt{2}$ among the rational numbers and, more generally, there can be no way of writing every real number using only a finite number of rational numbers and operations.
