Let me show you my resoning:
The inverse of addition is subtraction. To complete subtraction, we create the negative numbers $(-5)$.
The inverse of multiplication is division. To complete* division, we create the fractional numbers $(3/7)$.
The inverse** of exponentiation is root. To complete* root, we create the root numbers $(√2)$.
The inverse** of tetration is super-root. To complete* super-root, we create the super-root numbers $(Super√57)$.
And so on with every following hyperoperation.
*We still have division by zero and square roots of negative numbers, we need other numbers to complete them. **Also, there are other invseres, the logarithms.
We can then group up the numbers:
Naturals + Negatives = Integers
Integers + Fractionals = Rationals
Rationals + Roots = Rootionals
Rootionals + Super-roots = Super-rootionals...
Except that we don't do this for some reason. Instead, we make a cutoff at multiplication and say, "everything that isn't rational is irrational. That's it". I was wondering if anyone knows the logic behind such grouping of the numbers, as opposed to the one I expected (which I think is more intuitive).
The only possible reason I could think of, is that rational numbers can be written explicitly, while irrational numbers have infinite decimals so they can't. We can only write, "the number that exponentiatied to itself is $57$". Maybe that's very important to mathematicians, more so that keeping intuition?
To go further beyond the remark made by CyclotomicField in the comment, that’s not necessarily the most fruitful outlook on the study of numbers (though it’s been mostly considered throughout history). For instance, you made mention (in a footnote) of the necessity of other numbers “needed to complete [the square roots of negative numbers]”. This has been done—in the complex numbers.
The root numbers, as you call them, are all special cases of zeroes of polynomial equations, generally of the form $$a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0=0\,,$$ where the coefficients are (for now) integers. These are the “algebraic” numbers; for instance, your root number $\sqrt{2}$ is a solution to the equation $z^2-2=0$. If a number is never the solution to a polynomial equation as above, then they’re called “transcendental”—famous examples are $\pi$ and $e$ (the base of the natural logarithm).
This is a big story which continues today in special and broad subject areas —algebraic number theory, transcendental number theory, arithmetic algebraic geometry, etc.