It's known that imaginary numbers can be expressed as vectors such as $\alpha + \beta i$. My question is: Can you represent a computable real number with a similar algebraic definition?
For example does there exist some constants such that:
$$e =\alpha + \beta \sqrt{2}$$
Where $\alpha \in\Bbb{Z}$ and $\beta \in\Bbb{Q}$ and $e$ is Euler's number.
Computable reals are things that can be approached arbitrarily closely by an infinite sequence of rational numbers so is it possible to express any computable real in this way?
You can't have a single computable number that you can use to decompose all computable reals in that way.
You can't even have any finite number of "basis elements", even if you allow arbitrary rational functions of the basis elements rather than just the integer/rational combinations you describe.
This is because there exists a sequence of infinitely many algebraically independent computable numbers -- such as $$e^e, e^{e^2}, e^{e^3}, e^{e^4}, \ldots$$ thanks to the Lindemann-Weierstrass theorem. So the field of computable reals has transcendence degree $\aleph_0$ over $\mathbb Q$.