This question is hopelessly vague, but here goes:

Say I'm given some finite precision complex number, which I'm told is algebraic over $\mathbb{Q}$. Is there some well defined notion of arithmetic complexity which can allow me to deduce exactly what algebraic number this finite precision number represents?

For example, say I'm given $\sqrt{2}$ to 50 decimal places. Then what number this represents could be one of many things: the rational number which is $\sqrt{2}$ to 50 decimal places itself, or precisely $\sqrt{2}$, or other things. In some sense $\sqrt{2}$ is the "right" answer: it is definitely easier to describe than the other possibilities.

Arguably the general situation is not going to be as clear cut as in this case, but I'd love to know if there is any general theory along these lines that I haven't heard of. It seems like you can do a fair amount of stuff just working with a height function on number fields, and finding all elements agreeing up to that given precision within certain bounds for the height, but I feel there should be much more to say.

For example, given a number field $K$ of degree $d$ over $\mathbb{Q}$, we can define a height by $h(a) = \frac{1}{d} \sum_{v \in M_K} \log \left( \mathrm{max}(1,v(a) \right )$ where $v$ runs over the set $M_K$ of valuations. I don't know how good a choice this is, but it is explicit enough to allow to use it to show some of the above properties.

Edit: Actually, a definition that is easier to deal with is something like Kevin mentioned: set the height $H(a)$ to be the sum of the degree of $a$ and of the absolute values of the coefficients of its minimal polynomial. Then it is definitely true that there are only finitely many numbers with height $H$ lower than some given bound. This makes it very easy to deal with on a case by case basis (although it is obviously not computationally effective), so I'm more interested in what general type of theorems you can get.

A good example of the flavour of the theorems I have in mind would be the Thue-Siegel-Roth Theorem: in a way, it expresses how hard it is to approximate algebraic numbers with rational numbers. In particular, it gives an idea that it is hard to approximate an algebraic number of low complexity by a rational number of similarly low complexity. But in my mind this should be just the beginning.

(No idea what tags to use: I don't think complexity theory is adequate, but I can't find any better)