We know that for a function (adhering to certain rules). we can have a Fourier series, which gives us the function as the sum of sines and cosines. Thus, we can say that almost all functions (adhering to those rules) are "made up" of sines and cosines. We can say that these functions belong to a vector space with sines and cosines of integer angular frequencies as basis (in a lose sense as the dimension of such a vector space will be infinite).
Can we have such a thing for irrational numbers?
For example, $\sqrt{2}$ and $\sqrt{3}$ are two irrational numbers. Any number of the form $$a\sqrt{2}+b\sqrt{3}$$
where $a,b$ are integers, belongs to a vector space. Now, is there any way that if we were given a number from this vector space, in decimal form, then we could find $a$ and $b$.?
Also, lets say that now, we consider all irrational numbers which are rationally linearly independent, and form a vector space. Is there any way, that, if we have a number (in decimal form), then we can obtain the irrational numbers, used to produce it?
The answer is yes if your basis is made of algebraic numbers. Let $S=\{a_1,a_2,...,a_n\}$ be a set of algebraic numbers with heights $h_k$. Then the minimal polynomial for $\alpha=\sum_{k=1}^n s_k a_k$ has height less than $H<2n+\sum_{k=1}^n h_k$ (see here). This mathoverflow link is about using the LLL algorithm to find the minimum polynomial of a decimal expansion of an algebraic number, and $H$ is needed to make the algorithm work.