I'm trying to find the established name, if there is one, for what I'll call "rational connection" in this post.
So we would say some irrational number $x$ is "rationally connected" to another irrational number $y$ if $$x=a\times y^b + c$$ where $a$, $b$ and $c$ are rational numbers
For example:
The sum of the reciprocal of the squares is the irrational number $1.6449...$, and this number is "rationally connected" to $\pi$, as it can be represented in the form $a\pi^b + c$ where $a$, $b$ and $c$ are rational numbers, specifically $\frac{1}{6} \pi^2 + 0$.
One example of two irrational numbers that aren't "rationally connected" are $e$ and $\pi$.