Irrationality Preserving Operations

Question

We all know about rationality preserving operations. That is, if $r,s\in\mathbb{Q},$ we have that $r+s,r-s,rs,\frac{r}{s(≠0)}\in\mathbb{Q}.$ I was wondering if there are irrationality preserving operations. I know that addition isn't such an operation. To demonstrate this, we can just take $1+\sqrt{2},$ and $2-\sqrt{2}.$ Similarly, subtraction doesn't work either. Multiplication and division don't work either. A few examples would easily demonstrate this. Exponentiation doesn't work either. For this, we can consider $e^{\ln(2)}.$ This leaves me to wonder if there are any operations that preserve irrationality at all. All the operations I considered took two inputs, and I found no such irrationality preserving operations. So, my question is: Are there are irrationality preserving operations, and if there are, what is the minimum number of inputs they need?

Note: I am not considering the "do nothing operation" or operations that take only one input.

Answer 1

There certainly are, but most of them are impossible to describe. One that we can is digit alternation. Given $a,b$, express each in base $10$, using the terminating representation if there are two. Then create the result by alternating the digits of the two numbers each direction from the decimal point, so $f(3.14159,2.71828)=32.1741185298$. This will give a rational output if both inputs are rational and an irrational output if either input is irrational.

Answer 2

If $n$ is an integer greater than $1$, taking an $n$th root of an irrational number gives you an irrational number. If $a$ is irrational and $\sqrt[n]{a} = p/q$ were rational, then $a = p^n/q^n$ which would be rational, contradiction.