Is there any meaning in asking about the dot product of the Rationals and Irrationals?

Question

If we order both the Rationals and the Irrationals in the normal way (least to greatest), could it make sense to define some sort of "dot product" where you multiply each element of the respective sets as they appear? What if we order the sets by their absolute value from least to greatest, with ties being resolved by placing the positive number before the negative one? Is there any math that tries to make meaning of these kind of statements?

Answer 1

The statement, "let us index the irrationals" is impossible. This was proven using Cantor's diagonalization contradiction.

Now you could index some irrationals and make a vector, and you can certainly talk about a "dot product" of such objects. The space of infinitely long vectors of real numbers is denoted by $\mathbb{R}^\infty$.

There are some caveats. For example, not all vectors will have a finite dot product, unlike the finite dimensional spaces.