I’m trying to put in more rigorous math terms what it means to simplify a product of any two abstract things (in the sense that I mean it), and what makes it work.
Here’s an example: $2\times3=6$, but $2\times i$ remains $2i$ (i is imaginary). I can’t describe exactly what I mean by simplify, which is why it’s part of the question. So I intend on using this arbitrary example to give you clue of what I’m thinking about. In fuzzy terms, it’s like they somehow don’t ‘combine’. Btw, I only used the complex multiplication tag because general multiplication wasn’t an option.
You put in 2 and 3 and can get out something that’s neither (i.e. 6). But the same isn’t true of 2 and i, both are still present in the end form and still being multiplied. Sure, you could rename it c or something, or write 2 as $\sqrt{2}\times\sqrt{2}$ to make two different and seemingly combinable things look nicer and avoid the formalism of the problem, or change 2 to $1+1$ and i to $\sqrt{-1}$ to try to prove that the elements in the involved product has actually changed, but that doesn’t solve it, meaning it’s more about the ideas than notation.
You could say it’s because multiplication is closed over real numbers, but why assign 2 and 3 to the set real numbers? Since in fact, you could make the same argument with 2 and i as both part of the complex plane (also closed under multiplication). So what quality does make things combinable instead?
Feel free to ask for clarification (I’ve made my best effort to explain it), and thanks in advance!