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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!

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    $\begingroup$ $(\color{red}2 + \color{blue}0i) \times (\color{red}0 + \color{blue}1i) = (\color{red}0 + \color{blue}2i)$. In terms of visualizing complex numbers as tuples $(2,0)\otimes (0,1) = (0,2)$ where $\otimes$ is the multiplication of complex numbers: $(a,b)\otimes (x,y) = (ax - by, ay+bx)$. That simplified just the same way that $2\times 3 = 6$ did. $\endgroup$ – JMoravitz Oct 23 '18 at 22:19
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    $\begingroup$ You may be reaching for the concept of polynomial rings, and treating $i$ in your example as your variable. You have in polynomial rings that coefficients are distinct from variables. $\endgroup$ – JMoravitz Oct 23 '18 at 22:24
  • $\begingroup$ Numbers combine, letters stay. $\endgroup$ – Berci Oct 23 '18 at 22:51
  • $\begingroup$ @Berci sure, but i is a both a letter and number. And anyways, i combines with other quaternions, the other $2^n$-tonions, etc. $\endgroup$ – Benjamin Thoburn Oct 24 '18 at 10:07
  • $\begingroup$ I don't see what your notion of "combining" is from your question. 6 is not really not 2 or 3, as it is the product, and we have just assigned it a new symbol. I know you mention this in your question and that you could do this with any number and somehow factor it into a product, but since you acknowledge that fact, I fail to see what criteria you are using that allows you to differentiate between a situation like 2*3 = 6 and 2*i = i. If we decided to give 2i a new name, the situation would be the same. It would seem out of convenience people decided not to. $\endgroup$ – user1236 Jan 17 at 0:46

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