I haven't seen much literature on this question, and it seems to me that people are reluctant to talk about it.

To me, it seems that the idea behind is that we want our numbers to commute under multiplication. So, we simply define , which (according to your reasoning) we know how to compute.

For division vs. fractions, I think that it would be more natural to have division first. As soon as you introduce that operation, you realise that there are many holes in your number system: if you try to compute , you cannot do it. This certainly would motivate the idea of fractions to fill up those holes.