9.4

1. I'm assuming this is not part of the curriculum, so the most difficult part is irrelevant...? I don't know. I saw this construction in Math 190, so it's all good.

2. It talked about the definition of addition, multiplication, etc. being "motivated" by the definition in Z and Q. I'm wondering a few things: which came first, the numbers themselves and their requirements, or some abstract notion of quantity and relationships...I mean, we could define things a lot of different ways, right? But 1/2 of a pie plus 1/4 of a pie is 3/4 of a pie. I guess the numbers are placeholders for some notion of quantity anyway, and that notion has to be consistent and maybe all of math is just what follows from that. I might be kind of an engineer at heart, but I keep asking myself what immediate applications I can find for this (and I mean this specifically). Is there any kind of tangible example of quotient fields outside of the rationals?