First off, in your description of rational numbers, you're missing an e at the end if diverse. Right now I don't feel like that paragraph and mouse over work together. I'd work on removing the mouse over and then explain rational numbers similar to your explanation in the mouse over, and then continue to give examples like you've done.
I'd also make sentence that starts "Interestingly..." it's own paragraph.
I think you need more explanation going along with your examples of the Fn's. Can you write out the fractions in unreduced form first? I'd also like to see a different image than the one that's in this section, since their notation appears to be different from yours.
Somehow the math in the paragraph starting with "Terms that appear..." took me a second. It might help to rearrange it and label them as equations 1 and 2, and say, combining equations 1 and 2 gives us rq-ps=1.
Can you give an example of the mediant property with actual numbers? I think that will clarify that bit.
As I said before, I'm still not entirely sure what the "proof" of this sequence was. It seems like there are all sorts of interesting properties that can be proven, but I'm not seeing the proof you mention in your first sentence.
I think here " Think of three terms that are consecutive in a Farey sequence: p/q, p'/q', and p/q "you mean to have p''/q'' you might need to use the "nowiki" command there (just edit this page and see how I did that here).
Also, is there any way to get a diagram to go along with that last section? I think it would be really helpful to list the points and put in some arrows.
I still think that your description of a rational point needs some clarification. Maybe, you could actually list examples right there in that paragraph?
What exactly is the "proof" of the Farey sequence? It's not clear to me precisely what about it was proven.
This sentence : "Just insert each of the mediants such that q + q < n." doesn't make sense to me. Can you clarify it?
Also, a good picture in your last section could be really helpful.
I really like your explanation about how relatively prime makes more sense. You've convinced me!
Abram and I had a discussion about this. We aren't really sure which is more common, but we both agree that we like "relatively prime" better. My justification is that "coprime" to me means "prime together" and I have no idea what that would mean, whereas two things being relatively prime to each other makes more intuitive sense...
Though, I commented to Abram that we could have both been influenced by one professor at some point... But I'm pretty sure that I heard relatively prime in high school, and then I think it's what Fraleigh uses in his textbooks (and he wrote both my linear algebra and modern algebra textbooks).
But... I don't know. Really, I was just taking a break from transcribing interviews and looking for something to do :)
"Coprime" seems to sound more familiar to me but, I have a tendency to not remember terminology very well (I'm much better with concepts). So if you think "relatively prime" more common I can definitely change it! I trust you :)
I have one quick comment. I actually had never seen the word "coprime" before, and I think that the term "relatively prime" is much more common (at least it has been used more the in books that I've used). What are your thoughts on word choice here?