>>9282069

So then the statement "n/n = 1" is incomplete, or at least the "for n not equal to 0" is implicit. It seems like there is an analogy to the congruence relation in modular arithmetic, like a ≡ b (mod n), b ≡ c (mod n) then a ≡ c (mod n). Similarly f(n) = n/n = 1 = g(n), when we say f(n) = g(n) by the transitive property, we can't do so without maintaining the implicit restriction that they are equal WHEN n != 0, which arose when we set n/n = 1.

I've seen congruence modulo written with the modulus over the triple bar, so there is some sense in which the restrictions under which an equality holds are "attached" to the equality?

>>9281982

I think this question is hard to answer both well and concisely.

If you think a bunch about the rules of chess, you can come up with some theorems about how chess works. Maybe you can develop some new rules which follow logically from the "basic" rules which are set out in the definition of the game. Furthermore, maybe you can extend those idea outside of chess to tell you things about systems which are related to chess. Mathematics is about doing that, in general, for all kinds of different sets of rules.

You may find this channel interesting? https://www.youtube.com/watch?v=R7p-nPg8t_g