Peter Duveen writes: "To demonstrate that 1/-a = -(1/a): (1/-a) x -a = 1 definition of 1/-a eq.1 - - - (1/a) x a = -1 definition of 1/a - - -1 (1/a) x a = -1 ... " etc. etc. etc.
Peter, I get the impression you are trying to demonstrate these things to yourself in an attempt perhaps to understand things you learned by rote earlier. Furthermore, I get the feeling that you therefore think these "demonstrations" would automatically be a good way to introduce this material to first-timers.
Forgive me if I'm wrong. But if I'm at all close, don't you see a potential problem in that approach?