Re: why does a/(b/c) = a(c/b)?

I would argue that the whole point of a division is that it is the  
inverse of multiplication.

This is why, in noncommutative settings (eg. finite state machines),  
one is careful to distinguish A/B from B\A.

In the first case, (A/B)B = A, but in the second B(B\A) = A.

So I would say that if you have a formal system where a/(b/c) differs  
from a(c/b), it would be better to pick some other symbol than  
solidus for the operation.

(we know that people insist on using '+' for string concatenation  
despite the lack of commutativity, so this is obviously a personal  
opinion).

-Dave

An exercise: suppose your division is not an exact inverse (eg,  
python's '//', where (a//b)*b <= a): can we still say something  
useful about the relation between a//(b//c) and a*(c//b) ?

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