> As in, (x/y) * y=x is true either way only if y!=0. The only difference is whether when y=0 if it is not true because it is just illegal or if it is not true

> As in, (x/y) * y=x is true either way only if y!=0. The only difference is whether when y=0 if it is not true because it is just illegal or if it is not true because (x/0) * 0 = 0 for all x. Sure, I understand this claim. What I don't understand is the precise meaning of the claim "multiplication and division already are not entirely symmetrical." I don't know any existing technical meaning of "entirely symmetrical" for a pair of operations, but let's suppose that I switch the operations in your statement: > (x/y) * y = x is true [for all x] only if y != 0. Then I get: > (x * y)/y = x is true [for all x] only if y != 0. That's also true. What's the asymmetry?