Multiplication by x and division by x are inverses, except there is already the sole special case of x=0 where that isn't true.
The common way to resolve that is and be consistent is to axiomatically decide:
1) there is no inverse of * 0
2) 0 is not part of the range permitted for 1/x
The point of the article is that another way to resolve it and be consistent is to axiomatically decide:
1) there is no inverse of * 0
2) 0 is part of the range permitted for 1/x, and the resulting value is 0.
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.
> 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.
The common way to resolve that is and be consistent is to axiomatically decide:
1) there is no inverse of * 0
2) 0 is not part of the range permitted for 1/x
The point of the article is that another way to resolve it and be consistent is to axiomatically decide:
1) there is no inverse of * 0
2) 0 is part of the range permitted for 1/x, and the resulting value is 0.
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.