> So it is _not_ a theorem that a * (b / 0) = b * (a / 0)
As the article explains, the whole issue about dividing by 0 is the lack of multiplicative inverse. If you define division for 0 as something other than multiplication by the multiplicative inverse, there is no longer an issue doing the division. Hope that helps.
I'm curious what you mean by this: there is nothing "privileged" about the system of arithmetic that we usually use and this alternate definition.
The math that we all learn up through highschool is a system with a carefully chosen set of axioms that happen to be useful in most cases and has become the default system, but there isn't some intrinsic property of it that makes it "math" and alternate consistent set of axioms "not math".
If a hn title said 1*0=0 could you reply "Oh, so 1=0/0?"
The entire point of the article was suggesting you can define a consistent system without multiplication and division being entirely symmetrical (as they already are not)
Actually, that's the entire point of 0/0. It's indeterminate, meaning it can only take on a fixed value in the context of an operation to approach it (usually a limit).
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.
