> Does this really require knowing abstract algebra? Seems obvious to anyone doing any sort of multiplication that if the output is 0 then one of variables/functions has to be 0, if it is nonzero then the variable/function can be anything but 0.
Since you mention them specifically, it's not true for functions: multiply the function that is 1 for positive numbers and 0 elsewhere, by the function that is 1 for negative numbers and 0 elsewhere. (One can even produce continuous, or even smooth, examples with only a little more work.) A more traditional example is that it's not true for matrices: multiply the matrix ( ( 0 1 ) ( 0 0 ) ) by itself. (I just noticed sedeki https://news.ycombinator.com/item?id=15860883 pointed this out a few minutes earlier, noting that, for example, neither 2 nor 3 is congruent to 0 modulo 6, but their product is.) What I mean to say is: it often doesn't require knowing abstract algebra to think that things are obvious, but it may sometimes require knowing abstract algebra to figure out whether obvious things are true.
(Also, the last sentence you quote:
> Yet it is not the case that if f(x) * g(x) = 4, then either f(x) = 2 or g(x) = 2.
is, I would say, the important operational point. My students, especially in calculus, love to use this style of reasoning, even when specifically told that it doesn't work—although sometimes they change it (usually to conclude that f(x) = 4 or g(x) = 4). As Twain might have approximately said, it's not what's obvious that you don't know that gets you; it's what's obvious that ain't so.)
Regarding your students, well I don't dispute that there are students that think that - I just don't think that say a teacher knowing more about abstract algebra would be able to explain that better.
Since you mention them specifically, it's not true for functions: multiply the function that is 1 for positive numbers and 0 elsewhere, by the function that is 1 for negative numbers and 0 elsewhere. (One can even produce continuous, or even smooth, examples with only a little more work.) A more traditional example is that it's not true for matrices: multiply the matrix ( ( 0 1 ) ( 0 0 ) ) by itself. (I just noticed sedeki https://news.ycombinator.com/item?id=15860883 pointed this out a few minutes earlier, noting that, for example, neither 2 nor 3 is congruent to 0 modulo 6, but their product is.) What I mean to say is: it often doesn't require knowing abstract algebra to think that things are obvious, but it may sometimes require knowing abstract algebra to figure out whether obvious things are true.
(Also, the last sentence you quote:
> Yet it is not the case that if f(x) * g(x) = 4, then either f(x) = 2 or g(x) = 2.
is, I would say, the important operational point. My students, especially in calculus, love to use this style of reasoning, even when specifically told that it doesn't work—although sometimes they change it (usually to conclude that f(x) = 4 or g(x) = 4). As Twain might have approximately said, it's not what's obvious that you don't know that gets you; it's what's obvious that ain't so.)