I recently explained multiplication of (non-zero) complex numbers to my Mathematics Fundamentals students, the usual bit about “multiply their lengths, and add their angles”. Of course, there is always at least one student who wants to know why the angles are added instead of multiplied – after all, this is multiplication, right? – and so I go through a non-trivial example to establish that it is at least highly plausible that addition, not multiplication, of angles is what occurs. This quiets their objection, but it always gets me to wondering: Indeed, are angles EVER multiplied anywhere in Mathematics? I don’t recall ever having seen such a thing. And since this wonderful wonderland of MSE not exists, I will pass this question on to the community.
Of course, there are exercises of the form “If cos(f(x)) = sin(g(x)), solve for x.”, where this could happen, but of course what I am asking is whether this happens as part of a significant theorem, or as part of solving a problem of real physical interest. The non-multiplication of angles seems all the more counter-intuitive because they are actually dimensionless, in spite of the use of dimension-sounding phraseology such as “degrees” and “radians”. So, angles are just numbers, and surely numbers can be multiplied together, right?
(Of course, not being dimensionless is no barrier to getting multiplied by objects of the same/different type. After all, we deal with the square of seconds in regard to acceleration, and with the square of grams in regard to the (statistical) variance of weights, and so on. I’m just saying that it would seem all the more plausible that they would occasionally get multiplied together if they are dimensionless.)
I’m going to go out on a limb and conjecture that there is no such case. The basis of my conjecture is simply that I have never seen it happen, and also the fact that angles do not exist for vectors of zero length. I know this is quite tenuous, but what I’m guessing is that only quantities that behave well for 0 are eligible, so to speak, to be multiplied together.