I am curious whether in real numbers,
the multiplication as we know it is the only operation that
is distributive over addition and associative and commutative?
I heard it is. But I am not sure and is there any way to prove it?
Since the comments below shows that x*y=0 for all x, y and
x*y=r x y also satisfies all 3 laws, it is false that the multiplication is
the only operation that satisfies all 3 laws.
What about when the multiplication has an identity element which is 1,
which means that for all x, 1*x=x*1=x.
Sorry to add, but I would appreciate the proof!