Why an additive function in number theory is not the same thing as an additive function in algebra?

Question

I mean why in algebra and analysis the requirement is

$$f(x+y) = f(x) + f(y)$$

while in number theory it is

$$f(xy) = f(x) + f(y) \;?$$

Answer 1

Because the functions $f : \mathbb N \to \mathbb N$ satisfying $f(a+b)=f(a)+f(b)$ are linear...