Making numeric functions an instance of Num?

Question

I want to be able to compose numeric functions in haskell using binary operators. So, for example, with unary numeric functions:

f*g

should

Answer 1

an instance with generic arity

instance Num b => Num (a->b) where
    f + g = \x -> f x + g x
    f - g = \x -> f x - g x
    f * g = \x -> f x * g x
    negate f = negate . f
    abs f = abs . f
    signum f = signum . f
    fromInteger n = \x -> fromInteger n

Edit: As Christian Conkle points out, there are problems with this approach. If you plan to use these instances for anything important or just want to understand the issues, you should read the resources he provided and decide for yourself whether this fits your needs. My intention was to provide an easy way to play with numeric functions using natural notation with as simple an implementation as possible.