How to interpret this Haskell code that type checks?

Question

Studying the book Haskell Programming from First Principles, I've come across the following exercise in Chapter 6, "Typeclasses":

-- Hint: use some arithmetic operation to
-- combine values of type 'b'. Pick one.
arith :: Num b => (a -> b) -> Integer -> a -> b
arith = ???

I have come up with the following solution, and I call it a 'solution' because it compiles:

arith f _ x = f x

But I'm somewhat confused how to 'interpret' what I've just written above. My 'reading' of the type signature of arith is something like the following:

arith takes one function, one Integer, a parameter of type a, and returns a result that is of type b; moreover, the function that arith takes as the first parameter is a function that takes a parameter of type a and returns a value of type b, and that value has to be of type b whose typeclass is (or 'constrained') by Num.

and after that, I came up with the 'solution' above but I haven't used "some arithmetic operation to combine values of type 'b'". Somehow I think either the 'hint' is misleading, or it is perfectly fine, and I'm missing something, if that's the case, then what am I missing?

Answer 1

I think the intended solution is:

arith f i a = f a + fromInteger i

Or one with + replaced by - or *.

To recap the requirements: In this exercise you have to implement a function that:

1. Matches type signature arith :: Num b => (a -> b) -> Integer -> a -> b.

2. Uses arithmetic operation to combine values of type b.

You are given three values of types (a -> b), Integer and a. As a first step you need to convert them to two values of type b. You can get first one by applying (a -> b) to a. Then you are left with value of type Integer. When you notice that instances of typeclass Num support function fromInteger :: Integer -> a (where type a is type b in context of function we are defining), it becomes clear that you can obtain second value of type b by applying fromInteger to your Integer value. As a second step you combine two values of type b using one of arithmetic operations defined in Num typeclass.