Currying

Have you been wondering why the type for a function like plus listed below is Int -> Int -> Int rather than something like (Int,Int) ->Int ?
 

plus :: Int -> Int -> Int
plus x y = x + y

Well the expression plus 3 4 is really equivalent to ((plus 3) 4).  The result of plus 3  is then applied to the argument 4.  This means that the value of (plus 3) must also be a function! 

Indeed, we could define a new function to be the result of (plus 3) as follows:

plusThree :: Int -> Int
plusThree = plus 3

We would get the result that we expect when using this new function.

> plusThree 4
7

This method of applying functions to one argument at a time is called currying (after Haskell B. Curry).  Curried functions can be applied to one argument only, giving another function.  Sometimes these new functions can be useful in their own right.  Consider the following function:

twice :: (Int -> Int) -> Int -> Int
twice f x = f (f x)

The function twice takes as arguments a function and an integer and applies the function twice to the integer argument.  We could use the function resulting in using only the first argument to get the following new functions:

add2 = twice (+1)

quad = twice square

What would be the result of the expressions add2 3  and quad 2 ?