What's the difference between the classes Floating and Fractional in Haskell?

Question

What is the difference is between the Floating and Fractional classes in Haskell?

Answer 1

The definitions of Fractional and Floating can be found in the documentation of the Prelude:

class Num a => Fractional a where
    (/) :: a -> a -> a
    recip :: a -> a
    fromRational :: Rational -> a 

Fractional numbers, supporting real division.

[...]

class Fractional a => Floating a where
    pi :: a
    exp :: a -> a
    log :: a -> a
    sqrt :: a -> a
    (**) :: a -> a -> a
    logBase :: a -> a -> a
    sin :: a -> a
    cos :: a -> a
    tan :: a -> a
    asin :: a -> a
    acos :: a -> a
    atan :: a -> a
    sinh :: a -> a
    cosh :: a -> a
    tanh :: a -> a
    asinh :: a -> a
    acosh :: a -> a
    atanh :: a -> a

Trigonometric and hyperbolic functions and related functions.

[...]

So to translate that into English: A Fractional is any kind of number for which I can define a division:

(/) :: Fractional a => a -> a -> a

That can for instance be the case for floating point numbers, but also for fractions (where a fraction has a numerator and denominator). This is not the case for Int because if dividing an Int by an Int does not always produce an Int (well technically floating point division on a computer is not exact, but that is another story).

A subset of Fractional numbers are Floating numbers where trigonometric are defined. It is for instance impossible that the sin of a fraction always produces a fraction: a sin is defined as an sum over an infinite sequence. Only for a very limited number of cases (like sin 0) it holds. Basically the only numbers on a computer for which trigonometric are defined (approximatively) are floating point numbers.