Explanation of numbers in Haskell

Question

I would like a clear explanation of Num, Real, Integral, Integer, Int, Ratio, Rational, Double, Float.

Answer 1

This answer mostly assumes you know the difference between types and type classes. If that difference is hazy to you then clear up your understanding there before reading on.

Num

Num is a typeclass that includes all numeric types.

:info Num
class Num a where
  (+) :: a -> a -> a
  (-) :: a -> a -> a
  (*) :: a -> a -> a
  negate :: a -> a
  abs :: a -> a
  signum :: a -> a
  fromInteger :: Integer -> a
        -- Defined in ‘GHC.Num’
instance Num Word -- Defined in ‘GHC.Num’
instance Num Integer -- Defined in ‘GHC.Num’
instance Num Int -- Defined in ‘GHC.Num’
instance Num Float -- Defined in ‘GHC.Float’
instance Num Double -- Defined in ‘GHC.Float’

Real

Also a typeclass covering those types that can be represented as a real value (the Rational type).

:info Real
class (Num a, Ord a) => Real a where
  toRational :: a -> Rational
        -- Defined in ‘GHC.Real’
instance Real Word -- Defined in ‘GHC.Real’
instance Real Integer -- Defined in ‘GHC.Real’
instance Real Int -- Defined in ‘GHC.Real’
instance Real Float -- Defined in ‘GHC.Float’
instance Real Double -- Defined in ‘GHC.Float’

Integral

A type class for integrals, you know, ...,-2,-1,0,1,.... Types such as Integer (aka big int), Int, Int64, etc are instances.

:info Integral
class (Real a, Enum a) => Integral a where
  quot :: a -> a -> a
  rem :: a -> a -> a
  div :: a -> a -> a
  mod :: a -> a -> a
  quotRem :: a -> a -> (a, a)
  divMod :: a -> a -> (a, a)
  toInteger :: a -> Integer
        -- Defined in ‘GHC.Real’
instance Integral Word -- Defined in ‘GHC.Real’
instance Integral Integer -- Defined in ‘GHC.Real’
instance Integral Int -- Defined in ‘GHC.Real’

Integer

A type, not a type class such as what we've talked about till now, that can represent unbounded integers. So 2^3028 is a legal value.

Int

A fixed-width integral. In the GHC compiler this is 32 or 64 bits depending on your architecture. The Haskell language only guarantees this will be at least 29 bits.

Ratio

This is a type constructor, so you would say something like Ratio Integer to get a type for ratio's of two integers (mathematically a/b).

Rational

Well a rational is literally a ratio of two integers, understand ratio and you're good:

:i Rational
type Rational = Ratio Integer

Double

A type for double precision floating point values.

Float

A type for single precision floating point values.