NumHask.Algebra.Abstract

The abstract algebraic class structure of a number.

Mapping from Num

Num is a very old part of haskell, and a lot of different numeric concepts are tossed in there. The closest analogue in numhask is the Ring class, which magmaines the classical +, - and *, together with the Distributive laws.

No attempt is made, however, to reconstruct the particular magmaination of laws and classes that represent the old Num. A rough mapping of Num to numhask classes follows:

-- | Basic numeric class.
class  Num a  where
   {-# MINIMAL (+), (*), abs, signum, fromInteger, (negate | (-)) #-}

   (+), (-), (*)       :: a -> a -> a
   -- | Unary negation.
   negate              :: a -> a

+ is a function of the Additive class, - is a function of the Subtractive class, and * is a function of the Multiplicative class. negate is specifically in the Subtractive class. There are many useful constructions between negate and (-), involving cancellative properties.

   -- | Absolute value.
   abs                 :: a -> a
   -- | Sign of a number.
   -- The functions 'abs' and 'signum' should satisfy the law:
   --
   -- > abs x * signum x == x
   --
   -- For real numbers, the 'signum' is either @-1@ (negative), @0@ (zero)
   -- or @1@ (positive).
   signum              :: a -> a

abs is a function in the Signed class. The concept of an absolute value of a number can include situations where the domain and codomain are different, and size as a function in the Normed class is supplied for these cases.

sign replaces signum, because signum is a heinous name.

   -- | Conversion from an 'Integer'.
   -- An integer literal represents the application of the function
   -- 'fromInteger' to the appropriate value of type 'Integer',
   -- so such literals have type @('Num' a) => a@.
   fromInteger         :: Integer -> a

fromInteger is given its own class FromInteger

module NumHask.Algebra.Abstract.Group

module NumHask.Algebra.Abstract.Additive

module NumHask.Algebra.Abstract.Multiplicative

module NumHask.Algebra.Abstract.Ring

module NumHask.Algebra.Abstract.Field

module NumHask.Algebra.Abstract.Module

module NumHask.Algebra.Abstract.Action

module NumHask.Algebra.Abstract.Lattice