算術

數字型別層次結構

Num 位於數字型別層次結構的根。它的特徵操作和一些常見例項如下所示（預設載入 Prelude 加上 Data.Complex）：

λ> :i 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
  {-# MINIMAL (+), (*), abs, signum, fromInteger, (negate | (-)) #-}
      -- Defined in ‘GHC.Num’
instance RealFloat a => Num (Complex a) -- Defined in ‘Data.Complex’
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’

我們已經看過 Fractional 類，它需要 Num 並介紹了 division``(/) 的概念和數字的倒數：

λ> :i Fractional
class Num a => Fractional a where
  (/) :: a -> a -> a
  recip::a -> a
  fromRational::Rational -> a
  {-# MINIMAL fromRational, (recip | (/)) #-}
      -- Defined in ‘GHC.Real’
instance RealFloat a => Fractional (Complex a) -- Defined in ‘Data.Complex’
instance Fractional Float -- Defined in ‘GHC.Float’
instance Fractional Double -- Defined in ‘GHC.Float’

Real 級模型..實數。它需要 Num 和 Ord，因此它模擬有序數值域。作為一個反例，複數不是有序欄位（即它們不具有自然排序關係）：

λ> :i Real
class (Num a, Ord a) => Real a where
  toRational::a -> Rational
  {-# MINIMAL toRational #-}
      -- 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’

RealFrac 表示可以舍入的數字

λ> :i RealFrac
class (Real a, Fractional a) => RealFrac a where
  properFraction::Integral b => a -> (b, a)
  truncate::Integral b => a -> b
  round::Integral b => a -> b
  ceiling::Integral b => a -> b
  floor::Integral b => a -> b
  {-# MINIMAL properFraction #-}
      -- Defined in ‘GHC.Real’
instance RealFrac Float -- Defined in ‘GHC.Float’
instance RealFrac Double -- Defined in ‘GHC.Float’

Floating（暗示 Fractional）表示可能沒有有限十進位制擴充套件的常量和運算。

λ> :i Floating
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
  GHC.Float.log1p::a -> a
  GHC.Float.expm1 :: a -> a
  GHC.Float.log1pexp::a -> a
  GHC.Float.log1mexp::a -> a
  {-# MINIMAL pi, exp, log, sin, cos, asin, acos, atan, sinh, cosh,
              asinh, acosh, atanh #-}
      -- Defined in ‘GHC.Float’
instance RealFloat a => Floating (Complex a) -- Defined in ‘Data.Complex’
instance Floating Float -- Defined in ‘GHC.Float’
instance Floating Double -- Defined in ‘GHC.Float’

注意：雖然像 sqrt . negate::Floating a => a -> a 這樣的表示式完全有效，但它們可能會返回 NaN（“not-a-number”），這可能不是預期的行為。在這種情況下，我們可能想要處理 Complex 欄位（稍後顯示）。