Safe Haskell	None
Language	Haskell2010

NumHask.Algebra.Rational

Contents

$integral_functionality

Description

Integral classes

Synopsis

data Ratio a = !a :% !a
type Rational = Ratio Integer
class ToRatio a where
class FromRatio a where
fromRational :: (ToRatio a, FromRatio b) => a -> b
reduce :: (Ord a, AdditiveInvertible a, Signed a, Integral a) => a -> a -> Ratio a
gcd :: (Ord a, Signed a, Integral a) => a -> a -> a

Documentation

data Ratio a #

Constructors

!a :% !a

Instances

(Eq a, AdditiveUnital a) => Eq (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods (==) :: Ratio a -> Ratio a -> Bool # (/=) :: Ratio a -> Ratio a -> Bool #
(Ord a, Multiplicative a, Integral a) => Ord (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods compare :: Ratio a -> Ratio a -> Ordering # (<) :: Ratio a -> Ratio a -> Bool # (<=) :: Ratio a -> Ratio a -> Bool # (>) :: Ratio a -> Ratio a -> Bool # (>=) :: Ratio a -> Ratio a -> Bool # max :: Ratio a -> Ratio a -> Ratio a # min :: Ratio a -> Ratio a -> Ratio a #
Show a => Show (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods showsPrec :: Int -> Ratio a -> ShowS # show :: Ratio a -> String # showList :: [Ratio a] -> ShowS #
(Ord a, Signed a, Integral a, AdditiveGroup a) => AdditiveGroup (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods (-) :: Ratio a -> Ratio a -> Ratio a #
(Ord a, Signed a, Integral a, AdditiveInvertible a) => Additive (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods (+) :: Ratio a -> Ratio a -> Ratio a #
(Ord a, Signed a, Integral a, AdditiveInvertible a) => AdditiveInvertible (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods negate :: Ratio a -> Ratio a #
(Ord a, Signed a, Integral a, AdditiveInvertible a) => AdditiveCommutative (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(Ord a, Signed a, Integral a, AdditiveInvertible a) => AdditiveAssociative (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(Ord a, Integral a, Signed a, AdditiveInvertible a) => AdditiveUnital (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods zero :: Ratio a #
(Ord a, Integral a, Signed a, AdditiveInvertible a) => AdditiveMagma (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods plus :: Ratio a -> Ratio a -> Ratio a #
(Signed a, AdditiveInvertible a, AdditiveUnital a, Integral a, Ord a, Multiplicative a) => MultiplicativeGroup (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods (/) :: Ratio a -> Ratio a -> Ratio a #
(Signed a, AdditiveInvertible a, AdditiveUnital a, Integral a, Ord a, Multiplicative a) => Multiplicative (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods (*) :: Ratio a -> Ratio a -> Ratio a #
(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeInvertible (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods recip :: Ratio a -> Ratio a #
(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeCommutative (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeAssociative (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeUnital (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods one :: Ratio a #
(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeMagma (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods times :: Ratio a -> Ratio a -> Ratio a #
(Ord a, Signed a, Integral a, AdditiveInvertible a) => Distribution (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(Ord a, Signed a, Integral a, Multiplicative a, Ring a) => InvolutiveRing (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods adj :: Ratio a -> Ratio a #
(Ord a, Signed a, Integral a, Multiplicative a, Ring a) => CRing (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(Ord a, Signed a, Integral a, AdditiveGroup a) => Ring (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(Ord a, Signed a, Integral a, AdditiveInvertible a) => Semiring (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(FromInteger a, MultiplicativeUnital a) => FromInteger (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods fromInteger :: Integer -> Ratio a #
(Ord a, Signed a, Integral a, Multiplicative a, Ring a, AdditiveInvertible a) => LowerBoundedField (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods negInfinity :: Ratio a #
(Ord a, Signed a, Integral a, AdditiveInvertible a, Multiplicative a, Ring a) => UpperBoundedField (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods infinity :: Ratio a # nan :: Ratio a #
(Ord a, Signed a, Integral a, Multiplicative a, Ring a) => Field (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(Ord a, Signed a, Integral a, Multiplicative a, Ring a) => Semifield (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational
(Ord a, Signed a, Integral a, AdditiveGroup a) => Epsilon (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods nearZero :: Ratio a -> Bool # aboutEqual :: Ratio a -> Ratio a -> Bool # positive :: Ratio a -> Bool # veryPositive :: Ratio a -> Bool # veryNegative :: Ratio a -> Bool #
(Ord a, Signed a, Integral a, AdditiveInvertible a) => Signed (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods sign :: Ratio a -> Ratio a # abs :: Ratio a -> Ratio a #
FromInteger a => FromRatio (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods fromRatio :: Ratio Integer -> Ratio a #
ToInteger a => ToRatio (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Ratio a -> Ratio Integer #
(Ord a, Signed a, ToInteger a, Integral a, Multiplicative a, Ring a, Eq b, AdditiveGroup b, Integral b, FromInteger b) => QuotientField (Ratio a) b #
Instance details Defined in NumHask.Algebra.Rational Methods properFraction :: Ratio a -> (b, Ratio a) # round :: Ratio a -> b # ceiling :: Ratio a -> b # floor :: Ratio a -> b #
(Ord a, Integral a, Signed a, AdditiveGroup a) => Metric (Ratio a) (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods distanceL1 :: Ratio a -> Ratio a -> Ratio a # distanceL2 :: Ratio a -> Ratio a -> Ratio a # distanceLp :: Ratio a -> Ratio a -> Ratio a -> Ratio a #
(Ord a, Integral a, Signed a, AdditiveInvertible a) => Normed (Ratio a) (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods normL1 :: Ratio a -> Ratio a # normL2 :: Ratio a -> Ratio a # normLp :: Ratio a -> Ratio a -> Ratio a #

type Rational = Ratio Integer #

class ToRatio a where #

toRatio is equivalent to Real in base.

Minimal complete definition

toRatio

Methods

toRatio :: a -> Ratio Integer #

Instances

ToRatio Double #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Double -> Ratio Integer #
ToRatio Float #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Float -> Ratio Integer #
ToRatio Int #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Int -> Ratio Integer #
ToRatio Int8 #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Int8 -> Ratio Integer #
ToRatio Int16 #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Int16 -> Ratio Integer #
ToRatio Int32 #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Int32 -> Ratio Integer #
ToRatio Int64 #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Int64 -> Ratio Integer #
ToRatio Integer #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Integer -> Ratio Integer #
ToRatio Natural #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Natural -> Ratio Integer #
ToRatio Rational #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Rational -> Ratio Integer #
ToRatio Word #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Word -> Ratio Integer #
ToRatio Word8 #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Word8 -> Ratio Integer #
ToRatio Word16 #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Word16 -> Ratio Integer #
ToRatio Word32 #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Word32 -> Ratio Integer #
ToRatio Word64 #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Word64 -> Ratio Integer #
ToInteger a => ToRatio (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods toRatio :: Ratio a -> Ratio Integer #
(ToRatio a, ExpField a) => ToRatio (LogField a) #
Instance details Defined in NumHask.Data.LogField Methods toRatio :: LogField a -> Ratio Integer #

class FromRatio a where #

Fractional in base splits into fromRatio and MultiplicativeGroup

Minimal complete definition

fromRatio

Methods

fromRatio :: Ratio Integer -> a #

Instances

FromRatio Double #
Instance details Defined in NumHask.Algebra.Rational Methods fromRatio :: Ratio Integer -> Double #
FromRatio Float #
Instance details Defined in NumHask.Algebra.Rational Methods fromRatio :: Ratio Integer -> Float #
FromInteger a => FromRatio (Ratio a) #
Instance details Defined in NumHask.Algebra.Rational Methods fromRatio :: Ratio Integer -> Ratio a #
(FromRatio a, ExpField a) => FromRatio (LogField a) #
Instance details Defined in NumHask.Data.LogField Methods fromRatio :: Ratio Integer -> LogField a #

fromRational :: (ToRatio a, FromRatio b) => a -> b #

coercion of Rationals

fromRational a == a

$integral_functionality

reduce :: (Ord a, AdditiveInvertible a, Signed a, Integral a) => a -> a -> Ratio a #

reduce is a subsidiary function used only in this module. It normalises a ratio by dividing both numerator and denominator by their greatest common divisor.

gcd :: (Ord a, Signed a, Integral a) => a -> a -> a #

gcd x y is the non-negative factor of both x and y of which every common factor of x and y is also a factor; for example gcd 4 2 = 2, gcd (-4) 6 = 2, gcd 0 4 = 4. gcd 0 0 = 0. (That is, the common divisor that is "greatest" in the divisibility preordering.)

Note: Since for signed fixed-width integer types, abs minBound < 0, the result may be negative if one of the arguments is minBound (and necessarily is if the other is 0 or minBound) for such types.