NumHask.Algebra.Rational

Integral classes

Documentation

data Ratio a

Constructors

!a :% !a

Instances

Eq a => Eq (Ratio a)

Methods

(==) :: Ratio a -> Ratio a -> Bool

(/=) :: Ratio a -> Ratio a -> Bool

(Ord a, Multiplicative a, Integral a) => Ord (Ratio a)

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)

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)

Methods

(-) :: Ratio a -> Ratio a -> Ratio a

(Ord a, Signed a, Integral a, AdditiveInvertible a) => Additive (Ratio a)

Methods

(+) :: Ratio a -> Ratio a -> Ratio a

(Ord a, Signed a, Integral a, AdditiveInvertible a) => AdditiveInvertible (Ratio a)

Methods

negate :: Ratio a -> Ratio a

(Ord a, Signed a, Integral a, AdditiveInvertible a) => AdditiveCommutative (Ratio a)

(Ord a, Signed a, Integral a, AdditiveInvertible a) => AdditiveAssociative (Ratio a)

(Ord a, Integral a, Signed a, AdditiveInvertible a) => AdditiveUnital (Ratio a)

Methods

zero :: Ratio a

(Ord a, Integral a, Signed a, AdditiveInvertible a) => AdditiveMagma (Ratio a)

Methods

plus :: Ratio a -> Ratio a -> Ratio a

(Signed a, AdditiveInvertible a, AdditiveUnital a, Integral a, Ord a, Multiplicative a) => MultiplicativeGroup (Ratio a)

Methods

(/) :: Ratio a -> Ratio a -> Ratio a

(Signed a, AdditiveInvertible a, AdditiveUnital a, Integral a, Ord a, Multiplicative a) => Multiplicative (Ratio a)

Methods

(*) :: Ratio a -> Ratio a -> Ratio a

(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeInvertible (Ratio a)

Methods

recip :: Ratio a -> Ratio a

(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeCommutative (Ratio a)

(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeAssociative (Ratio a)

(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeUnital (Ratio a)

Methods

one :: Ratio a

(Ord a, Signed a, Integral a, AdditiveInvertible a) => MultiplicativeMagma (Ratio a)

Methods

times :: Ratio a -> Ratio a -> Ratio a

(Ord a, Signed a, Integral a, AdditiveInvertible a) => Distribution (Ratio a)

(Ord a, Signed a, Integral a, Multiplicative a, Ring a) => InvolutiveRing (Ratio a)

Methods

adj :: Ratio a -> Ratio a

(Ord a, Signed a, Integral a, Multiplicative a, Ring a) => CRing (Ratio a)

(Ord a, Signed a, Integral a, AdditiveGroup a) => Ring (Ratio a)

(Ord a, Signed a, Integral a, AdditiveInvertible a) => Semiring (Ratio a)

(FromInteger a, MultiplicativeUnital a) => FromInteger (Ratio a)

Methods

fromInteger :: Integer -> Ratio a

(Ord a, Signed a, Integral a, Multiplicative a, Ring a, AdditiveInvertible a) => LowerBoundedField (Ratio a)

Methods

negInfinity :: Ratio a

(Ord a, Signed a, Integral a, AdditiveInvertible a, Multiplicative a, Ring a) => UpperBoundedField (Ratio a)

Methods

infinity :: Ratio a

nan :: Ratio a

(Ord a, Signed a, Integral a, Multiplicative a, Ring a) => Field (Ratio a)

(Ord a, Signed a, Integral a, Multiplicative a, Ring a) => Semifield (Ratio a)

(Ord a, Signed a, Integral a, AdditiveGroup a) => Epsilon (Ratio a)

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)

Methods

sign :: Ratio a -> Ratio a

abs :: Ratio a -> Ratio a

FromInteger a => FromRatio (Ratio a)

Methods

fromRatio :: Ratio Integer -> Ratio a

ToInteger a => ToRatio (Ratio a)

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

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)

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)

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

Methods

toRatio :: Double -> Ratio Integer

ToRatio Float

Methods

toRatio :: Float -> Ratio Integer

ToRatio Int

Methods

toRatio :: Int -> Ratio Integer

ToRatio Int8

Methods

toRatio :: Int8 -> Ratio Integer

ToRatio Int16

Methods

toRatio :: Int16 -> Ratio Integer

ToRatio Int32

Methods

toRatio :: Int32 -> Ratio Integer

ToRatio Int64

Methods

toRatio :: Int64 -> Ratio Integer

ToRatio Integer

Methods

toRatio :: Integer -> Ratio Integer

ToRatio Natural

Methods

toRatio :: Natural -> Ratio Integer

ToRatio Rational

Methods

toRatio :: Rational -> Ratio Integer

ToRatio Word

Methods

toRatio :: Word -> Ratio Integer

ToRatio Word8

Methods

toRatio :: Word8 -> Ratio Integer

ToRatio Word16

Methods

toRatio :: Word16 -> Ratio Integer

ToRatio Word32

Methods

toRatio :: Word32 -> Ratio Integer

ToRatio Word64

Methods

toRatio :: Word64 -> Ratio Integer

ToInteger a => ToRatio (Ratio a)

Methods

toRatio :: Ratio 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

Methods

fromRatio :: Ratio Integer -> Double

FromRatio Float

Methods

fromRatio :: Ratio Integer -> Float

FromInteger a => FromRatio (Ratio a)

Methods

fromRatio :: Ratio Integer -> Ratio 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.