Data.Ratio

Standard functions on rational numbers

Documentation

data Ratio a

Rational numbers, with numerator and denominator of some Integral type.

Instances

Integral a => Enum (Ratio a)

Methods

succ :: Ratio a -> Ratio a

pred :: Ratio a -> Ratio a

toEnum :: Int -> Ratio a

fromEnum :: Ratio a -> Int

enumFrom :: Ratio a -> [Ratio a]

enumFromThen :: Ratio a -> Ratio a -> [Ratio a]

enumFromTo :: Ratio a -> Ratio a -> [Ratio a]

enumFromThenTo :: Ratio a -> Ratio a -> Ratio a -> [Ratio a]

Eq a => Eq (Ratio a)

Methods

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

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

Integral a => Fractional (Ratio a)

Methods

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

recip :: Ratio a -> Ratio a

fromRational :: Rational -> Ratio a

(Data a, Integral a) => Data (Ratio a)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Ratio a -> c (Ratio a)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Ratio a)

toConstr :: Ratio a -> Constr

dataTypeOf :: Ratio a -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Ratio a))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Ratio a))

gmapT :: (forall b. Data b => b -> b) -> Ratio a -> Ratio a

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Ratio a -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Ratio a -> r

gmapQ :: (forall d. Data d => d -> u) -> Ratio a -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Ratio a -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Ratio a -> m (Ratio a)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Ratio a -> m (Ratio a)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Ratio a -> m (Ratio a)

Integral a => Num (Ratio a)

Methods

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

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

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

negate :: Ratio a -> Ratio a

abs :: Ratio a -> Ratio a

signum :: Ratio a -> Ratio a

fromInteger :: Integer -> Ratio 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

(Integral a, Read a) => Read (Ratio a)

Methods

readsPrec :: Int -> ReadS (Ratio a)

readList :: ReadS [Ratio a]

readPrec :: ReadPrec (Ratio a)

readListPrec :: ReadPrec [Ratio a]

Integral a => Real (Ratio a)

Methods

toRational :: Ratio a -> Rational

Integral a => RealFrac (Ratio a)

Methods

properFraction :: Integral b => Ratio a -> (b, Ratio a)

truncate :: Integral b => Ratio a -> b

round :: Integral b => Ratio a -> b

ceiling :: Integral b => Ratio a -> b

floor :: Integral b => Ratio a -> b

Show a => Show (Ratio a)

Methods

showsPrec :: Int -> Ratio a -> ShowS

show :: Ratio a -> String

showList :: [Ratio a] -> ShowS

(Storable a, Integral a) => Storable (Ratio a)

Methods

sizeOf :: Ratio a -> Int

alignment :: Ratio a -> Int

peekElemOff :: Ptr (Ratio a) -> Int -> IO (Ratio a)

pokeElemOff :: Ptr (Ratio a) -> Int -> Ratio a -> IO ()

peekByteOff :: Ptr b -> Int -> IO (Ratio a)

pokeByteOff :: Ptr b -> Int -> Ratio a -> IO ()

peek :: Ptr (Ratio a) -> IO (Ratio a)

poke :: Ptr (Ratio a) -> Ratio a -> IO ()

type Rational = Ratio Integer

Arbitrary-precision rational numbers, represented as a ratio of two Integer values. A rational number may be constructed using the % operator.

(%) :: Integral a => a -> a -> Ratio a infixl 7

Forms the ratio of two integral numbers.

numerator :: Ratio a -> a

Extract the numerator of the ratio in reduced form: the numerator and denominator have no common factor and the denominator is positive.

denominator :: Ratio a -> a

Extract the denominator of the ratio in reduced form: the numerator and denominator have no common factor and the denominator is positive.

approxRational :: RealFrac a => a -> a -> Rational

approxRational, applied to two real fractional numbers x and epsilon, returns the simplest rational number within epsilon of x. A rational number y is said to be simpler than another y' if

• abs (numerator y) <= abs (numerator y'), and
• denominator y <= denominator y'.

Any real interval contains a unique simplest rational; in particular, note that 0/1 is the simplest rational of all.