base-4.18.1.0: Basic libraries

Copyright | (c) The University of Glasgow 2001
License | BSD-style (see the file libraries/base/LICENSE)
Maintainer | libraries@haskell.org
Stability | stable
Portability | portable
Safe Haskell | Safe
Language | Haskell2010

Data.Ratio

Description

Standard functions on rational numbers

Synopsis

- data Ratio a
- type Rational = Ratio Integer
- (%) :: Integral a => a -> a -> Ratio a
- numerator :: Ratio a -> a
- denominator :: Ratio a -> a
- approxRational :: RealFrac a => a -> a -> Rational

# Documentation

data Ratio a Source #

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

Note that `Ratio`'s instances inherit the deficiencies from the type parameter's. For example, `Ratio Natural`'s `Num` instance has similar problems to `Natural`'s.

#### Instances

Instances details

(Data a, Integral a) => Data (Ratio a) Source # | Since: base-4.0.0.0

Instance details

Defined in Data.Data

Methods

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

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

toConstr :: Ratio a -> Constr Source #

dataTypeOf :: Ratio a -> DataType Source #

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

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

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

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

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

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

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

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

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

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

(Storable a, Integral a) => Storable (Ratio a) Source # | Since: base-4.8.0.0

Instance details

Defined in Foreign.Storable

Methods

sizeOf :: Ratio a -> Int Source #

alignment :: Ratio a -> Int Source #

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

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

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

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

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

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

Integral a => Enum (Ratio a) Source # | Since: base-2.0.1

Instance details

Defined in GHC.Real

Methods

succ :: Ratio a -> Ratio a Source #

pred :: Ratio a -> Ratio a Source #

toEnum :: Int -> Ratio a Source #

fromEnum :: Ratio a -> Int Source #

enumFrom :: Ratio a -> [Ratio a] Source #

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

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

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

Integral a => Num (Ratio a) Source # | Since: base-2.0.1
Instance details Defined in GHC.Real Methods (+) :: Ratio a -> Ratio a -> Ratio a Source # (-) :: Ratio a -> Ratio a -> Ratio a Source # (*) :: Ratio a -> Ratio a -> Ratio a Source # negate :: Ratio a -> Ratio a Source # abs :: Ratio a -> Ratio a Source # signum :: Ratio a -> Ratio a Source # fromInteger :: Integer -> Ratio a Source #
(Integral a, Read a) => Read (Ratio a) Source # | Since: base-2.1
Instance details Defined in GHC.Read Methods readsPrec :: Int -> ReadS (Ratio a) Source # readList :: ReadS [Ratio a] Source # readPrec :: ReadPrec (Ratio a) Source # readListPrec :: ReadPrec [Ratio a] Source #
Integral a => Fractional (Ratio a) Source # | Since: base-2.0.1
Instance details Defined in GHC.Real Methods (/) :: Ratio a -> Ratio a -> Ratio a Source # recip :: Ratio a -> Ratio a Source # fromRational :: Rational -> Ratio a Source #
Integral a => Real (Ratio a) Source # | Since: base-2.0.1
Instance details Defined in GHC.Real Methods toRational :: Ratio a -> Rational Source #
Integral a => RealFrac (Ratio a) Source # | Since: base-2.0.1
Instance details Defined in GHC.Real Methods properFraction :: Integral b => Ratio a -> (b, Ratio a) Source # truncate :: Integral b => Ratio a -> b Source # round :: Integral b => Ratio a -> b Source # ceiling :: Integral b => Ratio a -> b Source # floor :: Integral b => Ratio a -> b Source #
Show a => Show (Ratio a) Source # | Since: base-2.0.1
Instance details Defined in GHC.Real Methods showsPrec :: Int -> Ratio a -> ShowS Source # show :: Ratio a -> String Source # showList :: [Ratio a] -> ShowS Source #
Eq a => Eq (Ratio a) Source # | Since: base-2.1
Instance details Defined in GHC.Real Methods (==) :: Ratio a -> Ratio a -> Bool Source # (/=) :: Ratio a -> Ratio a -> Bool Source #
Integral a => Ord (Ratio a) Source # | Since: base-2.0.1
Instance details Defined in GHC.Real Methods compare :: Ratio a -> Ratio a -> Ordering Source # (<) :: Ratio a -> Ratio a -> Bool Source # (<=) :: Ratio a -> Ratio a -> Bool Source # (>) :: Ratio a -> Ratio a -> Bool Source # (>=) :: Ratio a -> Ratio a -> Bool Source # max :: Ratio a -> Ratio a -> Ratio a Source # min :: Ratio a -> Ratio a -> Ratio a Source #

type Rational = Ratio Integer Source #

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 Source #

Forms the ratio of two integral numbers.

numerator :: Ratio a -> a Source #

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 Source #

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 Source #

`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.