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 |

Standard functions on rational numbers

# Documentation

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

(Data a, Integral a) => Data (Ratio a) | @since base-4.0.0.0 |

Defined in GHC.Internal.Data.Data 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 # | |

Integral a => Enum (Ratio a) | @since base-2.0.1 |

Defined in GHC.Internal.Real 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 # | |

(Storable a, Integral a) => Storable (Ratio a) | @since base-4.8.0.0 |

Defined in GHC.Internal.Foreign.Storable 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 # | |

Integral a => Num (Ratio a) | @since base-2.0.1 |

Defined in GHC.Internal.Real | |

(Integral a, Read a) => Read (Ratio a) | @since base-2.01 |

Integral a => Fractional (Ratio a) | @since base-2.0.1 |

Integral a => Real (Ratio a) | @since base-2.0.1 |

Defined in GHC.Internal.Real toRational :: Ratio a -> Rational Source # | |

Integral a => RealFrac (Ratio a) | @since base-2.0.1 |

Show a => Show (Ratio a) | @since base-2.0.1 |

Eq a => Eq (Ratio a) | @since base-2.01 |

Integral a => Ord (Ratio a) | @since base-2.0.1 |

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

, and`abs`

(`numerator`

y) <=`abs`

(`numerator`

y')

.`denominator`

y <=`denominator`

y'

Any real interval contains a unique simplest rational;
in particular, note that `0/1`

is the simplest rational of all.