Data.Ratio

Contents

Synopsis

# Documentation

data Ratio a Source

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

Instances

 Integral a => Enum (Ratio a) Eq a => Eq (Ratio a) Integral a => Fractional (Ratio a) Integral a => Num (Ratio a) Integral a => Ord (Ratio a) (Integral a, Read a) => Read (Ratio a) Integral a => Real (Ratio a) Integral a => RealFrac (Ratio a) (Integral a, Show a) => Show (Ratio a)

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 aSource

Forms the ratio of two integral numbers.

numerator :: Integral a => Ratio a -> aSource

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

denominator :: Integral a => Ratio a -> aSource

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 -> RationalSource

`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.

# Specification

``` module  Data.Ratio (
Ratio, Rational, (%), numerator, denominator, approxRational ) where

infixl 7  %

ratPrec = 7 :: Int

data  (Integral a)      => Ratio a = !a :% !a  deriving (Eq)
type  Rational          =  Ratio Integer

(%)                     :: (Integral a) => a -> a -> Ratio a
numerator, denominator  :: (Integral a) => Ratio a -> a
approxRational          :: (RealFrac a) => a -> a -> Rational

-- "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.
--
-- E.g., 12 `reduce` 8    ==  3 :%   2
--       12 `reduce` (-8) ==  3 :% (-2)

reduce _ 0              =  error "Data.Ratio.% : zero denominator"
reduce x y              =  (x `quot` d) :% (y `quot` d)
where d = gcd x y

x % y                   =  reduce (x * signum y) (abs y)

numerator (x :% _)      =  x

denominator (_ :% y)    =  y

instance  (Integral a)  => Ord (Ratio a)  where
(x:%y) <= (x':%y')  =  x * y' <= x' * y
(x:%y) <  (x':%y')  =  x * y' <  x' * y

instance  (Integral a)  => Num (Ratio a)  where
(x:%y) + (x':%y')   =  reduce (x*y' + x'*y) (y*y')
(x:%y) * (x':%y')   =  reduce (x * x') (y * y')
negate (x:%y)       =  (-x) :% y
abs (x:%y)          =  abs x :% y
signum (x:%y)       =  signum x :% 1
fromInteger x       =  fromInteger x :% 1

instance  (Integral a)  => Real (Ratio a)  where
toRational (x:%y)   =  toInteger x :% toInteger y

instance  (Integral a)  => Fractional (Ratio a)  where
(x:%y) / (x':%y')   =  (x*y') % (y*x')
recip (x:%y)        =  y % x
fromRational (x:%y) =  fromInteger x :% fromInteger y

instance  (Integral a)  => RealFrac (Ratio a)  where
properFraction (x:%y) = (fromIntegral q, r:%y)
where (q,r) = quotRem x y

instance  (Integral a)  => Enum (Ratio a)  where
succ x           =  x+1
pred x           =  x-1
toEnum           =  fromIntegral
fromEnum         =  fromInteger . truncate        -- May overflow
enumFrom         =  numericEnumFrom               -- These numericEnumXXX functions
enumFromThen     =  numericEnumFromThen   -- are as defined in Prelude.hs
enumFromTo       =  numericEnumFromTo     -- but not exported from it!
enumFromThenTo   =  numericEnumFromThenTo

(\r -> [(x%y,u) | (x,s)   <- readsPrec (ratPrec+1) r,
("%",t) <- lex s,
(y,u)   <- readsPrec (ratPrec+1) t ])

instance  (Integral a)  => Show (Ratio a)  where
showsPrec p (x:%y)  =  showParen (p > ratPrec)
showsPrec (ratPrec+1) x .
showString " % " .
showsPrec (ratPrec+1) y)

approxRational x eps    =  simplest (x-eps) (x+eps)
where simplest x y | y < x      =  simplest y x
| x == y     =  xr
| x > 0      =  simplest' n d n' d'
| y < 0      =  - simplest' (-n') d' (-n) d
| otherwise  =  0 :% 1
where xr@(n:%d) = toRational x
(n':%d')  = toRational y

simplest' n d n' d'       -- assumes 0 < n%d < n'%d'
| r == 0     =  q :% 1
| q /= q'    =  (q+1) :% 1
| otherwise  =  (q*n''+d'') :% n''
where (q,r)      =  quotRem n d
(q',r')    =  quotRem n' d'
(n'':%d'') =  simplest' d' r' d r
```