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numerator :: Integral a => 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 :: Integral a => 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

, andabs
(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.
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 = x1 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 instance (Read a, Integral a) => Read (Ratio a) where readsPrec p = readParen (p > ratPrec) (\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 (xeps) (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