The Haskell 98 Library Report: Rational Numbers

The Haskell 98 Report
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## 12`` Rational Numbers

`module Ratio ( Ratio, Rational, (%), numerator, denominator, approxRational ) where infixl 7 % data (Integral a) => Ratio a = ... type Rational = Ratio Integer (%) :: (Integral a) => a -> a -> Ratio a numerator, denominator :: (Integral a) => Ratio a -> a approxRational :: (RealFrac a) => a -> a -> Rational instance (Integral a) => Eq (Ratio a) where ... instance (Integral a) => Ord (Ratio a) where ... instance (Integral a) => Num (Ratio a) where ... instance (Integral a) => Real (Ratio a) where ... instance (Integral a) => Fractional (Ratio a) where ... instance (Integral a) => RealFrac (Ratio a) where ... instance (Integral a) => Enum (Ratio a) where ... instance (Read a,Integral a) => Read (Ratio a) where ... instance (Integral a) => Show (Ratio a) where ...`

For each `Integral` type t, there is a type `Ratio` t of rational pairs with components of type t. The type name `Rational` is a synonym for `Ratio Integer`.

`Ratio` is an instance of classes `Eq`, `Ord`, `Num`, `Real`, `Fractional`, `RealFrac`, `Enum`, `Read`, and `Show`. In each case, the instance for `Ratio` t simply "lifts" the corresponding operations over t. If t is a bounded type, the results may be unpredictable; for example `Ratio Int` may give rise to integer overflow even for rational numbers of small absolute size.

The operator `(%)` forms the ratio of two integral numbers, reducing the fraction to terms with no common factor and such that the denominator is positive. The functions `numerator` and `denominator` extract the components of a ratio; these are in reduced form with a positive denominator. `Ratio` is an abstract type. For example, `12 % 8` is reduced to 3/2 and `12 % (-8)` is reduced to (-3)/2.

The `approxRational` function, applied to two real fractional numbers `x` and `epsilon`, returns the simplest rational number within the open interval (`x`-`epsilon`, `x`+`epsilon`). A rational number n/d in reduced form is said to be simpler than another n'/d' if |n| <=|n'| and d <=d'. Note that it can be proved that any real interval contains a unique simplest rational.

### 12.1`` Library `Ratio`

`
-- Standard functions on rational numbers
module 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 "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
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 (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

The Haskell 98 Report
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December 2002`