yap-0.2: yet another prelude - a simplistic refactoring with algebraic classes

Data.YAP.Ratio

Description

Standard functions on rational numbers.

This version uses the same type as Data.Ratio, but with components generalized from `Integral` to `EuclideanDomain`. However using the same type means we have the old, more constrained, instances of `Ord`, `Show` and `Read`.

Synopsis

# Documentation

data Ratio a

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) EuclideanDomain a => Field (Ratio a) EuclideanDomain a => Ring (Ratio a) EuclideanDomain a => AbelianGroup (Ratio a) (Integral a, Integral a) => RealFrac (Ratio a) (Integral a, Integral a) => Fractional (Ratio a) (Integral a, Integral a) => Real (Ratio a) (Integral a, Integral a) => Num (Ratio a)

type Rational = Ratio Integer

Arbitrary-precision rational numbers, represented as a ratio of two `Integer` values. A rational number may be constructed using the `%` operator.

(%) :: EuclideanDomain a => a -> a -> Ratio aSource

Forms the ratio of two values in a Euclidean domain (e.g. `Integer`).

numerator :: EuclideanDomain 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 :: EuclideanDomain 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.