GHC.Real

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 Numeric.Natural.Natural's.

Real numbers.

The Haskell report defines no laws for Real, however Real instances are customarily expected to adhere to the following law:

Coherence with fromRational

if the type also implements

Fractional

, then

fromRational

is a left inverse for

toRational

, i.e.

fromRational (toRational i) = i

The law does not hold for Float, Double, CFloat, CDouble, etc., because these types contain non-finite values, which cannot be roundtripped through Rational.

Forms the ratio of two integral numbers.

raise a number to a non-negative integral power

raise a number to an integral power

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

gcd x y is the non-negative factor of both x and y of which every common factor of x and y is also a factor; for example gcd 4 2 = 2, gcd (-4) 6 = 2, gcd 0 4 = 4. gcd 0 0 = 0. (That is, the common divisor that is "greatest" in the divisibility preordering.)

Note: Since for signed fixed-width integer types, abs minBound < 0, the result may be negative if one of the arguments is minBound (and necessarily is if the other is 0 or minBound) for such types.

lcm x y is the smallest positive integer that both x and y divide.

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

Converts a possibly-negative Real value to a string.

Fractional numbers, supporting real division.

The Haskell Report defines no laws for Fractional. However, (+) and (*) are customarily expected to define a division ring and have the following properties:

recip gives the multiplicative inverse

x * recip x

=

recip x * x

=

fromInteger 1

Totality of toRational

toRational

is total

Coherence with toRational

if the type also implements

Real

, then

fromRational

is a left inverse for

toRational

, i.e.

fromRational (toRational i) = i

Note that it isn't customarily expected that a type instance of Fractional implement a field. However, all instances in base do.

Integral numbers, supporting integer division.

The Haskell Report defines no laws for Integral. However, Integral instances are customarily expected to define a Euclidean domain and have the following properties for the div/mod and quot/rem pairs, given suitable Euclidean functions f and g:

• x = y * quot x y + rem x y with rem x y = fromInteger 0 or g (rem x y) < g y

• x = y * div x y + mod x y with mod x y = fromInteger 0 or f (mod x y) < f y

An example of a suitable Euclidean function, for Integer's instance, is abs.

In addition, toInteger should be total, and fromInteger should be a left inverse for it, i.e. fromInteger (toInteger i) = i.

Extracting components of fractions.

General coercion to Fractional types.

WARNING: This function goes through the Rational type, which does not have values for NaN for example. This means it does not round-trip.

For Double it also behaves differently with or without -O0:

Prelude> realToFrac nan -- With -O0
-Infinity
Prelude> realToFrac nan
NaN

General coercion from Integral types.

WARNING: This function performs silent truncation if the result type is not at least as big as the argument's type.

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

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.