Data.Number.Transfinite

This module presents a type class for numbers which have representations for transfinite values. The idea originated from the IEEE-754 floating-point special values, used by Data.Number.LogFloat. However not all Fractional types necessarily support transfinite values. In particular, Ratio types including Rational do not have portable representations.

For the Glasgow compiler (GHC 6.8.2), GHC.Real defines 1%0 and 0%0 as representations for infinity and notANumber, but most operations on them will raise exceptions. If toRational is used on an infinite floating value, the result is a rational with a numerator sufficiently large that it will overflow when converted back to a Double. If used on NaN, the result would convert back as negativeInfinity.

Hugs (September 2006) stays closer to the haskell98 spec and offers no way of constructing those values, raising arithmetic overflow errors if attempted.

Documentation

class (Num a, Ord a) => Transfinite a where

Many numbers are not Bounded yet, even though they can represent arbitrarily large values, they are not necessarily able to represent transfinite values such as infinity itself. This class is for types which are capable of representing such values. Notably, this class does not require the type to be Fractional nor Floating since integral types could also have representations for transfinite values.

In particular, this class extends the Ord projection to have a maximum value infinity and a minimum value negativeInfinity, as well as an exceptional value notANumber. All the natural laws regarding infinity and negativeInfinity should pertain. Additionally, infinity - infinity should return notANumber (as should 0/0 and infinity/infinity if the type is Fractional). Any operations on notANumber will also return notANumber, and any equality or ordering comparison on notANumber must return False.

Minimum complete definition is infinity, isInfinite, and isNaN.

Methods

infinity :: a

A transfinite value which is greater than all finite values. Adding or subtracting any finite value is a no-op. As is multiplying by any non-zero positive value (including infinity), and dividing by any positive finite value. Also obeys the law negate infinity = negativeInfinity with all appropriate ramifications.

negativeInfinity :: a

A transfinite value which is less than all finite values. Obeys all the same laws as infinity with the appropriate changes for the sign difference.

notANumber :: a

An exceptional transfinite value for dealing with undefined results when manipulating infinite values. Since NaN shall return false for all ordering and equality operations, there may be more than one machine representation of this value.

isInfinite :: a -> Bool

Return true for both infinity and negativeInfinity, false for all other values.

isNaN :: a -> Bool

Return true only for notANumber.

Instances

Transfinite Double

Transfinite Float