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
necessarily support transfinite values. In particular,
Rational do not have portable representations.
For the Glasgow compiler (GHC 6.8.2), GHC.Real defines
0%0 as representations for
but most operations on them will raise exceptions. If
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
Hugs (September 2006) stays closer to the haskell98 spec and offers no way of constructing those values, raising arithmetic overflow errors if attempted.
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
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
as well as an exceptional value
notANumber. All the natural
negativeInfinity should pertain.
infinity - infinity should return
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
Minimum complete definition is
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
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.
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
Return true for both
false for all other values.
Return true only for