Minimal complete definition:
splitFraction or floor
There are probably more laws, but some laws are
(fromInteger.fst.splitFraction) a + (snd.splitFraction) a === a
ceiling (toRational x) === ceiling x :: Integer
truncate (toRational x) === truncate x :: Integer
floor (toRational x) === floor x :: Integer
If there wouldn't be Real.C a and ToInteger.C b constraints,
we could also use this class for splitting ratios of polynomials.
As an aside, let me note the similarities
between splitFraction x and x divMod 1 (if that were defined).
In particular, it might make sense to unify the rounding modes somehow.
IEEEFloat-specific calls are removed here (cf. Prelude.RealFloat)
so probably nobody will actually use this default definition.
New function fraction doesn't return the integer part of the number.
This also removes a type ambiguity if the integer part is not needed.
The new methods fraction and splitFraction
differ from Prelude.properFraction semantics.
They always round to floor.
This means that the fraction is always non-negative and
is always smaller than 1.
This is more useful in practice and
can be generalised to more than real numbers.
Since every T denominator type supports divMod,
every T can provide fraction and splitFraction,
e.g. fractions of polynomials.
However the ''integral'' part would not be of type class C.
Can there be a separate class for
fraction, splitFraction, floor and ceiling
since they do not need reals and their ordering?