G'day all.
Quoting Henning Thielemann
It seems we are at a point, where we have to define what is a 'number'.
For backwards compatibility, I'd say a Num is what it is at the moment. One of the proposals that comes up every so often is to allow the declaration of a typeclass instance to automatically declare instances for all superclasses. So, for example: class (Functor m) => Monad m where fmap f m = m >>= return . f instance Monad Foo where return a = {- ... -} m >>= k = {- ... -} fail s = {- ... -} This will automatically declare an instance of Functor Foo. Similarly, a finer-grained collection of numeric typeclasses could simply make Num a synonym for (Show a, Ord a, Ring a, Signum a). Declaring an instance for (Num Bar) declares all of the other instances that don't yet have a declaration.
More precisely: Can you tell me the difference between numbers and "more complex mathematical objects"?
Yes. A Num is anything which supports the common mathematically- significant operations which are supported by the basic built-in machine types such as Int and Double. It need not _be_ a built-in machine type, but it must support those operations. (Yes, some architectures support vector operations. This doesn't count as "basic". No, some architectures don't support Double or Word64 natively. I don't care.) Cheers, Andrew Bromage