A fundamental axiom of real numbers is that 0 /= 1. If you deny this axiom, then you get a set of real numbers that is a little more compact. Here is the definition. I think you'll agree that it has a certain elegance.

 instance Num () where
     () + () = ()
     () * () = ()
     negate () = ()
     abs () = ()
     signum () = ()
     fromInteger _ = ()

CategoryWhimsy