Unitary Arithmetic
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 _ = ()