In Haskell, is there infinity :: Num a => a?

Question

I\'m trying to implement a data structure where if I had the use of infinity for numerical comparison purposes, it would simply things greatly. Note this isn\'t maxBound/min

Answer 1

There is a more principled approach based on an idea from non-standard analysis. Given a totally ordered ring R of characteristic zero, you can consider the Laurent ring R[inf,1/inf] with the natural lexicographic total ordering. For example, you have:

for all x>0 in R,
.. -inf < -x < -d < -d^2 < .. < 0 < .. < d^2 < d < x < inf < inf^2 < .. 
where d = 1/inf.

This way the Laurent ring R[inf,1/inf] is again a totally ordered Z-algebra, i.e. an instance of Num, with other niceties you possibly want, including +/-infinity, +/-infinitesimal, second-order infinitesimals, etc.. But note that it's not Archimedian and induction will no longer work, which is a sort of second-order arithmetic. For implementation take a look at this example. As in the comment in the code this construction should work for other algebras, such as the list monad. You can think of lists where two elements are "infinitely close" "second-order infinitely far away" etc. (which leads to a generalization of rose trees.)