Half a tutorial on the Haskell number system

Dear nice Haskell people out there!

Thank you for your friendly and numerous reactions on my number system picture on different web locations. It seems, that many people feel the same pain when it comes to numbers in Haskell. Even so brilliant introductions like the Real World Haskell seem to capitulate with this idiosyncratic complexity and rather sum up the facts. As I said, I gave up on it as well. But your reactions are itching.

So please, allow me to show you at least the existing half of my tutorial. The missing part is the actual reconstruction of the type classes. At some point, I tried to combine that part of Haskell with a reconstruction of the mathematical evolution from natural, to integer, ... to complex numbers. Here is a glimpse of what I had in mind. I thought, that many programmers could need this kind of update, which is necessary knowledge if one really wants to understand the logic behind the type class zoo.

Originally, this tutorial started off as just a section of an introduction to Haskell itself, some kind of "Haskell for mathematicians", with the ambition of being "the first truely functional introduction to this functional language". What I missed in all the classic texts is a pure conceptional or semantic approach to the matter. For example, they explain "if..then..else.." as a language construct that needs proper alignment etc etc. But "in fact" (i.e. in a functional brain), this is a function of type "(Bool,a,a)->a" (that accidentally happens to have a non-default syntax). In other words, instead of forcing people to learn the language first, before they can decide if they want to think that way, I thought I could start with the philosophy right away before going into the formal details. I cut out the part of the original introduction that attempts to sketch the Haskell universe the way I try to approach it.