Ashley Yakeley wrote:
I considered doing something very like this for real (computable) numbers, but because I couldn't properly make the type an instance of Eq, I left it. Actually it was worse than that. Suppose I'm adding two numbers, both of which are actually 1, but I don't know that:
1.000000000.... + 0.999999999....
The trouble is, as far as I know with a finite number of digits, the answer might be 1.9999999999937425 or it might be 2.0000000000013565
...so I can't actually generate any digits at all. So I can't even make the type an instance of my Additive class.
You can, unless you are so ambitious that you want to have an ideal solution. Doing the stuff lazily means that you will have a thunk used in further computations, and the digits will be generated according to your needs. You *MAY* generate these digits physically ('1' or '2' in your case) if you permit yourself to engage in a possibly bottom-less recursive pit, which in most interesting cases actually *has* a bottom, and the process stops. Please look my "Pi" crazy essay. Once the decision concerning the carry is taken, the process becomes "sane", generative, co-recursive, until the next ambiguity. There are of course more serious approaches: intervals, etc. The infinite- precision arithmetic is a mature domain, developed by many people. Actually the Gosper arithmetic of continued fractions is also based on co-recursive expansion, although I have never seen anybody implementing it using a lazy language, and a lazy protocol. Anybody wants to do it with me? (Serious offers only...) Jerzy Karczmarczuk