I'm going to write a class library for .NET which provide an implementation of arbitrary precision arithmetic for integer, rational and maybe complex numbers. What best known approaches should I become familiar with?I tried to start with Knuth's TAOCP Vol.2 (Seminumerical Algorithms, Chapter 4  Arithmetic) but it's too complicated. At least I couldn't get the ideas in a relatively short period of time.
What are the most known arbitrary precision arithmetic implementation approaches?
algorithms;data structures
If you are using C# 4.0 or later, then you already have a BigInteger class. Starting from here and creating your own Rational class will save you a lot of time. If you want to build everything yourself, then you will have to implement some complex algorithms, especially for multiplication. You could start with the naive multiplication algorithm that has time complexity O(n). A lot of other algorithms will depend on multiplication (division, modular exponentiation, gcd, square roots, etc.). If you have have everything working and a solid set of unit tests, then you can replace the naive multiplication algorithm for something like 3-way Toom-Cook multiplication or an even more advanced algorithm.