We start by discussing the computer representation and fundamental arithmetic algorithms for integers and polynomials. We will keep this discussion fairly informal and avoid all the intricacies of actual computer arithmetic—that is a topic on its own. The reader must be warned that modern-day processors do not represent numbers and operate on them as we describe now, but to describe the tricks they use would detract us from our current goal: a simple description of how one could, in principle, perform basic arithmetic.

Although our straightforward approach can be improved in practice for arithmetic on small objects, say double-precision integers, it is quite appropriate for large objects, at least as a start. Much of this book deals with polynomials, and we will use some of the notions of this chapter throughout. A major goal is to find algorithmic improvements for large objects.

The algorithms in this chapter will be familiar to the reader, but she can refresh her memory of the analysis of algorithms with our simple examples.