In this chapter we look at ways of approximating irrational numbers by rationals. We are interested in ways of effectively finding ‘good’ approximations and ways of showing that one cannot do this too well. By this we mean that the there is a trade off between the accuracy of the rational approximation and the size of the integers that make the rational approximation. In our later applications we will combine this with theoretical diophantine approximation results to produce a practical method to solve a large class of diophantine equations. The results in this chapter provide the computational means by which the theoretical results of Baker and others are converted into practical results.

We start by looking at the one variable case, and then we consider a way of generalizing this to many variables. There are many ways to do this generalization, we present the one which has been the most successful in recent years, namely the algorithm of Lenstra, Lenstra and Lovász. This algorithm is usually referred to as the ‘LLL’ algorithm in honour of its inventors. It has proved a useful tool in much of computational number theory. For instance we can use it when computing units and class groups of number fields and it can also be used to factor rational polynomials. We shall concentrate on its use in solving diophantine equations.