'But it doesn't give you a way to add *infinitely many* elements together. The usual way to define infinite sums is as a limit of finite sums. But a group structure is not enough to be able to define limits; you need an additional, topological, structure, to do that. And even once you have such a topological structure, that doesn't guarantee that *every* series or *every* sequence converges. Some do, some don't. If your topological structure comes from a metric structure (The standard topological structure on Q does), and the metric space is complete (Q isn't), then any Cauchy sequence will converge. Since the Reals are complete, this guarantees that the series for zeta(2) converges to some Real number (but not necessarily to a number in Q, since Q is not complete).' So how do we know that the incomplete nature of Q produces Reals, or indeed how can we determine the 'completetion' of an incomplete group (R is the completion of Q). Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com