Kirby says: >OK, so repeated addition is OK up to the point where instead of 1/2 of something, you want 1/pi of it, and that something happens to be irrational already. That's when repeated addition breaks down,
Well, some people like to hold on up to the bitter end. One way to approach irrational numbers is as the limits of certain sequences - and define the product of two such limits as the limit of the products of corresponding terms of rationals. That way, you can keep your repeated addition going infinitely. Or indefinitely. Or maybe indinfinitely.