Just wanted to say, Jonathan, that I do notice a lack of rigor even in my own understanding of number. I do often try to review with my (high school) students the laws governing operations between numbers, the different types of numbers, and what justifies each type of operation. I often remind students that there are counting numbers, positive and negative numbers, zero, rational and irrational, and then one must demonstrate that the rules governing addition, subtraction, multiplication and division can be extended to each type of number, from counting to all the reals. I am not actually an expert in this, so I have to fumble my way through, but at least I recognize the importance of doing so. A while back, it occurred to me that there are operator numbers which operate on other numbers, and since these seemed to be different from other numbers, and it never occurred to me before that there might be different types of numbers, each with different properties. so 5 x 75 is an! operation performed on 75, and thus, the "5" is a different type of number from 75. It sort of stopped me dead in my tracks, since I had never explored such an idea before. Wonder if you have ever explored such an idea. I would also like to know a simple source whereby I could review what you seem to indicate is "Indian" mathematics where the expansion of the number system and the operations involved are more carefully elaborated.