I was discussing a potential problem with a physicist, and there is a bit of a question that I have, about modular number systems.
Consider a modular system of integers. That is we can consider the integers mod n to be {0,1,2,...,n-1}. Addition and multiplication can be defined just fine, as can the additive inverse, subtraction. But what about multiplicative inverse (division)?
It is certainly possible to define fractions for some modular integer systems. For example, 1/2 is defined as 'the number such that, when multiplied by 2, equals 1'. In integers mod 7, we have that 1/2=4, as 4 multiplied by two equals 8=1. However, in the integers mod 8, 1/2 cannot exist, as it would requide double an integer to be equal to nine.
When then brings us onto the main question. What about a modular system of real numbers? I've only ever seen modular systems of integers before, never seen people work with modular real numbers. There do exist systems that people work with that are modular, and can accept real numbers (such as eix), but working with modular real numbers (that is, a half-open interval [0,k)) I have not seen done.
Certainly, over the real numbers, it would be possible to define addition, multiplication, and subtraction, but what about division?
The real question which I want to answer, it is it possible to define a vector field over such a space? A vector field requies modular reals to be a field, which requires multiplicative inverses to exist.
EDIT: I have had a last minute brainwave about this. It turns out that division can only exist on the interval (k-1,k). However, what is to stop us modifying our definition of addition so the new addition has an additive identity of k-1? We would then have a field that is analogous to [0,k-k-1), or have I gone wrong somewhere?