Systems on time scales incorporate continuous- and discrete-time dynamical systems. They can serve as models of real systems from various areas. Positive systems on time scales are studied here in more detail. Such systems appear in biology, medicine and chemistry. The language of time scales allows for natural description of time discretization in continuous-time models. During such discretization some properties of the system may be lost, while other may be preserved. Invariance under discretization of positivity and uniform exponential stability is investigated, both for linear and nonlinear systems. One of the results says that if discretization preserves positivity of the system, then it also preserves uniform exponential stability.