Discrete time control is fundamental for processes in many fields such as economics or biology as well as continuous processes where digital controllers must be implemented. Among the several approaches proposed in the literature, hereafter, we present two methodologies, namely the algebraic approach and the geometric approach both introduced in the ’80s to deal with nonlinear discrete time systems. These methodologies are illustrated with respect to a classical control problem, the linearization problem, which enables various characteristics of discrete time systems to be illustrated.