We propose a new number representation and arithmetic for the elements of the ring of integers modulo p. The socalled Polynomial Modular Number System (PMNS) allows for fast polynomial arithmetic and easy parallelization. The most important contribution of this paper is the fundamental theorem of a Modular Number System, which provides a bound for the coefficients of the polynomials used to represent the set Z_p. However, we also propose a complete set of algorithms to perform the arithmetic operations over a PMNS, which make this system of practical interest for people concerned about efficient implementation of modular arithmetic.