Arithmetic Applied Mathematics deals with the deterministic theories of particle mechanics using a computer approach. Models of classical physical phenomena are formulated from both Newtonian and special relativistic mechanics with the aid only of arithmetic. The computational power of modern digital computers is highlighted, along with simple models of complex physical phenomena and solvable dynamical equations for both linear and nonlinear behavior. This book is comprised of nine chapters and opens by describing an experiment with gravity, followed by a discussion on the two basic types of forces that are important in classical physical modeling: long range forces and short range forces. Gravitation and molecular attraction and repulsion are considered, along with the basic concepts of position, velocity, and acceleration. The reader is then introduced to the N-body problem; conservative and non-conservative models of complex physical phenomena; foundational concepts of special relativity; and arithmetic special relativistic mechanics in one space dimension and three space dimensions. The final chapter is devoted to Lorentz invariant computations, with emphasis on the arithmetic modeling and analysis of a harmonic oscillator. This monograph will be of interest to mathematicians, physicists, and computer scientists.