Recent attempts to utilize residue number systems in digital computers have raised numerous questions about adapting the techniques of numerical analysis to residue number systems. Among these questions are the fundamental problems of how to compare the magnitudes of two numbers, how to detect additive and multiplicative overflow, and how to divide in residue number systems. These three problems are treated in separate chapters of this thesis and methods are developed therein whereby magnitude comparison, overflow detection, and division can be performed in residue number systems. In an additional chapter, the division method is extended to provide an algorithm for the direct approximation of square roots in residue number systems. Numerous examples are provided illustrating the nature of the problems considered and showing the use of the solutions presented in practical computations. In a final chapter are presented the results of extensive trial calculations for which a conventional digital computer was programmed to simulate the use of the division and square root algorithms in approximating quotients and square roots in residue number systems. These results indicate that, in practice, these division and square root algorithms usually converge to the quotient or square root somewhat faster than is suggested by the theory.