The objective of this study was to determine ways of attaining a working competence in the arithmetic of the full real number system, and to explore means of removing some ineffective practice employed in instruction. A secondary objective was to determine the feasibility of developing infinite decimal arithmetic directly, without building on the arithmetic of fractions. The experimental testing of the classroom materials took place at the Middle School of the Laboratory Schools of the University of Chicago during 1964-65. Among the principal conclusions of this study are (1) working mastery of the real number system by appropriate use of its infinite decimal representation as a vehicle, and (2) this arithmetic can be built on or developed independently of fraction facts. (RP)