This book is written as an introduction to the theory of error-free computation. In addition, we include several chapters that illustrate how error-free comΒ putation can be applied in practice. The book is intended for seniors and firstΒ year graduate students in fields of study involving scientific computation using digital computers, and for researchers (in those same fields) who wish to obtain an introduction to the subject. We are motivated by the fact that there are large classes of ill-conditioned problems, and there are numerically unstable algorithms, and in either or both of these situations we cannot tolerate rounding errors during the numerical computations involved in obtaining solutions to the problems. Thus, it is important to study finite number systems for digital computers which have the property that computation can be performed free of rounding errors. In Chapter I we discuss single-modulus and multiple-modulus residue number systems and arithmetic in these systems, where the operands may be either integers or rational numbers. In Chapter II we discuss finite-segment p-adic number systems and their relationship to the p-adic numbers of Hensel [1908]. Each rational number in a certain finite set is assigned a unique Hensel code and arithmetic operations using Hensel codes as operands is matheΒ matically equivalent to those same arithmetic operations using the corΒ responding rational numbers as operands. Finite-segment p-adic arithmetic shares with residue arithmetic the property that it is free of rounding errors.