Computer addition is a groupoid. If an additive identity exists it is unique. If(and only if) addition is defined with the compute through the overflow (CTO) property, then a finite ring of integers is the homomorphic image of the computer number system and addition. Stated another way, the necessary and sufficient condition for CTO is a congruence relation on the integers. Also, if the number system has CTO capabilities for addition, it also has extended CTO properties for addition. A technique is presented for determining the correct sum in the extended compute through overflow (ECTO) mode of computation.

This paper surveys some recent contributions to computer arithmetic. The survey includes floating-point arithmetic, nonstandard number systems, and the generation of elementary functions. The design objective for these areas is computation that is both accurate and fast.

Overflow detection and overflow recovery imposed no particular requirements on the structure of (X, X1< X1, fx). In particular, if f is associative and commutative, overflow recovery is obtainable even if fxis neither associative or commutative.

This paper classifies error correcting codes for arithmetic and provides the necessary and sufficient conditions for the various code classes. Particular attention is given to the arithmetic properties of the codes. It is shown that the Brown codes and the Henderson codes are examples of a general class of nonseparate, nonsystematic codes. Henderson's systematic code is shown to be the only systematic, nonseparate code. Finally, conditions are given for the existence of code classes for both radix complement and diminished radix complement coded arithmetic systems.

A model is presented which, when modulo addition is used, can be used to derive multiplication correction schemata for operands expressed in either the radix complement or the diminished radix complement code.

A novel number system called the residue number system is developed from the linear congruence viewpoint. The residue number system is of particular interest because the arithmetic operations of addition, subtraction and multiplication may be executed in the same period of time without the need for carry. The main difficulties of the residue code pertain to the determination of the relative magnitude of two residue representations, and to the division process. A discussion of the arithmetic operations and the conversion process required to convert from a residue code to a weighted code is given. It is concluded that in its present state the residue code is probably not suitable for general purpose computation but is suitable for a special class of control problems. Further research in both components and arithmetic is required if a residue code suitable for general purpose computation is to be obtained.

The usual definition given for the parity check is unwieldy and not particularly suited for the analysis or the study of the arithmetic properties of parity checking. The definition of parity by means of congruences provides a convenient mathematical basis for the concepts of the parity check. In this paper congruence notation is used to generalize the concepts of parity to include nonbase two number systems. Consideration is given to the cases where the check base is equal to the number base, and where it is not equal to the number base. The arithmetic properties of each case are considered by means of congruences.