Interval Arithmetic over Finitely Many Endpoints

To my knowledge all definitions of interval arithmetic start with real endpoints and prove properties. Then, for practical use, the definition is specialized to finitely many endpoints, where many of the mathematical properties are no longer valid. There seems no treatment how to choose this finite set of endpoints to preserve as many mathematical properties as possible. Here we define interval endpoints directly using a finite set which, for example, may be based on the IEEE 754 floating-point standard. The corresponding interval operations emerge naturally from the corresponding power set operations. We present necessary and sufficient conditions on this finite set to ensure desirable mathematical properties, many of which are not satisfied by other definitions. For example, an interval product contains zero if and only if one of the factors does. The key feature of the theoretical foundation is that “endpoints” of intervals are not points but non-overlapping closed, half-open or open intervals, each of which can be regarded as an atomic object. By using non-closed intervals among its “endpoints”, intervals containing “arbitrarily large” and “arbitrarily close to but not equal to” a real number can be handled. The latter may be zero defining “tiny” numbers, but also any other quantity including transcendental numbers. Our scheme can be implemented straightforwardly using the IEEE 754 floating-point standard.