Mathematica automatically uses sophisticated algorithms to track the precision of approximate numbers. Particularly for some verification applications, however, it is convenient to maintain explicit numerical intervals. All Mathematica's basic mathematical functions and operations automatically operate on Interval objects.
Interval — represent an interval to be used in mathematical operations
IntervalMemberQ — test whether a number lies within an interval
IntervalUnion — combine intervals together
IntervalIntersection — find common parts of intervals
RootIntervals — give lists of isolating intervals for polynomial roots