Interval Arithmetic

Interval arithmetic is a method of numerical computation in which numbers are represented by intervals that bound the possible values, and operations are performed on these intervals rather than on point estimates. Instead of computing that x = 3.14, interval arithmetic computes that x lies in [3.13, 3.15]; instead of adding point values, it adds intervals, producing a new interval guaranteed to contain the true result. The guarantee is the point: interval arithmetic does not estimate; it encloses.

The method was developed by Ramon Moore in the 1960s as a response to the catastrophic accumulation of rounding errors in finite-precision computation. In standard floating-point arithmetic, each operation introduces a small error; over millions of operations, these errors compound unpredictably. Interval arithmetic replaces the illusion of precision with the discipline of bounded uncertainty. The result is wider than a point estimate but honest: it never claims more certainty than the computation justifies.

Interval arithmetic is the computational counterpart to the admissible heuristic in search and to the branch-and-bound bound in optimization. All three share the same logical structure: a conservative estimate that is guaranteed never to be wrong in the dangerous direction. An admissible heuristic never overestimates the cost to the goal; an interval bound never underestimates the range of possible values; a branch-and-bound lower bound never exceeds the true optimum. Each is a form of epistemic humility encoded in mathematics.

The technique has found applications in rigorous global optimization, computer-assisted proof verification, and robust control theory. In each domain, the challenge is the same: compute with uncertainty without losing the guarantee. Interval arithmetic is not a replacement for probabilistic methods. It is a complement: where probability quantifies uncertainty through distribution, interval arithmetic bounds it through containment.