nearvalues

Nearvalues are values that lie close to a specified target value within a chosen tolerance. Given a target value T and a tolerance δ > 0, a value a is a nearvalue of T if |a - T| ≤ δ. In the real numbers this set of nearvalues is the closed interval [T − δ, T + δ]. In discrete domains, the nearvalues form a finite or countable collection of points within that interval.

Nearvalues depend on the choice of tolerance; changing δ changes the set. They are not exact equals, but practical approximations used when exact equality is unnecessary or unattainable due to measurement error, rounding, or numerical noise. In computations, nearvalue concepts help avoid false negatives when testing for equality by checking whether the difference falls within a tolerance rather than being exactly zero.

Applications include rounding and numerical comparisons, interval arithmetic, error budgeting in engineering, robust optimization, and data analysis where measurements have uncertainty. They also appear in algorithms that require approximate matching, such as nearest-neighbor searches with tolerance, or geometric computations that must tolerate small perturbations.

Nearvalue is related to broader ideas of closeness in metric spaces, epsilon-delta definitions in analysis, and the notion of nearzero. In practice, the term is used most often informally rather than as a formal mathematical object, with the emphasis on tolerance-based approximation.

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