nearequal

Nearequal, often rendered as approximately equal, is a relation used to indicate that two numbers are close to each other within a defined tolerance rather than exactly equal. It is widely used in numerical analysis, scientific measurement, and computer arithmetic to account for rounding errors, measurement uncertainty, and other sources of imprecision.

A common formalization defines two numbers a and b to be nearequal if the difference |a - b| is within a specified tolerance. Many frameworks use two tolerances: an absolute tolerance (abs_tol) that bounds the allowed absolute difference, and a relative tolerance (rel_tol) that bounds the difference relative to the size of the numbers. A typical criterion is |a - b| <= max(abs_tol, rel_tol * max(|a|, |b|)). Some contexts also reference unit in the last place (ULP) or machine epsilon for setting tolerances appropriate to floating-point representation.

In notation, nearequal is often represented by the symbol ≈ or by phrases like “is approximately equal to.” The concept is distinct from exact equality and from strict equivalence relations in algebra. The nearequal relation is generally symmetric and reflexive, since every number is nearequal to itself, but it is not transitive in the simplest formulations: if a is nearequal to b and b is nearequal to c, a may not be nearequal to c under a fixed tolerance.

Applications include comparing floating-point results, validating numerical algorithms, and handling measurement data. Practitioners must choose tolerances carefully based on data scale, units, and the expected error, balancing robustness against the risk of obscuring genuine differences.

See also: approximately equal, isclose, numerical tolerance, floating-point arithmetic.