Approximates

In mathematics and related fields, an approximation (often called an approximant) is a value that is close to a quantity of interest but easier to compute or observe than the exact value. Approximations are used when exact results are unknown, computationally costly, or unnecessary for a given task.

Error measures quantify closeness. The absolute error is the difference between the approximation a and the true value x, e = |x − a|, while the relative error is e_rel = |x − a| / |x|. Convergence describes the behavior of a sequence of approximations a_n as n increases, with a_n approaching x in the limit.

Common objects and methods labeled as approximants include partial sums of a series, which provide successive approximations to a function’s value; continued fractions and their convergents, which yield rational approximations to irrational numbers; and Padé approximants, which approximate a function by a ratio of polynomials. Rational approximations and Chebyshev approximations are among other well-known approaches used to balance accuracy and computational efficiency.

Approximation often involves truncation, rounding, interpolation, or extrapolation. Truncation discards terms beyond a certain point; rounding replaces a quantity with a nearby value of limited precision. Numerical analysis studies accuracy, stability, and error propagation in approximations, aiming to understand how computational choices affect results.

Approximations are pervasive across disciplines, including the numerical computation of constants, simulation of physical systems, and engineering design, where exact solutions are rare or impractical but sufficiently accurate approximations are essential.