Approximations

Approximations are representations or estimates of quantities that are close to the exact value but not identical. They arise when exact computation is impractical, impossible, or unnecessary, and they are fundamental in science, engineering, and mathematics.

Approximations can take many forms: numerical values obtained by rounding or truncation; and functions or models that imitate a more complex target over a domain.

Common methods include series expansions such as Taylor and Fourier series, polynomial interpolation and regression, numerical methods for integration and differentiation, and iterative methods for solving equations. Geometric and algebraic simplifications replace complex objects with simpler ones. Probabilistic methods, including Monte Carlo simulations, estimate quantities via random sampling.

Error is assessed with absolute and relative error, often expressed as the difference between the approximation and the true value. Convergence describes whether successive approximations approach the target, with the rate of convergence and error bounds providing quantitative guarantees. Stability concerns how small input perturbations affect the result.

Approximations enable tractable modeling, simulation, and computation across disciplines. They carry limitations, including potential bias, loss of exact properties, and misleading conclusions if used beyond their domain of validity. Historical milestones include Archimedes' pi bounds, polynomial interpolation, Gaussian quadrature, and the Newton–Raphson method for finding roots.