numerically

Numerically is an adverb derived from the adjective numerical, used to indicate that something relates to numbers, quantities, or numerical methods rather than symbolic algebra. The term is common in mathematics, science, and engineering to describe actions performed with numbers or through numerical procedures. When a problem is solved numerically, its solution is obtained by approximate computations using numbers rather than by algebraic simplification or closed-form formulas.

Numerical computations are a central focus of numerical analysis, which studies algorithms for estimating quantities with controlled accuracy. Key concerns include error estimation, stability, convergence, and computational cost. Techniques described as numerical include solving equations by iterative methods, numerical integration and differentiation, interpolation, and solving differential equations with time-stepping schemes such as Euler or Runge-Kutta methods.

A common distinction is that analytic or analytical results provide exact expressions, while numerical results are approximate and expressed as digits. In practice, numerical methods are chosen when exact solutions are unavailable or impractical due to complexity, dimensionality, or computer constraints. Numerically computed data appear in simulations, data analysis, engineering models, and scientific research, where precise symbolic solutions may be infeasible.

Etymology generally traces numerically to the Latin root for number, with the English word formed by the adjective numerical and the adverbial suffix -ly.