infinitesimal

An infinitesimal is a quantity with magnitude smaller than any positive real number, yet not equal to zero. Historically used in the development of calculus by Newton and Leibniz, infinitesimals allowed informal reasoning about quantities that change by infinitely small amounts. Although appealing, their rigorous status was questioned, leading to the development of limits as the foundation of calculus in the 19th century and the rejection of actual infinitesimals in standard analysis. Nevertheless, the intuition persists in the notation of differentials (dx, dy) and in the idea of an infinitesimal change in a function.

Nonstandard analysis, formulated by Abraham Robinson in the 1960s, provides a rigorous framework in which infinitesimals exist as actual numbers within the hyperreal extension of the real numbers. The transfer principle ensures that many real-number properties extend to hyperreals, and the standard part map associates a finite hyperreal with its real value.

In geometry and algebraic geometry, infinitesimals appear in the study of tangent spaces and in infinitesimal thickening, where one considers nilpotent or dual numbers to capture first-order behavior.

In physics and applied mathematics, infinitesimals appear in differential equations, perturbation theory, and in the heuristic notation of differentials, though precise justification often relies on limits or nonstandard frameworks.

Infinitesimals remain a foundational and interpretive tool across schools of mathematics, serving both as a heuristic device and, in nonstandard analysis, as a rigorous element of number systems.