Infinitesimally

Infinitesimal refers to a quantity with magnitude smaller than any positive real number, yet not zero. In historical calculus, infinitesimals allowed intuitive reasoning about derivatives and integrals. In modern standard analysis, there are no real infinitesimals; calculus is built on limits. The adverbial form infinitesimally is used in ordinary language to mean to an extremely small degree.

The concept originated with 17th-century thinkers such as Leibniz and Newton, who used infinitesimals as a formal tool in developing differential and integral calculus. Their notation, including dx and dy, helped describe instantaneous rates of change and accumulation. Philosophical objections and the subsequent development of epsilon-delta rigor by Cauchy, Bolzano, Weierstrass, and others led to a limit-based foundation that does not require actual infinitesimals.

In the 20th century, nonstandard analysis provided a rigorous framework in which infinitesimals exist as hyperreal numbers. In this setting, an infinitesimal is a nonzero hyperreal whose absolute value is smaller than 1/n for every standard natural number n. Derivatives and integrals acquire intuitive definitions via infinitesimal changes, for example f'(x) = st(Δy/Δx) for an infinitesimal Δx, where st denotes the standard part. The term infinitesimally remains in everyday language to describe very small quantities or distinctions.