An infinitesimal is an intuitively vanishing quantity — smaller than any finite positive number — yet treated as distinct from zero. Infinitesimals have been used informally since ancient times to describe processes that involve quantities that shrink without ever reaching zero. In modern mathematics they appear in different, rigorously defined forms, serving as tools to express instantaneous change, differentials, and limiting behavior.

Characteristics and basic idea

Informally, an infinitesimal x satisfies the property that for every positive real number ε, |x| < ε. That informal characterization cannot hold inside the real numbers (the only real number with that property is 0), so different frameworks have been developed to give infinitesimals precise meaning. In practice mathematicians distinguish between the heuristic use of infinitesimals (as in early calculus) and formal objects such as elements of a field of hyperreal numbers.

Historical development

Infinitesimal reasoning was central to the 17th–18th century development of calculus by Newton and Leibniz, who used different infinitesimal-style methods to compute derivatives and integrals. Critics later challenged the lack of a rigorous foundation, leading to the 19th-century reformulation of calculus in terms of limits and epsilon-delta definitions by Cauchy and Weierstrass.

Modern formalizations

  • Limits and epsilons: The standard approach replaces infinitesimals with limit processes that avoid introducing new number types.
  • Nonstandard analysis: Introduced in the 20th century, constructs the hyperreal numbers to include actual nonzero infinitesimals and infinitely large numbers while preserving the usual rules of arithmetic.
  • Smooth infinitesimal analysis: Uses intuitionistic logic and nilpotent infinitesimals in a categorical setting for differential geometry and synthetic approaches.

Uses and important distinctions

Infinitesimals reappear in physics, engineering, and differential geometry as a convenient language for rates of change (dx, dy, differentials) and local linear approximations. The careful reader should note the distinction between the informal phrase "infinitesimal" and specific formal implementations: different frameworks offer different properties and proof techniques, and only some treat infinitesimals as actual numbers rather than shorthand for limiting behavior.