infinitesimal

An infinitesimal is the intuition of a quantity that is as small as you like yet not actually zero — smaller than any ordinary positive number you could name, but still somehow there. It is the picture early calculus ran on: a curve is built from infinitely many infinitely short straight bits, an area is a pile of infinitely thin slices. The dx in dy/dx and in an integral is a fossil of this idea.

Newton and Leibniz used such 'vanishingly small' quantities to invent calculus in the late 1600s, and the method worked spectacularly. But it sat on shaky logic: in a single calculation an infinitesimal had to behave like a nonzero number (so you could divide by it) and then like zero (so you could drop it). Critics, most famously Bishop Berkeley, mocked these as 'ghosts of departed quantities'. The eventual rigorous fix, built in the 1800s by Cauchy and Weierstrass, replaced the infinitesimal with the limit: instead of an actual infinitely small number, you study what a quantity approaches as it shrinks toward zero.

So in standard modern calculus there is no infinitesimal number on the real line — it is a guiding intuition made precise by limits, not an object you compute with. Honesty requires one footnote, though: in the 1960s Abraham Robinson built nonstandard analysis, a rigorous system that does contain genuine infinitesimals. It is logically sound but specialized; the limit-based approach remains the standard foundation taught and used almost everywhere.