An infinitesimal is only a notional quantity - there exists no infinitesimal real number. This can be shown using the least upper bound axiom of the real numbers: consider whether the least upper bound *c* of the set of all infinitesimals is or is not an infinitesimal. If it is, then so is 2*c*, contradicting the fact that *c* is an upper bound. It it is not, then neither is *c*/2, contradicting the fact that among all upper bounds, *c* is the least.

The first mathematician to make use of infinitesimals was Archimedes. See how Archimedes used infinitesimals.

When Newton and Leibniz developed the calculus, they made use of infinitesimals. A typical argument might go:

- To find the derivative
*f*'(*x*) of the function*f*(*x*) =*x*², let d*x*be an infinitesimal. Then f '(*x*) = (*f*(*x*+d*x*)-*f*(*x*))/d*x*= (*x*²+2*x**d*x*+d*x*²-*x*²)/d*x*= 2*x*+d*x*= 2*x*, since d*x*is infinitesimally small.

It was not until the second half of the nineteenth century that the calculus was given a formal mathematical foundation by Karl Weierstrass and others using the notion of a limit, which obviates the need to use infinitesimals.

Nevertheless, the use of infinitesimals continues to be convenient for simplifying notation and calculation.

Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson. In this theory, the above computation of the derivative of *f*(*x*) = *x*² can be justified with a minor modification: we have to talk about the *standard part* of the difference quotient, and the standard part of *x* + d*x* is *x*.

Alternatively, we can have synthetic differential geometry.

See also: