In mathematics, an infinitesimal is a number greater in absolute value than zero yet smaller than any positive real number. A number x ≠ 0 is an infinitesimal iff every sum |x| + ... + |x| of finitely many terms is less than 1, no matter how large the finite number of terms. In that case, 1/x is larger than any positive real number.
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 2c, 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.
- To find the derivative f '(x) of the function f(x) = x², let dx be an infinitesimal. Then f '(x) = (f(x+dx)-f(x))/dx = (x²+2x*dx+dx²-x²)/dx = 2x+dx = 2x, since dx 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) = x² 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 + dx is x.
Alternatively, we can have synthetic differential geometry.