In any ring, sometimes nilpotent elements (that is, those nonzero elements that can be made zero by raising them to some power) are also known as infinitesimal, especially when the ring minus these nilpotent elements is a field. There is good reason for this terminology. An example illustrates this best.

Consider the set Cn[ 0,1 ] of all n-times continuously differentiable real-valued functions in the closed interval [ 0,1 ]. To each of these functions, we may associate its n-degree Taylor polynomial. Suppose now that we are interested in taking sums and products of these functions. Ordinarily, this can be done with no difficulty, but say that we consider the Taylor n-degree polynomial approximations to be good enough to use instead of the actual functions, higher powers being more accuracy than we need. This amounts to working in the formal ring of polynomials R[x] of the real field with its ordinary algebraic operations, except that we throw out any power of x higher than n when we multiply. If we decide on n = 5, for instance, the following familiar-looking polynomials

1 - x2/2! + x4/4!

x - x3/3! + x5/5!,

have for a product the polynomial

x -(1/3! + 1/2!)x3 + (1/5! + 1/(3! 2!) + 1/4!) x5,

where we have already thrown out the higher powers of x, because they are more precision than we need; x6 is already "negligible", and it is safe to ignore it. Here we say that the element x in R[x] is infinitesimal. It is "small": its sixth power is too small to even be worthy of mention.

This can now be abstracted, generalised, and made rigorous. Observe that the interpretation of R[x] as Taylor polynomials is quite incidental, and we do not need it. We may proceed perfectly algebraically instead. Take any field F, such as R. If we wish to adjoin an infinitesimal element x to F, we first form the ring of polynomials F[x], and then we quotient by the ideal generated by xn, denoted (xn). The quotient ring


is then nothing more but the field F with an infinitesimal element x attached to it, since xn will be in the ideal (xn), and therefore zero in the quotient.

Here comes a technical apologia. Infinitesimal elements of the kind discussed in this writeup do have their uses, such as the Taylor polynomial interpretation. They are not, however, anything like the sort of thing that Leibniz and Newton had in mind when they were talking about infinitesimals, because the ring F[x]/(xn) is not a field. The element x is a zero divisor, and a field cannot have such creatures. It is curious to observe that both Newton and Leibniz would sometimes perform calculations as if they were working in R[x]/(xn), except that they would denote their infinitesimal elements by dx or by o. In order to salvage the intuitions and interpretations of the founders of modern calculus and form a bona fide ordered field with infinitesimal elements requires considerably more work than forming quotient rings. We are thus led to non-standard analysis.