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Most mathematicians throw a conniption fit if you happen to mention the unfortunate term "infinitesimal".

This actually happened to me in my senior year of high school, when the head of the math department was a substitute teacher in Calc class one day. The merest mention of the word "infinitesimal" and we were subjected to another month of limit theory.

The inventors of calculus imagined differentials as infinitesimals, and used them in their work.

Unforunately, the notion of a quantity that is infinitely small causes some pretty severe paradoxes in arithmetic.

Because of the problems associated with infinitesimals, calculus was reformulated by 18th century mathematicians to be based upon the theory of limits.

The father of modern set theory, Georg Cantor, called infinitesimals "the Cholera-bacillus of mathematics." This may have due to the fact that the existence of infinitesimals would render his Continuum Hypothesis false.

First of all, in the usual model of the real line there are no infinitesimal numbers (i.e. positive numbers that are smaller than 1/n, for any positive integer n). They don't exist.

Before the calculus was properly formulated infinitesimals were very popular for a very good reason. They make all the proofs a lot easier! For example Newton and Leibeniz used these "ideas". Eventually, after the work of Weierstrass the familiar epsilon-delta arguments of mathematical analysis that we use today were introduced, and infinitesimals were relegated to the dustbin of history. Although, in high school mathematics infinitesimals are still often employed in plausibility arguments (I will not call them proofs).

All that changed in the 1960s when Abraham Robinson invented the idea of nonstandard analysis and hyperreal numbers. The basic idea is that instead of working with the usual set of real numbers R one adjoins some extra elements to R to form *R the hyperreal line. This contains all the usual real numbers but it also contains infinitesimals (and their reciprocals which are infinitely large).

After doing this it is possible to set the clock back and give the same kind of arguments that Newton gave but now in a completely rigorous way. Those high school arguments also become possible to justify.

This is quite a fun idea and I'm a bit surprised that it hasn't caught on as a way to teach analysis. Probably the reason why it hasn't is that although things are nice once you have the hyperreal line, its construction is a bit abstract and nonintuitive and that perhaps scares people off.

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

F[x]/(xn)

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.

## Definition of infinitesimals in Robinson's non-standard analysis

Infinitesimals are merely the (non-standard) objects whose standard approximation is .

In other words: Form the set Φ(x) of all formulae

0 < x & x < 1/n
for every (standard!) natural numbers n. As R is archimedean, you could equally well take all rational numbers or all real numbers n, it makes no difference.

This set is finitely satisfiable in the real world, hence it is satisfiable in the non-standard world. Any x satisfying all of Φ(x) is a (positive) infinitesimal.

Let O be the non-standard set of all infinitesimals. As (R,+) is a topological group, it turns out that the standard approximations for any real a are precisely a+O -- take a, and add an infinitesimal. This meshes nicely with our intuitions.

The notion of standard approximation is thus easier to use than infinitesimals (it also works in any metric space). Ironically, Robinson's non-standard analysis tends not to use infinitesimals by their name, but rather that wider related concept.

In`fin*i*tes"i*mal (?), a. [Cf. F. infinit'esimal, fr. infinit'esime infinitely small, fr. L. infinitus. See Infinite, a.]

Infinitely or indefinitely small; less than any assignable quantity or value; very small.

Infinitesimal calculus, the different and the integral calculus, when developed according to the method used by Leibnitz, who regarded the increments given to variables as infinitesimal.

In`fin*i*tes"i*mal, n. Math.

An infinitely small quantity; that which is less than any assignable quantity.

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