# How small is an infinitesimal quantity?

When speaking of infinitesimals, I see some mathematicians say that it represents an "extremely small" element, such as an infinitesimal area on a manifold.

What bothers me about this naive definition is how small is, an infinitesimal area, for example, supposed to be for it to be called infinitesimal?

• Really, really, really small. – tparker Jul 12 '17 at 14:47
• @tparker actually, half of that. – Carl Witthoft Jul 12 '17 at 14:55
• Infinitesimally small. – user428517 Jul 12 '17 at 15:34
• The way I see it, is that an infinitesimal is just a construct we use to represent taking a limit to zero. This got me through most of the conceptual difficulties. – 1010011010 Jul 12 '17 at 15:37
• "Sufficiently small" – MC ΔT Jul 12 '17 at 16:22

There are, I think, three main senses.

The first sense (sometimes called "true" infinitesimals) is when you have an easily identified collection of things that are not infinitesimal, and some sense of comparing "size". The infinitesimals are those objects that are smaller than every non-infinitesimal.

A typical example is the hyperreals from nonstandard analysis: an infinitesimal hyperreal is a number whose magnitude is smaller than the magnitude of every nonzero (standard) real number.

The second sense is somewhat metaphorical; where you have objects that represent some infinitesimal-like notion. So while you don't have any "true" infinitesimals, you can still use the metaphor to do many of the things you wanted to use infinitesimals for anyways.

For example, the real line has no (nonzero) infinitesimals, but we can talk about its tangent bundle: the set of pairs of real numbers $(x,y)$ where $x$ denotes a point on the real line and $y$ is imagined as the scale of some infinitesimal displacement from $y$. Then, to do calculus with these, we say that if $f$ is a differentiable function, we also treat it as a function on the tangent bundle too, with $f(x,y) = (f(x), f'(x) y)$.

This sort of thing is very important to differential geometry.

We can actually make the tangent bundle into an algebraic structure called the dual numbers in a similar fashion to how the complex numbers are defined: we interpret a real number $x$ as the point $(x,0)$, let $\epsilon = (0,1)$. Addition is defined in the obvious way, and multiplication by setting $\epsilon^2 = 0$. (rather than $i^2 = -1$ as we do with complex numbers)

Repeating the above, if $f$ is differentiable, we set $f(x + y \epsilon) = f(x) + f'(x) y \epsilon$. Note the appealing similarity to the notion of a "differential approximation".

In this sort of algebraic setup, we say that $\epsilon$ is a "nilpotent infinitesimal" (to distinguish from "true" infinitesimals). Nilpotent is an adjective that means you get zero by raising it to a sufficiently large power.

The third sense is approximate; "infinitesimal" is used as a shorthand for the idea of being "approximately infinitesimal", which means something is small enough for whatever purpose you need.

In physics, the notion of an infinitesimal quantity or area is used extremely informally, to indicate roughly anything much smaller than some given reference quantity. To give a mathematically acceptable description of the infinitesimal is a more serious undertaking.

"Infinitesimal" means, formally, "Smaller than any positive ordinary number." Practically, that means smaller than any positive number you can explicitly name, such as $.1,.01,.001,...$. This seems paradoxical, and was taken historically, as in the philosophical work of Berkeley in the 18th century, to indicate that the very notion of infinitesimal is incoherent. While these arguments had a significant influence on the formalization of the subject of analysis in the late 19th century, they were not the last word, as Robinson in the 1960s rehabilitated the notion of infinitesimal back into mathematical respectability. Vaguely, the starting point is to observe that there's nothing immediately contradictory about positing the existence of a positive number smaller than any (say) positive rational number, and to axiomatize your way into a situation where such numbers do indeed exist. The most traditional way to introduce such numbers is the study of hyperreal number systems. But for geometrical applications, the theory of synthetic differential geometry may be more useful.

Infinitesimals were used in the genesis of analysis which was appropriately called at the time infinitesimal analysis or infinitesimal calculus. Infinitesimals were used fruitfully for several centuries. Around 1870 certain foundational developments led to the mathematicians jettisoning the infinitesimals. The language they used still exploited the intuitive terminology of infinitesimals but having no precise mathematical counterpart for them, they had to be eliminated when their arguments were formalized.

This resulted in the notorious epsilon-delta paraphrases, the nightmare of a majority of undergraduate math majors. In 1961 infinitesimals were restored to respectability by Abraham Robinson but old habits die hard. The precise meaning of an infinitesimal $\epsilon$ is a fixed number that is less than $\frac12$, less than $\frac13$, less than $\frac14$, all the way down.

Since you mentioned "manifolds" I would like to point out the existence of an approach to differential geometry via infinitesimals where one can naturally talk about infinitesimal area elements (rather than paraphrases in terms of differential forms); see this recent publication for details.

• Why can't an infinitesimal be thought of as the number defining the distance between zero and the nearest point to zero on the real line? Strictly speaking, due to density of the reals, there is no nearest point, but what I mean is that an infinitesimal could probably be though of as the size of one point. I think that's what is happening during Riemann integration - the integral goes over each point, "one by one". – sequence Jul 18 '17 at 22:08
• @sequence, one way of implementing the intuition about "the smallest number" (i.e., the one closest to zero) is to use a hyperfinite partition and limit one's choices of points to elements of the partition. – Mikhail Katz Aug 9 '17 at 16:43

Engineers must use a practical concept of "infinitesimal" because we work with both very large and very small numbers. To us, "infinitesimal" is basically the same as "irrelevant," "ignorable," or "existing within the limit of statistical noise," which we often define generally as less than 3% of the whole. Thus, (a) an infinitesimal fraction of 2pF (two pico-Farads) is anything less than (0.03)*2pF or 60fF (60 femto-Farads) and (b) an infinitesimal fraction of 2MV (two mega-volts) is (0.03)*2MV or 60KV (60 kilo-volts).

• Thanks for adding an engineer's perspective. But this is math.se and the OP's question is about mathematics. – Ethan Bolker Jul 12 '17 at 21:08
• Point well taken, but engineering is math. Let me ask a clarifying question. If I take the time to re-aquaint myself with the mathematics behind metastable exclusion around an inflection point, which is where the generalized 3% comes from, would that help? Or are mathematicians only worried about a quantity zero such that I can still divide by that number and derive a limited, rational result? – JBH Jul 12 '17 at 21:18
• In the selected answer, my answer conforms with definitions #1 and #3. Definition #2 is the only mathematically precise answer. Had that answer not included definition #2, would it also have been deemed inadequate? I'm just curious. I'm a ways away from participating in Meta, so I can't ask it there. – JBH Jul 12 '17 at 21:44
• Forgive another intrusion from an engineer. I dont think that the question "How small is an infinitesimal?" has any good and simple mathematical answer, but I always give engineering students an answer that I like to think of as close to the mathematical spirit "It is a quantity so small that if it were made any smaller, you couldn't tell the difference", And likewise for infinity. – Philip Roe Jul 13 '17 at 12:36
• To me it is clear that "engineers" (and "scientists") do math. That math may not meet some "formal requirements" of self-styled mathematicians, but I'm not sure that that exclusionary criterion is the best criterion for whether one is doing math or not. And, e.g., heuristics for interesting things are better than scrupulous proofs of boring things. :) – paul garrett Jul 13 '17 at 21:26

From my experience, the term infinitesimal is generally synonymous with non-Archimedian, compared with some other elements in the algebraic structure (e.g. a field), such that no finite combination of infinitesimal elements is larger (in an absolute value) than any (non-zero) non-infinitesimal.

Extend the set of rational numbers Q to the set of sequences of rational numbers Q*N (I think it is the closest we can think of as Euler's concept of real number) an infinitesimal is then any of these sequences that converges to zero and it is as small as a null sequence is.

How to found analysis in this way see here:

http://homepages.math.uic.edu/~kauffman/InfinitesimalsHenle.pdf

I think this is the best approach to analysis.

• Welcome to Math.SE. Please use MathJax for math symbols. Also please take care of proper writing, e.g. use capitals and punctuation. – M. Winter Jul 13 '17 at 21:29

Before you ask what any mathematical construct is, you need to specify in what set you are seeking it. If that is the set $\Bbb R$ of real numbers, then the answer is simple: There are no infinitesimals in $\Bbb R$. That is, given any nonzero (say positive) real number, then adding finitely many copies of it together can make a number as large as you like, and there is always a fraction $1/n$, for some positive integer $n$, smaller than it.

To make sense of the idea of "infinitesimal", you have to go beyond the real numbers, as discussed in the other answers. Infinitesimals play as big a role in properly-done everyday real analysis as ghosts do in everyday life—namely, not a very big role. Some people think that you can informally sneak them in and out of the real numbers without causing confusion. I don't. Learn proper real analysis first. When you have mastered that, you will have the background to read Robinson's book, or perhaps an easier later book such as Nonstandard Analysis for Dummies.

• (I didn't downvote, but) I disagree with a claim of irrelevance of infinitesimals and potential confusion, for at least two reasons. First, in most heuristic situations the intention and physical intuition is very clear and suggests the right answer (which is why many scientists and engineers continue to think/talk in those terms). Second, specifically, the "standard part" notion (see Robinson, and also Nelson), going back to Leibniz, I think quite well addresses Bishop Berkeley's complaints about cheating: nothing which was non-zero becomes 0: no ghosts of departed quantities. – paul garrett Jul 13 '17 at 21:30
• Thanks to all, you've all been very informative. – Steven Aug 17 '17 at 3:31