infinitesimal
/ĭn″fĭn-ĭ-tĕs′ə-məl/
adjective
- Immeasurably or incalculably minute.
- (Mathematics) Capable of having values approaching zero as a limit.
nonzero positive ‘number’ smaller than any positive real number, formalizable in a number of ways (surreals, hyperreals etc.)
Explaining the concept of an infinitesimal...how would you go about it?
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Yesterday, my girlfriend asked me an interesting question. She's getting a PhD in pharmacology, so she's no dummy, but her math education doesn't extend past calculus.
She said, "There's a topic in P Chem that I never understood. Like dx, dy. What does that mean? Those are just letters to me."
My response was, "Well, you've taken calculus, so you may remember the concept of a limit? When we talk about a finite value we refer to it as delta y, so y2-y1 for example. But if we are talking about an infinitesimal, like dy, then we are referring to the limit as delta y approaches zero."
She said, "That just seems like witch craft. Like you're making it up."
I said, "Infinitesimals are just mathematical objects that are greater than zero but less than all Real numbers. They're infinitely small, but non-negative."
I struggled to explain it to her in a way that seemed rigorous. Bare in mind, I'm studying Chemical Engineering so I'm not mathematician. I've just taken more math than she has so she thought I should be able to answer.
What would you guys have said?
TLDR: Girlfriend asked me to explain infinitesimals to her, but my explanation wasn't satisfactory.
The real numbers $\mathbb{R}$ is an example of a field, a space where you can add, subtract, multiply and divide elements. In addition, $\mathbb{R}$ is an example of an ordered field, i.e. for any $a, b \in \mathbb{R}$ we have either $a < b$, $a = b$, or $a > b$. Note, there are some further conditions on the interaction between inequalities and the field operations.
A positive infinitesimal in an ordered field is an element $e > 0$ such that $e < \frac{1}{n}$ for all $n \in \mathbb{N}$. A negative infinitesimal is $e < 0$ such that $-e$ is a positive infinitesimal. An infinitesimal is either a positive infinitesimal, a negative infinitesimal, or zero.
In $\mathbb{R}$ there is only one infinitesimal, zero - this is precisely the Archimedean property of $\mathbb{R}$. So while people use the word infinitesimal to convey intuition, the real numbers don't have any non-zero infinitesimals, so their explanation is flawed.
In the early development of calculus by Newton and Leibniz, the concept of an infinitesimal was used extensively but never defined explicitly. The way this has been rectified through history is via the introduction of limits which still capture the intuition, but are in fact defined perfectly well.
It should be noted that other ordered fields do have non-zero infinitesimals. You might even try to find an ordered field which contains all the real numbers that you know and love, but also has non-zero infinitesimals. Such a thing exists! Abraham Robinson first showed such an ordered field exists in $1960$ using model theory, but it can actually be constructed using something called the ultrapower construction. This is called the field of hyperreal numbers and is denoted ${}^*\mathbb{R}$. With the hyperreals at hand, you can take all the ideas that Newton and Leibniz used and interpret them almost literally. Calculus done in this way is often called non-standard analysis.
Infinitesimals are a natural product of the human imagination and have been used since antiquity, so I would not describe them as "unthinkably small". One can think of them and even represent them graphically using the pedagogical device of microscopes, as in Keisler's classic textbook Elementary Calculus.
In my experience teaching infinitesimals in the classroom, students tend to think of infinitesimals as quantities tending to zero, or in terms of "variable quantities" as they were often described by the pioneers of the calculus like Leibniz and Cauchy. This is a useful intuition that should be encouraged, but ultimately they have to be constructed as constant (or as you say "stationary") values if they are to be formalized within a modern mathematical framework.
The "infinitesimal error" you are referring to seems to be the type of technique that occurs for example in the calculation of the derivative of $y=x^2$, where $\frac{\Delta y}{\Delta x}$ is algebraically simplified to $2x+\Delta x$ and one is puzzled by the disappearance of the infinitesimal $\Delta x$ term that produces the final answer $2x$; this is formalized mathematically in terms of the standard part function.
To answer your question about the applications of infinitesimals: they are numerous (see Keisler's text) but as far as pedagogy is concerned, they are a helful alternative to the complications of the epsilon, delta techniques often used in introducing calculus concepts such as continuity. The epsilon, delta techniques involve logical complications related to alternation of quantifiers; numerous education studies suggest that they are often a formidable obstacle to learning calculus. Infinitesimals provide an alternative approach that is more accessible to the students and does not require excursions into logical complications necessitated by the epsilon, delta approach.
In fact, I did a quick straw poll in my calculus class yesterday, by presenting (A) an epsilon, delta definition and (B) an infinitesimal definition; at least two-thirds of the students found definition (B) more understandable.
To respond to the recent comment, a difference between our approach and Keisler's is that we spend at least two weeks detailing the epsilon-delta approach (once the students already understand the basic concepts via their infinitesimal definitions). Thus the students receive a significant exposure to both approaches. Our educational experience and the student reactions to our approach are detailed in this recent publication.
I was wondering, since infinitesimal/ly means small, is saying "infinitesimally small" a plenoasm? Otherwise, if I mean that something is not relevant would I say it's then infinitesimally relevant?