TITLE: What is the meaning of infinitesimal?
QUESTION [24 upvotes]: I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It cannot be a stationary value because if so then a smaller value on real number line exist, so it must be a moving value. Moving value towards $0$ so in most places we use its magnitude equal to zero but at the same time we also know that infinitesimal is not equal so in all those places were we use value of infinitesimal equal to $0$ we are making an infinitesimal error and are not $100\%$ accurate, maybe $99.9999\dots\%$ accurate, but no $100\%$! So please explain infinitesimal and its applications and methodology in context to the above paragraph or elsewise intuitively please.

REPLY [7 votes]: 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.