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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.

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  • $\begingroup$ a non-zero quantity can be thought infinitesimally small (large) if its half equals itself. $\endgroup$ – giorgi Jul 30 '13 at 13:57
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    $\begingroup$ @giorgi Your example describes zero... $\endgroup$ – Džuris Jul 30 '13 at 14:01
  • $\begingroup$ @Juris thank you. This example appears to be useful for infinitesimally large quantities. $\endgroup$ – giorgi Jul 30 '13 at 14:06
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    $\begingroup$ Your mistake is right here: "It cannot be a stationary value because if so then a smaller value on real number line exist...". Infinitesimals are not real numbers, and therefore don't live on the real number line in the first place. They are part of an extension of the real numbers, just as the real numbers are an extension of the rational numbers, and the rational numbers are an extension of the integers. $\endgroup$ – WillO Jul 30 '13 at 14:23
  • $\begingroup$ Possibly helpful: math.stackexchange.com/questions/1991575/… $\endgroup$ – Ethan Bolker Apr 21 at 23:38
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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. 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.

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    $\begingroup$ About nonstandard analysis: perhaps a little surprisingly, such methods find use in functional analysis. For an example see this post by Haskel Curry: math.stackexchange.com/a/263966/30222 $\endgroup$ – tomasz Jul 30 '13 at 14:37
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    $\begingroup$ Hah, you should put quotes around "Haskel Curry," since I don't think that is the poster's real name, but rather an homage to the real Haskel Curry. @tomasz $\endgroup$ – Thomas Andrews Jul 30 '13 at 14:40
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    $\begingroup$ You say that infinitesimals must be greater than zero but then go on to state that zero is an infinitesimal for the real numbers. This is clearly a contradiction. $\endgroup$ – Cameron Williams Jul 30 '13 at 14:48
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    $\begingroup$ @CameronWilliams: I said a positive infinitesimal is greater than zero, I didn't say an infinitesimal is greater than zero. $\endgroup$ – Michael Albanese Jul 30 '13 at 14:50
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    $\begingroup$ Oops I missed the part where you included zero. My mistake. I was not reading closely enough. $\endgroup$ – Cameron Williams Jul 30 '13 at 14:59
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In general, it is better to think of infinitesimals as an intuition or motivation, rather than as something that actually exists. In the standard theory of the real numbers, there is no such thing as an infinitesimal.

In the early days of calculus, a lot of the ideas were defined in terms of an intuitive idea of infinitesimals, but in the 19th century, as mathematics became more and more driven to make sure the foundations of mathematics made sense, they found problems with infinitesimals, and a way to do calculus without needing the infinitesimal numbers, and therefore discarded them.

In calculus, the "motivating idea" of infinitesimals remains in some of the notation:

$$\frac{dy}{dx}$$ is not a fraction, but we represent it as a fraction of infinitesimals. The key is to remember it is not actually a fraction, even though it often acts like a fraction. Same with the notation:

$$\int_a^b f(x)\;dx$$ the $dx$ is again representing an intuitive idea of an infinitesimal, but it is not an actual number, but notation.

More modern mathematics can give a rigorous foundation which includes infinitesimals. This is non-standard, and probably more complicated than you need.

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  • $\begingroup$ $\frac{dy}{dx}$ is indeed a fraction, for $dy=f'(x)\Delta x$ and $dx=\Delta x$, then $\frac{dy}{dx}=f'(x)$ and $dy=f'(x)dx$. Learn more in this book $\endgroup$ – iMath Mar 2 '17 at 3:20
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    $\begingroup$ No, $dx\neq \Delta x$. I think I've learned enough calculus, thanks. @iMath $\endgroup$ – Thomas Andrews Mar 2 '17 at 3:54
  • $\begingroup$ books.google.com.hk/… the textbook explained why dx=Δx $\endgroup$ – iMath Mar 10 '17 at 6:13
  • $\begingroup$ @ThomasAndrews, the usual convention with regard to the independent variable $\Delta x$ is to assign $dx=\Delta x$, whereas for the dependent variable $\Delta y =f(x+dx)-f(x)$ there is of course a difference between $dy=f'(x)dx$ and $\Delta y$. $\endgroup$ – Mikhail Katz Mar 20 '17 at 16:29
  • $\begingroup$ Sorry, but in the notation $\frac{dy}{dx}$, $dx$ does not mean $\Delta x$. The same is true for most other occurrences of $dx$ in integrals, differential forms, and the like. $dx$ is not a number in these senses. It is possible to develop a theory of the real line with infinitesimals, but it is non-standard, and not usually done. $\endgroup$ – Thomas Andrews Mar 20 '17 at 16:41
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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.

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  • $\begingroup$ It may be that (A)+(B) is more understandable than either on its own. $\endgroup$ – Evgeni Sergeev Aug 14 '17 at 10:52
  • $\begingroup$ Hi @Evgeni, I responded in the body of my answer. $\endgroup$ – Mikhail Katz Aug 15 '17 at 8:18
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I believe that modern mathematics mostly stays away from infinitesimals. We prefer to speak in terms of limits and in sentences like, "For all numbers $\epsilon$, however small, the following property holds."

I think non-standard analysis defines the infinitesimal: http://en.wikipedia.org/wiki/Non-standard_calculus

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  • $\begingroup$ If you consider algebraic geometry to be a branch of modern mathematics, you will have to reconsider your claim that modern mathematics stays away from infinitesimals. $\endgroup$ – Mikhail Katz Jan 7 '16 at 13:14
  • $\begingroup$ @user72694 Point well taken. I hope to understand it better soon. $\endgroup$ – Eric Auld Jan 7 '16 at 17:58
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Imagine a number which has a smaller absolute value than the absolute value of any nonzero real number. It is an infinitesimal number. This is how I understand it.

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    $\begingroup$ @ThomasAndrews I don't think so. They aren't equal by this definition (although their standard part is always equal to zero) and they can define derivatives rigorously (via nonstandard analysis). All real numbers are finite; this doesn't make them equal. $\endgroup$ – Ian Mateus Jul 30 '13 at 15:41
  • $\begingroup$ Oh, I missed the "real number" part, never mind, I remove my objection. :) $\endgroup$ – Thomas Andrews Jul 30 '13 at 15:42
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Consider Infinity. Is it a "stationary value"? Where is it in the number line? Infinity is a concept. It has a value larger than any value you can imagine.

Likewise, Infinitesimal is a concept; its value is smaller than any value you can imagine.

Check out this video and you will appreciate why Infinity and Infinitesimal cannot be "explained" to someone seeking to find "applications" / "methodology".

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    $\begingroup$ $+\infty$ is the upper endpoint of the number line. Infinity stopped being "just a concept" over a century ago, infinitesimals at least 50 years ago. Both may have gone back much further. $\endgroup$ – Hurkyl Oct 19 '13 at 15:30
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    $\begingroup$ I wish I could downvote comments. $\endgroup$ – Mark C Sep 5 '18 at 23:27

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