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For the definition of Infinitesimal, wikipedia says

In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, but not zero in size; or, so small that it cannot be distinguished from zero by any available means.

MathWorld says

An infinitesimal is some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity.

BUT I met some definition of Infinitesimal in textbooks says

If $\lim_{{{x}\to{x}_{{0}}}} f{{\left({x}\right)}}={0}$, then we call $f(x)$ is an infinitesimal when ${x}\to{x}_{{0}}$.

I found the textbooks's definition conflict with the above two definitions..

Obviously, $f(x)=0$ satisfy the textbooks's definition ,then can we call 0 an Infinitesimal ?

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    $\begingroup$ What type of textbook was this definition in? That is, what was the topic of the textbook? $\endgroup$
    – KSmarts
    Dec 31, 2014 at 15:08
  • $\begingroup$ I think so far my answer is the only one to give a precise definition of "infinitesimal". ${}\qquad{}$ $\endgroup$
    – Michael Hardy
    Dec 31, 2014 at 16:24
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    $\begingroup$ No, zero cannot qualify for being infinitesimal. Infinitesimals were precisely introduced to solve the indeterminacies caused by zero and avoid quantities like $0/0$. They are "the last station before $0$". $\endgroup$
    – user65203
    Dec 31, 2014 at 18:45
  • $\begingroup$ @KSmarts The book is about mathematical analysis(a.k.a. advanced calculus). $\endgroup$
    – iMath
    Jan 1, 2015 at 3:16

6 Answers 6

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$0$ is infinitesimal.

Natural language is a bad reference for mathematical definitions; it's 'optimized' for quickly conveying meaning in 'natural' settings, not for expressing things precisely. There are all sorts of conventions like if if you ever hear someone talk about a "small number", you're supposed to assume there's a good reason for using that phrase instead of "zero", and thus should assume that the number is, in fact, nonzero, despite the fact zero is a small number.

For a nonnumeric example of this phenomenon, if I told you I lived near Paris, you would infer that I do not live in Paris.

With that in mind, I am not surprised to find that the English meaning of infinitesimal excludes zero.

However, that makes for a bad mathematical definition. The typical mathematical usage of infinitesimal is in a sense where 0 would be included; e.g. in nonstandard analysis, if $f$ is a standard, continuous function with $f(0) = 0$, then we would like to say "$f(x)$ is infinitesimal whenever $x$ is infinitesimal". Being able to say that requires that we consider $0$ to be infinitesimal; if we did not, then we would have to say something more awkward, like "$f(x)$ is either infinitesimal or zero whenever $x$ is infinitesimal".

The definition you mention from your textbook doesn't really make sense when taken literally, at least when taken out of context like this.

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  • $\begingroup$ The textbooks definition does show 0 is an Infinitesimal,satisfy what you mean here,I think it make sense in Mathematics ,but perhaps not in Natural language, because it's 'optimized' for quickly conveying meaning in 'natural' settings, not for expressing things precisely as you said ! Thanks for your clarification ! $\endgroup$
    – iMath
    Jan 1, 2015 at 13:37
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The only context I can think of where an infinitesimal makes sense is in a formalization of nonstandard analysis. In this context,

$\epsilon \in {}^*\mathbb{R}$ is called infinitesimal if $|\epsilon| < r$ for all positive real numbers $r$.

The set of infinitesimals $I$ is an ideal in the ring of finite hyperreal numbers. Modding out by the ideal $I$ one gets the original set of real numbers $\mathbb{R}$. In fact, the important standard part function is the natural ring homomorphism from the finite hyperreals to the reals induced by taking this quotient.

All ideals contain $0$, so in this context one always wants $0$ to be infinitesimal. Also, most definitions involving infinitesimals (e.g., $f(x) - f(a)$ is infinitesimal whenever $x - a$ is infinitesimal) require that $0$ is infinitesimal. In cases where we don't want to include $0$, we should explicitly ask for a nonzero infinitesimal.

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  • $\begingroup$ are you sure the $0$ is defined as an infinitessimal? I thought that a goal of nonstandard analysis was to represent a derivative as the quotient of 2 values, and that would be much simpler if zero was excluded. I don't know a good reference myself, do you? (Also I think your vocabulary is going to be difficult to understand for something asking this sort of question.) $\endgroup$
    – DanielV
    Jan 1, 2015 at 14:11
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    $\begingroup$ @DanielV, zero is indeed an infinitesimal by the definitions of all versions of NSA I'm familiar with. In internal set theory, zero is the only standard infinitesimal, which makes it easy to distinguish from the others. When necessary, we talk about positive or non-zero infinitesimals. $\endgroup$
    – pash
    Jan 1, 2015 at 19:34
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    $\begingroup$ @DanielV Think about $dy / dx$. We need $dx$ to be nonzero, yes--but $dy$ should certainly be allowed to be zero. Indeed, the definition of a derivative is the value $\text{st} \left(\frac{f(x + \epsilon) - f(x)}{\epsilon} \right)$, for some nonzero infinitesimal $\epsilon$, and the definition of differentiable is that this value is the same for all nonzero infinitesimals $\epsilon$. $\endgroup$
    – Caleb Stanford
    Jan 1, 2015 at 19:37
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    $\begingroup$ @DanielV Basically it ends up making more sense to exclude zero in the cases where it should be excluded, then to explicitly include it where it should be included. Stylistically speaking, "nonzero infinitesimal" is much less clunky than "infinitesimal or zero". $\endgroup$
    – Caleb Stanford
    Jan 1, 2015 at 19:38
  • $\begingroup$ @Goos Thank you for your response, that makes a lot of sense. $\endgroup$
    – DanielV
    Jan 2, 2015 at 4:57
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That $\varepsilon$ is an infinitesimal means that the sum $$ |\varepsilon|+\cdots+|\varepsilon| $$ remains less than $1$ no matter how large is the finite number of terms being added.

By that definition, $0$ is an infinitesimal. But the term is seldom used except when talking about non-zero infinitesimals.

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    $\begingroup$ Your answer conveys some property that "infinitesimals" intuitively should have, but cannot claim to be "a precise definition of `infinitesimal'". $\endgroup$
    – Christian Blatter
    Dec 31, 2014 at 19:51
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    $\begingroup$ @ChristianBlatter : It's precise if "finite" is precise. $\endgroup$
    – Michael Hardy
    Dec 31, 2014 at 23:09
  • $\begingroup$ @MichaelHardy When an infinite number of terms being added , does the expression remain less than 1 ? $\endgroup$
    – iMath
    Jan 1, 2015 at 13:38
  • $\begingroup$ I have to agree with Christian Blatter, for example, you would probably also assume that $2\epsilon > \epsilon$, than $\epsilon_1 + \epsilon_2 \in ~^{*}\mathbb R$, that $n\cdot(\epsilon_1 + \epsilon_2) = n\cdot \epsilon_1 + n\cdot \epsilon_2$...there is a litany of possible sets of properties that could be definitive, just the 1 property is unlikely to be sufficient to be definitive. $\endgroup$
    – DanielV
    Jan 1, 2015 at 14:07
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    $\begingroup$ @DanielV : Your comment is unclear. You say $2\varepsilon>\varepsilon$, but that should hold only if $\varepsilon>0$, and my answer carefully included both positive and negative infinitesimals. I'm guessting "than" is a typo for "then". As far as ${}^{*}\mathbb R$ goes, I certainly was not assuming we were working in that context at all. How could the distributive property possibly be part of a definition of infinitesimal? The distributive law holds for numbers that are not infinitesimal. To me your comments and those of Christian Blatter appear to have no merit. $\endgroup$
    – Michael Hardy
    Jan 2, 2015 at 2:09
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The Leibniz definiton of infinitesimal says that "they are lower than any real number but still greater than 0". So i think that $f(x)=0$ can't be considered an infinitesimal.

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According to most dictionaries and in everyday context infinitesimal just means "extremely small".

But in mathematics, the meaning of "infinitesimal" depends on the context. It is mostly used for the concept of Limits. When writing $x \to 0$ one always means that value of $x$ are smaller than any other real number. Here, the value of $x$ approaches so close to zero that we cannot differentiate with the zero.

Since these numbers (infinitesimals) are very small than most of the real numbers but not equal to zero, these numbers fall into a unique category of real numbers "Hyperreal Numbers" which tend to either zero or infinity.

So, the answer is maybe Yes.

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    $\begingroup$ In the context of nonstandard analysis, $0$ should certainly be infinitesimal. Even in the limit case, you write "as $x \to 0$ $f(x) \to a$" and you intend this to hold in the case that $f(x)$ is constant equal to $a$. Therefore you need $0$ to be considered infinitesimal if you want $f(x) \to a$ to be a formalization of infinitesimals. $\endgroup$
    – Caleb Stanford
    Dec 31, 2014 at 20:23
  • $\begingroup$ yes , I agree with @Goos 's opinion, if you take 0 as an Infinitesimal , it doesn't conflict with the mean of $x \to x_0$. $\endgroup$
    – iMath
    Jan 1, 2015 at 13:40
  • $\begingroup$ @Goos Is my answer relevant now. If yes you can reverse your downvote. (But, do not upvote.) $\endgroup$
    – Dheeraj Kumar
    Jan 1, 2015 at 13:46
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Wikipedia and Mathworld are correct, and the textbook's definition is incorrect and nonsensical. For which specific values $x$ does the textbook definition claim that $f(x)$ is an infinitesimal? None. It just says

when $x\to x_0$

which tells one nothing in this context.

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    $\begingroup$ A textbook may use any term it wants with any meaning it wants, as long as it defines it first. So the textbook is not "incorrect and nonsensical", merely unorthodox. It calls certain functions infinitesimal. $\endgroup$
    – GEdgar
    Dec 31, 2014 at 16:43

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