Is $0$ an Infinitesimal?

Question

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 ?

Answer 1

$0$ 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".

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

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.