What is the meaning of infinitesimal?

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.

Solution 1:

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.

Solution 2:

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.