Can calculus be used for real number?

Question

I recently learned that there is no positive infinitesimal real number. The only infinitesimal real number is 0.

For calculus integral, dx is always interpreted as infinitesimal small but non-zero which contradict with the property of real number.

And I don't think zero would be a valid infinitesimal number for calculus as real numbers divided by zero is undefined.

So my questions are: 1) Can we use calculus for real number in strictly speaking? 2) Is it correct to say the infinitesimal concept for calculus doesn't exist under real number system?

Answer 1

1) Can we use calculus for real number in strictly speaking?

Yes, one normally does not rely on infinitesimals in calculus. One defines limits using epsilon-delta definition. One uses a similar approach when defining integrals (either using limits or infimum/supremum)

2) Is it correct to say the infinitesimal concept for calculus doesn't exist under real number system?

Yes, the concept of infinitesimal requires you to extend beyond real numbers. There are a number of extensions that allow for infinitesimals, these will contain the set of real numbers as a subset.

However these extensions require some rather advanced tools to pull off which means they don't simplify things, at least not the first grade calculus student.

Answer 2

I'm gonna elaborate slightly on what others have said.

1) Yes. When mathematicians started taking issue with the idea of infinite and infinitesimal numbers, the $\epsilon-\delta$ approach was proposed. As a rule, any statement that's apparently about infinite or infinitesimal numbers can be converted to a statement about the real numbers in the standard framework of $\epsilon-\delta$ and the related way we think about limits of sequences. These are typically unintuitive at first, but they are the way mathematicians have understood these concepts for over a century.

Interestingly, calculus was founded on the ideas of infinitesimals, as it's still often taught in early calculus courses. In fact in German, the word for calculus to this day translates as "infinitesimal calculation". Though the usage of infinitesimals was not wholly accepted at the time, it's of note that the math done by using them was by and large correct, despite the weak grounding. It was the discovery in the mid-20th century of the hyperreals and subsequent work on them that showed why this was (roughly speaking, it sheds light on how we can translate statements about "infinitesimals" into statements in $\epsilon-\delta$ and vice-versa and still preserve truth). But more on that below.

2) In the real numbers? Yes. Depending on your definition of infinitesimal, this is exactly what it means to be infinitesimal. However, there are well-known ways to extend the real numbers in such a way that they include infinitesimals and infinite values. The construction of these "hyperreals" requires a bit of mathematical maturity to understand, which some people take as a reason not to teach it to calculus students. Here it is worth noting that the same can be said of real numbers themselves, a construction which also takes a degree of mathematical maturity (and some constructions of the hyperreals have very similar intuitions to those of the reals).

Though few mathematicians actually question the validity of the construction of the hyperreals, not all see it as a useful exercise. This is a matter of significant debate. But as of now, it'd still behoove a young math student to understand the $\epsilon-\delta$ principles. The revolution may or may not come, but it is definitely not yet at hand.

Answer 3

It takes a bit of mathematical maturity to understand the rigorous construction of the real number system in a set-theoretic context. Therefore in most universities, freshman calculus courses don't dwell on such technical details. Instead, such courses take the basic properties of the desired number system for granted, and proceed from there to the techniques and procedures of infinitesimal calculus.

The idea of extending a number system to accomodate the needs of new applications is ubiquitous throughout our educational system. First students are introduced to natural numbers. At the university level the students usually discover that the latter form a set denoted $\mathbb N$. The needs of elementary arithmetic motivate the extension of the number system further, to $\mathbb Z$, the set of integers. To accomodate the solution of general linear equations, the number system is extended further to $\mathbb Q$, the rationals. Even elementary geometric applications like diagonal of a unit square and area of a unit circle require a more sophisticated number system, leading to a further extension $\mathbb {Q}\hookrightarrow \mathbb R$.

The needs of infinitesimal calculus are best accomodated by extending the number system further, to a number system containing infinitesimals: $\mathbb{R}\hookrightarrow{}^\ast\mathbb R$. Neither the set $\mathbb R$ nor the set ${}^\ast\mathbb{R}$ is typically developed in a set-theoretic context in freshman calculus. Keisler's wonderful infinitesimal calculus textbook entitled Elementary Calculus does a beautiful job explaining the calculus based on such an infinitesimal-enriched number system. The educational advantages of this approach have been recently analyzed in this article. The students themselves vote with their feet in favor of this approach.

Answer 4

The standard calculus track includes many tricks for circumventing the need for actual infinitesimals.

For example, one of the main principles at work is that sufficiently small real numbers behave like infinitesimals. So, if you consider real numbers at all possible scales, you ensure that your argument involves sufficiently small scales — thus the $\epsilon$-$\delta$ argument that is so important for doing low-level arguments in calculus.

Another is to rescale quantities so that they aren't infinitesimal anymore. If $y$ is a function of $x$, while one might ostensibly be more interested in the quantities $\Delta y$ and $\Delta x$, we instead consider their ratio $\frac{\Delta y}{\Delta x}$ which 'cancels' out the infinitesimal scale to produce quantities large enough that we can study in terms of the standard reals; thus the definition of the derivative

$$ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} $$

You see these two basic themes:

• Argue in terms of the 'sufficiently small' or 'suffiicently large'
• Adjust the notions you use so as to cancel out infinitesimal or infinite scales

all throughout standard analysis.