What's the importance of "infinitesimally small" whenever calculus is explained

First of all, i just like reading and understanding things related to math, but NOT at all any expert in math. So, I apologize if the question seems dumb.

It always puzzles me whenever a basic calculus is explained, it always make use of "infinitesimally" "very very small" etc. Why is it so important to use this term in calculus?

Similarly with "limits". The answer always seems blurry.. as it ends up with the notion of "the answer is say, as and as we go from point $a\to b$" etc.

Why can't there be a "clear and perfect" answer when doing problems in calculus. Like as i said, why something unclear like "infinite" or symbols like "$x \to 2$" (limits) etc needs to be used in Calculus for solving the problems.

V.

• Check out the accepted answer here: math.stackexchange.com/questions/118/… It might help. Jan 23, 2013 at 20:37
• ok thnx for the link.. gone through the answer there. So for example i take a problem, as mentioned there "finding area under a curve" ? Is this an uncertain question ? Calculus tries to give answer having same level of uncertainity ? How to decide the correctness of the answer ie. how much close my answer is to the exact "area" under the curve ? Especially because, the calculation has been done using the "infinitesimally" small rectangles. And "smallness" may vary from one person to another. Jan 23, 2013 at 21:01
• It's not an uncertain question for curves that obey a few rules. If a curve obeys the rules, we call it "integrable" and that means we can find the 100% exact area under that curve. The way we find it actually turns out to be a straighforward, mechanical calculation. The proof that the area we calculate is correct is the part that involves imagination and infinities and stuff like that. We use our imagination to come up with a way to solve the problem. We use logic to check our imagination. Then we use the process that we created with logic and imagination to calculate the answer. Jan 23, 2013 at 21:27
• Also, we don't let the "smallness" vary from one person to the next. That's why we attach numbers to the smallness like $\epsilon$ and $\delta$. In books and lectures we hardly ever actually give a specific number for $\epsilon$, we just imagine the kinds of numbers it could be. There are books where the first limit they actually use numbers like $.001$ and $.00001$ and so on to lead your imagination onward. How many zeros can you imagine between the $.$ and the $1$ in those small numbers? Jan 23, 2013 at 21:34
• math.stackexchange.com/questions/822664/…
– user117644
Jun 6, 2014 at 10:46

I see calculus as being built from these "unclear" concepts, rather than forcing the use of them. The limit is the first and arguably most important concept in calculus, and the rest of the field is built on limits. Limits can be hard to completely understand, partly because they often involve very small ("infintessimal") and very large ("approaching infinity") numbers.

My mental picture for limits, infintessimals, and infinity is based on two people having a conversation about numbers. When something "increases without bound", "goes to infinity" or (poorly worded) "equals infinity", I imagine that anytime one of my imaginary people says a number, the other one says a larger number, then the first says an even larger number and so on. They will never stop going back and forth, and that is how I see it. I see infintessimals analogously with smaller and smaller numbers.

For limits, remember that there is a formal definition for a limit, which usually includes something like "for all $\epsilon$ there exists a $\delta$ such that," (now paraphrasing) when $x$ gets really close to $a$, $f(x)$ gets really close to some number $L$. How close is "really close"? That's what $\epsilon$ and $\delta$ are for. Now we go back to my two people. Here's the conversation:

A: This gets really close to $L$.

B: How close?

A: How close do you want it? Pick any number greater than zero.

B: Ok, how about this number $\epsilon$ I happen to have in my pocket?

A: That works. Every time you pull any $\epsilon$ out of your pocket, I will give you a $\delta$. You take my $\delta$ and push $x$ until it is less than $\delta$ far away from $a$. Once you've done that, I guarantee that $f(x)$ will be closer than $\epsilon$ distance away from $L$.

So person B keeps pulling out smaller and smaller epsilons and finding no matter how small they are, person A can always give back a $\delta$ that works.

In other words, no matter how close person B wants $f(x)$ and $L$ to get, person A can always guarantee the closeness, as long as person B is willing to put $x$ a certain distance closer to $a$. Their conversation continues forever, just like them going back and forth with larger and larger numbers to approach infinity. Therefore, person A has "proven" the closeness of $f(x)$ and $L$ to person B. Since you and I can't talk forever, I let my two imaginary friends do the talking for us and just skip to the end of their conversation.

Calculus is the part of math where we leave behind ideas that have one, simple answer. We are not asking "what is $513+2138756$?" We are asking more general questions like "if this goes on forever, where will it end up?" Once we start going to those imaginary places like "forever", we have to get some new concepts that are not as simple as the old ones.

Does that help at all?

• So how the answer is decided, when the f(x) is close to L ? If the answer is uncertain, how come solutions from "Calculus" are useful. Are they useful because the "Errors" prodcued by their uncertainity, too are "Infinitesimally Small" and hence don't affect us ? Jan 23, 2013 at 20:11
• I am tempted to reply to your comment by saying "there is no answer, so we don't have to worry about how it is decided", but I don't think that will be helpful. I think I'm trying to say that we agree that $f(x)$ is close enough to $L$ (that is, the limit of $f(x)$ is $L$) after we both imagine A and B talking forever and B agreeing after an infinite conversation. Then we decide we won't wait as long as B and instead trust B to only be convinced if it's true. I haven't figured out a way to do calculus without using my imagination. Jan 23, 2013 at 20:17

I think this old but very entertaining book will help. The book is also freely available on the internet.

Silvanus Phillips Thompson (1851-1916), Calculus Made Easy, 2nd edition, MacMillan and Company, 1914.

• ah!.. yes, have read this book. It had really helped me a lot to atleast grasp some of the basic understanding. I remember how clearly proved some of the basic formulae of calculus, a^n = n*a^(n-1), with the formula of (a+b)^2 . I noted he has also focussed on "infinitesimally small likelihoods" but still could never get the answer... that what will happen, if those quantities are not removed. No matter how small they are. I think, only the equation would look bigger! Jan 23, 2013 at 19:43
• @Vishwas Gagrani: I don't have time now to try to sort out these matters with you (I'm at work ...), and no doubt others will be doing this anyway, but in case you don't know about it, I think you will find a kindred spirt in George Berkeley's The Analyst. Jan 23, 2013 at 20:21

I'll narrow the scope of my answer to just be about "Differential Calculus". The proper objects of study in differential calculus are the derivatives of functions. Intuitively, we can say that differential calculus is the study of the "local" behavior of a function.

How can we define the concept of "local"? Clearly, we want to look at points close to the point of interest. But to make this definition precise, we must use some kind of limit. Using a fixed-size neighborhood just will not cut it, because we can zoom in arbitrarily and make any fixed-size neighborhood arbitrarily large.

So in the end, to discuss anything related to differentiation, we must use the concept of a limit in order to get a local picture of a function. To get a precise understanding of limits requires some technical mathematics, but I'm afraid that's the only way to build a solid foundation for calculus.

This concept of "local analysis" is part of a broad theme in Mathematics: to achieve a global understanding of an "object" by stitching together all the local pictures.

In my opinion those symbols are clear and unambiguous. The problem is you probably weren't given a definition about what they mean, so you give them an intuitive meaning and that's why you find'em blurry. The same can be said about sentences like "very, very small" or "when $n$ is big". Those sentences have a clear and well defined mathematical meaning.

For instance in the context of sequeces, one often hears that "$\displaystyle \frac{1}{n}$ is essentialy $0$ when $n$ is big enough". The mathematical meaning of that sentence is $\displaystyle \left(\forall \epsilon \in \mathbb{R}^+\right) \left(\exists p\in \mathbb{N}\right) \left(\forall n\in\mathbb{N}\right) \left(n\ge p \Longrightarrow \left\vert \frac{1}{n}-0\right\vert\ <\epsilon \right)$ and it is often abbreviated by $\displaystyle \frac{1}{n} \to 0$, or by "$\displaystyle \frac{1}{n}$ converges to $0$". There's nothing unclear about this.

In my view of opinion if u divide anything into infinite parts u get zero for each part.This is because u will never get zero for each part if u divide anything by any large finite number.Also if u keep dividing anything endlessly u will never reach the infiniteth division.

From the above two sentences we can imagine that the infiniteth division (which u will never reach) will give u zero as each part.

If this is true,its converse is also true,i.e.if u add zero infinite times u get any finite number.

Its hard to imagine but just think of a scale being divided into infinite parts(u get zero for each part) and then u add all the zeros (infinitesimally parts),u get the whole scale.

My brain tells this is the basic of integration.i.e. u add up infinite zeros u get a finite number.

• Your first two statements are (almost) converses, and there is no logical connection between them. You seem to be operating under a very common misconception about how Riemann integration works. See my answer here for more detail. Oct 31, 2013 at 12:40
• @CameronBuie, I undid the "down" vote not because I think the answer is particularly insightful but to make the basic point that Riemann integration can indeed be treated in terms of infinite sums. More precisely, the integral is the standard part of an infinite sum over a hyperfinite partition of the domain of integration. Nov 1, 2013 at 8:47
• @user72694: I'm not sure who downvoted. (When I do, I tend to say so.) The result you mention is highly non-trivial, and I still feel that the "intuition" about Riemann integration can be very misleading, especially for beginners. Nov 1, 2013 at 14:04
• @CameronBuie, especially for beginners, the intuitively appealing approach to integration via infinite Riemann sums is a pedagogical goldmine. This is not based on my "feelings" but rather on classroom experience. Nov 3, 2013 at 13:08
• In my opinion, you will be taken greatly more seriously if you take the time to write you instead of u, and generally, grammatical English. I understand you may not be a native speaker, but please at least try. Nov 23, 2013 at 13:19