Why Do We Need Limits and Infinitesimals?

So many math courses jump into limits, infinitesimals and Very Small Numbers (TM) without any context. But why do we care?

This is a companion discussion topic for the original entry at http://betterexplained.com/articles/why-do-we-need-limits-and-infinitesimals/

Hi Kalid! =) Wonderful to have those insights as always!

Just one very unimportant correction; you said:
"Video shows still images at 24 times per second"
However the correct would be to say that Film shows the frames at 24 times/second. Even when converted to digital, it is sped-up to 25 FPS (PAL video standart) or “telecined” to 29.97 FPS (NTSC standart, USA’s video)
Note: It’s not so simple, there are other standarts and variants, I was refering to SD video (not HD, which is commonly the double FPS) and also: digital video on a computer can have any FPS, even floating numbers and variable frame-rate (a totally complex thing to handle, easier to get in than to get out of it)

And as I’m talking about video, I’d like to suggest an idea for an article: how digital video and image works. It’s a fabulous bunch of AHA! moments when you get the concept behind all of those blocky artifacts and color schemes other than ye olde RGB.

And I loved to think on infinitesimals in a new way! Thanks! =D

Hi Kalid,

Thanks for the wonderful post. I have totally forgotten all my math and have been thinking of re-learning it (especially from a computer science perspective).

I found your post very useful, and I think it will also give that little push I needed to get started.

BTW, I too share your passion for helping others learn. I have aggregated various open computer science course videos on my website.

Hi…This one is as good as the previous posts! I appreciate ur enthusiasm in promoting the interest in Math among young readers! I enjoyed every bit of the article man…Thank you very much…

@Camilo: Ah, thank you for the clarification! I’ll change ‘video’ to ‘film’ :).

That’d be a really cool article – I don’t know too much about the video formats, but know that MPEG has some really neat technology to make it compress well.

@Parag: Great, glad you enjoyed the article! Checking out your site now, thanks for collecting all those links – I’m hoping to go back and refresh a lot of my cs knowledge also :).

@Murugesh: Thanks for the support, I’ve had a lot of fun trying to get my brain around these concepts again, but being able to ask “Wait, what does it really mean to me?”. Glad it was useful for you!

I found a few typos:

Paradox of zero: “slices so thin we can’t seem them” (see)
Summary: “I feel like I’m talking from boths ides of my mouth” (both sides)
Summary: “Here’s the key concepts” (Here are)

Thanks for another great article!

@Anonymous: You’re welcome, and thanks for the corrections! I just made them now.

Again I started reading and did not realized that I finished a long article! It was very entertaining. Thank you very much for your efforts.

@nanoturkiye: You’re welcome! Glad you enjoyed the article.

Hi Khalid, props for this great series which I just found out recently and reading your posts has been a daily habit for me.

Just one confusion in this topic, can you elaborate more on this:

Around 0, sin(x) looks like the line “x”.

I think of x as the x-axis in the plane that was demonstrated. It could also be just the variable in the equation. But neither makes sense. I know x/x is 1, but how come sin(x) is x?

thanks! and more power!

Hello, Kalid,
Very well-written and descriptive. Thank you for giving me a good and pleasant read on things past and nearly forgotten!

I could only wish that more people like you were teaching in high schools and universities. Around here, the tutors are often skilled in their field, but regularly and gravely fail to convey the meaning behind the definitions, theorems and proofs they teach - only the items themselves; and the educational process plummets.

Around 0, sin(x) looks like the line “x”.
I believe this means the line “y = x”. Thus y_1 = sin(x), y_2 = x and y1 ~= y2 for x -> 0.

@Arbie: Wow, I like that functional representation of it! Yes, integrations are a general “applying” one function to another, vs. some static multiplication just to find area (area just limits our creativity/intuition I think).

Ah, I should be more clear about that… the I meant the line “y = x”, that is, a 45 degree line extending from the origin. So the equation y = sin(x) looks very similar to y = x for very small numbers (sin(x) extends 45 degrees from the origin when it first starts off).

Hope this helps!

@mcmlxxxvi: Glad you enjoyed it, and thanks for the comment. I too wish there was more emphasis on true understanding vs. the “let’s learn enough to pass the next test” mentality. Learning the intuition may take a bit longer than memorizing in the short term, but in the long run it gives you a more flexible set of knowledge, and not to mention it’s way more fun. I sometimes see grades as a curse because rather than being an indication of knowledge, they become an end in itself vs. the learning it should represent. It’s very hard to test intuition – it’s a gutcheck you need to ask yourself. But with no grades there’s no “incentive” (carrot or stick) – I don’t know the answer, but I too wish there was another way.

This paper offers similar views about mathematics education as well as a criticism of the cultural opinion of mathematics that you might like. http://maa[dot]org/devlin/devlin_03_08.html

@Anonymous: Thank you – I’ve seen the essay and really like it :).

Smooth Infinitesimal Analysis handles infinitesimals better than Non-Standard Analysis:

In intuitionistic math, the law of excluded middle is rejected (i.e. not not A doesn’t imply A) so you must provide an algorithm for constructing all your objects.

There is no general procedure for detecting whether or not 2 objects are equal. You must explicitly provide an algorithm for showing 2 objects are equal.

The trichotomy law (a b, a = b) doesn’t hold in general.

All functions are continuous. Piecewise functions are nonsensical.

In other words, the continuum is unbreakable into points. Functions transform the continuum onto the continuum.

With this as our basis, Smooth Infinitesimal Analysis introduces an object called epsilon.

There is no algorithm to tell whether or not epsilon != 0 or epsilon = 0. This avoids the first problem entirely.

epsilon^2 = 0 though which gives us a way to get rid of them from our formulas.

So I view infinitesimals as the glue that makes the continuum unbreakable and there is no algorithm to decide if the expression “epsilon = 0 or epsilon != 0” is true (see why we have to reject the law of excluded middle to make this work?).

@asdf: Wow, really interesting stuff! I like that insight of infinitesimals as the “glue” that makes the continuum unbreakable. Great analogy.

Hey, Kalid, I’ve just got a quick question to ask.

If you learn calculus via the use of infinitesimals, is it possible to then make the leap over to using limits? While I doubt it would happen, I’d like to be an amateur mathematician in the vein of Fermat some time and develop proofs (more as a beauty thing, to be honest), but writing in a fashion that is contrary to the norm is rather like handing out Spanish pamphlets in an English neighborhood- they might understand, but they won’t like it.

So, yeah, can you jump from infinitesimals over to limits? From what I can tell, limits are mainly used because they’re easily to rigorously define an to keep the constructivist camp from yelling at you.

@Dave: Great question. I can’t say I’m completely comfortable with limits, but I think you can jump back and forth (the Keisler Calculus book has some examples like this I believe). I think the bigger goal is to figure out what is being said, i.e. “What does this equation equal, within some level of tolerance?”. Limits and infinitesimals are two ways to define that tolerance threshold, but infinitesimals are “easier” in that it’s built in (and you don’t need to explicitly define epsilon, delta, etc.).

Hello, i have silly question. How intuitively explain that cos x/x is undefind?
There is graf> http://www.wolframalpha.com/input/?i=Plot[{cos[x]%2C+x}%2C+{x%2C+-1.0%2C+1.0}]