@werterber: Not a silly question at all! In my head, it’s saying “what’s the ratio of width [cos(x)] to distance traveled (x)”.
As our distance traveled goes to 0 (we aren’t moving from the starting point), cos(x) tends towards 1 – we’re pretty much at the same width. So it becomes “1 / 0” in my head.
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Hey, Kalid … You hold a marvelous scape valve from the montains of unintuitive theorems and corolaries contained in every text-book.Outside, our memory rests in peace, and the big picture awakes our deep passions about math.Oh, precious and full of insight scape valve.
This comes down to this: we can’t possibly describe what we can’t possibly imagine. That’s why it must always be “small enough rectangles” of a sort…
Interestingly, Brian Greene in his “Elegant Universe” gives to understand, that the “superstring theory”, along with expected resolution of some fundamental problems, must bring about radical change in mathematical modes, so that you can’t decrease the size of those “small rectangles” down to infinity, but that it must have its limit somewhere around the level Plank constant ~10^-34. After which further decrease will actually mean increase.
Now every theory serves for some convenience. Therefore, aren’t we free to take such approximation with those rectangles, as will serve our purpose the best? And not bother any more than we can help? Cause that’s what we do anyway.
Thanks Kalid,
Your articles did help me a lot.
By the way, what software do you use to illustrate examples in your articles (like this one)? Thanks
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Thanks Tue – I use PowerPoint 2007 to make the diagrams.
Hey khalid plz i am getting a doubt !
You said that infinitesimal are the values which we cannot measure ! My question is can we imgaine infinitesimal ?? According to me , humans can think only of finite values …so whenever we try to assign a value to infinitesimal it woud be of finite digits and tat would be against the defination of infinitesimal … So according to me due to the limited scope of human brain we can never think of what value wud be of infinitesimal … Am i correct plz ???
Hi khalid plz i hav a doubt ??
My question is can we think about wat number would be infinitesimal ??
According to me we humans hav a limited horizon of thinking and so we can just think of finit numbers… So even if we assign any value to infinitesimal it would be some fiite value and a value smaller than it will still exist… So is the limit which we are talking about is the limit of our brains to comprehend such small amounts ??? Plz help ??
@kalid
@ kalid
Hi kalid plz i hav a doubt ?? My question is can we think about wat number would be infinitesimal ?? According to me we humans hav a limited horizon of thinking and so we can just think of finit numbers……. So even if we assign any value to infinitesimal it would be some fiite value and a value smaller than it will still exist……… So is the limit which we are talking about is the limit of our brains to comprehend such small amounts ??? Plz help ??
very nice, i loved the way, you taught us. Very interesting!
@Dave
post 17
Regarding your question, “If you learn calculus via the use of infinitesimals, is it possible to then make the leap over to using limits?”, I suppose it is possible for I have (in a way) done it, though I never knew I was learning infinitesimals.
I must admit that prior to reading this post I have never even heard about the defined mathematic concept of ‘infinitesimal’. I also never took a formal Calculus course. I originally learned Calc in my AP physics class in high school. Our teacher (one of the few who truly loved the craft of teaching and had a passion for what she did) had both the constraint of putting her Physics class on hold to teach Calc to those who have never seen it, and also the freedom that brevity provided; she was free to teach the idea of calculus without the strict procedural rigor that a formal class drags its pupil through. We learned the basic idea of the integral before the derivative, heresy in Calc101. Here it is 21 years later and I can still hear her voice saying 'Taking the integral just means add up a whole bunch of things, and ‘taking a differential element of’ just means cut the thing into really teenie weenie chunks." We learned the idea of a derivative as slope of a function without being given 2 points, just one point and an interval to the next. After seeing what happened as the interval got smaller we finally visualized ‘slope at a point’. Only afterward were we shown the ‘official’ formula with a limit in it. I saw it as a perfectly nice piece of legal-eez that made the rest of the world happy for me to have learned the ‘right way’, and I was enormously grateful our teacher taught us the intuitive way.
Fascinating article Kalid!
This is something new for me. After reading this post I started some research on infinitesimals, and quickly re-affirmed how valuable your common sense approach is by comparison to an army of equations, lemmas, and theorems.
My great ‘a-ha’ moment was your description of infinitesimals as another dimension, similar to the way imaginary numbers are another dimension to reals. In a strange way, that may not be obvious at first, it reminded me of a conundrum I faced learning the history of physics. It seems that every time we define what an ‘element’ is -the smallest indivisible component of a thing- some clever lad comes along later and figures out way to break that ‘element’ into something smaller. This means, of course, that the old thing never was a true element, we just thought it was. But then what about this new ‘element’, how can we know it is the smallest thing?
A revelation came when I realized that in order to be an ‘element’ we don’t really need it to be true that you can’t break it apart, it just means that if you do break it down further then it is no longer the same stuff. Thus the element is really just the smallest possible piece of a thing WHICH can still be the same thing. E.g. an ‘element’ of water (H2O) can be broken down, but it is no longer water, just hydrogen and oxygen atoms. An atom can be broken down into protons, neutrons, and electrons, but it is no longer the same stuff as the original atom. A little chunk of matter (a superstring exhibiting one class of vibration in 10 dimensions) can indeed be broken down, it is just no longer matter. It is also not exactly energy, but when the ‘stuff’ comes back together in a different pattern (the superstring having the same vibration just in a different dimension) it appears to us as a little chunk of energy.
It seems natural to me to take a cue from the physical world to comprehend numbers. When we look at an element and it appears we’ve ‘hit the limit’ in terms of breaking it up, but we can go further it just means we have to view it in a different dimension. Why then could we not do the same with numbers? Here’s a rational number you can only break it apart but to a certain extent and no smaller. I know you may object and say ‘take that number and divide by 2, it is smaller and still rational’. But take notice of the irrationals, like sqrt(2). It does exist, sitting there staring us in the face. It is in between rationals. So how does there exist any space between rationals? How can the rationals be broken down finer than it is possible to break them down? Imagine thinking you understand that atoms are elementary particles, then this clown Rutherford comes along and experimentally identifies this object (nucleus) in the middle of an atom.
I say the best way forward is to take as true those things that must be true and re-evaluate our preconceived notions that have pigeon-holed us into an apparent paradox. It is difficult and un-nerving. You can be guaranteed you’ll get it wrong a few times before you make some progress, but some progress is far better than the certainty of smaller minds.
Thanks Eric, that’s a really thought-provoking comment.
I think the element analogy is apt, we’re able to function at a certain level (water molecules) and while we can go to a deeper level (individual subatomic particles) those details presumably don’t change the measurements we’re making at the macro level. In the same way, infinitesimals can bounce around in funny ways but not effect the numbers one level up. (I.e., when we switch domains, the infinitesimal part goes to 0.)
Trying to “fill in” the number line with rationals is another great example. We have a smooth continuum on the number line, but the rationals are so sparse they’ll never complete it! There must be another way to get to those in-between numbers, and it isn’t by dividing the ones we have into smaller bits.
Kalid
Leibnitz and Newton originated calculus in the 17th century, long before imaginary numbers were around. Can’t we just say that limits are paradoxical but they work and leave it at that?