If there’s anything you’d like me to elaborate on, please let me know. I’m not close to the level of 3Blue1Brown, but perhaps I can still offer something that will lead to a new insight.
An interesting perspective–thanks. I have only the vaguest notions of most of real analysis: as you suggest, for me, calculus is a tool I use in in engineering and science. But math is taught by the math department and they have their own motivations.
Not that I want calculus to be taught in a crank-the-handle sense, either–that’s part of the motivation behind this thread. But if calculus is going to be taught to people using it as a tool, then the foundation that’s used to build it should reflect this, and not be selected just because it’s a stepping stone to some other math.
All I see in Nava’s post is mention of limits. Limits are useful but they aren’t used (in pre-calc) to build up the real number system. Anyway, my issue with your comment was just with the word “formally”. Very little is done formally at the time (which is totally fine, of course).
I agree with all this. My suggestions in this thread only apply to first-year calculus, and go away when students are comfortable with more abstract manipulations. But it’s always nice to provide a foundation that students can fall back on when other methods fail, or maybe just when they are trying to really understand some process at an intuitive level. This is a reason why I really enjoy 3Blue1Brown videos in general.
I totally agree and the fact that Kiesler actually worried about the construction (though only in an appendix, I think) may have contributed. BTW, the Cauchy reals are probably a bit easier to understand than the Dedekind. There is also a construction due to Emil Artin. But none of this is or should be presented in Calc I. And I think another construction of infinitesimals would be formal Laurent series in a variable called h, ordered in such a way that h is infinitesimal. Then the reals are the power series and the ordinary part of a power series is its constant term. The main difficulty is how to extend functions to a power series. What is sin of a power series? It can be done by using the power series of sin, but is not pleasant.
So small positive X small positive = 0? Not in an integral domain it doesn’t.
OR
You’re saying 1.000000000000000000000000000000000000000000000000000001 = 1?
I’m not sure what you’re after here.
If you want a in-depth explanation of how infinitesimals are handled, cites have already been provided in this thread.
A quick-and-dirty explanation is all you’ll get here, at least from me. An infinitesimal is non-zero, so you can divide by it without breaking math like dividing by zero would do. But simultaneously it’s small enough to act like zero in other situations, so that it sometimes can be ignored. And if you multiply it by itself, the product is so super-duper small it can pretty much always be ignored. Confusing? Sure, can be. That’s the whole reason real analysis as it developed in the 19th century ignored infinitesimals and relied on limits instead – while still preserving Leibniz’s intuitive notation even as they changed the explanation behind the notation.
But there’s a damn good reason infinitesimals came “first”, if not quite in rigorous form. It’s just a really natural formulation for some people, including – it should be repeated – the two guys who discovered the idea in the first place. So it’s no surprise that a lot of people even today hear this way of thinking and go “Yeah I get that” in a way they just don’t when we talk limits.
But not everybody responds well to this. The foundations of calculus were pretty damn mysterious to people for quite a long time. A math prof friend of mine said the whole thing was a building made of brick with no mortar: people could see the building was standing but couldn’t see what was holding it together. It wasn’t until the 20th century that the rigorous theory of infinitesimals was finally put together. If you want the rigorous version, rather than the simple story, you’ll need to look into non-standard analysis.
I’m in the US and we learned it in the same order you did and then moved on to integration. As to why implicit differentiation is taught when it is, I couldn’t answer. I’m not sure why anything is taught when it is to be honest.
So, if the SDMB were going to test infinitesimals vs limits for teaching calculus would you volunteer as a guinea pig?
I was thinking about this further, and think it’s a good insight. And there is a more concrete connection than one might think. I have a friend who was tutoring a kid using Common Core techniques and we’d talked about some of the strategies.
One of them was expressing multiplication of sums geometrically. So for instance, one might ask what (5+2)(3+1) is. Well, one way is to reduce it to 74. But another, equivalent way, is to look at the sums as sides of a rectangle and sum up the four sub-rectangles: in this example, it would be 53+51+23+21. Same answer, different method–and good for deep insight.
Consider a big square, say 10001000. If we increase the side length by 1, how much bigger does it get? Well, one answer is 10011001-10001000=2001. But another way is to look at as four rectangles: the original 10001000 square, two 11000 rectangles, and finally a tiny 11 square in the corner. We add up the three new rectangles and get 2001 as before.
We might do the same for a 1000000*1000000 square, which gives an answer of 2000001. It’s always double the side length plus 1. If we keep getting bigger and bigger, the difference in area grows but that little corner square doesn’t. Eventually, we can ignore it.
This observation is exactly equivalent to the fact that the derivative of x[sup]2[/sup] is 2x. We have a square that’s growing on two of its edges, and there’s also a little extra corner piece but it’s so small that you can ignore it. So the derivative ends up being twice the side length.
Hopefully those students exposed to Common Core techniques are able to apply this type of insight to their calculus education even without infinitesimals.
I’m not a good guinea pig. (Cuz I’m just so unique, you know:))But what are they paying? Just so I know.
Sorry you think so, it was not my intent. Please see my reply in the Great Debates thread for more detail.
Actually, starting in the 1980s there was a movement to reform the way Calculus was taught. (Disclaimer: I learned math in the 1980s and have been teaching it since, and have never been in a situation where “reform calculus” was directly discussed in any detail, though I have certainly seen it referred to.)
I tried, without a whole lot of success, to find a user-friendly definition or description of the Reform Calculus movement. I did come across this 1997 article: “Reform Calculus” Has Been a Disaster, Critics Charge; judging by that article, both the approach taken, and the criticisms against, Reform Calculus remind me of those of Common Core math.
However, one idea that has stuck (judging from my experience) is that, in math/calculus instruction, whenever possible, concepts should be presented and looked at from multiple points of view: algebraically, graphically, numerically, and verbally.
Whenever I’ve tutored someone (most often, in chemistry), I’ve begun by checking what and how were they taught. If we had time and they had the inclination, we went beyond the requirements - but that’s an IF. The target was to help them pass the course (often despite the teacher not knowing an electron from an elephant), not to turn them into chemists.
Real numbers were 6th grade, limits were 10th…
Doctor Jackson, I’ll agree with your point that you can live a successful and rewarding life without calculus (as distinct from, say, algebra, which is beneficial to everyone). And that’s a good argument for not requiring calculus in schools. But, well… Calculus isn’t required in schools. It’s an option, only taken by those students who have some reason to be interested in it (maybe because they’re planning on going into engineering or some other field that will use it, maybe because they want to impress college admissions boards, maybe just because they find math beautiful and worth study for its own merits). And there are certainly some people for whom calculus is useful or otherwise worthwhile. To those people, it would be a great loss if it weren’t taught at all.
Concur. In high school, we did differentiation first. Integration came afterwards. This would have been in 1992. I also took an accelerated calculus class (actually, I guess it would have been calculus III/multivariable calculus) in college, and the “review” portion of the class (maybe the first week-ish) also went in that order.
I don’t live near my nephews, so it’s unlikely that course-specific tutoring would be very efficient. But when the time comes, I may be able to fill in certain gaps in their knowledge.
I’ve forgotten just about everything I was simply taught. But I’ve retained almost everything I understand the principles of. When I know how to derive something, I know it forever.
To be honest, I hardly remember what they were teaching in 6th grade math. I was reading about complex numbers and quaternions in my spare time.
What exactly did you go over in 6th grade regarding the reals? I have a somewhat hard time believing it was very deep. Prove the existence of irrationals, such as showing that sqrt(2) is irrational? Cantor’s diagonalization proof? That the reals have the least-upper-bound property while the rationals do not? I’m pretty certain that US students don’t do anything like that. We had “the number line” and some other things, but nothing that could distinguish the rationals from the reals. We had spent a lot of time on the rationals, of course–fractions are a big deal in US math education.
∂x/∂y certainly is an infinitesimal change called ∂x divided by an infinitesimal change called ∂y.
The only trouble is, we have this terrible notational convention for partial differentiation which gives the same name to different things at different times, causing all kinds of confusion.
In ∂x/∂y, the ∂x represents change in x when y is allowed to change but z is held fixed.
In ∂x/∂z, the ∂x represents change in x when z is allowed to change but y is held fixed.
These are, alas, different things being called ∂x at different times.
If x is a function of y and z, then there is a total differential dx = x(y + dy, z + dz) - x(y, z). And under ordinary conditions, this can be split by linearity of derivatives into dx = dx_{dy = 0} + dx_{dz = 0}, where dx_{dy = 0} = x(y + 0, z + dz) - x(y, z), and so on.
What you are calling (∂x/∂y) is then dx_{dz = 0}/dy, but NOT the total differential dx/dy. And so on.
And the expression (∂x/∂y)(∂y/∂z)(∂z/∂x) amounts to (dx_{dz = 0}/dy) * (dy_{dx = 0}/dz) * (dz_{dy = 0}/dx). Of course, this wouldn’t show any nice automatic cancellation. None of the numerators match any of the denominators.
But none of this is the fault of treating infinitesimals as arithmetic quantities. This is all just the fault of our terrible notation for partial derivatives having confused us into distinct infinitesimals were the same as each other.
I don’t remember how I got here, and now I depart.