Infinitesimals and calculus

[IllegalArgument]

Here I'm after all these years and bad memories, learning calculus.

Last time I tried, sometime just after the earth cooled, it went very badly, more to do with my poor algebra skills than calculus.

The good news is that I seems to have gotten it, at least basic derviatives. The key was understand the concept of infinitesimal, which is the foundations of limits.

What did it for me was the showing that, 1/3 + 2/3 == .333... + .666... == 1

Having said all that, infinitesimals feel like a hack, you can't divide by zero but if you get really really close, then let's just pretend it's zero.

It's a really elegant hack, but does anyone else feel that way?

 

[pgwenthold]

Here I'm after all these years and bad memories, learning calculus.

Last time I tried, sometime just after the earth cooled, it went very badly, more to do with my poor algebra skills than calculus.

The good news is that I seems to have gotten it, at least basic derviatives. The key was understand the concept of infinitesimal, which is the foundations of limits.

What did it for me was the showing that, 1/3 + 2/3 == .333... + .666... == 1

Having said all that, infinitesimals feel like a hack, you can't divide by zero but if you get really really close, then let's just pretend it's zero.

It's a really elegant hack, but does anyone else feel that way?

No, but because that's not the way it is.

You never "pretend it's zero."

As you note, it is all based on the concept of a limit. Limits are not restricted to what happens when something goes to zero or to infinity. You can do a limit using infinitesimals to approach any number.

 

[psionl0]

Having said all that, infinitesimals feel like a hack, you can't divide by zero but if you get really really close, then let's just pretend it's zero.

It's a really elegant hack, but does anyone else feel that way?

No it's about realizing that no matter how small the number is, it is not zero.

The basic idea is that if two functions are the same except (perhaps) when x = a then they both have the same limit at x = a.

For example, y = 2x/x and y = 2 are the same function except when x = 0 (2x/x is meaningless if x = 0). So 2x/x has a limit of 2 when x = 0.

 

[Big Al]

No, but because that's not the way it is.

You never "pretend it's zero.".

I agree with the first sentence but not the second. I was taught inifinitesimals at O level, and didn't learn the more sensible limit method until A level.

I did express doubts, "So this is the smallest interval imaginable, smaller than anything else, but not zero... but when you square it, it becomes zero???? But doesn't that mean you divided by zero earlier?"

Did the teacher fess up and drop the gen (limits aren't any harder to understand than infinitesimals, and they put calculus on a sound mathematical footing)?

No. "Well, this is the way we're going to do it today."

Both Newton and Leibnitz knew it was a kludge, but they knew it gave the right answers. Limits didn't come 'til later (although a long, long time before I went to school!)

 

[IllegalArgument]

No, but because that's not the way it is.

You never "pretend it's zero."

As you note, it is all based on the concept of a limit. Limits are not restricted to what happens when something goes to zero or to infinity. You can do a limit using infinitesimals to approach any number.

Good, I'm going to learn something from the thread!

I was aware that you can approach any number infinitesimally close. I was just using h->0, since it's the standard limits for derivatives at my level.

As Big Al said, in beginning it seems a lot like you're pretending it's not zero, until right at the end of the derivative calculation, you can drop all h because h can be treated like it's zero. I understand from his comment that this might be corrected by more advanced limits theory later.

This seems to be common confusion amoung people I have talked in my class.

 

[edd]

Did the teacher fess up and drop the gen (limits aren't any harder to understand than infinitesimals, and they put calculus on a sound mathematical footing)?

Infinitesimals can be just as sound, and not a kludge. However, I'd suggest that when they are made sound they are harder to understand - http://en.wikipedia.org/wiki/Non-standard_analysis

 

[icebear]

Here I'm after all these years and bad memories, learning calculus.

Last time I tried, sometime just after the earth cooled, it went very badly, more to do with my poor algebra skills than calculus.

The good news is that I seems to have gotten it, at least basic derviatives. The key was understand the concept of infinitesimal, which is the foundations of limits.

The key to grasping derivatives is understanding that if you take secant lines through points which get closer and closer together on a curve, they start to look like a tangent line.

 

[Big Al]

Infinitesimals can be just as sound, and not a kludge. However, I'd suggest that when they are made sound they are harder to understand - http://en.wikipedia.org/wiki/Non-standard_analysis

Wow! Not an easy read. Thanks for educating me, Edd.

It's still kludgy in my humble opinion. However, IMHO, this is just giving infinitesimals a definition that leads into limits. It does seem to me that the concept of limits is inherent in this rationalisation, and trickier to calculate than limits,

As presented to me, infinitesimals were smaller than the smallest things ever until you multiplied them together, and then they became zero. Or at least my maths teacher never bothered to correct me on this misapprehension, even though he could have taught us about limits instead.

 

[IllegalArgument]

The key to grasping derivatives is understanding that if you take secant lines through points which get closer and closer together on a curve, they start to look like a tangent line.

I totally got that part.

It's the whole as the limit approaches zero part when taking a derivative, I can mechanically follow the rules, get the right answer.

Trying to understand the underlying theory, how you can remove all the h variables from the equations even though h never actually reaches zero, it's just infinitesimally close.

I'll look over edd's link.

 

[ddt]

No it's about realizing that no matter how small the number is, it is not zero.

The basic idea is that if two functions are the same except (perhaps) when x = a then they both have the same limit at x = a.

Nitpick: provided they have a limit. And to be fair, all the functions with simple recipes you are going to encounter do have limits (even if it's sometimes +/- infinity).

 

[IllegalArgument]

Infinitesimals can be just as sound, and not a kludge. However, I'd suggest that when they are made sound they are harder to understand - http://en.wikipedia.org/wiki/Non-standard_analysis

Thanks edd. I don't think I'll be able to understand those concepts for awhile.

For now, I'll have to accept for now that it's mathmatically sound, just not easily explainable.

 

[Crossbow]

Here I'm after all these years and bad memories, learning calculus.

Last time I tried, sometime just after the earth cooled, it went very badly, more to do with my poor algebra skills than calculus.

The good news is that I seems to have gotten it, at least basic derviatives. The key was understand the concept of infinitesimal, which is the foundations of limits.

What did it for me was the showing that, 1/3 + 2/3 == .333... + .666... == 1

Having said all that, infinitesimals feel like a hack, you can't divide by zero but if you get really really close, then let's just pretend it's zero.

It's a really elegant hack, but does anyone else feel that way?

Well, I feel somewhat the same way and here is what I do to resolve the issue:

When possible, use the actual fractions as opposed to the decimal equivalents thereof. So instead of using:

0.3333... + 0.6666... = 1

Use instead:

1/3 + 2/3 = 1

And by doing so one does not have to deal with the irrational numbers.

Note: this method is often practical when one is working with the more form of abstract forms of mathematics, but it is often not practical when one is working with applied forms of mathematics.

I hope this helps.

 

[Vorpal]

The good news is that I seems to have gotten it, at least basic derviatives. The key was understand the concept of infinitesimal, which is the foundations of limits.

No. If you're doing limits, you're free to avoid infinitesimals completely, and you can do all of calculus without having them show up in your mental concept-space.

I agree with the first sentence but not the second. I was taught inifinitesimals at O level, and didn't learn the more sensible limit method until A level.

I did express doubts, "So this is the smallest interval imaginable, smaller than anything else, but not zero... but when you square it, it becomes zero???? But doesn't that mean you divided by zero earlier?"

Did the teacher fess up and drop the gen (limits aren't any harder to understand than infinitesimals, and they put calculus on a sound mathematical footing)?

No. "Well, this is the way we're going to do it today."

Meh. Numbers with such a property aren't any harder to model than complex numbers, there's nothing inconsistent about it.

Say you have some numbers (e.g., real numbers) with an involution operation x^* (e.g., the identity x^* = x; in this example it will do nothing). Pick any number λ and define multiplication of ordered pairs by

(a,b)·(c,d) = (ac + λd^*b, da + bc^*)​

For example, if λ = -1, then this is just multiplication of complex numbers a+bi and c+di: the element (0,1) squares to be (1,0) and we can just say i = (0,1) and i² = -1 after identifying real numbers x with (x,0). The rest of arithmetic can be likewise be defined.

In general, though, (0,1) is a square root of (λ,0), and it works exactly like complex numbers do, except with a different square instead of -1. If we pick λ = 0, then ε = (0,1) is a nonzero number with ε² = 0, and for any smooth real function,

f(x+ε) = f(x) + f'(x)ε.​

As a side-note, this allows a pretty cute way of representing a Galilean boost: (t',x') = e^-vε(t,x) = (t,x-vt). The Galilean transformation between inertial frames is a "rotation in spacetime" analogous to the Euclidean rotation (x',y') = e^-iθ(x,y). A similar thing holds for Lorentz boosts of special relativity (pick λ = +1).

 

[IllegalArgument]

Here's an book from 1912, Calculus made easy.

It shows how to calculate a few basic derivatives w/o mentioning limits.

http://www.gutenberg.org/files/33283/33283-pdf.pdf

 

[Soapy Sam]

Here I'm after all these years and bad memories, learning calculus.

Last time I tried, sometime just after the earth cooled, it went very badly, more to do with my poor algebra skills than calculus.

The good news is that I seems to have gotten it, at least basic derviatives. The key was understand the concept of infinitesimal, which is the foundations of limits.

What did it for me was the showing that, 1/3 + 2/3 == .333... + .666... == 1

Having said all that, infinitesimals feel like a hack, you can't divide by zero but if you get really really close, then let's just pretend it's zero.

It's a really elegant hack, but does anyone else feel that way?

Yes. Absolutely. I remember telling my high school maths teacher the whole thing seemed like a shameful bodge. That was around 1970. It still seems like a bodge.
I think when you teach kids mathematics, the words you use can be critically important.

 

[shadron]

Yes. Absolutely. I remember telling my high school maths teacher the whole thing seemed like a shameful bodge. That was around 1970. It still seems like a bodge.
I think when you teach kids mathematics, the words you use can be critically important.

The problem there is assuming the teacher understands the concept.

 

[IllegalArgument]

Yes. Absolutely. I remember telling my high school maths teacher the whole thing seemed like a shameful bodge. That was around 1970. It still seems like a bodge.
I think when you teach kids mathematics, the words you use can be critically important.

Read the first few chapters of the book, it's done in non-formal way that's a joy to read.

 

[wrs]

Here I'm after all these years and bad memories, learning calculus.

Last time I tried, sometime just after the earth cooled, it went very badly, more to do with my poor algebra skills than calculus.

The good news is that I seems to have gotten it, at least basic derviatives. The key was understand the concept of infinitesimal, which is the foundations of limits.

What did it for me was the showing that, 1/3 + 2/3 == .333... + .666... == 1

Having said all that, infinitesimals feel like a hack, you can't divide by zero but if you get really really close, then let's just pretend it's zero.

It's a really elegant hack, but does anyone else feel that way?

Interesting, I was taught epsilon, delta in simplified form in Freshman calculus and then learned it from the ground up in Real Analysis as a grad student. I was required to take Real Analysis because the formal math used for Measure Theory and most of System Theory requires a good foundation of limits, convergence and set theory, all of which you will get in a good Real Analysis course. That is a senior level undergrad math course that anyone wanting a BS in math must take at UT. It was a hard course but it was worth the effort. It clarified a lot of things that were glossed over in my other math courses.

Math gets more and more abstract when it's actually applied to real world problems. Metric spaces are common in even the most basic problems such as optimal function approximation. So what happens in a graduate course in EE where you will be using this kind of concept, you end up needing the background gained in a Real Analysis course. Lot's of other grad students were in that course as well, it was more grad students than undergrad.

 

[Mark6]

If I am not mistaken, Newton developed calculus without the concept of limit -- that came later. In Fluxional Calculus he used more or less the same hack.

 

[IllegalArgument]

If I am not mistaken, Newton developed calculus without the concept of limit -- that came later. In Fluxional Calculus he used more or less the same hack.

You're correct, limits weren't formally worked out until the 1800s. As one math blogged I follow put it.

Most calculus classes cover “limits, derivatives, integrals” in that order because… why?
Limits are the most nuanced concept, invented in the mid-1800s. Were mathematicians like Newton, Leibniz, Euler, Gauss, Taylor, Fourier and Bernoulli inadequate because they didn’t use them?
(Conversely: are you better than them because you do?). Most courses are too timid (or oblivious) to question the strategy of covering the most elusive, low-level topic first.

Math Explained Better
http://betterexplained.com/