is $dx$ greater than $\frac{dx}{2}$?

Question

I wanted to ask if $dx$ is greater than $\frac{dx}{2}$? i will make conclusions i am sure they are wrong : a) if yes then why in integration we do not use smaller than $dx$ like its half ? b) if you said they are equal then does it mean $1 = \frac{1}{2}$? some may say you can't divide by $dx$ but we do it a lot in solving differential equations ? so who is greater? might seem low question because i am not math major like you

$dx$ is not a number, so I don't know what you mean by "greater than". This is the source of your confusion, I believe. — Chris Gerig, May 21 '15 at 23:27
In classical analysis $dx$ is just a symbol, not a number. In infinitesimal analysis (which has only become a rigorous subject in the last ~50 years), if $dx$ is an infinitesimal then you do indeed have $dx>dx/2$. But the result in integration a la infinitesimal analysis would be the same, perhaps up to an infinitesimal difference, so it doesn't really matter. — Ian, May 21 '15 at 23:27
is the operation of < and > not defined ? but we learnt in physics dx is an interval and we can compare intervals ?? — Mohamed Osama, May 21 '15 at 23:28
In practice in physics, it doesn't really matter which infinitesimal $dx$ actually is, the point is that for any given $dx$ you have some property. That is, you never work with particular infinitesimals, instead just with arbitrary ones. For instance when you write $dy=f'(x) dx$, that is really understood as an equation which holds for all infinitesimal $dx$. — Ian, May 21 '15 at 23:29
but we can compare functions at a certain number or limited domain ? — Mohamed Osama, May 21 '15 at 23:30
In physics, we usually say that $(x, x+dx)$ is an interval -- not $dx$. Can you provide a source where you saw $dx$ defined as an interval? — , May 21 '15 at 23:30
$dx$ is NOT an infinitesimal in mathematics -- it's a linear function. In physics, it's usually considered just a very, very small number. For instance, if you're talking about an ideal gas with approximately Avogadro's number of particles, then adding or subtracting $1$ particle is relatively speaking a increasing or decreasing the density by a very, very small amount. — , May 21 '15 at 23:33
@Bye_World That's not entirely correct; see my original comment. (For technical details, look up Robinson nonstandard analysis or smooth infinitesimal analysis.) — Ian, May 21 '15 at 23:34
Even if $dy = f'(x),dx$, somehow considered numbers, it could be that $dx$ is negative, and then $dx/2$ is greater than $dx$. — GEdgar, May 21 '15 at 23:35
@Ian OK, but obviously I was talking about standard analysis. — , May 21 '15 at 23:36
In that image one is not meant to understand $dx$ and $dt$ as separately meaning anything; $v$ and $\frac{dx}{dt}$ are just both symbols to refer to the limit that was described above. This is the way that $dx$ and friends are used in classical analysis. As for the use of "intervals", that's just a quick of English: we use "interval" both to mean $(a,b)$ and $b-a$. But mathematically, the relevant interval is $(t,t+\Delta t)$, whose width is $\Delta t$. — Ian, May 21 '15 at 23:37
But in that context the physics consider dx as an interval ? or i misunderstood ? — Mohamed Osama, May 21 '15 at 23:38
@MohamedOsama $\Delta x$ is an interval -- not $dx$. We define the derivative of $y=f(x)$ as the limit: $$\frac {dy}{dx} = \lim_{\Delta x \to 0} \frac {\Delta y(x, \Delta x)}{\Delta x}$$ $dx$ is not a separate entity here: $\frac {dy}{dx}$ is $1$ symbol. — , May 21 '15 at 23:38
The two quantities do not make sense on their own. Therefore they are not necc. comparable. Avoid writing it! — OKPALA MMADUABUCHI, May 21 '15 at 23:39
but we separate them in differential equations , how we separate one entity ? — Mohamed Osama, May 21 '15 at 23:40
The separation of differentials in solving separable differential equations is a notational shorthand. Classically, if you are solving $\frac{dy}{dx} = f(x) g(y)$, you write $\frac{1}{g(y)} \frac{dy}{dx} = f(x)$, then you integrate both sides with respect to $x$, then you change variables to compute the integral on the left side. — Ian, May 21 '15 at 23:41
anyway thanks it seems engineering books and physics one don't say much about those subtle things — Mohamed Osama, May 21 '15 at 23:44
But i think infinity is some how different from dx , you can't say they have same properties — Mohamed Osama, May 21 '15 at 23:44
calculus is referring to the fact that in some number systems -- like the hyperreal numbers -- different types of infinities and different types of infinitesimals are treated on the same footing as real numbers. — , May 21 '15 at 23:46
It´s sure not the same. But $\infty$ is also not a number. So there is an analogy. — callculus42, May 21 '15 at 23:46
@MohamedOsama There are several questions on this site about what differentials really are (I even wrote an answer to one of those questions). I recommend you spend some time searching through several of them and see if that clears up your misconceptions. If not, figure out exactly what you don't understand and either reword this question or make a new one. — , May 21 '15 at 23:50
the problem is i use math a tool not rigorous so i do not know the basics but the question poped inside of my head — Mohamed Osama, May 21 '15 at 23:51
Here are some good questions from math.SE that might help: 1, 2, 3, 4, 5, ... — , May 21 '15 at 23:58
The question is irrelevant: they are both indistinguishable from zero by definition and so can be neglected if they remain at the end of a calculation. The definition implies a resolution independent of purely theoretical considerations. This is assuming you want the instantaneous rate of change of course, if you didn't they would be larger. — , Dec 01 '15 at 19:45

score 3 · Answer 1 · answered May 22 '15 at 00:05

3

To fully answer this question would require a many-volume narrative of the history of mathematics/physics since Newton and Leibniz! :)

But/and I would say that the question is eminently reasonable, rarely addressed directly in textbooks, and, indeed, subtle to answer "correctly".

As a ridiculously short sketch of what humans know about this, to the best of my own knowledge (and I am interested in such things for some years now):

Newton and Leibniz did argue/think genuinely in terms of "infinitesimals", and, yes, in that context, $dx/2$ is half as large as $dx$. (Yes, $dx$ is itself problemmatical in modern terms... though not at all impossibly so, in various ways, as "differential form", or as Skolem-Robinson-Nelson "infinitesimal").

Yes, tangential to foundational issues, differential equations can be solved by treating the various $d(whatever)$ as things existing in their own rights, without explaining what they are. That is, a heuristic succeeds in producing outcomes that are checkable.

The last 150 years of didactic tradition has been in a different direction, for somewhat artifactual reasons. That is, the popular style of calculus makes an exaggerated show of disparaging "infinitesimals" (despite Skolem-Robinson-Nelson's complete legitimization of them!), and of disparaging the symbol-manipulations that ... jeez! resolved zillions of questions over at least two centuries!

In short, the question is profoundly reasonable... but/and the accumulation of some centuries' artifacts about accepted mathematics does, indeed, seriously confuse anyone's understanding of ... for example... eminently reasonably heuristics in physics texts...

The operational answer is: try to think not in terms of "rules", but that the mathematics is mostly, and, certainly, initially, exactly a narrative, a description, of things. Then we hope that our subsequent manipulations of this description give us further information.

That is, no, we cannot deduce by pure logic what the minimum legal parking distance away from a fire hydrant might be. But we can easily understand that there is some reasonable distance.

... sorry, yes, a seemingly vague answer, but, so far as I know, after some experience, maybe to the point.

answered May 22 '15 at 00:05

paul garrett

52,465

it is very vague specially for a non math major :) but the answer i get is you can't treat them as numbers hence you can't compare is it right or read your answer again? – Mohamed Osama May 22 '15 at 00:09
@paulgarrett There's a reason that mathematicians since Cauchy have spent time emphasizing that $dx$ and $dy$ are not infinitesimals -- because the infinitesimals that Newton and Liebniz and even Euler used were not consistent mathematical objects. It was only with Robinson being very careful was it discovered that in fact one could make infinitesimals rigorous, but that's a relatively new part of mathematical history and as most students won't ever need to learn nonstandard analysis, it's still worth emphasizing that in standard analysis $dx$ is NOT an infinitesimal. – May 22 '15 at 00:13
Can you define infinitesimal more clearly please ? is it the limit of a difference when it goes to zero ? – Mohamed Osama May 22 '15 at 00:14
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Dear Mohamed, I think the point is two-fold: your curiosity is completely reasonable, but/and the ultimate issues are subtle. That is, we can drive a car without understanding the thermodynamics of an internal combustion engine... we can use the internet without knowing how it works, or even how a transistor works. Yes, it is honorable to be curious about those things... but it would be foolish to refuse to use the internet until one knew how transistors, ... worked. The point is that the question is substantial, fully "weighted"... and in fact cannot be trivially answered. :) – paul garrett May 22 '15 at 00:15
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@paulgarrett that's way i said dear that i just use them in physics and engineering without understand them :) majoring math is kinda far from me but the question just poped but it seems it is far beyond my level thanks alot for your answer. – Mohamed Osama May 22 '15 at 00:17
@Bye_World, well, ok, if you insist. I myself find that viewpoint too negative. Mileages vary, and all that. I've been amused over the years by a few "colleagues"'s complaints that I'd made things "understandable by weaklings". I don't take the hard-line Spartan viewpoint, insofar as I think that reduction of difficulty probably benefits all, "not only the weaklings". :) – paul garrett May 22 '15 at 00:19
@MohamedOsama it's a number of I remember correctly that has the properties of the reals, in addition to some, including but not limited to: $$a+dx=a$$ $$adx\neq0$$ $$0 < dx << \mathbb{R}$$ but don't quote me on those, it's been awhile. I would check Wikipedia for nonstandard analysis and or infinitesimals. Cheers. – Shinaolord May 22 '15 at 05:39
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@user57404 If $a+dx = a$ for all $a\in \Bbb R$, then $dx = 0 + dx = 0$. So while I've never studied nonstandard analysis, I doubt that's one of the properties of infinitesimals. I also think you mean $adx \ne 0$ for $a\ne 0$ and $|dx| \lt \Bbb R^+$, not $dx \lt \Bbb R$. Again, those are just my guesses, but if they're not the case then this system loses most of the useful properties of the reals -- to the point where I have a hard time believing they form a consistent arithmetic. – May 22 '15 at 16:45
@Bye_World I had a professor explain to me the infinitesimal number is a number that is so small that no real number can represent it, no matter how many zeros you put aftee the decimal. I inferred most of those properties from that statement, although that statement may have been simplified and it may have contained assumptions that are not present in the whole of nonstandard analysis. My apologies if they are wrong. Your statement of how it works does seem to be closer to what nonstandard analysis probably defines them as, I believe. – Shinaolord May 22 '15 at 17:00

score 0 · Answer 2 · answered May 22 '15 at 00:04

In indefinite integral it is just a symbol. But it exist because indefinite and definite integral by Riman has very cool relationship as Newton-Leibniz theorem said.

In definite integral by Riman's definition it's small subarea in function domain that has infitine small diametr and take part in infinite summation. So in fact split your domain in which you're interesting as you wish for such dx-s... BUT: that parts shouldn't overlap, and max(length(dx)) in limit should go to measure_of_length(point)=0....

Theorems which use integration are based only on such definition. So split as you want.

Do define definitte classical integral you need to know classic definition of the limit. And define: function, function domain, how to measure area in function domain, what is a length, what is an operation of summation

Beside Riman's integral defintion, it is also exist Lebega integral which is used in theory of probability. As Software Engineer I know only that two integrals.

Riemann integration does not sum infinitesimal lengths. It's defined as the limit of the lower and upper Riemann sums as the norm of the partition goes to $0$. — , May 22 '15 at 00:06

is $dx$ greater than $\frac{dx}{2}$?

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