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I think the title pretty much explains what I want to ask. Bassically is $$\lim_{x\to 0}(x)$$ different from $dx$?

Anoher way to put this would be, how wouldth equation: $$F=adm$$ Be different from $$F=a\lim_{m\to 0}(m)$$ I think the second can be written as $$F=0$$ Whilst the first we cannot, but I cannot explain why?

Quantum spaghettification
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    Whatever else $dx$ is, it is not a real number. The limit of $x$ as $x \to 0$ is a real number. These two quantities cannot be equal. – Simon S Feb 20 '15 at 19:28
  • @SimonS sorry why is $dx$ not a real number? Or equivlently If I have an infitismal bit of mass $dm$ why is this not a real number? – Quantum spaghettification Feb 20 '15 at 19:29
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    I was hoping that the title didn't explain well what you wanted to ask. –  Feb 20 '15 at 19:29
  • Joseph, how do you define an infinitesimal? How do you define arithmetic operations on those? Now, do these definitions fit with the usual real numbers definitions? This should answer your question. – bartgol Feb 20 '15 at 19:32
  • @bartgol I get your (and Simon S's) point but I can't see when we are using it to define real quantites, such as an infitimsal bit of mass, or an infitismal length of string, how it cannot be a real number? – Quantum spaghettification Feb 20 '15 at 19:34
  • @Joseph: If $dx$ is a real number, what is its decimal expansion? Or alternatively, what is an example of a Cauchy sequence of rationals that converges to $dx$? – WillO Feb 20 '15 at 19:36
  • An infinitesimal $\rho$ is an object with the property $0 < \rho< x$, for all $x \in \Bbb R_{>0}$. Do we have such an object in $\Bbb R$? – Ivo Terek Feb 20 '15 at 19:37
  • You might want to look at http://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio – quid Feb 20 '15 at 19:49
  • As a response to @Joseph's comment: why do you think an infinitesimal bit of mass/string is a 'real thing'? I've certainly never seen one, and if I know my physics, the notion runs counter to atomic theory. The idea of an infinitesimal is a useful approximation that seems to give us results that work, but I wouldn't call it real. –  Feb 20 '15 at 20:24

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It is confusing because the way derivatives are taught today are different from how it was done back in the 1600s. Back then a derivative was $dy/dx$, where $dy$ and $dx$ were infinitesimal quantities. Today, we introduce derivatives using limits. The confusing part is that learn differential calculus by limits but use the old notation.

Nicolas Bourbaki
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