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When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this means. In particular, I gather the validity of treating a ratio of differentials is a subtle issue and I'm not sure I get it. Can someone explain the difference between $dx$ and $\Delta x$?

EDIT:

Here is a related thread:

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

I read that and this is what I don't understand:

There is a way of getting around the logical difficulties with infinitesimals; this is called nonstandard analysis. It's pretty difficult to explain how one sets it up, but you can think of it as creating two classes of real numbers: the ones you are familiar with, that satisfy things like the Archimedean Property, the Supremum Property, and so on, and then you add another, separate class of real numbers that includes infinitesimals and a bunch of other things.

Can someone explain what specifically are these two classes of real numbers and how they are different?

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Stan Shunpike
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  • By itself, $\mathrm{d}x$ means nothing, while $\Delta x = x_1-x_0$. In certain contexts like measure theory or differential geometry a meaning is given to the symbol $\mathrm{d}x$, but they are just defined in order to adapt to this notation. – hjhjhj57 Apr 28 '15 at 03:10
  • Okay, thanks for replying. I updated to clarify what I didn't follow from that link. A good link! – Stan Shunpike Apr 28 '15 at 03:13

2 Answers2

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Back in the days of non-standard analysis, the idea of differentiation was (informally) defined as the ratio between two infinitesimal values.

The real number system (often denoted as $\mathbb{R}$) can be defined in terms of a field. It is a field with properties such as total ordering (basically means every element in it has an order), Archimedean property (which states that every two elements are within an integer multiple of each other). $\mathbb{R}$ can, however, be extended. One example is to allow the existence of imaginary number, $\sqrt{-1}$, in which case you would have complex numbers (and that is also a field).

Extending $\mathbb{R}$ by introducing the element infinitesimal to it would make it lose the Archimedean property.

So when Arturo Magidin talked about "two classes of real numbers", basically he is referring to $\mathbb{R}$ and an ordered field containing all elements in $\mathbb{R}$ and also infinitesimal, a "number" defined as greater than 0 but smaller than any integer unit fraction.

Archy Will He 何魏奇
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    Non-standard analysis is still very much a thing. I think you're thinking of the nonrigorous methods of Euler and the like, who used infinitesimals in a nonrigorous manner (which prompted Weierstraß to come up with the epsilon-delta definition). Non-standard analysis came around in the 20th century (EDIT: 1960s, according to that article) as an alternative to Weierstraß's approach, giving a rigorous (and kind of complicated) foundation to infinitesimals. They showed that the formulations are equivalent. – Akiva Weinberger Apr 28 '15 at 18:36
  • This is incorrect. Back in the day, they used infinitesimals with no foundation. Non-standard analysis is newer than the real number system, both logically and historically. – Christopher King Mar 04 '16 at 23:37
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The term "infinitesimal" was used by Leibniz. This was at a time before the concept of limits, as we know it today. He still thought of $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ as a quotient with $\mathrm{d}y$ & $\mathrm{d}x$ being very small.

Today $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ is not a quotient but is notation for the limit after the limit has been applied, i.e. the whole thing is notation for the derivative.

Another way of looking at your formula is

$$\frac{\mathrm{d}y}{\mathrm{d}x}=\lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

The current notation can be very misleading

sai-kartik
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Stephen P
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