# How is an infinitesimal $dx$ different from $\Delta x\,$? [duplicate]

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:

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?

## marked as duplicate by Rolf Hoyer, Newb, Claude Leibovici, kjetil b halvorsen, Daniel W. FarlowApr 28 '15 at 14:00

• 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 3:10
• 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 3:13

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.