Infinitesimally small area $(dx)^2$ as a point

Question

I read in one book that infinitesimally small area $(dx)^2$ is a point. But point has no length in general sense. It's dimensionless.

How is it possible? And how to understand it?

Thanks.

Answer 1

As the comments note, this is very muddled language. There is a way to work with things like $dx$ somewhat like we work with numbers, but only if you put in the work to understand nonstandard analysis, which was developed to make the vague notion of "infinitesimal" rigorous.

You'll see this kind of argument a lot in Physics courses or numerical analysis, where we use the heuristic that $dx > (dx)^2 > (dx)^3$ etc as a ways of properly "cancelling out" powers of differential signs when we do formal symbolic manipulation of things like power series. But it's a heuristic.

I'm not sure what book this is but based on the typesetting it looks older. There are plenty of modern texts in calculus that don't stray into this territory and instead focus on the usual limit-based arguments.

If you want a rigorous introduction to single variable calculus, I'd recommend Ross' Elementary Analysis: The Theory of Calculus

Answer 2

Would you be so kind as to identify the source for this passage? It seems to be a reprint of a 19th century calculus or analysis textbook. Indeed, the explanation that the author provides is insufficient. It does not explain anything to claim that "$dx^2$ is a point". A proper explanation involves the standard part.