infinitesimals

Dual numbers use nilpotent infinitesimals (which are not invertible), while non-standard (Robinson) infinitesimals are not nilpotent and are invertible.

Compared to hyperreal infinitesimals, nilpotent infinitesimals are much easier to construct.

In order to work with infinitesimals in any reasonable way, you have to define how they work.

There is a reasonable way to extend the real numbers in a larger field containing infinitesimals.

Yes, it's basically dual numbers (or power series -- when considering inverses of infinitesimals, you get Laurent series) with different terminology.

My favorite are nilpotent infinitesimals which give rise to synthetic differential geometry[3] and dual numbers[4].

Calculus with infinitesimals works by discarding (symbolic) terms known to be (numerically) below epsilon.

To reason about this properly I think you need some sort of theory of infinitesimals and continuous motion.

Using infinitesimals is logically valid (alternative real analysis), useful for physics and other practical calculations but not at all helpful proving theorems.

Here[5] is a gentle introduction to nilpotent infinitesimals and intuitionist logic, and here[6] is a very good book on synthetic differential geometry.

However, even in these fields which admit infinitesimals, there is no definition for x/0.

For example, we can consider adding infinitesimals to the real numbers, allowing us to do calculus without taking limits.

My understanding of infinities and infinitesimals is that they are "not numbers," but are essentially a short hand notation for the processes involved in taking limits.

" My understanding of infinities and infinitesimals is that they are 'not numbers,' " Nope, they're perfectly good numbers.

But the infinitesimals invite treating them as "quantities" in an algebraic "game".

"An equation such as ∫ 3x^2 dx = x^3+C says that we add together uncountably many infinitesimals, and we get a medium-sized number." Yelp!