TITLE: Is non-standard analysis worth learning?
QUESTION [149 upvotes]: As a former physics major, I did a lot of (seemingly sloppy) calculus using the notion of infinitesimals. Recently I heard that there is a branch of math called non-standard analysis that provides some formalism to this type of calculus. 
So, do you guys think it is a subject worth learning? Is it a branch that is growing and becoming interconnected with other branches of math? Does it make calculus any easier?

REPLY [3 votes]: As a student of calculus that had the perennial trouble with epsilon-delta proofs and manipulations before trying my hand at non-standard calculus, perhaps I can provide a bit of perspective:
I still have trouble with epsilon-delta proofs of for derivative formulae. I think that arises from a lack of familiarity with the rigorous handling of intervals, neighborhoods, sequences and the like and I imagine that a more in-depth treatment of the subject would remedy this.
Infinitesimal calculus (as I have seen it so far) seems like a straightforward extension of the properties of "regular" numbers to "really small" numbers (infinitesimals) and "really big" numbers (infinites). The principles used for working with hyperreal numbers pretty much guarantee that an infinitesimal will behave in the same way you'd expect a "really small" real number to behave, while infinites behave as you'd expect "really big" real numbers to behave.
This makes the proofs of the basic differentiation formulae much easier both to understand and to read.
I also disagree with the complaint that one needs a good grasp of set theory and mathematical logic to be able to work with hyperreals. In this sense I think of hyperreals the same way I think of imaginary numbers, or sausages. While it can be illuminating to know how the sausage is made, one does not need to know that to find the sausage tasty. As a tutor of lower-level mathematics I can say that many, if not most of the students can handle imaginary numbers just fine without knowing how to construct the complex number system.
In short, I find non-standard calculus worth learning because it makes the arguments more understandable, easier to follow and more intuitive. However, if you are already comfortable with epsilon-delta calculus I don't know if you'd find it of much use.