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> You can just work directly with infinitesimal values on a sound theoretical footing. This goes under the name "nonstandard analysis". https://en.wikipedia.org

> You can just work directly with infinitesimal values on a sound theoretical footing. This goes under the name "nonstandard analysis". https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis This is an oversimplification. Nonstandard analysis, the hyperreals, are one way of adding in infinitesimals to the reals, and definitely not the one I'd recommend for all use cases (although going from context, they may be appropriate here). There are plenty of ways to make a number system that add in infinitesimals to the reals, such as yes the hyperreals, and also I'd count the dual numbers among them, but there's also e.g. the surreal numbers: https://en.wikipedia.org/wiki/Surreal_number https://en.wikipedia.org/wiki/Surreal_number So, why am I kind of down on the hyperreals? Well, thing is, as I understand it, nonstandard analysis isn't really the study of the hyperreals; it's the use of the hyperreals to study the reals. I mentioned above that one big use of passing to a larger number system is that it reflects on the smaller number system; however, as best I can tell, the hyperreals are pretty much used purely in this way. They're used pretty much entirely as a tool for proving statements about the real numbers, rather than an object of study in their own right. And there's a reason for that; people often talk about "the" hyperreals, but actually, they're not uniquely defined. There's not really the system of hyperreal numbers, so much as there are potentially different systems of hyperreal numbers, which is annoying, but not if you only want to use them as a tool to study the reals, because their (relevant) relation to the reals is all the same. It's a bit icky. So yeah if you want to do analysis or calculus -- which might be the case, given that the earlier context was dual numbers, and that's what one would typically use dual numbers or -- then sure, use hyperreals. But if you just want to play around with a nifty number system that includes both reals and infinitesimals... eh, they're not great. You're likely to have more fun with the surreals. (More generally, of course, it's worth remembering that there's no need to stick to well-known systems of numbers... you can invent your own! Like, if for some reason you need infinitesimals, but you don't want them to square to zero like in the dual numbers, but you also don't want all the stuff that's in the hyperreals or surreals, there's nothing wrong with using R[ε] (or R(ε), or other variants depending on exactly what you're doing) to get a sort of minimal reals-with-infinitesimals...) [Edit: Is there no way to do bold anymore? Those R's in the above paragraph were supposed to be bold, to indicate the real numbers...]

thaumasiotes 3y ago
> [if] you also don't want all the stuff that's in the hyperreals or surreals, there's nothing wrong with using R[ε] (or R(ε), or other variants depending on exactly what you're doing) to get a sort of minimal reals-with-infinitesimals...) > [Edit: Is there no way to do bold anymore? Those R's in the above paragraph were supposed to be bold, to indicate the real numbers...] I believe you can do bold in Unicode. (see: 𝐛𝐨𝐥𝐝) As far as I know HN has never supported bold as a markup style. Standard number sets can be done the same way; I would represent the reals as ℝ. My standard method is to go to the wikipedia page for "blackboard bold" and copy the letter I want. I'd like to know more about the sets you refer to, ℝ[ε] and ℝ(ε), but I don't recognize them. Do they have names I can search for?
- Sniffnoy 3y ago
  > I believe you can do bold in Unicode. (see: 𝐛𝐨𝐥𝐝) As far as I know HN has never supported bold as a markup style. Oh, I think you're right, I'd forgotten. Grr, HN's limited subset of Markdown is quite annoying sometimes. Well, I'm going to be lazy and just write "R". > I'd like to know more about the sets you refer to, ℝ[ε] and ℝ(ε), but I don't recognize them. Do they have names I can search for? No, they don't, that's part of my point -- that you don't need to use recognized systems with names, you can use the usual constructions (or unusual constructions...) to make your own in the way that mathematicians always do. I mean I guess they sort of have names in that the notation would be pretty understandable to most everyone, although pronouncing it is annoying in that they'd both most typically just be pronounced "R adjoin epsilon", although I guess you could say "ring-adjoin" or "field-adjoin" to disambiguate. But note that there's plenty of other variants one could make as well, not just these. Basically go learn some abstract algebra, is what I would say. Or go read about ring adjunction (and polynomials) or field adjunction (and rational functions). Btw, I wouldn't describe these as "sets", that's not really the appropriate word to use here. When we talk about systems of numbers, we are, well, talking about systems, or algebraic structures -- rings, fields, ordered rings or fields, topological rings or fields, etc. To say "set" implies that what is important is the elements of these things, the contents; but these elements have no meaning on their own, they're given meaning by the structure -- the permitted operations, relations, etc. (Addition, multiplication, negation, less than or equal to...) Formally, an algebraic structure is a tuple, with the base set being just the first element of that tuple, even if we typically abuse notation and use the same symbol to refer to the set and the structure on that set. Um, hope that's helpful?

thaumasiotes 3y ago

> [if] you also don't want all the stuff that's in the hyperreals or surreals, there's nothing wrong with using R[ε] (or R(ε), or other variants depending on exactly what you're doing) to get a sort of minimal reals-with-infinitesimals...) > [Edit: Is there no way to do bold anymore? Those R's in the above paragraph were supposed to be bold, to indicate the real numbers...] I believe you can do bold in Unicode. (see: 𝐛𝐨𝐥𝐝) As far as I know HN has never supported bold as a markup style. Standard number sets can be done the same way; I would represent the reals as ℝ. My standard method is to go to the wikipedia page for "blackboard bold" and copy the letter I want. I'd like to know more about the sets you refer to, ℝ[ε] and ℝ(ε), but I don't recognize them. Do they have names I can search for?

Sniffnoy 3y ago
> I believe you can do bold in Unicode. (see: 𝐛𝐨𝐥𝐝) As far as I know HN has never supported bold as a markup style. Oh, I think you're right, I'd forgotten. Grr, HN's limited subset of Markdown is quite annoying sometimes. Well, I'm going to be lazy and just write "R". > I'd like to know more about the sets you refer to, ℝ[ε] and ℝ(ε), but I don't recognize them. Do they have names I can search for? No, they don't, that's part of my point -- that you don't need to use recognized systems with names, you can use the usual constructions (or unusual constructions...) to make your own in the way that mathematicians always do. I mean I guess they sort of have names in that the notation would be pretty understandable to most everyone, although pronouncing it is annoying in that they'd both most typically just be pronounced "R adjoin epsilon", although I guess you could say "ring-adjoin" or "field-adjoin" to disambiguate. But note that there's plenty of other variants one could make as well, not just these. Basically go learn some abstract algebra, is what I would say. Or go read about ring adjunction (and polynomials) or field adjunction (and rational functions). Btw, I wouldn't describe these as "sets", that's not really the appropriate word to use here. When we talk about systems of numbers, we are, well, talking about systems, or algebraic structures -- rings, fields, ordered rings or fields, topological rings or fields, etc. To say "set" implies that what is important is the elements of these things, the contents; but these elements have no meaning on their own, they're given meaning by the structure -- the permitted operations, relations, etc. (Addition, multiplication, negation, less than or equal to...) Formally, an algebraic structure is a tuple, with the base set being just the first element of that tuple, even if we typically abuse notation and use the same symbol to refer to the set and the structure on that set. Um, hope that's helpful?