This question already has an answer here. That answer is abstract. Could you help me with some not-so-abstract examples of what the answerer is talking about? For example, give examples of hyperreal numbers which are written as numbers, if that is possible.

Another examples that I would like to understand are these statements:

Hyperreal numbers extend the reals. As well, real numbers form a subset of the hyperreal numbers.

I've not yet studied mathematics at university level.

The usual construction of the hyperreal numbers is as sequences of real numbers with respect to an equivalence relation. For example, the real number 7 can be represented as a hyperreal number by the sequence $(7,7,7,7,7,\ldots)$, but it can also be represented by the sequence $(7,3,7,7,7,\ldots)$ (that is, an infinite number of 7s but with one 7 replaced by the number 3). Any real number $x$ can be represented as a hyperreal number by the sequence $(x,x,x,\ldots)$. An example of an infinitesimal is given by the sequence $(1,1/2,1/3,1/4,\ldots)$, which happens to be a sequence of numbers converging to $0$. An example of an infinite number in the hyperreals is given by the sequence $(1,2,3,4,\ldots)$.

The equivalence relation is a bit complicated so I won't tell you how it works but just tell you one exists, and that for any sequence of numbers there are many other sequences that correspond to the same hyperreal number. This is analogous to the construction of the rational numbers as numbers of the form $a/b$ where $a$ and $b\neq 0$ are integers, where for instance $2/6$ and $1/3$ are considered equivalent as rational numbers. The equivalence relation for rational numbers is quite simple, but I'll mention that the equivalence relation for hyperreal numbers is not constructive (it uses the axiom of choice), so it is not in general possible to tell whether two sequences are equivalent as hyperreal numbers.

When actually working with hyperreal numbers however, how they were constructed is not important (whether by the method of identifying different sequences of real numbers as above, or otherwise), and the real number 7 is simply called 7 in the hyperreal numbers, and whenever an infinitesimal is needed, one might simply call it $\varepsilon$ with no regard for which specific infinitesimal it is.

• I think this is slightly misleading - there isn't one equivalence relation! Any nonprincipal ultrafilter yields an equivalence relation which works, and different ultrafilters yield different equivalence relations. For example, there's no way to say at the outset whether $(1, 1, {1\over 2}, {1\over 2}, {1\over 3}, {1\over 3}, {1\over 4}, {1\over 4}, ...)$ is less than $(2, {1\over 2}, 1, {1\over 3}, {1\over 2}, {1\over 4}, {1\over 3}, {1\over 5}, ...)$, since they alternate being larger and smaller than each other. – Noah Schweber Feb 8 '17 at 18:32
• I didn't say there was only one, but I could probably have been clearer. Good point. – Samuel Feb 8 '17 at 18:36
• You're quite right, you didn't (and your answer is really good - I upvoted) - I just wanted to point it out explicitly since it's really weird. In particular, in the sentence "The equivalence relation is a bit complicated so I won't tell you how it works but just tell you one exists", the second clause indicates that there is more than one, but the first suggests that there is a unique one by using "the". – Noah Schweber Feb 8 '17 at 18:38
• This answer strongly reminds me of the section on "real numbers" in Wildberger's paper, with the difference that you don't seem to recognize the imprecision involved in hiding an arbitrary infinite unspecifiable sequence in "...". – Wildcard Feb 9 '17 at 7:32
• @Wildcard: It is a standard convention in mathematics to write ellipses after a unique sequence if the pattern is considered obvious. – Samuel Feb 9 '17 at 8:02

Unfortunately, there is no "concrete" description of the hyperreals. For instance, there is no way to give a concrete description of any specific infinitesimal: the infinitesimals tend to be "indistinguishable" from each other. (It takes a bit of work to make this claim precise, but in general, distinct infinitesimals may share all the same definable properties. Contrast that with real numbers, which we can always "tell apart" by finding some rational - which is easy to describe! - in between them; actually, that just amounts to looking at their decimal expansions, and noticing a place where they differ!)

Similarly, the whole object "the field of hyperreals" is a pretty mysterious object: it's not unique in any good sense (so speaking of "the hyperreals" is really not correct), and it takes some serious mathematics to show that it even exists, much more than is required for constructing the reals.

While the hyperreals yield much more intuitive proofs of many theorems of analysis, as a structure they are much less intuitive in my opinion, largely for the reasons above.

To answer your other question, yes, the reals are (isomorphic to) a subset of (any version of) the hyperreals; that's what's meant by saying that the hyperreals extend the reals.

• "Any time I have two positive infinitesimals, they have all the same properties." This doesn't seem to make sense — if $f(x)=\sin(1/x)$, and $f^*$ is its natural extension to the hyperreals, then there exist two infinitesimals $\epsilon_0$ and $\epsilon_1$ such that $f^*(\epsilon_0)=0$ and $f^*(\epsilon_1)=1$. This distinguishes them. – Akiva Weinberger Feb 9 '17 at 17:03
• :: there is no "concrete" description of the hyperreals. This is a common delusion. Paper by Kanovei-Shelah in JSL 69 (2004), 159-164, presents an explicit construction of hyperreals with no reference to unspecified choice of anything (say an ultrafilter), as "concrete" as say the Dedekind construction of reals themselves. – Vladimir Kanovei Mar 6 '17 at 8:57
• @VladimirKanovei I am of course aware of your paper, and I believe I've mentioned it in previous answers on this site. However, I strongly disagree with your comment. It is not concrete in my opinion (while being really mathematically interesting!), and certainly not nearly as concrete as the Dedekind construction of the reals: I can explain the Dedekind construction (informally) to a sharp high-school student, but are you really telling me that if a student asked, "I get that we suppose the existence of a hyperreal field, but what is it?", you'd respond with that? (cont'd) – Noah Schweber Mar 6 '17 at 13:42
• I know that the sequences $(0, 1, 0, 1, ...)$ and $(1, 0, 1, 0, ...)$ correspond to different hyperreals, but which is bigger? At the end of the day, the student is forced to accept a high level of abstraction - individual hyperreals are things that are fundamentally mysterious, in a way that real numbers are not. This is something I've never heard a good answer to, regardless who I ask: how does one respond to this question from a student? (Meanwhile, the epsilon-delta definition is not part of the definition of reals, but rather of continuity, so I don't see its point here.) (cont'd) – Noah Schweber Mar 7 '17 at 13:35
• And in fact I can make this mathematically precise, and prove it. Both real numbers and representative sequences for hyperreals can be effectively coded as infinite sequences of zeroes and ones (in the latter case, by "zigzagging" through the array of the sequences corresponding to the terms of the representative sequence). Actually, this goes the other way: each infinite sequence codes a number of the appropriate type. Now let's look at the reals. There is a single Turing machine $T$ such that, if I give it as oracles sequences $\alpha$ and $\beta$ corresponding to distinct reals, (cont'd) – Noah Schweber Mar 7 '17 at 13:44

Assuming you mean your question in the practical sense rather than about doing logical foundations... picture in your mind the real numbers: that picture is exactly how the hyperreal numbers look.

I'm not even exaggerating. The hyperreal numbers, along with the rest of the nonstandard model of mathematics they're contained in, is carefully designed to have exactly the same properties that the real numbers do within the standard model.

In fact, in some philosophical approaches to the subject (e.g. how one might interpret internal set theory), it's the hyperreals within the nonstandard model that is actually what mathematicians have been studying for the past few millenia.

Except for a few esoteric applications, one only considers the hyperreals in the context of nonstandard analysis, which is all about making comparisons between standard model and a nonstandard model. One can't get the flavor of NSA by asking "what do the hyperreals look like?" — one has to ask the question "how do the reals and hyperreals compare?".

In the usual formulations, the main distinctive feature is that every real number is also a hyperreal number, and that all of the finite hyperreals can be partitioned according to their standard part.

That is, if $x$ is a finite hyperreal, there is some real $r$ such that $r-x$ is an infinitesimal hyperreal.

Phrased conversely, to each real number $r$ there is a halo of hyperreals surrounding $r$ that are an infinitesimal distance from $r$, and these halos partition the finite hyperreals.

If you are willing to continue on to the extended real numbers (i.e. $\pm \infty$), then the nonfinite hyperreals lie in the halos* of $\pm \infty$ depending on whether they are positive or negative.

Keisler's book uses the analogy of a microscope. At one level, you're studying the standard reals, and at any time you may pick a real and "zoom in" to look at its halo of nonstandard reals with that standard part.

*: Be careful that some sources define halo in a different way so that this statement is no longer true.

• @Noah: The monad of $+\infty$ is the intersection of all standard open sets containing $+\infty$, which is precisely the set of all positive nonfinite hyperreals. ($N^2$ might not lie in the monad of $N$, but AFAIK the same goes for $\epsilon^2$ and $\epsilon$; it's sort of unfair to treat the nonfinite numbers differently in this regard) – user14972 Feb 9 '17 at 4:20
• That seems to be a point of contention - I learned (and e.g. math.brown.edu/~ysolomon/NSA.pdf confirms) that the halo of a hyperreal is the set of hyperreals infinitesimally close to it, but it looks like some other sources use the definition you give. I think, given this, that that paragraph is still misleading; it's probably worth saying what a halo is, given that there seem to be multiple definitions. – Noah Schweber Feb 9 '17 at 4:47
• @Wildcard: Right. In the nonstandard model, we have nonstandard decimal expansions too, whose places are numbered by the hyperintegers. Let $N$ be a hyperinteger that is larger than any standard integer. Then, any decimal expansion that starts with $N$ zeroes to the right of the decimal place, followed by arbitrary digits is, indeed, smaller than $10^{-N}$. This is a perfectly ordinary hyperreal, but when we compare to the standard model it is infinitesimally in size, since it is smaller than any positive standard real number! – user14972 Feb 9 '17 at 17:58
• 'Keisler's book uses the analogy of a microscope. At one level, you're studying the standard reals, and at any time you may pick a real and "zoom in" to look at its halo of nonstandard reals with that standard part.' Are the nonstandard reals here synonymous with hyperreal numbers? – Andreas Feb 9 '17 at 21:41
• @Panthéon: Yes; I am using "hyperreals" synonymously with "the real numbers of the nonstandard model under discussion". – user14972 Feb 11 '17 at 1:07

I am glad to see from your question here that you are using Keisler's wonderful book on infinitesimal calculus.

We have been using this textbook at our university for the past three years to train about 400 freshmen, an educational experience we have reported on in this recently published article. The results of opinion polls indicate that students overwhelmingly prefer infinitesimal definitions of key concepts of the calculus like continuity, derivative, and convergence. Once the students have understood those key concepts in the intuitive (and rigorous) language of infinitesimals, we also present the epsilon-delta paraphrases of those concepts, so that by the end of the semester the students are ready to enter a follow-up course using either method.

Next year we will be teaching calculus using this approach to an estimated 200 students.

We present the procedures of infinitesimal calculus in a rigorous way. Just as in non-infinitesimal approaches, the construction of advanced number systems is postponed to more advanced courses in analysis.

Besides, the constructions of the real number field $\mathbb R$ and the hyperreal number field ${}^{\ast}\mathbb{R}$ are not that dissimilar. Here $\mathbb R$ can be defined as the quotient of the ring consisting of Cauchy sequences of rational numbers. The kernel in this case is a suitable maximal ideal MAX (namely the ideal of sequences converging to zero). Similarly, ${}^{\ast}\mathbb{R}$ is a quotient of the ring consisting of all sequences of real numbers. The kernel in this case is similarly a suitable maximal ideal MAX. This second MAX is of course different from the first MAX mentioned above, but the point is that the existence of a maximal ideal extending a given one is standard material of an undergraduate algebra course.

In this context, an infinitesimal is generated by the concrete sequence $1,\frac12,\frac13,\ldots$ namely the sequence $(\frac{1}{n})$.

At any rate these theoretical developments don't really affect the practice of freshman calculus.

For those interested in these subtler issues at the research level, I would like to refute the canard that the hyperreals are a more mysterious object than the reals. Namely, in Nelson's axiomatic approach, the infinitesimals are found within the real number line itself. Nelson's approach involves enriching the language of set theory by the introduction of a one-place predicate st$(x)$, which reads "$x$ is standard". In this approach an infinitesimal is a number $\epsilon\in\mathbb{R}$ such that $\neg \text{st}(\epsilon)$ and $\epsilon<r$ for every positive standard real number $r$. In this way, clinging to non-infinitesimal versions of analysis merely amount to a commitment to the details of a particular set-theoretic foundation, specifically designed to exclude infinitesimals.

These issues are explored in more detail in this recent article due to appear in the journal Real Analysis Exchange; no relation to SME :-)

Infinitesimals are not assignable, as the founders of the calculus understood. But this does not mean that infinitesimals are indistinguishable from one another. If x is an infinitesimal that 2x clearly differs from x. In Edward Nelson's Internal Set Theory, infinitesimals reside within the standard real line.

Some editors feel that in order to understand the hyperreals, a student needs background in algebra, set theory including cardinalities, and analysis. Users with sufficient reputation score can view one such (deleted) answer below.

I feel that Andreas' question should be understood in the context of his interest in learning the calculus following Keisler's textbook Elementary Calculus. So the question practically speaking is whether the infinitesimal approach is a practical way of learning the calculus. One can't expect a student to learn algebra, set theory, and analysis (!) before he learns calculus.

There is a further deeper observation that you might want to take to heart. Edward Nelson developed an original approach to Robinson's infinitesimals. Here instead of extending the real numbers, we enrich the language spoken by set theory by adding a one-place predicate "st" (meaning "standard") together with suitable axioms governing the interaction of "st" with the other set-theoretic axioms. In this approach, infinitesimals are found within the standard real line, as I pointed out in my answer (this was also mentioned in another answer).

From this point of view, requiring a student to learn analysis on $\mathbb R$ before he learns calculus on $\mathbb R$ is particularly paradoxical. What the existence of Nelson's framework reveals is that clinging to the non-infinitesimal way of teaching calculus amounts merely to an apriori commitment to the details of a particular set-theoretic framework at the expense of another.

• Would you like to include this text into your first answer? :-) Or which are your reasons to have written two answers? – Andreas Mar 8 '17 at 12:36
• @AndreasAlmgren, this is mainly a response to the user who posted an answer that contains some interesting points, but then went on to delete it. I feel my answer above deals with a different aspect of the story and should appear separately. – Mikhail Katz Mar 8 '17 at 12:37