The Hyperreal Number System

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The hyperreal number system is a way of treating infinities and infinitesimals in a rigorous way that is consistent with the way that we treat ordinary numbers.

I recently posted a short video introducing the concept in a simple way. I thought some of you might be interested.

I find the hyperreals interesting for a number of reasons. First of all, I think that infinities and infinitesimals are somewhat of the equivalent of Intelligent Design for mathematics. Infinitesimals were essentially banned from mathematics in the 1800s because it was said that they were inconsistent and non-rigorous (this is why calculus switched from infinitesimals to limits). This move was largely philosophically motivated, with Hilbert and others trying to naturalize mathematics.

However, in the 1960s, infinitesimals were proved to be equally mathematically rigorous as other standard mathematical entities.

In any case, from a practical side, infinitesimals make calculus, limits, and other sorts of mathematical ideas a LOT simpler to work with. In fact, using these types of numbers, Calculus becomes pretty much identical with Algebra.

13 Replies to “The Hyperreal Number System

  1. 1
    kairosfocus says:

    JB, yay! KF

    PS: As I reflect on a lot of hand waving by teachers and lecturers or textbook writers over the years, I have been led to a radical conclusion: the quantity and structure sets that are most useful and are implied in how we think, are the hyperreals and as mileposts in them hyperintegers, with the 1/x catapult function as a powerful means to lock the thus mileposted continuum together from infinitesimals to transfinites. And of course around any particular number in the domain R* we can shift a cloud of infinitesimally close numbers from the cloud around 0. The density of the hypercontinuum is astonishing, on reflection.

  2. 2
    kairosfocus says:

    PPS: I suggest, posit some H as a hyper integer in Z+* beyond any n in N, but such that 1/H –> h such that h is smaller than any 1/n , n a natural counting number. Where, as any specific n is such that n+1, n+2 etc are setting out further +1 increments, we cannot reach to H by any stepwise finite process from 0 or n. This means H is infinitely larger than any natural, and h is infinitely smaller than any reciprocal of a natural. Which obviously extends to reals as the naturals are mile posts along the continuum of reals.

  3. 3
    johnnyb says:

    Kairos –

    Have you read our preprint, “Hyperreal Numbers for Infinite Divergent Series“?

  4. 4
    kairosfocus says:

    Good. Will look through. I suggested the H approach as allowing the next level to be articulated to those too “sophisticated” for the hand waving.

  5. 5
    EricMH says:

    The really fascinating upshot of all this is that naturalist prejudice permeates even to the supposedly completely neutral discipline of mathematics, and once again we can make major progress just by removing the naturalism blinders. Brilliant work JohnnyB!

  6. 6
    kairosfocus says:

    JB & EMH,

    Of course, philosophy and ideology will be everywhere.

    But it seems the issue of the unreliability of naively conceived infinitesimals was a genuine one, similar to the paradoxes of early set theorising. We need not even more than mention the problem of taming the transfinite. The epsilon delta limit approach allowed for a cautious way to build a confident knowledge of Calculus (and broader linked areas) which was important.

    However, after Abraham Robinson’s breakthrough in 1968, we do have a way to confidently deal with tamed hyperreals.

    I observe that the typical handwaving points to how the relevant domain for learning about structured quantities is R*, with Z* as mileposts. Where, we can define an integer part function for every r* in R*, Int [r*] –> closest integer whose modulus is smaller, in effect an extension of the whole number part. That allows us to see that we cannot bridge from the transfinite realm to any definable specific finite counting numbers [and their additive inverses] in unit scale steps. Notice, how I have used Dr Carol Thomas’ discussion as a springboard for going H –> 1/x –> h, where h is of smaller modulus than any 1/n where n is any identifiable or particular represented value reachable in stepwise +1 increments from 0 in N. All of this came out in the long debate here on a claimed transfinite causal-temporal past.

    Now, I find that thinking in terms of logic of being and possible worlds allows us to see that the requisite of distinction that sets apart a PW W from a close neighbour W’, some A in W but not W’, gives us a partition W = {A|~A}, thus allowing us to identify nullity as present in the partition element, a simple unity for A and a complex one for ~A thus duality. Applying the von Neumann construction of successive sets from that which collects nothing, we see:

    {} –> 0
    {0} –> 1
    {0,1} –> 2
    . . .
    {0,1,2 . . .} –> w, a first identifiable transfinite ordinal

    We then can use additive inverses to deliver from N so identified, Z, then Q, R and C. The hyper-move allows going to Z* and R* etc.

    All of this means, in any PW, such abstracta are present, integral to its framework, i.e. these are necessary beings. This of course means that a core part of Mathematics will be universally applicable; a powerful result and one that has been remarked on with amazement by Wigner et al on observing it physically.

    Naturalists are of course extremely suspicious of any abstract entities. These days, that needs not detain us further than noting their ideological peculiarities.


  7. 7
    kairosfocus says:

    H’mm, could we say that Calculus becomes an application and elaboration of the Algebra as applied to the hyperreals? KF

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    johnnyb says:

    KF –

    Yes, Calculus = Hyperreals + Basic Algebra. I’ve thought of having a second calculus textbook that taught it this way, but currently still focusing on the one I have 🙂 Still trying to figure out if it is less cognitive work to start from instantaneous slopes and infinite sums and move to hyperreals to provide a more solid foundation, or start with hyperreals and do calculus as a mere extension of that number system. It’s a tough question. My own Calculus from the Ground Up dives straight in to slopes, introducing hyperreals at the end. Calculus Made Easy starts with differentials, but doesn’t do hyperreals. Keisler’s Elementary Calculus starts with hyperreals, but doesn’t take them seriously enough. He basically uses them to simplify limits but then switches back to the reals for everything else. He has some problems which are truly simple in the hyperreals, but which he shrugs at because they aren’t solvable in the reals.

    So far, my approach has worked well enough. However, I always wonder if teaching hyperreals first would be better.

    As to whether they could have used hyperreals earlier. The fact is that the objections to infinitesimals actually weren’t that strong. Most of the objections had already been answered, even as early as Leibniz. The primary objection was having this entity which was simultaneously zero and not-zero. However, Leibniz had already accounted for that by using a different equal sign for something that was only equal up to an infinitesimal. The most that was lacking was an explicit rounding function to remove infinitesimal parts.

    It is true that Robinson has defined the hyperreals in a way that is more conducive to moderns, but the *reason* why it is more conducive to moderns is because moderns only believe in elements which are true by construction, which is a philosophical carryover from materialism.

    I am in the camp of Eugenia Cheng, who states that the only requirement for math is that the entities you propose don’t contain explicit contradictions. The requirements to build numbers in ZFC or some other framework should only be thought of as a tool to help prevent contradictions, not as a golden rule.

    The one other set of numbers I have been wanting to explore are Conway and Knuth’s Surreal numbers, but have found a lack of good introductory resources.

  9. 9
    johnnyb says:

    By the way, a really fun book is Henle and Kleinberg’s Infinitesimal Calculus. What’s fun is that it has a main text, but also a sidebar text, which is almost a stream-of-consciousness discussion from the author. It’s not really footnotes, it’s just anything the author found interesting and wanted to talk about more. While there is a point at which such authorial intrusions can get annoying, I wish that it were more common in textbooks. Giving commentary, history, etc., with a personal tone, is what really hooks readers/students into the subject. Students can only take dry facts for so long until their brain turns off.

  10. 10
    PaV says:


    x^2 – 25/x – 5= x + 5, so if you evaluate it as x approaches 5, this yields, 5 + 5 = 10. So, the hyperreal system gives us the correct answer. My suggestion is that you include this portion in the video as a ‘check’ on the system.

    (I’ve looked briefly at analysis, and I found Cauchy sequences a bit objectionable–it’s been so long ago I can’t quite remember the exact reason why. Weyl, in his book “The Continuum,” if I’m not mistaken, also has problems with it. Could you please comment on how the hyperreal system treats such sequences?)

  11. 11
    PaV says:


    Students can only take dry facts for so long until their brain turns off.

    Amen. When I read texts, my main thought is: “What’s the importance here? Why should I be interested? Why tax my brain for no reason at all?” Motivating the text, IMO, is essential to keeping the reader’s interest.

  12. 12
    kairosfocus says:

    JB & PaV,

    we have an interesting discussion developing here.

    My thought is, that Wigner posed a really really powerful challenge in asking why is Math so effective. My sense is, logic of being, possible worlds and necessary, world framework beings are key pointers as I discussed months back. As above, certain quantities and structures tie into the identity of any distinct possible world, opening up the logical structures and patterns that flow from it. Thus, bare distinct identity and coherence focussed on quantities will not cause things by force of self-action but by being constraints on being they lay out what can or must be or cannot be or happens not to be. So, we see abstract logic model worlds that we may construct that then lead to key entities that if necessary are framework to any possible world. And if just part of the contingencies of some PW that is close enough to our own, they can speak again to us.

    So, I am finding that axiomatic constructions of systems using framing propositions is going to be helpful.

    I forgot, we also have a built up body of known facts that constrain model building in subtle ways through the community and through the sense of intuition of the individual.

    Above, we are seeing, go to the Hyperreals and voila, things start to come out of algebraic manipulations that tie to sequences, series, limits, thus rates and accumulations of change. Hence, Calculus pops out. And on this the body of knowledge becomes a compendium that is then useful. I find getting back to dy and dx and dt or curly versions not being “mere” symbols but infinitesimal hyperreals as symbolised is a reassuring bridge between Physics and Math. Though I am very aware that in Physics we get away with some sloppy mathematical trickery as we have fairly well behaved frameworks that keep away from pathologies.

    Hey, here is something that popped up in digging around:

    Christopher Fenech

    How mathematical ontology informs ‘the mathematical absolutism versus mathematical fallibilism debate’ in Contemporary Western Philosophy of Mathematics, 2014

    [ . . . ]

    I will present various prominent schools of thought within a historical
    context and try to categorise the various positions held by their proponents under either the heading
    of ‘mathematical absolutism’ (the view that mathematics is discovered and is considered to be
    universal, objective and certain) or that of ‘mathematical fallibilism’ (the view that mathematics is
    invented and always being developed; being historical and social) . . . .

    The realist in ontology believes that mathematical entities (as well as mathematical relations and
    structures) exist objectively and their existence is necessary. The realist also believes that
    mathematical objects are abstract objects i.e. they do not reside in space and time, and have no
    connection with humans. Like the fact that physical objects exist independently of us, according to
    the philosophical realist, so too do mathematical objects according to the platonist. Mathematical
    truths, according to this view, are discovered, not invented. This is the absolutist view par
    excellence . . . .

    Some philosophers of mathematics are realists in a different way than realists in ontology. Instead
    of focusing on the existence of mathematical entities, they are concerned with the objectivity of
    mathematical truth. Realists in truth-value argue that every accepted mathematical statement has a
    distinctive and objective truth-value that is independent of our knowledge or whether it abides by
    the logic that arises from the mathematical theories that we know of. Truth-value realism also tends
    to entail the idea that most mathematical statements that are held as true are actually true. Therefore,
    truth-value realism is a methaphysical position, but is not an ontological view. On the other hand,
    platonism is an ontological view, i.e. it is concerned with the existence of abstract objects and
    mathematical concepts. Although truth-value realism holds that mathematical statements have
    uniquely distinct and objective truth-values, it does not claim that these truth-values follow from an
    ontology of mathematical entities . . . .

    David Hilbert’s work . . . was seminal to the school of formalism. He shared the intuitionists’
    view that natural numbers are basic in mathematics, but disagreed with them on the matter that the
    natural numbers are mental constructions. For Hilbert, natural numbers can be considered to be
    symbols. [– counters in the game? Sounds naturalistic]

    Hilbert never took a revisionist view towards the entire edifice of mathematics, as opposed
    to the intuitionists. His view of higher mathematics (such as number theory and topology) was that
    it was just a formal game. The point of this game, argued Hilbert, is to prove statements of basic

    When it comes to the Surreals, I found the key relevance as a way to construct the “zoo” of numbers by stepwise iterations. Yes, find ways to do operations etc is fine but the focus for me was the diagram that visually structured and showed for instance the reals line as filling in after w steps, etc. See if this link goes through:

    Of course, there are demonstrations of isomorphisms.

    Now, I am extremely right brained and tend to see things visually. Your mileage may vary.



  13. 13
    kairosfocus says:


    could I take a try?

    I found that rates and accumulations of change are crucial and widespread. Accordingly we have motivating context galore. Change, flow, rates point to TIME as a logical first independent variable. As a spiral curriculum thinker I tend to go with an opening shot key case study that a learning activities loop hits on lines that run out to a cluster of key ideas as anchor points. Think, spiral spider’s web.

    For this, water running into a cylindrical glass, so rates and accumulations come all together. Steady flow, linear plots, slopes and areas fall into place. But what of varying flow — I use a more or less Gaussian impulse and its sigmoid growth curve . . . big applications all over the place, even in marketing, economics, dynamics of movemennnnts etc.

    How do you define a slope of a curve? How do you get area under a curve?

    Chord-tangent and parallel strips, thus infinitesimals.

    Thence, fundamental theorem of the Calculus and mutually inverse operations.

    Then we can go to space variables and more abstract cases.


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