Good Math, Bad Math

The Surreal Reals

The Surreal Reals

I was reading Conway’s Book, book on the train this morning, and found something I’d heard people talk about, but that I’d never had time to read or consider in detail. You can use a constrained subset of the surreal numbers to define the real numbers. And the resulting formulation of the reals is arguably superior to the more traditional formulations of the reals via Dedekind cuts or Cauchy sequences.

First, let’s look at how we can create a set of just the real numbers using the
surreal construction. What we want to do is get a notion of the simplest surreal number that satisfies some condition. We need to toss out a few definitions first, to work our way towards it.

Suppose you have a surreal number x={XL|XR}. The members of XL are called the left options of x, and the members of XR are called the right options of x. The elements of the union of the left and right options are called the options of x.

So, take the surreal number x={XL|XR}. Suppose we have another number z which satisfies the following two conditions:

  1. ∀a∈XL,z>a
  2. >∀b∈XRz<b.

If none of the options of z also satisfy the two conditions above, then we say that
x=z, and z is the simplest number equal to x.

There’s a bit of a trick hidden in there. Why do those conditions mean that z is the simplest number? If no options of z can satisfy the rules, then that means that z is the number with the earliest birthday that fits between XL and XR. Think of an example: {1,2|4,5}. What number is that? It’s greater than 2, but less than 4. 7/2 obviously matches the two conditions: (7/2) is greater than 1 or two; and it’s less than four or five. But 7/2 is {3|4} is surreals; and 3 also satisfies the condition. But no options of 3 can possibly satisfy them. So that final restriction about the options of z guarantees that we get the number with the earliest possible birthday as the simplest number.

Next, we can define the integers in terms of the surreals. An integer is a surreal number whose simplest form has only integers as options, and for which at least one of its options sets is empty. So {1,2|} is an integer; {|-2,-3} is an integer, etc.

The set of real numbers then, consists of all numbers x={XL|XR} such that:

  • There exists an integer z such that -z<x<z
  • XL={a | a=x-(1/n)}n>0
  • XR={a | a=x+(1/n)}n>0

It should be pretty easy to see why that defines the reals. It’s bounded by the set of integers – so ω, which is beyond the range of integers is excluded. And it clearly includes the irrationals – the definition above is very similar to Dedekind cuts; you can clearly defined π in terms of a series of numbers getting ever closer on either side, so that only π is left in the gap.

So we’ve got a very simple definition of the real numbers in terms of the surreals. Why is this a good thing?

Conway makes a convincing argument, based on how you teach numbers. When you teach numbers in an abstract way, trying to build up to our common understanding of the reals, you end up working through a lot of proofs and a lot of arguments. The pain of many of those arguments is the amount of case-based reasoning you need to work with. For example, when you define the real numbers using Dedekind sections over the rationals, you need to consider four cases when you define multiplication (based on the signs of the numbers being multiplied). The more complex you get, the more cases: the associative law has 8 cases! Screwing up the cases, or missing a case, is the bane of advanced math students everywhere.

In the surreals, there’s no real difference between positive numbers and negative numbers. There’s no essential difference between integers and rationals. The distinctions become no less important, but less consequential in discussions of
the surreal reals: the case-based reasoning can go away, because in all cases, the definitions reduce to questions about set membership and empty sets.

I’m not saying that I’m convinced that the surreal numbers are the right way to teach reals. I’d want to actually try teaching all about abstract numbers from the perspective of surreals, and see how it goes. But I find the argument compelling enough that I’d be willing to try it, given the chance.

Comments

  1. #1 .mau.
    April 6, 2007

    I have to admit that I never quite understood the surreal numbers (nor the games), so I may be dead wrong. But wouldn’t be simpler to use for left and right options
    X_L={a | a=x-(1/(2^n))}, n>0
    X_R={a | a=x+(1/(2^n))}, n>0
    ? It’s clear that the two sets are infinite anyway, but in this way we would use only numbers generated in finite days.

  2. #2 malpollyon
    April 6, 2007

    “An integer is a surreal number whose simplest form has only integers as options”
    Surely you mean an integer is a surreal number whose simplest form has only a *finite number* of integers as options. Otherwise {1,2,3,…|} = ω is an integer, which isn’t what you want.

  3. #3 Blake Stacey
    April 6, 2007

    Knew you’d get around to it soon! :-)

  4. #4 .mau.
    April 6, 2007

    (gosh, am I stupid! since x is not an integer, my construction is useless)

  5. #5 billb
    April 6, 2007

    Isn’t there a bug in your description of X_L and X_R for the surreal reals? Your definition seems to be self referrential. x is defined in terms of X_L and X_R which are themselves defined in terms of x. Maybe that should be z in the definition of X_L and X_R?

  6. #6 ej
    April 6, 2007

    I own a copy of Donald E. Knuth’s “Surreal Numbers” but it loses me very quickly. I’ll try reading it again with the help of your explanation.

  7. #7 Enigman
    April 7, 2007

    One pedagological problem with defining the real numbers in terms of the surreals may be how that goes via defining the integers in terms of the surreals; i.e. students might be put off by the counter-intuitiveness of “There’s no essential difference between integers and rationals.” I suspect that students might find the idea of defining the integers, which are (intuitively) signed counting numbers, slightly off-putting.

    Would they think that we could define the counting numbers, or would they think that we describe what we find them to be? Is it not the crucial virtue of the Peano axioms that they correctly describe the counting numbers (in second-order logic)? We find that such numbers have nice algebraic properties, and that is all well and good; but we find them first, and then describe them. Having them, it makes sense to get the rational numbers (and maybe even the real numbers) from them, because when applying real numbers to the real world, they are applied to a world of numerous things. The counting numbers do not just connect higher math with the students’ intuitions, in an accessible way, but also with the coherent world of science, in a reliable way.

  8. #8 moshez
    April 8, 2007

    Actually, there’s another formulation which does not require either, and doesn’t require such baroque structures like surreal numbers (well, it’s sort of like surreal numbers, but much easier to formulate): if you take an aleph-null product of Q and divide by an ultrafilter. You get a field, with a natural injection of Q. If you take the ring of finite elements (all elements smaller than some element of Q), and divide by the infinitesimal element (all elements smaller than all non-zero elements of Q), you get the real numbers. That’s a lot like what Conway does with the surreal, except you get the “field with infinite/infinitesimal elements” much more easily (at the cost of using a constrained axiom of choice, I guess).

    I read about this method in some book about non-standard analysis, and fell in love with it.

  9. #9 Jonathan Vos Post
    April 8, 2007

    I agree with two things said above. If I may summarize what I think I read:

    (1) Surreal numbers are a cool way to unify reals with infinitesimals, making a particularly neat kind of nonstandard analysis possible;

    (2) The original approach (Newton-era) to infinitesimals was so inconsistent and wacky as to drive Analysis into the arms of epsilon-delta arguments which may be rigorous, but are not intuitive to many students.

    Hence, I am in favor of letting individual Math teachers and textbooks explore the space of possible explanations in an adventurous and creative way. Some students may benefit from unusual approaches.

    The purpose of Mathematics is insight.

  10. #10 Doug
    April 8, 2007

    Thanks for the reference to this Conway book [ONAG].
    Conway is really interesting for numbers, games and Monstrous Moonshine.

    I have found this book, hyped as the first textbook on the subject, which appears to be an extension of dynamic noncooperative game theory:

    Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications (Princeton Series in Applied Mathematics) (Hardcover)
    by Bernd Heidergott, Geert Jan Olsder, Jacob van der Woude (Authors) “In this book we will model, analyze, and optimize phenomena in which the order of events is crucial…”

    The authors use a Kelvin-like numbering system, with the smallest number -oo represented by epsilon and treated as the zero element. The unity element is e.
    [The number 0 and 1 are simply numbers and not treated as special elements].
    There are two operators:
    the MAX function represented by a circled_+
    and addition represented by circled_x.
    Both operators are associative and commutative with circled_x distributive over circled_+.
    The circled_+ is idempotent.

    Graph theory is utilized although vertices are referred to as nodes and edges as arcs. Eigenvalues, eigenvectors and eigenmodes are most often in matrix form.

    Although civil engineering examples tend to be used in this book, the authors [and other websites] note that many types of applied mathematicians, engineers and econometricians have used this type of algebra, which appears to be very powerful. Use of the MIN function and +oo, stochastics and continuous [PDE] synchronized networks are briefly discussed. There are 213 pages with 85 references, searchable at Amazon.

  11. #11 Flaky
    April 9, 2007

    The first I read about surreals (it was probably on this very same blog), I though that due to the captivating simplicity of their definition, they’d be good for the basis of a theory of reals in some automatic theorem proving/verification application, such as proof-carrying code. This post has convinced me that it’s an idea definitely worth trying out.

  12. #12 Craig Helfgott
    April 9, 2007

    Doug,

    Conway is also interesting for his work on polytopes and on lattices (among many, many other things). Both of those subjects are related to finite groups — which brings us back to the Monster.

  13. #13 Doug
    April 14, 2007

    Mark and Craig,

    I have started reading the Conway ONAG 2ed 2001.

    In chapter 0, Conway explains the use of von Neumann numbers rather than Cantor or Dedekind numbers [p 12-13].

    This is within a discussion of the [Number] “Creation Story” started on page 10, diagrammed on page 11.

    Has anyone read the multivolume ‘Winning Ways for Your Mathematical Plays’ by ER Berlekamp, JH Conway, RK Guy (Authors) referred to in the ONAG preface? Amazon lists vol 1,2,4 in paperback. Pages total nearly 1,000.

  14. #14 Peter Ripota
    July 10, 2010

    I’m afraid I’m a little bit late, but could anybody explain how to get to the surreal squareroot of any number, say of 2? Or of omega? Or of 1/omega? Conways series I can’t understand, and nobody else does explain it.
    -Peter Ripota, Munic (Germany)-

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