This is a review of two things: the mathematical system of surreal numbers, and Surreal Numbers by Donald Knuth, a book which describes them. Surreal Numbers is a strange book. You could describe it as a hybrid of a novella and a textbook, but its goal is pretty different than either of those. The plot, such as there is, involves two students on vacation (Alice and Bill) discovering the depth of their love for each other and the depth of mathematics. The mathematical showpiece is a number system, developed by John Conway, which smoothly integrates infinite and infinitesimal numbers into a system of more familiar numbers, like the integers, rationals, and reals. Let’s start by going over surreal numbers as the book describes them; later, we’ll discuss the strengths and weaknesses of the book itself.

The rules of surreal numbers

Unlike a textbook, the aim of Surreal Numbers is to convey the excitement of discovery involved in building a full system of mathematics from a few simple rules, as Alice and Bill do over the course of the novella. At the beginning of the book, they find a stone tablet listing two basic rules for generating numbers. The “tablet” writes these rules in pseudo-biblical prose from “J. H. W. H. Conway,” but I’ll paraphrase them here.

I. To create a number, you need a left set and a right set of already existing numbers, such that no number in the left set is greater than or equal to any number in the right set.

II. For two numbers x and y, x is less than or equal to y (x ≤ y) if no member of x‘s left set is greater than or equal to y, and no member of y‘s right set is less than or equal to x.

The tablet does not define what “greater than or equal to” (≥) means, but Alice and Bill reasonably declare that y ≥ x when x ≤ y.

Our heroes soon realize that the tablet describes how the number 0 pops out of nothing. If we put no numbers in the left set, and no numbers in the right set, this fulfills condition I, since there aren’t any numbers in either set to serve as counterexamples. Mathematicians write surreal numbers using braces, with a vertical bar separating the left and right sets, like this:

0 = { | }

0 is the only number created on day 0. (Days are numbered by the largest number created on that day.) On day 1, now that we have 0, we can use it to build new numbers:

1 = { 0 | }, -1 = { | 0 }

Note that { 0 | 0 } is not a valid number, since it violates rule I: 0 is in the left set, and 0 is in the right set, and 0 ≥ 0.

Equality and fractions

With 3 numbers, we have even more new possibilities on day 2, which Alice and Bill enumerate:

{ -1 | }, { -1,0 | }, { 1 | }, { 0,1 | }, { -1,1 | },{ -1,0,1 | },
{ | -1 }, { | -1,0 }, { | 1 }, { | 0,1 }, { | -1,1 }, { | -1,0,1 },
{-1 | 0 }, { -1 | 0,1 }, { -1 | 1 }, { 0 | 1 }, { -1,0 | 1 }

This is a lot of numbers! Luckily, several of these represent the same underlying value. Two numbers x and y are equal (x = y) if both x ≤ y and y ≤ x. Alice and Bill are able to simplify this mess by proving basic rules of equality, for example that any number is equal to an earlier-created number if and only if that number is greater than every element of its left set and less than every element of its right set. With this rule and others, they deduce that there are only four unique new numbers created on day 2, and the simplest representations of them are:

{ | -1 }, {-1 | 0 }, { 0 | 1 }, { 1 | }

That is, one new number on each end and one new number between each existing pair of adjacent numbers.

Alice and Bill later find another piece of the tablet, explaining how to do arithmetic with surreal numbers. With the rules for addition, they are able to prove that 1 + 1 = { 1 | }, and { 0 | 1 } + { 0 | 1 } = 1. That is, { 1 | } represents 2 and { 0 | 1 } represents one half. Similarly, { | -1 } represents -2 and { -1 | 0 } represents minus one half.

So as the integers get created, the fractions in between them slowly get filled in—at least, fractions with powers of 2 in the denominator (called “dyadic fractions”). We can write one fourth as { 0 | ½ }, and three fourths as { ½ | 1 }.

Day infinity

As the days go on, we get more and more integers and dyadic fractions. But what about, say, ⅓, or π, or the square root of 2? Well, nothing in the rules says that we can only have a finite amount of numbers in our left and right sets. These numbers get created after we have all the integers and dyadic fractions, by putting all the dyadic fractions below them on one side and all the dyadic fractions above them on the other. So ⅓ is represented like this:

{ 0,¼,⁵⁄₁₆,²¹⁄₆₄… | 1,½,⅜,¹¹⁄₃₂… }

This can be derived from its binary representation (0.01010101…).

But the other fractions and the irrationals aren’t the only numbers created on this day. What if we put all the integers on the left side and nothing on the right? We’ve created our first infinite number, represented with a lowercase Greek letter omega:

ω = { 0,1,2,3,4,5,6… | }

We can also put 0 on the left and a series that tends toward zero on the right. This number must be smaller than any positive rational (or real) number, but greater than 0. It’s infinitesimal, represented by a lowercase Greek letter epsilon:

ε = { 0 | 1,½,¼,⅛,¹⁄₁₆… }

Keeping our day naming scheme, the non-dyadic rationals, irrationals, and these infinite and infinitesimal numbers all get created on day ω.

After day infinity

Alice and Bill keep going from there. The day after day ω, we create

ω+1 = { ω | }

After another infinite number of days, we get ω+ω, or 2ω. At this point, the sky’s the limit.¹

There’s a truly mind-boggling amount of surreal numbers out there—bigger than any infinity you can consistently describe. In the surreals, infinity plus one is actually a meaningfully different number, as is the square root of infinity, so long as you specify which infinity you mean.

Mathematically, the surreal numbers are the largest totally ordered class. Class just means that it’s a collection of things. Totally ordered means that for any pair of surreal numbers x and y, either x ≤ y or y ≤ x, or both, and if x ≤ y and y ≤ z then x ≤ z. The surreal numbers in a deep sense contain all other totally ordered classes, like the integers and reals as we’ve already seen, or the Levi-Civita field which has a simpler collection of infinite and infinitesimal numbers. The surreal numbers can do this because they have basically no extra rules to create them beyond that they must be in order. If you can slot it into the ordering, it’s allowed.

And yet somehow, the surreals still have sensible rules for doing arithmetic. You can take roots and logarithms, and they have their usual definitions. This works thanks to splitting the creation into different days: these give you a way to define operations recursively, working your way up from day 0 to whatever day your inputs and outputs were created. But not everything works the same. Limits are weird, for example: in the surreals, there are infinitely many numbers between 0.99999… and 1, such as 1-ε.

The Book

Again, Surreal Numbers is a strange book. There is no point in reading it for the plot, though the plot does provide a nice break in and among the math. Unlike the math, the characters have no depth, and it’s easy to forget whether it’s Alice or Bill speaking about ε seconds after reading the little “A.” or “B.” at the start of the line. Though as a gigantic math nerd, I do appreciate some of the mathematical jokes and quips.

Frankly, if someone said this to me in earnest, I might just marry them on the spot

Does it serve the purpose Knuth desired? In the postscript, Knuth describes how he wants to remedy a problem in education where students don’t get to see how new math is invented until well into graduate school. Surreal Numbers does in fact showcase characters inventing math that is new to them. It does not exactly allow a reader to do the same, but it’s certainly a start.

The most similar book to Surreal Numbers that I have read is a book called The Number Devil by Hans Magnus Enzensberger, from 1997. The Number Devil is similarly an exploration of math, building from very basics and emphasizing the sense of discovery, with a layer of plot on top. But The Number Devil is much more accessible and presents much simpler math, though it’s certainly still interesting. And The Number Devil full of illustrations and colorful diagrams where Surreal Numbers has formulas and proofs. Surreal Numbers does have a lovely piece of cover art, though, drawn by Knuth’s wife Jill.

The Number Devil is an instant recommendation from me for anyone interested in the basics of number theory, of the sort a bright nine-year old could understand. Surreal Numbers is a more qualified recommendation. If you like thinking about numbers, understand how a mathematical proof works, don’t mind being confused sometimes, and have the time to take your learning at a more leisurely pace, this is the book for you. This happens to be me, so I enjoyed reading it. But if not, you may find the experience to be rather surreal.

Coming Soon: The best empire of each century since 500 BCE

---

¹ Well, that and the rules of set theory