Surreal Numbers

An invention of the mathematician JohnHortonConway. From the preface of his (beautiful) book OnNumbersAndGames [ISBN 1568811276 ], which introduces the SurrealNumbers (though not by that name):

This book was written to bring to light a relation between two of its author's favourite subjects - the theories of transfinite numbers and mathematical games. [...] we obtain a theory at once simpler and more extensive than Dedekind's theory of the real numbers just by defining numbers as the strengths of positions in certain games.

The SurrealNumbers are defined recursively (or, if you prefer, "inductively"). The definition resembles three things: (1) Dedekind's construction of the real numbers out of the rationals by means of "cuts", (2) von Neumann's construction of the ordinal numbers, and (3) a simple and elegant definition of the notion of "game", which you will also find in OnNumbersAndGames.

The resulting structure includes the real numbers, and also the ordinals (a kind of generalization of the natural numbers, including lots of infinite numbers - see BigOmega). It accordingly also includes a whole host of infinitesimals and other wonderful beasts.
A little novella entitled:

Surreal Numbers By DonaldErvinKnuth

ISBN 0201038129 Telling of two people finding an ancient document all about the creation of all numbers out of nothing by a being called simply JHWC :-).

I would like to see some examples if possible -- ChanningWalton

OK. I need to explain a bit about how the theory works before I can give any examples.

It's easier to start with the idea of a game; more accurately, with the idea of a position in a game. You know all there is to know about a game if you know the following things: (1) how to tell when the game is over; (2) which player won when that happens; (3) for each position, what positions each player can move to.

It's convenient to simplify this a bit, and declare that (1) the game is over when the player whose move it is has no legal move, and (2) when that happens, the player whose move it is loses. When you've done this, the above amounts to the following definition: A position is a pair of sets of positions (saying what positions each player can move to from that position).

The theory of these things is very rich and beautiful. One special part of it is particularly interesting; it's the theory of games with the following property. In all positions of the game, you get a better position by letting your opponent move than you do by moving yourself. These special games (or, strictly, their starting positions) are called numbers. There are elegant recursive definitions for the usual arithmetic operations on them, which result in a structure that includes the real numbers and a whole lot of other things besides.

Here are some simple games that happen to be numbers.

The simplest game of all is one in which neither player has a legal move. We write it like this: {|}. In general, we write a game as { positions one player can move to | positions the other player can move to }. It's usual to call the players Left and Right, in that order. Anyway, the game {|} is also called "0".

Then there are the games {0|} and {|0}. They're called "1" and "-1" respectively. So, for instance, in the game called "1", Left can move to a position where neither player has a move; Right can't move at all. You can think of this as Left having one "free move". In "-1", Right has one free move.

This talk of "free moves" probably sounds rather strange. It will make more sense if I explain an important operation on games. It's called addition, and when the games happen to be numbers it corresponds exactly to ordinary addition. To add two games G and H, you play them both at once, according to the rule: To move in G+H, you move in either G or H. This kind of thing comes up quite often in some games, like Go, when independent things are happening in two different parts of the board -- you have to choose whether to play here or there.

You might like to convince yourself that if you have any game then "adding" 0 to it (remember that 0 is the boring game in which neither player has any moves) doesn't change its outcome. That sort of explains why 0 is called that.

The game {0|0} isn't a number, because Left and Right can both move to 0, which contradicts the condition that Left's moves are always worse for Left than Right's moves are. (0 isn't worse than 0.)

What about the game {0|1}? That means that Left can move to a position in which no one has any moves, and Right can move to a position in which Left has one move. The game tree looks like this:

       /           /            0         1

This game is called "1/2". You can prove that 1/2 + 1/2 = 1 in the following sense: if you take any other game G, then 1/2 + 1/2 + G and 1 + G have the same outcome whoever moves first.

You probably won't be surprised to learn that {0,1|} is called "2", and {0,1,2|} is called "3", and so on. Continuing further, {0,1,2,...|} is a perfectly good game; it's called "omega" (which I'll write as "w"). If you know about "ordinals" (see BigOmega again), you'll recognize it. We can continue and get {0,1,2,...,w|} = w+1, and so on and so on. Lots of infinite numbers.

On the other hand, we can continue filling in the spaces between numbers. For instance, {0|1/2} is 1/4, and {1/2|1} is 3/4. All rational numbers with powers of 2 on the bottom can be constructed in this sort of way; and then you can say things like {0,1/4,1/4+1/16,...|1/2,1/4+1/8,...} = 1/3. This is rather like the construction of the reals from the rationals by "Dedekind sections".

And then we can combine all these things and get things like sqrt(w^2+3) + 1/w^w - 6.

I think I've probably bored you more than enough now. If not, go read:

On Numbers and Games, by JohnHortonConway

ISBN 1568811276 Conway explores the strengths of various positions in several games and arrives at a new class of numbers, called surreal numbers, which include both real numbers and ordinal numbers; these surreal numbers are applied in the author's mathematical analysis of game strategies


ISBN 1040991622

Winning Ways Berlekamp, Conway, Guy

ISBN 1568811306

"To move in G+H, you move in either G or H."

Is there a similar intuitive definition for multiplication?

Thanks for bringing back the magic of this book. Every time I clean out my bookshelves, it manages to hang in there, even though I don't really read it any more. Good synopsis. I wonder if it actually means something to someone who hasn't read the book. -- AlistairCockburn

It does. But I have to think about it a bit before it really sinks in. -- AnonymousDonor

Actually, it makes me think of something... Russell derived 'number' from sets of sets, Church derived 'number' from powers of recursive functions, Conway derives 'number' from 2-person positional games. We don't understand this simple thing called 'number', so we derive it from something we clearly can't have a clue about??? -- AlistairCockburn

This is very common in mathematics, as you noticed. Cayley's Theorem in GroupTheory? says that all groups are isomorphic to a group of permutations. This sounds very cool and very powerful until you realize this just means we don't know very much about permutation groups.

I think the hope is that by getting insights into any one of these derivations, we can gain insights into the others. -- GrahamHughes

It's also a more interesting version of nonstandard analysis. In Abraham Robinson's formulation of nonstandard analysis, he uses the fact that (in first-order logic), the axiom system for the real numbers R cannot have a unique model; there must be others. He constructs the non-standard real numbers *R (the * should be a superscript, but oh well) by adjoining an infinite number he calls c. Why isn't this a contradiction? Because, in this model, c is not really part of the language of the model; the only theorems and proofs in the model cannot use c in their statements. It's strange; I spent some of my undergraduate career wrapping myself around this.

Conway's SurrealNumbers are a more natural construction of a nonstandard analysis. Actually, the definitions of multiplication and division don't seem very natural, but you can't have everything. -- EricJablow

For the curious, there is a Perl implementation of SurrealNumbers at (created so that Yet Another program to compute 2+2 could be written). Translations into other languages are invited.

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