I was reading xkcd last night (John has made me a fan). Then I found a youtube video of author of the comic, Randall Munroe, talking at Google (here). Donald Knuth was in the audience and asked a question. I had heard of Knuth before and knew he’s a big figure in computer science, so I went searching for a little more information about him. I noticed on his Wikipedia entry that he had a written a book related to surreal numbers, which I had never heard of. Intrigued, I discovered an introduction to the subject by a Danish IT consultant, Claus Tondering (here). Though he is not a mathematician, the document looks well written. There is also the wiki entry (here), of course, and this podcast featuring the inventor/discoverer of surreal numbers John Conway (here). In 1996, then high school student Jacob Lurie won an award for doing research on the subject (here).

I’ve only started reading Tondering’s introduction. The surreal numbers are an entirely different system from the reals we all know and love, built up from set theory. Their definition is very strange. A surreal number number consists of a pair of sets, a left set and a right set. These sets contain other surreal numbers. For a pair of sets to be well-formed and thus a surreal number, none of the surreal numbers in the right set can be less then or equal to any member of the left set. So what does “less than or equal to” mean for surreal numbers? To quote Tondering: “A surreal number x is less than or equal to a surreal number y if and only if y is less than or equal to no member of x’s left set and no member of y’s right set is less than or equal to x.” So less or equal to is defined in terms of itself. How do we get anywhere with this? We need one surreal number to start off with, the surreal number with no elements in either its left or right set. After showing that the number is well-formed, call this zero and go from there.

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August 10, 2008 at 6:58 pm

jsalvatiThat doesn’t sound remotely useful, but I guess that intuition can be misleading.

August 10, 2008 at 10:06 pm

cdfoxI was in a bit of a rush, so I didn’t go into why these numbers are useful. Of course, just the idea of a different number system is interesting, as is the way the numbers are recursively defined. But the surreal numbers also, in some sense, let you work with infinity and infinitely small numbers in ways you couldn’t ordinarily (at least that’s about the state of my understanding of it). But your intuition isn’t too far off. There hasn’t been a great deal of interest in these numbers, as far as I can tell, outside of the area for which John Conway came up with them — combinatorial game theory.

Also, I need to take a look at a couple of other systems, the hyperreals and the superreals. (Though I think the surreals have the coolest name.)