Re: MATH: Surreal numbers?

Steve Witham (
Mon, 10 Nov 1997 18:31:59 -0400

>...Surreal Numbers...
>...crazy notions about numbers such as infinity + 1, and was
>calling them surreal numbers. Has anyone else heard of such a thing...?

Surreal numbers are strings of zero or more up-arrows and down-arrows.
The empty string (I'll write {}) means zero. Up-arrow (^) means loosely,
"the next obvious higher number," and down-arrow (v), "the next obvious
lower number." So,
{} = 0
{^} = 1, {^^} = 2, {^^^} = 3...
{v} = -1, {vv} = -2, {vvv} = -3...
{^v} = .5, {^vv} = .25, {^^v} = 1.5, {^v^} = .75 (Smiley arithmetic?)...

So far so good. But, when you allow strings whose lengths are up to aleph-0,
for instance, all real numbers are covered. When the strings are allowed
to be longer than that?...ah. Various infinities, numbers between the
real numbers, etc. Pretty soon you're talking surreal numbers, to
paraphrase Everett Dirkson(*).

Lots of these numbers (including "transfinite" and "transifinitessimal"
numbers--sums of much too big and much too small numbers) were already
conceived of. Surreal numbers are just a single notation that covers
a lot.

This is what I remember from an introductory web page whose address is
probably around here somewhere, on one of my computers, in one of my
copies of Netscape... Darn, I need a bookmark file to locate my
bookmark files!

It also said people had worked out how to multiply the things but not
add them...!?

And what about octonions!?

(*) "A billion here, a billion there, pretty soon you're talking real money."

--    Steve Witham under deconstruction
"...when activated, it pops a message off the bag
    and recurs with the tail of the bag."
             --Vijay Saraswat and Patrick Lincoln