Surreal Numbers
goodmath | May 3, 2007 So, today we're going to play a bit more with nimbers - in particular, we're
going to take the basic nimbers and operations over nimbers that we defined last time, and
take a look at their formal properties. This can lead to some simpler definitions, and
it can make clear some of the stranger properties that nimbers have.
The first thing we did with nimbers was define nimber addition. The way that we did it was iterative: take a set of existing nimbers that exist at ordinal stage N, and use a simple procedure to try to create as many nimbers as we could by adding pairs of stage-N nimbers.…
goodmath | April 30, 2007(A substantial part of this post was rewritten since it was first posted. I managed to mangle things while editing, and the result was not particularly comprehensible: for example, in the original version of the post, I managed to delete the definition of "mex", which continuing to use mex in several other definitions. I've tried to clear it up. Sorry for the confusion!)
This is actually a post in the surreal numbers series, even though it's not going to look like one. It's going to look like an introduction to another very strange system of numbers, called nimbers. But nimbers are a step on…
goodmath | April 23, 2007 Finally, as I promised a while ago, it's time to look at the sign-expanded forms of infinites in the surreal numbers. Once you've gotten past the normal forms of surreal numbers, it's pretty easy to translate them to sign-expanded form.
Suppose you've got a surreal number in normal form: Σωyry. Basically, it's going to be
formed from a concatenation of the sign-expansions for each &omegayry, with one restriction. The sign expanded
form needs to be generated in descending order of y's. To make this
work, we need to distinguish between relevant and irrelevant signs in the sign expansion…
goodmath | April 12, 2007 When I left off yesterday, we'd reached the point of being able to write normal forms
of surreal numbers there the normal form consisted of a finite number of terms. But
typically of surreal numbers. that's not good enough: the surreals constantly produce
infinites of all sorts, and normal forms are no different: there are plenty of surreal
numbers where we don't see a clean termination with a zero term.
For me, this is where the surreal numbers really earn there name. There is something distinctly surreal about a number system that not has a concrete concept of infinity, but allows you to…
goodmath | April 11, 2007 On the way to figuring out how to do sign-expanded forms of infinite and infinitesimal numbers, we need to look at yet another way of writing surreals that have infinite or infinitesimal parts. This new notation is called the normal form of a surreal
number, and what it does is create a canonical notation that separates the parts of a number that fit into different commensurate classes.
What we're trying to capture here is the idea that a number can have multiple parts that are separated by exponents of ω. For example, think of a number like (3ω+π): it's not equal to 3ω; but there's no real…
goodmath | April 10, 2007 When I first read about the sign-expanded form of the surreal numbers, my first thought was "cool, but what about infinity?" After all, one of the amazing things about the surreal numbers is the way that they make infinite and infinitessimal numbers a natural part of the number system in such an amazing way.
Fortunately, it turns out to be very easy to play with infinities in sign-expanded form: you just need to use exponents of ω. Fortunately, exponents of ω are really cool! Getting to the point where we've really captured the meaning of exponents of infinity, so that we can talk about…
goodmath | April 9, 2007 In addition to the classic {L|R} version of the surreal numbers, you can also describe surreals using something called a sign expansion, where they're written as a sequence of "+"s and "-"s - a sort of binary representation of surreal numbers. It's fully equivalent to the {L|R} construction, but built in a different way. This is a really cool, if somewhat difficult to grasp, construction.
It's based on ordinals (which we also called birthdays) Remember, ordinals are the numbered generations of surreal numbers. Ordinal 0 contains the value 0; ordinal one adds the values +1 and -1; ordinal…
goodmath | April 5, 2007The 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…
goodmath | April 4, 2007 Coming back from games to numbers, I promised earlier that I would define
division. Division in surreal numbers is, unfortunately, ugly. We start with
a simple, basic identity: if a=b×c, and a is not zero, then c=a×(1/b). So if we can define how to take the reciprocal of a surreal number, then division falls out naturally from combining it the reciprocal with multiplication.
This is definitely one of my weaker posts; I've debated whether or not to post it at all, but I promised that I'd show how surreal division is defined, and I don't foresee my having time to do a better job of explaining…
goodmath | March 29, 2007 Late last summer, shortly after moving to ScienceBlogs, I wrote a couple of posts about Surreal numbers. I've always meant to write more about them. but never got around to it. But Conway's book actually makes pretty decent train reading, so I've been reading it during my new commute. So it's a good time to take a break from some of the other things I've been writing about, and take a better look at the surreal numbers. I'll start with an edited repost of the original articles, and then move into some new stuff about them.
So what are surreal numbers?
Surreal numbers are a beautiful set-…