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.
(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 the path from
surreal numbers to games and game theory.
Nimbers come from a very old game called Nim. We’ll talk more about Nim later, but it’s one of the oldest strategy games known. The basic idea of it is that you have
a couple of piles of stones. Each turn, each player can take some stones from one of the piles. Whoever is left making the last move loses. It seems like a very trivial game. But it turns out that you can reduce pretty much every impartial game to some variation of Nim.
Analyzing Nim mathematically, you wind up finding that it re-creates the concept of ordinal numbers, which is what surreals are also based on. In fact, creating nimbers can end up re-creating the surreals. But that’s not what we’re going to do here: we’re going to create the nimbers and the basic nimber addition and multiplication
operations.
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.
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 have an infinite hierarchy of infinities, resulting in numbers that have, as their simplest representation, and infinite number of terms, each of which could involve numbers which can’t be written in a finite number of symbols. It’s just totally off the wall, insane, crazy, nuts… But fun!
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 multiplier that you can apply to 3ω that captures the difference between the two.
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 general infinities in terms of sign expansion for is going to take a bit of work. So as a bit of motivation, and to give you a first taste, since 1/2 has a sign-expanded form of “+-“, (that is, integer part=0, binary fractional part or 0.1=1/2), ω/2 = +ω–ω.
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.
The Surreal Reals
I was reading Conway’s Book
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 it in a reasonable time frame.. So my apologies if this is harder to follow than my usual posts.
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-based construction that allows you to create and represent all real numbers in a simple elegant form that has the necessary properties to make them behave properly. In addition, the surreal number system allows you to create infinitely large and infinitely small values, and have them behave and interact in a consistent way with the real numbers in their surreal representation. And finally, it makes the infinitely large numbers as natural a part of the number system as any other number: there’s nothing about the construction of an infinitely large number that makes its construction any different from a perfectly reasonable real number like 1/3.