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!
So to work our way up to the general normal form, we need to think about the normal form expansion in terms of ordinals. Following Conway's notation, we'll use lowercase greek letters for ordinals: α, β, ...
So, suppose we're looking at a surreal number N. We'll build its normal form in steps. For step one, we find ωy1, the leader of the commensurate class of N, and r1, the multiplier for ωy1 that gets closest to N in that commensurate class. Putting them together as ωy1*r1, we get a term of the normal form of N called it's 1-term, or N1
Next, the 2-term of N is the 1-term of N-N1. And so on - the α-term of N is the simplest number N-(Σβ<α(ωyβ*rβ).
Ok. We're almost there. The main catch that we're left with is proving that the normal form "terminates" (for an odd definition of terminates - that is, there is some ordinal number α, where α could be larger than ω...).
The way that we can prove that is pretty simple. Remember that every surreal number N has a birthday: it is a member of some class of numbers whose birthday is an ordinal. In the worst possible case, N will have a term for every ordinal up to its birthday. But by the definition of how numbers are constructed, for a number with birthday α, there's no way to define a number which contains a β-term where
β>α with a birthday of α - it's a nonsensical idea. Every part of
N has to be constructed before its birthday - and since the normal form requires everything to be built from the simplest possible numbers, nothing in the normal form at
an ordinal γ<α before the α-term could have introduced anything with a birthday later than γ; so the normal form must terminate
no later than the ordinal α of N's birthday.
So, finally, we can say, for a surreal number N, with birthday ordinal α, the normal form of N is the simplest set of terms such that for some β≤α: Σγ≤β(ωyγ*rγ).
And voila! There we are. The general normal form of an arbitrary surreal number, spanning as many classes of infinities and infinitesimals as we want. Now, finally,
we're at a point where we can talk about the sign-expanded form of infinites and infinitesimals - which will be the subject of the next post.