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 of the ordinal y.
What’s an irrelevant sign in the sign-expansion of y? It’s anything in a term of y that
would imply that that term of y is larger than some other ordinal term that preceeded it.
In other words, if the sign expansion of y is the sequence of signs
[Y0,Y1,…Yδ,…], yδ is irrelevant
if and only if there is some ε<δ such that
[Y0,Y1,…Yε,…] is greater than or equal to the
ordinal x of some term ωxrx in the normal form of Y, and
So, to generate the sign expansion of y, we’ll concatenate the sign-expansions of each of the non-zero terms in its normal form, omitting the irrelevant signs.
Given a term t=ωxrx, where:
- The relevant sign expansion of x is [xδ]δ<α;
- the complete sign expansion of rx is [rx0,rx1,…]
- ei is the total number of “+” signs among the various xi where i<δ.
Then the sign expansion of the term t is [x0rx0ωe0+1,
And… The sign expansion of the full number T whose normal form is the list of terms t0,…,tα is the concatenation of the sign-expanded forms of each of the terms.
In practice, what does this hairy mess mean? It’s actually amazingly simple in some ways. Suppose you’ve got a number like 3/4. In sign-expanded form, that’s “+-+”. So: 3/4×ω = +ω–ω+ω.
Suppose you’ve got (1/2)ω2 + ω. Since 1/2=”+-“, then
for the (1/2)ω term, you’d have “+ω2–ω2“, and for the ω term, you’d have “+ω“, so the full expansion would be “+ω2–ω2+ω“.
Nifty? I think so. And it’s even useful in some ways. But that’s a topic for another day.