{"id":396,"date":"2007-04-23T21:29:05","date_gmt":"2007-04-23T21:29:05","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/04\/23\/sign-expansions-of-infinity\/"},"modified":"2007-04-23T21:29:05","modified_gmt":"2007-04-23T21:29:05","slug":"sign-expansions-of-infinity","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/04\/23\/sign-expansions-of-infinity\/","title":{"rendered":"Sign Expansions of Infinity"},"content":{"rendered":"

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<\/a>, it’s pretty easy to translate them to sign-expanded form.<\/p>\n

<\/p>\n

Suppose you’ve got a surreal number in normal form: Σωy<\/sup>ry<\/sub>. Basically, it’s going to be
\nformed from a concatenation of the sign-expansions for each &omegay<\/sup>ry<\/sub>, with one restriction. The sign expanded
\nform needs to be generated in descending<\/em> order of y’s. To make this
\nwork, we need to distinguish between relevant<\/em> and irrelevant<\/em> signs in the sign expansion of the ordinal y. <\/p>\n

What’s an irrelevant sign in the sign-expansion of y? It’s anything in a term of y that
\nwould imply that that term of y is larger than some other ordinal term that preceeded it.
\nIn other words, if the sign expansion of y is the sequence of signs
\n[Y0<\/sub>,Y1<\/sub>,…Yδ<\/sub>,…], yδ<\/sub> is irrelevant
\nif and only if there is some ε<δ such that
\n[Y0<\/sub>,Y1<\/sub>,…Yε<\/sub>,…] is greater than or equal to the
\nordinal x of some term ωx<\/sup>rx<\/sub> in the normal form of Y, and
\nx>y.<\/p>\n

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.<\/p>\n

Given a term t=ωx<\/sup>rx<\/sub>, where:<\/p>\n