Sign Expansions of Infinity

Apr 23 2007 Published by under Surreal Numbers

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
x>y.

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,
x1rx0ωe1+1, ...
rx1ωeα,
rx2ωeα,...]

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.

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  • Doug says:

    This comment relates to the JH Conway 'On Numbers & Games' [ONAG] 2ed, 2001 often used as reference for this topic.
    In chapter 4, Conway discusses complex and imaginary numbers in the Ring No[i]. He is one of the few game theorists to discuss such numbers.
    Yet his Fig 0 contains no lower case "i", but does have "e", "pi" and "omega".
    This can readily be resolved.
    Merely take an Euler Identity Circle and orthogonally match +1 and -1 with those of Fig 0.
    There will be an "i" perpedicular to the top of the new figure and a "-i" perpendicular to the bottom of this figure.
    One may then construct a Fig 0_i such that it is
    a - labeled with each Fig 0 having an "i" concatenated to each number and
    b - placed orthogonal to the two previous figures with "i" and "-i" matching those of the Euler Identity Circle.
    The resultant diagram has a parabaloid appearance at the outermost arcs.
    Wiki has an Euler Identity Circle
    http://en.wikipedia.org/wiki/Euler%27s_identity
    Caspar Wessel in about 1797 demonstrated that "i" exists [ie NOT a mathematical construct] and is rotated 90 degrees counterclockwise, one unit distant from the real line.
    http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Wessel.html
    Alternatively, in lieu of the 3D Riemann Sphere, one might use a 3D spindle torus.
    The spindles respectively representing the infinity of infinity at oo and the infinities of [1/infinity] at zero.
    http://functions.wolfram.com/ElementaryFunctions/ArcSinh/visualizations/9.html
    http://mathworld.wolfram.com/SpindleTorus.html

  • Doug says:

    Hi Mark.
    Within 3 days of my post #1, I found other game theorists using complex numbers in Max-Plus [sometimes called Tropical] Algebra.
    'Max-Plus Algebra for Complex Variables and Its Application to Discrete Fourier Transformation'
    Authors: Tetsu Yajima, Keisuke Nakajima, Naruyoshi Asano
    (Submitted on 26 May 2005)
    Abstract: A generalization of the max-plus transformation, which is known as a method to derive cellular automata from integrable equations, is proposed for complex numbers. Operation rules for this transformation is also studied for general number of complex variables. As an application, the max-plus transformation is applied to the discrete Fourier transformation. Stretched coordinates are introduced to obtain the max-plus transformation whose imaginary part coinsides with a phase of the discrete Fourier transformation.
    http://arxiv.org/abs/nlin/0505056
    2 - Nature v446, 26 April 2007 has two interesting math [physics and NP] articles.
    a - p1053-55 von Neumann relation generalized from 2D to 3D.
    Article: 'The von Neumann relation generalized to coarsening of three-dimensional microstructures'
    Robert D. MacPherson & David J. Srolovitz
    Abstract: Cellular structures or tessellations are ubiquitous in nature: examples include foams and crystalline grains in metals and ceramics. In many situations, the cell/grain/bubble walls move under the influence of surface tension (capillarity), with a velocity proportional to their mean curvature. As a result, the cells evolve and the structure coarsens. Over 50 years ago, the Hungarian-born mathematician John von Neumann derived an exact formula for the growth rate of a cell in a two-dimensional cellular structure. Now the much-sought extension of this result into three (or more) dimensions has been found. The formula should lead to predictive models for various industrial and commercial processes, from the heat treatment of metals to controlling the head on a glass of beer.
    doi:10.1038/nature05745
    Abstract | Full Text | PDF (188K) | Supplementary information
    http://www.nature.com/nature/journal/v446/n7139/edsumm/e070426-09.html
    b - p992-3 NP complete problem and zero-knowledge proofs
    Computing: The security of knowing nothing p992
    'Zero-knowledge' proofs are all about knowing more, while knowing nothing. When married to cryptographic techniques, they are one avenue being explored towards improving the security of online transactions.
    Bernard Chazelle
    doi:10.1038/446992a
    Full Text | PDF (167K)

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