This is *way *outside my area of competence -- to the extent that I still have one -- but I remember reading about Conway's Surreal numbers<http://en.wikipedia.org/wiki/Surreal_number>, which may be of interest.
*-- Russ* On Tue, Jan 24, 2012 at 10:21 AM, Joshua Thorp <[email protected]> wrote: > Thanks Roger, interesting paper. > > I have always been fascinated at the relationship between the language of > a mathematics and corresponding science that can be described with it. > > --joshua > > On Jan 23, 2012, at 11:43 PM, Roger Critchlow wrote: > > http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf > > -- rec -- > > On Mon, Jan 23, 2012 at 5:38 PM, Owen Densmore <[email protected]>wrote: > >> Integers, Rationals, Reals .. these scalars seemed to be enough for quite >> a while. Addition, subtraction, multiplication, division all seemed to do >> well in that domain. >> >> But then came the embarrassing questions that involved the square root of >> negative quantities and the brilliant "invention" of complex numbers (a + >> bi) where i = √-1 which allows the fundamental theorem of algebra .. i.e. >> that a polynomial of degree n has n roots .. but the roots must be allowed >> to be complex. >> >> The obvious question is "what next"? I.e. if we look at complex numbers >> at 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more >> that are needed for operations beyond polynomial equations? >> >> This led me to think of linear algebra .. after all, there we are >> comfortable with n-tuples and we can apply any algebra we'd like to them >> (likely limiting them to be fields). >> >> Wikipedia shows this: >> >> >> http://en.wikipedia.org/wiki/Complex_numbers#Matrix_representation_of_complex_numbers >> >> which illustrates an interesting job of integrating complex numbers into >> matrix form, not surprising 2x2, although the matrices are the primitives >> in this algebra, not 2-tuples. >> >> 3D transforms do get us into quaternions which wikipedia >> >> >> http://en.wikipedia.org/wiki/Complex_numbers#Generalizations_and_related_notions >> >> considers a generalization of complex numbers. >> >> So the question is: are there higher order numbers beyond complex needed >> for algebraic operations? Naturally n-tuples show up in linear algebra, >> over the fields N,I,Q,Z and C. But are there "primitive" numbers beyond C >> that linear algebra, for example, might include? >> >> What's next? And what does it resolve? >> >> -- Owen >> >> ============================================================ >> FRIAM Applied Complexity Group listserv >> Meets Fridays 9a-11:30 at cafe at St. John's College >> lectures, archives, unsubscribe, maps at http://www.friam.org >> > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org > > > > ============================================================ > FRIAM Applied Complexity Group listserv > Meets Fridays 9a-11:30 at cafe at St. John's College > lectures, archives, unsubscribe, maps at http://www.friam.org >
============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
