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
