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
