Consider Baez on Octonions - talks about what the issues are. Beyond me
for now. Suspect you are about to pop out of algebra and end up
someplace else as interesting.
Carl
On 1/23/12 5:38 PM, Owen Densmore 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
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============================================================
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
