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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