Talk:Commutative algebra
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[edit] On binary commutating structures
Binary numbers forming a commutating field are the complex numbers, but there are also other commutating binary (or two component) numbers, which form rings, rather than fields. (The inverse does not exist for all nonzero elements)...but does exist for most.
For complex numbers
Z=x+ty where tt = -1
For ring of dual numbers
Z=x+ty where tt = 0
For perplex numbers
Z=x+ty where tt = 1
A nifty property of the ring of dual numbers is to express a function over this domain as f(z) = f(x+ty) = f(x)+ty df/dx where tt=0. This follows from Taylor. Gives a nice algebraic definition of derivative.
Generalized Cauchy Riemann relations also exists for functions over the new domains.
Questions: Is this delving into homological algebra? what can the interplay of algebra and analisis be called? How is this classified?
astarfish
