Re: Why is there no *really* useful Algebra beyond complex numbers?

> (ax + by)z = a(xz) + b(yz) and x(ay + bz) = a(xy) + b(xz).
Does that define multiplication once addition is established?
> If
> x/y
>
> is defined for all y not equal to (0, 0, ..., 0) then the algebra is
> said to be a _division_ algebra.
Would it give us more possibilities if we only required that say triple
expressions are 1? i.e. xy/z
> Now here's the rub: only for certain n can can an n-dimensional real
> algebra have the above properties:
>
> (1) Only for n = 1 and 2 are there n-dimensional real commutative
> division algebras.
>
> (2) Only for n = 1, 2 and 4 are there n-dimensional real associative
> division algebras. For n = 4 the algebra is called the quaternions.
They are useful for rotations? What else?
> (3) Only for n = 1, 2, 4 and 8 are there composition algebras with
> units. For n = 8 the algebra is called the octonions.
They are non-associative?
In maths it's all about inverting operations and isolating an expression on
one side of an equation? So it must be difficult to deal with
non-associative structures??
> Note that already in the case n = 2 (the complex numbers) there is no
> order as there is on the reals. So one looses something even with the
> complex numbers.
So what is so special about complex numbers to allow them things like
contour integrals?

Is there any non-algebraic mathematical system that can be calculated with?
For example for applications in physics, game theory, system theory or
whatever?

Anton
.

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