Commutative Hypercomplex Numbers

A reader writes: "The Generalized Number System (N+) implements commutative hypercomplex arithmetic to provide an alternative to vectors for processing multivariate data in three or more dimensions. Because of similarities between N+ and the complex number system, software for processing multivariate signals is readily derived from that for processing complex (or real) numbers. The derivation involves replacing operators on complex (or real) numbers with corresponding operators on hypercomplex numbers similar to the way in which steel replaced bronze as the ingredient for making swords during the Renaissance. In both cases, improved performance and capabilities of the product are attributed to properties of the new ingredient while many aspects of making and using the product remain the same. N+ and its application to signal and image processing is described on the website at www.hypercomplex.us ".

Re:Breathing important?

[Jerf]

Actually, every time you write a proof and somewhere along the way, you switch the order of the operands for some addition or multiplication, you are using commutivity. So a whole fucking lot of proofs end up using it, even if they don't specify it directly. Same for associativity.

Nor are these "trivial" uses, either; if you couldn't use commutivity as part of the equation re-writing process, many very common transforms become impossible... even the simple act of dividing "3x+2" out of an equation becomes difficult to set up if you can't re-order anything. (Remember that if you have x * 3 and you don't know multiplication is communitive, you can't rewrite that as 3*x, and thus you couldn't use that as part of 3x+2.)

In fact one would be hard pressed to find a non-trivial proof where commutivity isn't used implicitly, and you may find it very challenging (possibly even beyond your skill or downright impossible) to correctly re-write the proof without using commutivity.

(I speak in this post of "traditional" math, such as a normal person sees in school, somewhere up through low-level Calc. As others have pointed out, as you get higher into math, you encounter number and symbol systems where communitivity does not always hold. You typically meet one, "Matrix Math", in high school.)

Re:It's renaming

[exp(pi*sqrt(163))]

They're just a shorthand for a sub-ring of the matricies

Now that's bullshit if I ever read it. Any algebra that has a faithful matrix representation can be considered a sub-ring of a matrix algebra. You might as well dismiss the complex numbers, the quaternions and a whole host of other systems because they are all for "a sub-ring of the matrics". Hell, why not dismiss the discrete fourier transform. That's just a matrix.

In fact, even just looking at the web site, you can see how from a computational point of view they aren't just matrices. A dimension N hypercomplex number is represeted by an NxN matrix. An O(N) algorithm isn't the same as an O(N^2) one. Etc.