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 ".

It's bullshit

[RachaelAnne]

There's a bunch of handwavy stuff about higher dimension number systems and getting communtative multiplication (needed for a bunch of signal processing algorithms and most other "real" math applications), but not proof.

Number systems with more than 2 coordinates are treated like matrices, because no one is able to find rules of muliplication and addition for say a 5-coordinate system that makes the numbers associative, commutative and distributive. Abstract algebra is basically only concerned with figuring out which of those properties hold for different sets. But in their stuff, they don't once show how their "N+" system will allow a 10-coordinate number (for instance) to be commutative with another one. From their site:

The Generalized Number System (N+) eliminates this disadvantage to extend the domain of signal processing to all dimensions in hyperspace. It provides several advantages compared to alternate hypercomplex number systems to include quaternions, octonions, and sedenions: The associative, commutative, and distributive laws from the arithmetic of real and complex numbers hold. Number dimension is fully programmable. Gauss-Jordan elimination is applicable as required to solve many types of inverse, least-squares, and optimization problems. Definitions exist for elementary functions to include sin, cosine, logarithm, and exponential.

That sure sounds like they've found a system form making quaternions commutative. But quarternions aren't. I read everything in the left two columns of links. And there wasn't more than vague promises of N+ solving signal processing problems and basic descriptions of real number field theory. But nothing saying how their N+ numbers are associative, commutative and distributive. So this is just bullshit.

Rachael

It's renaming

[MarkusQ]

Number systems with more than 2 coordinates are treated like matrices, because no one is able to find rules of muliplication and addition for say a 5-coordinate system that makes the numbers associative, commutative and distributive.

I read everything in the left two columns of links. And there wasn't more than vague promises of N+ solving signal processing problems and basic descriptions of real number field theory. But nothing saying how their N+ numbers are associative, commutative and distributive. So this is just bullshit.

I had the same reaction, but after digging a little deeper on their site [hypercomplex.us] (hurah Google) I did turn up a explanation, and guess what? They're just a shorthand for a sub-ring of the matricies.

-- MarkusQ

Re:Commutativity important?

[lirkbald]

Theorem: All finite, abelian groups are isomorphic to a group of the form Z_n1 X Z_n2 X ... X Z_ni for some n1,n2,... ni. Apologies for lousy formatting, as I don't think slashdot lets you do subscripts. X == direct product, Z_n == cyclic group of order n. Also apologies if I didn't get that quite right, as I don't have an Abstract Algebra text in front of me.

And if you don't know what what I just said means, then you shouldn't be commenting on the existence of 'important' theorems, should you?