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 ".

Comments

Too Deep For /. ?

[grantdh]

Wow, you mean this item was so deep, so intense and so amazing that it's wiped out even the most intense urge to claim First Post in the crowd???

Maybe it was that, by the time they got to the end of the text, they were too zonked out to post.

Congrats to Hemos for finding & posting this one - that's one hell of a subject/concept to get yer head around over a midnight snack :)

Re:Too Deep For /. ?

[Zerth]

They had me right up to the bit where they made the metallurgy analogy. Then I started laughing, looked at the site, said "oh, this is one of those ad-stories", and almost closed the window until I saw your post and had to get in on the early post action:)

I love how they'll sell you the necessary software to incorporate "this wonderful new way to compute".

No sir, I don't like it

[0x0d0a]

WTF? Is this a lousy article or what?

corresponding operators on hypercomplex numbers similar to the way in which steel replaced bronze as the ingredient for making swords during the Renaissance...

How the hell did Hemos let this one by?

Let's rewrite the article to be useful instead of stupid:

"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".

becomes:

The Generalized Number System (N+) is numerical processing software that uses commutative hypercomplex arithmetic to solve multivariate data problems in more than two dimensions. This outperforms the more traditional vector-based approach.

I mean, all the info thrown out here can be comfortably mentioned on the website, and it doesn't look like a blatant attempt to get "wow, that's complicated" comments.

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

Commutativity important?

[Euphonious Coward]

I don't know of any important theorems that depend on commutativity. Do you?

Re:Commutativity important?

[j.e.hahn]

Actually, yes, I do. Many. Go read any graduate level Algebra text. There are a LARGE number of theorems which rely on the commutativity (or non-commutativity) of the ring in question. Some of them are even important.

Commutativity is (one of the things) that distinguishes R^(N^2) from GL(N). It's a fundamentally important property. Commutative rings are by their very nature smaller than non-commutative rings, given the same set of generators.

I could go on, but...

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?

Re:Commutativity important?

[Euphonious Coward]

j.e.hahn wrote, "I could go on, but..."

Please do. I was hoping to learn something. It seems like most extensions from scalars into multiple dimensions (matrices, quaternions) shed commutativity like last year's skin.

Can somebody please give an example of an important and comprehensible theorem that depends on commutativity? That is, what is a consequence that would be meaningful to a non-mathematician?

Re:Commutativity important?

[trixillion]

Rather than give you examples of non-commutation in mathematics (play with any interesting matrices lately?), I'll give you some important examples of real world applications of non-commutation.

Angular Momentum: Yup, the stuff that keeps you upright while on a bike or motorcycle, anti-commutative. The Coriolis force, a corollary of angular momentum, this is the same effect that causes tornados and hurricanes (but not toilettes) to always rotate in the same direction in the northern hemisphere. To understand or prove any of these effects you must minimally know about the cross product, which is (as you might guess) non-commuting.

Maxwell's equations have an inherent twist in them, consider Ampere's right-hand and left-hand rule. All motors and generators rely on this effect, and without it there could be no propagation of light.

Polarized sunglasses - that's right, photons have polarity and polarizing filters are caused by non-commutating interactions with materials.

The list of everyday phenomenon that require non-commutation to prove is much longer than this, but I hope this gives you a taste of how boring the world would be without this quirk of mathematics.

Re:Commutativity important?

[Euphonious Coward]

The examples quoted seem to me to answer the opposite to the question I asked. For example, all the standard electromagnetic results may be expressed with quaternions, which do not support commutativity.

I asked about examples of results for which commutativity is an essential quality. I don't doubt there are some, or plenty, but what are a few of them?

Re:Commutativity important?

[Euphonious Coward]

lirkbald wrote, "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?"

If you don't know the answer, just say so, and let somebody else speak up.

Re:Commutativity important?

[spiro_killglance]

huh, Quaterions and Octerions aren't commutative
but you have well defined inverses. And you can
even invert Matrixes of Octerions. P.S. I've
miss spelled Quaterions and Octerions, its late.

I'm just a dumbass

[orthogonal]

I'm just a dumbass who doesn't "get" higher math at all. But let me try to get this straight, anyway:

Dude develops a "new" system of mathematics. Dude isn't looking for a Nobel prize, or (excuse me!) (is he under 40?) a Fields Medal. He doesn't (prominently enough to notice) post any link to citations in any journal, let alone a peer-reviewed journal.

But he is willing to sell you a MATLab add-on for $250.00?

Does he sell tinfoil hats and perpetual motion machines too?

Re:I'm just a dumbass

[orthogonal]

Picky note: Dude better not be looking for a Nobel prize. There is no Nobel prize in Mathematics [mathforum.org].

Sigh. Yeah, that's why I said, "(excuse me!) Fields Medal".

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 (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: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.

Re:It's renaming

[MarkusQ]

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

exp(pi*sqrt(163)) (aka Mr. 262537412640768744-epsilon): 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.

Agreed. But that isn't anything new. I wasn't disputing their claims, just their hype. What is it that's so wonderful here?

• That, for any integer N>=1 you can find N orthagonal matricies to form the basis of a sub-ring?
• That the sub-ring so formed will have the properties you'd expect of a ring?
• That, knowing (choosing) the form of your basis, you can find computational shortcuts?

I guess it feels to me as if Bob Shmoe anounced the discovery of a new group of order 163, which he dubed "Shmo", and looking at it I saw it was {f^163=I}. I'd say "hey, that's just C-163. That wouldn't mean I was dismissing all modulo arithmetic.

-- MarkusQ

P.S. If you think I'm missing something here, I'd be glad to know what. I think I read through more of their web pages than most posters, but I did it with a 13-month old on my lap, so I may well have bounced past some key point.

Re:It's renaming

[exp(pi*sqrt(163))]

What is it that's so wonderful here?

I have no idea. I'm just pointing out that the fact that the hypercomplex numbers have a faithful linear representation doesn't tell you there isn't smart going on. For example using quaternions can really simplify computations with rotation matrices even though they don't really add anything new (well, not in the context of computer graphics anyway).

However, I, like you, suspect that the whole thing is hype anyway.

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.)

Excuse me?

[peacefinder]

I'm only a minor-league math geek, likewise only a minor-league history nut. But even though I have only a dim understanding of what they are claiming, the phrase:

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.

looks like bullshit to me.

For starters, the change from bronze to steel was so huge, in terms of technological consequences, that any analogy in mathematics has to be something as novel, powerful, and versatile as "calculus". Furthermore, any bronze sword in the Renaissance would have been centuries out of date; iron and steel had been available in those parts for a long time.

Did they invent anything nearly as important as calculus? Not likely. Did they understand history well enough to properly state their analogy even if their invention is that important? No.

I guess nothing livens up a press release like a heaping helping of hubris. Credibility, meet toilet.

Hmmm

[jefu]

I'm not at all sure what to make of this except that I don't see any interesting applications on his site (Yes, I've looked at the "applications" column.) What do these things actually do and why should I care? I also don't see any math (theorems, conjectures at least) that looks interesting either. (However, I"m still looking at the associated russian site .)

For the computational I'd expect to see the one and for the mathematical I'd expect to see the other.

Worse yet, there's mention of a patent. On the MATLAB toolbox, I'd expect. Certainly not on the mathematics, since that is not allowed in US Patents (I'm carefully hiding any laughter at the thought of something being patented that should not be).

On the plus side the russian side (see above) does seem to have some math and that doesn't look (at first glimpse) to be the kind of junk thats often generated like this.