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Number line -> Complex plane -> ? Space

I was reading up de Brange's 'apology' (1) and I got thinking. Just as the square root of -1 extended the number line into a 'plane' and brought about so much, extending the plane into a '3D space' should help solve many of today's mathematical problems and usher in some new stuff as well.

Anyone know if someone has broached it, or any research on the subject. Google fails me (<rant>and, of late, has been failing me!</rant>)

(1) http://zdnet.com.com/2100-1104_2-5229702.html?tag=zdfd.newsfeed

KayJay
Friday, June 11, 2004

A friend and I actually toyed with a similar idea quite a few years ago, attempting to wrap such concepts up in a neat form.  While it showed some promise, it's actually relatively difficult to handle and doesn't provide any additional information - in fact it obfuscates some information.

So nice line of reasoning, but it probably won't float.

...
Friday, June 11, 2004

The 'number space' solves no interested problems that can't be done with Complex numbers. Sorry.

Quaternions (which are the hyperspace equivalent of what you propose) have enjoyed some popularity as a representation of rotations among games programmers.

Mr Jack
Friday, June 11, 2004

Just checked Quaternions out at Mathworld.com (1). Skimming through it, any geometric representation remains within the plane (to my not so mathematical brain). Am I right? If so, any complex function can be translated into a quaternion one and vice versa with ease (or unease as the case may be).

(1) http://mathworld.wolfram.com/Quaternion.html

KayJay
Friday, June 11, 2004

Woo! I remember thinking like this when I first found out about complex numbers. I thought maybe you could take the square root of 'i' and have this as a third dimension, stuff like that (of course you can't, it's just another complex number). I think I really just wanted to make a 3-dimensional mandlebrot set or something. No Joy. :-(

When you study the algebraic structure of fields more you kinda see why extending the complex numbers to 3 dimensions wouldn't help / be possible in the sense you imagine. I'll attempt to explain why:

The main advantage complex numbers give over real numbers is that they let you solve more polynomial equations.

Eg. x^2 + 1 = 0 has no solution in the reals, but obviously has solutions in the complex numbers. It turns out that /every/ polynomial over the complex numbers, has as many solutions in the complex numbers as it's possible for it to have - the complex numbers are 'algebraicly closed'. You can't 'extend' them any further in the same way, as there are no polynomials left without solutions.

You might then start thinking about looking for solutions of infinite power series to extend the complex numbers with. These are the transcendental numbers, and it turns out they're already in the complex numbers too, whenever it's meaningful to talk about an infinite power series having a solution. This is infact how numbers like Pi are commonly defined, and follows from the completeness axiom for the reals.

When you look at it from an algebraic point of view you'd tend to start out with just the rationals, then throw in all the solutions of polynomials (giving you the 'algebraic closure' of the rationals) and only then throw in the transcendental numbers, as algebraists tend to be snobs about using any calculus/limit constructions unless they absolutely have to.

Either way, the complex numbers are demonstrably the 'biggest' field you can get, in a lot of senses.

There are interesting algebraic structures with dimension 4, 8 (maybe even 16?) over the reals. Quarternions are 4-dimensional, think the 8-dimensional ones are called Octernions or something like that. The problem is that these structures aren't 'fields', which means that the usual laws of algebra we're used to don't necessarily hold in them. Quarternions don't have commutative multiplication, so are a Ring but not a Field. I think the Oction-thingies don't even have associative multiplication, and you loose more and more algebraic structure the higher you go.

Of course you can have a vector space over the reals of any dimension you like - eg the space we live in can be seen as a three dimensional space in this way. The problem is that this doesn't give you much algebraic structure on the points in the space. You can add them (adding vectors) and multiply them by scalars, but you can't multiply them commutatively like you can the complex numbers. There is something called a vector product on a three-dimensional space, this isn't commutative however. It's used in physics and mechanics quite a bit.

Matt
Friday, June 11, 2004

We could also start talking about tensors, and matrices where the elements themselves are matrices, and, and, and...Whee!!!

anon
Friday, June 11, 2004

I _think_ I am begining to understand why such a space makes no additional contributions. I shall read up more and tear up even more paper before I shoot my mouth off again! Thanks and regards.

KayJay
Friday, June 11, 2004

Ok, here's what you do...take the square root of i.  You'll get (sqrt 2) + (sqrt 2)i which on a unit circle is halfway between i and 1.  i is at 90 degrees or pi/2 radians (don't wanna se that fight again), and since i is 1 unit from the origin, we stay one unit away.  Taking the square root cuts the angle in half.  Now try it with -i.  -i is at 270 degrees, or 3/2 pi radians.  The square root is thus at 135 degrees (3/4 pi radians) and again one unit away, or -(sqrt 2) + (sqrt 2)i on the complex plane.

Incidentally, those are the primary solutions that emerge.  All square roots have two solutions 180 degrees (pi radians) from the primary - to get the secondary solution just multiply the above results by -1.

It's even more fun when you talk about third roots, fourth, and beyond.  Just divide the angle by 3, 4, or whatever.  Additional solutions are 120 degrees (2/3 pi radians), 90 degrees (1/2 pi radians), and so on.

anon
Friday, June 11, 2004

I left out what happens if the maginitude of the number you are taking a root of is anything other than one.  That's easy:  Just take the nth root of the magnitude and that's the new magnitude.

Bet you never thought polar coordinate systems would be useful in taking roots of numbers, eh?

anon
Friday, June 11, 2004

No worries, it's a good question to ask! thinking about how to generalise and extend things, as opposed to just taking them for granted and moving on, is ones of the hallmarks of a good mathematician I'd say. Although can lead you up some blind alleys of course.

Yep, matrices might interest you too - for example the 2-by-2 matrices form a 4-dimensional space over addition, and also a ring, since you can multiply them.

Again the multiplication isn't commutative, but still. It turns out the complex numbers can be represented as a particular subset of the 2-by-2 real matrices, whose multiplication is commutative. Quarternions can also be represented in a similar way

Matt
Friday, June 11, 2004

>> The square root is thus at 135 degrees (3/4 pi radians) and again one unit away, or -(sqrt 2) + (sqrt 2)i on the complex plane.

Ah! Ok. Learn one thing a day and learn it well. Thanks.

KayJay
Friday, June 11, 2004

Glad to.  It's a thing I think is deeply cool and was astounded by the simplicity of it once I grokked it.

Oh yeah, whoever it was that got annoyed by the word "grok" (as being a "made up" word) can just deal.  All words  (those that are not onomonopias, at least) are made up.  By people.

Sorry, went off on a tangent there.  You're quite welcome.

anon
Friday, June 11, 2004

>> Bet you never thought polar coordinate systems would be useful in taking roots of numbers, eh?

In fact, I always thought so. "Geometry <-> Alegbra" has always been my crutch as well as my anchor!

KayJay
Friday, June 11, 2004

Matt -

Spaces and rings were just at the edge of what I dealt with in number theory...I just barely grasp them.  Sigh...sometimes I long for the college days, when I had time and freedom to pursue coeds - I mean, abstract math concepts.  :)

Actually, I took pretty much enough math for quantum mechanics (which I guess is pretty far, but not to true math theory), so I'm doin' alright.

anon
Friday, June 11, 2004

It may be worth looking into why the complex space was "invented", as Matt did. People said: _what if_ we could just take the square root of -1, and call it "i". Now if we take the square of i, we have 1 again, so in that sense, i does exist. And 8i, and 8i+4.
But how would you create yet another dimension, say "j", Kay Jay? There must be a way to get into that 3rd number dimension, and back to the i-plane again. There is no math function that I know of that has this property. But of course I'm a CS guy, and not a mathematician (always sounded like magician to me :-) ... Maybe someone will come up with something sometime.

Janonymous
Friday, June 11, 2004

IIRC, it was the other way around. People saw square root of -1 popping up in solutions and wondered what such a creature  actually looked like. Will just have to wait and see what unforeseeable creatures crop up in solutions nowadays.

KayJay
Friday, June 11, 2004

Oh Quantum mechanics is pretty hardcore, I mean it incorporates a lot of fairly pure maths aswell as being useful for physics.

You must've looked at Hilbert spaces to do any quantum mechanics right? those being infinite-dimensional complex vector spaces (woah! haha) whereas we're only talking 3 real dimensions here. If you looked at linear operators on a Hilbert space these form a ring, also.

The terminology of rings fields etc sounds kinda abstract and scary but really understanding what a ring or a field or a group or a vector space 'is' is pretty easy... they're just generalisations of some of the properties of more concrete algebraic structures we're used to, eg the integers (a ring), the rational numbers (a field), the permutations of a set (a group), etc)

Proving things about their structure can be a bit of an abstract head-fuck though. Can get a bit too abstract for me even ;)

Matt
Friday, June 11, 2004

Well, you can do quantum mechanics pretty fine in "just" six dimensions - three spatial and three momentum - but my advisor was working on minimal energies of systems in infinite dimensional space, which led directly to what became my thesis work.  (Global optimization of functions, if you are curious...or even if you're not.)

I didn't do much study on the properties of infinite dimensional spaces themselves, I just used them...minimal comprehension required.

anon
Friday, June 11, 2004

(Actually, introductory q-mech uses just two dimensions, one spatial, one momentum, but it's not a very realistic problem doman, is it?)

anon
Friday, June 11, 2004

And...to tie this back to the original discussion of imaginary numbers and wtf are they for anyway?...

It turns out that if you have imaginary energies in a q-mech system, they actually represent a sort of average amount of time it takes for a particle to tunnel through something, if it can at all.  Yes, imaginary energies and indeed even imaginary time do have relevance to real world systems.

anon
Friday, June 11, 2004

Oh okay... well I've not done any quantum mechanics myself but I've done the more pure Hilbert space theory and was told it's pretty much used as the basis for quantum mechanics...

Matt
Friday, June 11, 2004

Yup, it is the basis set for it...;)

anon
Friday, June 11, 2004

Infact I think the lecturer referred to quantum mechanics as 'a good introduction to hilbert space', heh, kind of a put-down to the physicists who would probably see it as being the other way round...

Matt
Friday, June 11, 2004

I've noticed that, too.  Theoreticians tend to look down on experimentalists, and the more abstract the subject matter, the more they look down on the less abstract.  Guilty of it myself, frequently.

A saying I like to use:

"In theory, theory equals reality.
In reality, theory does not equal reality."

anon
Friday, June 11, 2004

On the flip side, "The most practical thing in life is a good theory". Nothing of any note has ever been _done twice_ without a good theory.

KayJay
Friday, June 11, 2004

Oh yes pure mathematicians love to sneer at anything useful, hehe. Usually they're a bit tounge in cheek about it though.

By that measure though, set theorists would probably get to lord it over everybody. Or perhaps philosophers...

Matt
Friday, June 11, 2004

Hmm...I think anyone who can use "discrete manifold" in a sentence and know what the hell they are talking about is one up on me...or more than one up, at least.

"Engine manifold"...now that's a different beast, and one I'm just about as familiar with.

anon
Friday, June 11, 2004

And linguists!

KayJay
Friday, June 11, 2004

Thinking about it...set theorists might be the right one.  The only math class I ever had to drop was basically along the lines of "Start with set theory and prove calculus to be true."  Ugly ugly ugly.

Then I had to drop a second semester solid state physics class in grad school, mostly because I hadn't taken the first semester.  I swear my head was going to explode from that.

anon
Friday, June 11, 2004

To the OP, if you're really interested this paper seems to be the definitive work on the subject of systems of complex-like numbers (it calls them 'hypercomplex') in higher dimensions:

http://arxiv.org/PS_cache/math/pdf/0011/0011044.pdf

It does outline a three-dimensional hypercomplex number system, which appears to be a commutative ring, but not an integral domain (two non-zero numbers can multiply to give zero).

Matt
Friday, June 11, 2004

aaaaaand my brain shuts off.  :)

Maybe I'll get to this one later...like after I eat, nap, and such.  Or maybe after I finish healing from surgery...or after I send my son to college...or...

lol

anon
Friday, June 11, 2004

Matt, I actually am reading that document now. Got there from Wolfram and then onto a page for Clifford Algebra and then onto....You get the picture. I am currently collecting documents and bookmarks. Will study them once I pay off this month's bills (most likely next month that is!).

KayJay
Friday, June 11, 2004

Ah cool. Yeah that's how I found it too :-)

Seems to state a lot of results without proving anything in detail, but interesting as an outline.

Like I said though the 3-dimensional space there looks like it's not an integral domain, so certainly not a field.

It's actually not possible to have a field that's a degree 3 extension of the reals, as this would imply there was an irreducible polynomial over R of degree 3, which there isn't - all cubics have a real root. Any course with material on algebraic field extensions should give you the tools to prove this/see why it's true... so that's something else to google on if you're interested.

Matt
Friday, June 11, 2004

So much for the great theory that math is a game for the young.

anon
Friday, June 11, 2004

KayJay and anon are in completely different dimensions in this discussion.

Observer
Friday, June 11, 2004

And as expected. One, so obviously a learner and the other, apparently, quite learned.

KayJay
Saturday, June 12, 2004

I was posting as anon most of the time in this thread (tho not that last time).  I tend to be anonymous when posting from work...not sure why I bother, because we work in network security and it'd be easy for anyone working with me to see that I was posting.  Maybe I should just post as me at work, too.

Aaron F Stanton
Saturday, June 12, 2004

You want geometric algebra =) This is intimately related to clifford algebras.

A famous name to google for is David Hestenes,
http://modelingnts.la.asu.edu/GC_R&D.html. He is mostly into physics it seems.

For computer graphics, check out a neat introduction at http://www.science.uva.nl/~leo/clifford/CGA3.pdf. In computer graphics, you are recommended to use a different dot product than Hestenes uses, for more details, see their page at http://carol.wins.uva.nl/~leo/clifford/.

As the proponents of GA remark, if schools used it to teach geometry from an early point, yours and everybody else's geometry skills would be much more advanced when you graduate, since it is a much more expressive mathematical notation.

gnrv
Sunday, June 13, 2004

Information regarding N-dimensional (N is user defined) commutative-associative hypercomplex numbers and corresponding algorithms may be found on the web pages at www.hypercomplex.us.

Thomas Jewitt
Wednesday, June 23, 2004

I should also mention that you can download a white paper providing an excellent overview of this topic, and an evaluation copy of a toolkit providing hypercomplex numerical computing capability for the Matlab numerical computing environment. Because Gauss-Jordan elimination applies, the toolbox provides the unique capability to solve many types of inverse problems for multidimensional, multivariate signals in user-defined dimensions.

Thomas Jewitt
Thursday, June 24, 2004

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