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.
KayJay
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.
...
The 'number space' solves no interested problems that can't be done with Complex numbers. Sorry.
Mr Jack
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).
KayJay
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. :-(
Matt
We could also start talking about tensors, and matrices where the elements themselves are matrices, and, and, and...Whee!!!
anon
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
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.
anon
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.
anon
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.
Matt
>> 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.
KayJay
Glad to. It's a thing I think is deeply cool and was astounded by the simplicity of it once I grokked it.
anon
>> Bet you never thought polar coordinate systems would be useful in taking roots of numbers, eh?
KayJay
Matt -
anon
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.
Janonymous
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
Oh Quantum mechanics is pretty hardcore, I mean it incorporates a lot of fairly pure maths aswell as being useful for physics.
Matt
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.)
anon
(Actually, introductory q-mech uses just two dimensions, one spatial, one momentum, but it's not a very realistic problem doman, is it?)
anon
And...to tie this back to the original discussion of imaginary numbers and wtf are they for anyway?...
anon
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
Yup, it is the basis set for it...;)
anon
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
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.
anon
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
Oh yes pure mathematicians love to sneer at anything useful, hehe. Usually they're a bit tounge in cheek about it though.
Matt
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.
anon
And linguists!
KayJay
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.
anon
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:
Matt
aaaaaand my brain shuts off. :)
anon
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
Ah cool. Yeah that's how I found it too :-)
Matt
So much for the great theory that math is a game for the young.
anon
KayJay and anon are in completely different dimensions in this discussion.
Observer
And as expected. One, so obviously a learner and the other, apparently, quite learned.
KayJay
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
You want geometric algebra =) This is intimately related to clifford algebras.
gnrv
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
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
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