Friday, October 05, 2007

The Complex Plane

And what's on your notepad today?



[A tribute to LdV, see also roots of unity]

23 comments:

Philippe said...

I guess many people feel pushed into this complex plane simple exp(iwt). This drawing begs the question : In which phase are you ?

Bee said...

I'm not in an eigenstate, and my overall phase is non-observable anyway ;-)

Garrett said...

I like it. Hmm, I'm not sure "Fatalism" is best for the negative real axis though. Maybe "Fantasy"?

Bee said...

Isn't Im(Fantasy) \neq 0 ?

Rae Ann said...

You guys are too smart for me!

Bee said...

Hi Rae Ann: I meant to add a minimum of explanation, but I was too tired yesterday. R is the 'Real' axis (where all the numbers are you find on your bank statement), I is the Imaginary axis. You introduce it so you can solve all polynomial equations (the equation x^2 = -1 does not have a solution in R but on I). If you draw a vector in the plane from the origin to the unit circle, it is called a pure 'phase' meaning its length equals one. You can multiply these phases by adding the angles, i.e. it's a rotation. If you know this, you can easily take the n'th root of 1: it's the point on the unit circle with an angle that - when added n times - brings you back to one. I've always been a big fan of complex numbers :-) If you chose an arbitrary point in the complex plane, you can assign to it a Real and an Imaginary part, that are the projections on the two axis. The Imaginary part is usually denoted with Im( something ), the real part with Re( something ) What I meant above with Im(Fantasy) neq 0 is that, at least as far as I am concerned, fantasies imply a non-vanishing amount of imagination ;-)

Best,

B.

Uncle Al said...

There you go giving Bush the Lesser ideas for his many sub rosa European torture prisons. As they say in the US Department of Justice,

"Albanians gotta eat too!"

Aaron said...

Hahahaha! I'm totally going to print this out on a ferromagnetic backing and use it like a "how are you feeling today?" poster. :)

Plato said...

Mersenne Prime

It looks as though primes tend to concentrate in certain curves that swoop away to the northwest and southwest, like the curve marked by the blue arrow. (The numbers on that curve are of the form x(x+1) + 41, the famous prime-generating formula discovered by Euler in 1774.)

A mandalas of sorts?

One we like to design to help us draw "the relationships" in our life?

Could it be so simple that underlying the everyday life that a geometrical pattern would have also come with it?

Just use Bee's:)

The universality is in the understanding that "all life precedes in such a fashion" when you reduce the chaotic events of the day to such a "geometrical substance?"

That is wo/man's nature?

Cynthia said...

Uncle Al, I guess imaginary numbers are a bit shrouded in secrecy... And if I'm reading you right, I completely agree that this couple standing back-to-back is trapped in a complex wheel, appearing as though they're in a medieval torture chamber of sorts!;~)

CapitalistImperialistPig said...

Bee,

It worries me that both your optimism and pessimism see equally unrealistic ;)

CapitalistImperialistPig said...

see -> seem, I meant.

Arun said...

CIP,

Optimism and pessimism are states of mind, not of reality. I think Bee is quite correct.

Arun said...

Which explains "It is clear, then, that the term 'reality' (which in this context means 'reality as a whole') is not properly to be regarded as part of the content of thought. Or, to put this in another way, we may say that reality is no thing and that it is also not the totality of all things, i.e. we are not to identify 'reality' with 'everything'."

Optimism, pessimism are part of the content of thought, but not of reality. We cannot identify reality with everything (e.g., optimism). I don't get "reality is no thing" unless it means "no single thing".

Bee said...

Hi CIP, Hi Arun,

Yes, the positive real axis doesn't say 'reality' but 'realism', I was thinking about it much like Cynthia says, being trapped in a wheel pulled to all sides. The positive real axis is probably the way to go?

I don't get "reality is no thing" unless it means "no single thing".

It's probably impossible to get it without actually reading the full chapter, since it's all about things and no things and thoughts and non thoughts. I mean, I read the chapter and didn't get it ;-) Bohm roughly argues (my interpretation, my words) that during the evolution of man it became necessary to distinguish between ideas or thoughts that were nothing than such, and the rest which he calls non-thoughts. As far as I can see he is more or less dancing around defining 'reality', building up on the fact that most of us share some kind of ability to distinguish reality from ideas. Then he argues that non-thoughts can be thoughts because our brains are real (this is one of the points that annoy me about books like this, I mean, it's the obvious point, why do I have to read like 20 pages to get there just so he explains that the previous 20 pages don't make sense?). And if it is of any help the outcomes of all the thought and non-thought is

"We have indeed already suggested in this connection that the term 'reality' indicates an unknown and undefinable totality of flux that is the ground of all things and of the process of thought itself, as well as of the movement of intelligent perception. But this does not basically alter the question, for if reality is thus unknown and unknowable, how can we be certain that it is there at all? The answer is, of course, that we can't be certain."

(As far as my flux from thought to non-thought is concerned I just noticed I can type blindly even though I couldn't tell you where the letters are on the keyboard.)

What he means to say with the other quotation I don't know. I would have said thought is part of reality, thus if you want to separate thoughts from things, then reality is no thing. Otoh, to me the division into both doesn't make sense to begin with. Also, he seems to argue mostly with 'things' and thinking about 'things', whereas I have the impression we are more and more thinking about thoughts, so maybe it's virtual reality missing here (but then, the book is from 1980).

Best,

B.

Neil' said...

OK, the psych graph is interesting, but the roots of unity involve a very real problem IMHO:
There are n nth roots of unity, right? That also means, four roots for 1^(3/4) etc. (ie, the four fourth roots of unity each taken to the third power, so (1, -1, i, -i) each to the third, and so for the roots of 1^(7/23) etc. So, what are the roots of the perfectly meaningful exponent 1^(sqrt 0.5) etc? If there are as many roots as the denominator of the exponent, well here we have an irrational number that no fraction can represent. So, is the solution set of the latter, the entire circle of points of modulus one, in the complex plane? Just asking.

Bee said...

"perfectly meaningful exponent 1^(sqrt 0.5)"

Well, for it to be perfectly meaningful you'll have to define it, and it's not clear to me what you actually mean, i.e. what is your definition for sqrt in the complex plane. The general definition for z^w goes via the logarithm as exp(w*ln(z)). What you are asking for is a solution to the equation z^something = exp ( something* ln(z) ) = 1. For the complex logarithm there are in general infinitely may possible values (on different branches, so they don't actually lie all in the 'same' plane), though for the special case where w = 1/integer there are finitely many roots, and one can use the nice graphical solution of the problem.

Best,

B.

paul valletta said...

Hi Bee, first thing that comes to mind here, your missing Stefan!

This definately rates the best notepad "doodling", I have come across, littered with sexual Innuendo's ?

Stefan is on your notepad as well as your mind? ;)

best wishes to you both, paul.

Neil' said...

Bee, the number in the exponent was meant to be the normal definition, so that would be around 0.7071... The ambiguity was supposed to be in the full solution set for expression of 1^0.7071... not in the number giving the exponent itself. The trouble is, 1^(707/1000) should have 1000 roots, 1^(7071/10,000) should have 10,000 roots (per my straightforward argument earlier) and so on, to "infinitely many" for an infinite irrational decimal expression. That is the question/problem. (I suppose, the solution would be a continuous circle, or would it be a categorical "sieve" of some kind?)

Bee said...

Neil, I understand your question but you didn't understand my answer. The 'normal' definition for sqrt doesn't help you because you need a sensible definition in the complex plane, I have given you the appropriate one above. I have also explained that these solutions lie on different branches of the ln and not actually in the same plane. You can find all of this in any textbook on functional analysis. I have no idea why you think this is a problem, except that you can't construct these cases graphically as you can for an integer. Best, B.

Anonymous said...

Neil is asking what points on the circle represent unity to the power of an irrational number.

Anonymous said...

The answer to Neil's question is a countable infinity of points that is dense on the unit circle.

Bee said...

okay, thanks, if that was the problem then I misunderstood the question. But he seems to know this answer, so I don't get where the problem is?