# Epsilon Pi's ideas on imaginary numbers

1. Sep 27, 2004

### Epsilon Pi

It is known that it was Descartes the one that gave the symbol i the connotation of imaginary; in electrical engineering there is the concept of apparent power(MVA)
S = P + i Q
where
P(MW)=generation or consumed power
and
Q(MVAR) = reactive power
and they both can be measured, so they exist, they are not "imaginary".

In fact the symbol i, can be taken as a symbol to differentiate two different realities that in general cannot be reduced the one to the other, as P is quite different from Q, even in its physical behavior, and a reason of those rules explained in this thread too for its mathematical operations, as we must not confuse pearls and apples, when working with it.

Regards
EP

2. Sep 27, 2004

### Gonzolo

In the physics of waves (electromagnetics, sounds, strings, quantum mechanics etc.), cos(x) and sin(x) are ever present. Using Euler's equation allows us to work with $$e^{ix}$$ instead of cos(x). The special properties of $$e^x$$ simplify problems and theories so much that frankly I don't even want to know or attempt to know what they would look like using only cos(x).

So if you have a cos(x), you can add isin(x) to it to make $$e^{ix}$$, do a whole bunch of math with this $$e^{ix}$$, and then retrieve a physical, meaningful answer by dropping the isin(x) again, leaving only the rational number cos(x).

3. Sep 27, 2004

### Epsilon Pi

It is also known that it was Leonhard Euler with its well-known Euler relation
i(theta)
e = cosine(theta)+ i sine(theta)
found in 1745, when studying infinite series the one who finally gave status to complex numbers. As a matter of fact it took 100 years for the complex plane to be accepted, and even such a powerful and brilliant mind as that of Gauss was reluctant to accept it, so it is not strange that they are not used in its full power yet, except in EE.
Regards
EP

4. Sep 27, 2004

### mathwonk

I am curious as to the evidence of Gauss' reluctance to accept complex numbers, especially since he is credited with the first proof, at the age of 22, of the fundamental theorem of algebra, using them. Indeed he is sometimes credited with having helped place complex numbers on firm ground.

5. Sep 28, 2004

### Epsilon Pi

I am sorry, I did not say complex numbers, but the complex plane, and I talked about reluctancy about accepting it, in fact:

"Wessel's memoir appeared in the Proceedings of the Copenhagen Academy for 1799, and is exceedingly clear and complete, even in comparison with modern works. He also considers the sphere, and gives a quaternion theory from which he develops a complete spherical trigonometry. In 1804 the Abbé Buée independently came upon the same idea which Wallis had suggested, that [itex]\pm\sqrt{-1}[itex] should represent a unit line, and its negative, perpendicular to the real axis. Buée's paper was not published until 1806, in which year Jean-Robert Argand also issued a pamphlet on the same subject. It is to Argand's essay that the scientific foundation for the graphic representation of complex numbers is now generally referred. Nevertheless, in 1831 Gauss found the theory quite unknown, and in 1832 published his chief memoir on the subject, thus bringing it prominently before the mathematical world. "

And my point was that it was completly accepted(the complex plane, not complex numbers) at the end of XIX century, with the works of Oliver Heaviside, Maxwell and Steinmentz when they found applications for it, putting in this way the mathematical foundation of EE on firm ground.

Regards
EP

6. Sep 28, 2004

### matt grime

"In fact the symbol i, can be taken as a symbol to differentiate two different realities that in general cannot be reduced the one to the other, as P is quite different from Q, even in its physical behavior, and a reason of those rules explained in this thread too for its mathematical operations, as we must not confuse pearls and apples, when working with it." - epsilon pi

but this is maths, if we want to do philosophy of physics about what is an apparent reality then we'd be doing that instead. the sooner mathematics removes this kind of attempt to explain itself the better.

7. Sep 28, 2004

### Epsilon Pi

sorry, I thought maths were just a tool or a language we use to represent reality "out there" to make easier and more realistic its applications.

Regards
EP

8. Sep 29, 2004

### matt grime

well, it isn't. grothendieck duality don't do a whole lot to decribe what's out 'there', though someone may have other ideas as to what it describes. maths is not a tool, it is a beautiful abstract subject. it may be used for practical things, but that is not how one should think about it anymore since it will rapidly only serve to confuse people. look at the standard confusions that arise from learning about imaginary numbers. even the name is a poor choice, though we can't do anything about that.

9. Sep 29, 2004

### Tom Mattson

Staff Emeritus
That's simply false. In fact, I have never met an electrical engineer who knew as much about complex analysis as a mathematician or physicist. Engineers typically don't use contour integration or conformal mapping to solve problems in electrodynamics, but physicists learn it as undergraduates. Use of the Euler relation does not even come close to using complex variables "in its full power".

10. Sep 29, 2004

### matt grime

there is one electrical engineer who might know more than a mathematician, but he is the exception: Raul Bott. Apart from that, I've never heard of an EE'er even knowing what the germs of continuations of analytic functions are.

11. Sep 29, 2004

### Epsilon Pi

Well Mr Mattson, I really was wondering if you are willing to have a serious discussion and my apologies if I seem so direct. As you most probably know because of our preview discussions, my point is not quantity, who knows more or less; if you have read what I promised you and you claimed then you will know what is my point.
I know maths as a pure science has many possibilities, but should we not make it simple at the moment of applications?
My claim is the same: can you put the fundamental equations of physics under a unified framework? As far as I know you can't even with the best you have. And this can be done for sure with Euler relation and the basic unit system derived from it, this is my real point.

Best regards
EP

12. Sep 29, 2004

### Epsilon Pi

confusion is not a result, precisely because, there is not anymore a unified framework when coping fundamental matters? why are you really so sure we can't do anything?
Regards
EP

13. Sep 29, 2004

### matt grime

who said we can't do anything. we can do a lot. i will strenuously maintain that the best way to do mathematics is to do so in the abstract where an object *is* its properties: the reals are an ordered complete field, the complex numbers are are degree two fied extension of R, the kernel of a surjective homomoprhism is the obeject such that every map which factors as zero factors through the kernel.... and so on.

thinking that i *is* the imaginary electromotive force or whatever it was you said it is gets you nowhere since it doesn't even imply that the complex numbers are a division algebra.

14. Sep 29, 2004

### Epsilon Pi

In the same way we can make pure philosophy, of course, you have all the right to make pure mathematics, but for sure even from the point of view of pure maths, there are another ways of presenting the matter, as it is not at all a closed field, is it?

sorry I did not get what you meant; i is just a symbol to differentiate two different entities that must not be reduced one to the other, a reason of why we have those different rules for mathematical operations as far as I know, and was taught.
Regards
EP

15. Sep 29, 2004

### matt grime

And where in you alternate view point did you define what i is, and its properties? You didn't. You just said it was a symbol to differentiate between different realities that cannot be reduced to one another. That is flowery prose, but acutally almost an entirely vacuous sentence, and meaningless in terms of mathematics. And as I keep saying, this is my opinion on the best way to teach and learn mathematics. I am a mathematician and a teacher of mathematics.

16. Sep 29, 2004

### Epsilon Pi

Are not the properties defined by the way mathematical
operations are done with it? Is it not a property of complex numbers to remain with the same form with differentiation and integration except by a rotation of +/- 90 degrees? Is not this another result of the properties of them?
The sentence that you call flowery prose, has specific applications when defining the complex power in EE, that as a matter of fact can be measured.
Thank you so much for your time!
Regards
EP

17. Sep 29, 2004

### matt grime

"Is it not a property of complex numbers to remain with the same form with differentiation and integration except by a rotation of +/- 90 degrees? Is not this another result of the properties of them?"

erm, that is meaningless as far as i can tell.

"has specific applications when defining the complex power in EE, that as a matter of fact can be measured."

but does not actually define the complex numbers. it describes a physical phenomenon that can be accruately represented with complex numbers. that isn't the same thing at all.

18. Sep 29, 2004

### Tom Mattson

Staff Emeritus
I was wondering the same thing about you. You made several false statements in our previous discussions, and exhibited a number of severe misconceptions. I made the necessary corrections and referred you to a source or two, but you wouldn't listen.

This remark serves no other purpose than to dodge what I am saying. You claim that only electrical engineers use complex numbers to their full power. You are wrong. Physicists use the Euler relation in their equations (indeed, the differential equations of EE are equations of physics), and in addition to that we use the other techniques I mentioned.

It should be made as simple as possible, but no simpler.

By the way, this is the second time you have introduced an undefined idea into this thread. First, when you say that EE's use the complex number system "in its full power", you give no indication of what that means. I assumed it meant that EE's use the theory of complex analysis in its entirety, and I pointed out that that is false. You now say that it isn't about quantity, so one can only speculate as to what you mean.

Now, you introduce your undefined notion of "simple". Most people--and I suspect you are included in this--mean that "simple" should mean "elementary" or "easy to understand". But most scientists and mathematicians use the word "simpler" to mean "relies on fewer axioms".

And here is undefined concept #3. What does it mean to be "unified"?

You have made it perfectly clear in our previous discussion that you don't understand very much about the equations of physics. During our entire interaction, you steadfastly held onto the false idea that QM and SR offer competing ideas on space and time, and that those of QM should be preferred over those of SR (or at least that the picture of spacetime in SR should be modified by that of QM).

Last edited: Sep 29, 2004
19. Sep 29, 2004

### Tom Mattson

Staff Emeritus
I split this off from the thread i, as the discussion is off topic, to say the least.

20. Sep 29, 2004

### juju

Hi all,

Mathematics can exist in an abstract space all its own. It need only refer to other forms within this space for consistancy and completeness.

Physical reality is another story. Only those mathematical tools that can explain the perceved reality (qualitatively and quantitatively) are valueble is this area.

Complex numbers, for instance, when used to explain reality are valueble since the results map onto the perceived reality.

The operations in the complex plane give real valued answers to real physical problems. The might also give answers that have no physical reality. These latter answers do not map. This is the confusion. Mathematics can give two answers to the same question, where one maps and one doesn't.

juju

21. Sep 29, 2004

### Epsilon Pi

You mean that you even read any of my references?
How can we discuss something then?
Regards
EP
PS: Thank you for having moved the thread, I would have done the same!

22. Sep 29, 2004

### Epsilon Pi

Thank you, juju, as I said we have all the right to make pure maths, but when dealing with applications and physical reality there must be correspondence between the two, or else how can we make predictions and measurements?
Regards
EP

23. Sep 29, 2004

### Epsilon Pi

To Mr Mattson

Here is a copy of the posts I thought you have read,

Mr Mattson and everyone interested,

Here is a paper I have prepared specially for this forum and a consequence of the dialogues I have had in it:

Is the Pendulum an Open Dynamic System?
Abstract.
In this paper the pendulum and its approximation factor, that can be validated
with what is observed in the reality "out there", is presented by using the
complex basic unit system concept based on Euler relation. This paper is a
result and a promise made in Physics Forum, in its sub forum Theory
Development, where I have been participating under the pseudonym Epsilon Pi.
Here I want to show that it is possible to cope the fundamental equations of
physics from a point of view or framework that includes the third, which means
mathematically speaking, by using complex numbers.
Comments: 7 pages, 1 figure, 1 table and equations.
The url is:
http://www.geocities.com/paterninae...rs/Pendulum.pdf
The next paper will be:
The Schrodinger's wave equation and the rationalization of duality

Best regards
EP

Second post:

--------------------------------------------------------------------------------

Mr Mattson and everyone interested,

As you made so serious claims in this post, I am presenting here the second paper of four:

The Schrodinger's wave equation and the rationalization of duality.

http://www.geocities.com/paterninaedgar/QM.pdf

In this paper the SWE is presented not as a postulate but under the concept of the basic unit system.

Abstract. In this second paper of four the Schrodinger's wave equation is presented under the concept of the basic unit system. Again it is too, a result and a promise, because of those dialogues in Physics Forum in its TD sub forum. By using complex numbers we find that the duality of time and space cannot be dropped out just by taking the square of a complex equation as in this way we drop out not just one part of that complex equation but, we do not rationalize duality of time and space, of wave-particle, anymore.

There you will find why I do not consider the Klein and Gordon's equation a consistent solution to the problem of duality of time and space, wave-particle.

In my next paper I will present a non relativistic point of view of the Lorentz Transformation Group by using the same basic unit system concept.

My best regards
EP

24. Sep 29, 2004

### Tom Mattson

Staff Emeritus
I have read "Complex Thinking" and "The Schrodinger's wave equation and the rationalization of duality". But reading them is not necessary to evaluate your claims as false.

You should look in the mirror. The fact of the matter is that we cannot discuss anything until you look at the references that I cited. Your entire case in the "Schrodinger" paper is based on two misconceptions:

1. That a complex wave equation is somehow more correct than one that is not complex and...
2. That there is no complex wave equation that is compatible with SR.

Both are wrong. And I referred you at least twice to the Dirac equation, which explicitly shows that #2 is wrong. The Dirac equation is a complex wave equation that is covariant under Lorentz transformations.

As for #1, one has only to look at experimental results to see that it is wrong. The Klein-Gordon theory agrees with reality (that's "observed" reality, not "Epsilon Pi's preferred reality") than does the Schrodinger equation.

As for this question from the Schrodinger paper:

"Is it not true that if we can put them both (edit: Schrodinger and SR) under a kind of same conceptual mathematical roof, we have solved those incapabilities mentioned above?"

The answer is: Yes, we could, if we could put them under the same roof. But we can't do that, which is why relativistic quantum mechanics is needed.

25. Sep 30, 2004