Epsilon Pi's ideas on imaginary numbers

[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

Sorry if this isn't the right forum, I didn't know so I just went to general.

Could someone explain how this i (imaginary numbers) thing works? I know i is supposed to be a number which is the sqrt of a negative number, which isn't supposed to exist, but what's its use? And yeah...really any information would be appreciated.

 

[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).

 

[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

Sorry if this isn't the right forum, I didn't know so I just went to general.

Could someone explain how this i (imaginary numbers) thing works? I know i is supposed to be a number which is the sqrt of a negative number, which isn't supposed to exist, but what's its use? And yeah...really any information would be appreciated.

 

[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.

 

[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 <math>\pm\sqrt{-1}<math> 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 completely 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

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.

 

[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.

 

[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

"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.

 

[matt grime]

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

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.

 

[quantumdude]

so it is not strange that they are not used in its full power yet, except in EE.

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".

 

[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.

 

[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

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".

 

[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

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.

 

[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.

 

[Epsilon Pi]

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...

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?

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.

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

 

[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.

 

[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.
And I am sorry if I am contradicting your opinion!
Thank you so much for your time!
Regards
EP

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.

 

[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.

 

[quantumdude]

Well Mr Mattson, I really was wondering if you are willing to have a serious discussion and my apologies if I seem so direct.

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.

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.

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.

I know maths as a pure science has many possibilities, but should we not make it simple at the moment of applications?

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".

My claim is the same: can you put the fundamental equations of physics under a unified framework?

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

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.

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).

 

[quantumdude]

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

 

[juju]

Hi all,

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

 

[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!

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).

 

[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

Hi all,

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

 

[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 Schrödinger's wave equation and the rationalization of duality

Thanks in advance for your time, comments or criticism.
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 Schrödinger'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 Schrödinger'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.
Comments: equations included.

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

 

[quantumdude]

You mean that you even read any of my references?

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

How can we discuss something then?

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 "Schrödinger" 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 Schrödinger equation.

As for this question from the Schrödinger paper:

"Is it not true that if we can put them both (edit: Schrödinger 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.

 

[matt grime]

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"./QUOTE]


But no one said that P or Q are imaginary. you're talking rubbish which has thankfully been moved to TD.

 

[Epsilon Pi]

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

What about the Pendulum as an open dynamic system? There you have a table that can be validated with what is observed. Please note its deduction is obtained under the same roof, in which I obtained the SWE, and those equations associated with the LTG and gravitational fields, I can present later specially for the sake of this discussion.

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 "Schrödinger" 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.

1. The the SWE is a complex equation, that is a fact; a complex equation describing the behavior of an entity such as the electron, again it is not a quantitative matter.
2. From your point of view, which is of course of modern physics, there is not a complex equation for describing the fundamental equations of physics, and I say, yes, there is. Up to know I have presented two examples, but will you pay attention if I present the others two, I mean, that of the LTG, and those describing the behavior of gravitational fields?

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 Schrödinger equation.

If I have a better way to represent things why should I look to one that has even been qualified, not precisely by me , as cumbersome?

As for this question from the Schrödinger paper:

"Is it not true that if we can put them both (edit: Schrödinger 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.

Yes, of course, we can put them, and this a complex mathematical assertion. I have already done it, and if this thread is not locked before, I will present it in here. But please note this will not be a TOE, but a complex mathematical methodology to see the whole thing, we are talking about, under a same roof.


Regards
EP

 

[Epsilon Pi]

Was not you the one who said the term "imaginary" was unfortunate?
Certainly if I have another point of view, not dualistic anymore as the prevailing paradigm; anything that cannot be included in it will be called "rubish", but the important thing is that we have high standards in our discussions, and the best way to do it is criticizing something flaw in the mathematical equations.
Regards
EP

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"./QUOTE]


But no one said that P or Q are imaginary. you're talking rubbish which has thankfully been moved to TD.

 

[Epsilon Pi]

Hi all,

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

Hi juju and all,

Well I think you have touched a point that seems to me fundamental: with pure mathematics we can have results that do not map to physical reality.
I asked myself some time ago sort of philosophical question: is Reality simple or complex? and the answer I obtained that best matches with the reality out there is that it is complex; but then complexity should be taken to a minimum and this can be done only with complex numbers, where the symbol i makes it possible to take complexity to a minimum so we can manage it properly, and represents a minimum structure.
Complex numbers are then a starting point for representing a complex reality and above all having in mind the fact that with them it is posible to recover simplicity and elegancy in our mathematical representations of reality.
For example in examining the pendulum I found that not only we have a different conception of time, but most importantly with the Bus concept it is posible to obtain an approximation factor for the pendulum that can be validated with what is observed, moving in this way from a metaphysical theory, as we have a real way to invalidate the whole thing, not by arguing but by confronting facts.

Regards
EP

 

[krab]

Why are you bringing the pendulum into this discussion? The pendulum is already understood, the equations of motion solved to any precision, or the motion written explicitly in terms of Jacobian elliptic functions. EP, you also said

If I have a better way to represent things why should I look to one that has even been qualified, not precisely by me , as cumbersome?

So by your own rules, why should I look to your solution which appears to me to be unnecessary, cumbersome, and inaccurate?

 

[Epsilon Pi]

The pendulum is certainly understood, as it were, classically, but:
"How else are we account for Galileo's discovery that the bob's period is entirely independent of amplitude, a discovery that the normal science stemming from Galileo had to eradicate and that we are quite unable to document today?" T.S.K.
The reason for bringing it is that, for the case of the pendulum, we have found its exact formula with an approximation factor, that in good science, can be confronted with what is observed, and it is this same mathematical methodology the one that is used to find fundamental equations even that of the Lorentz Transformation Group, so all those fundamental equations under a same roof, a main aim that for me worths by itself all this "empresa quijotesca" of mine.

If you say inaccurate you should give, in good science, a reason why.Did you find any flaw in the mathematical reasoning? or you just simply will not read those papers presented, what is, a personal position that I consider valid but in this case there is certainly nothing to argue.
Of course I have looked to all those solutions tried, a reason why I was motivated to present my own, but another more important reason is the need I have had over the years to see the whole thing in a more congruent way, and more akin with other fields of human activity, which is exposed in my paper, Physics, Edgar Morin and Complex Thinking. I really hate to be in sort of cocoon.

Regards
EP

Why are you bringing the pendulum into this discussion? The pendulum is already understood, the equations of motion solved to any precision, or the motion written explicitly in terms of Jacobian elliptic functions. EP, you also said So by your own rules, why should I look to your solution which appears to me to be unnecessary, cumbersome, and inaccurate?