Here's a very simple example of the real world. The motor can rotate left or right. When turning to the right is a positive frequency, then rotate to the left is a negative frequency. What do you think about this?

I don`t think that this is a good example.

Remember the definition of the term "frequency". It is simply the number of periodic events per second (without any "direction"). This is equivalent to the reciprocal of the time for one period.

More than that, in your example you should use the term "revolution per second" rather than the term "frequency".

The author evidently confuses the real world with the mathematical model used as a tool for its analysis. Perhaps a simpler question would be: "do negative numbers really exist?" I don't think that they really do.

Moreover, even in this mathematical model there are actually no negative "frequencies". What is negative there is the harmonic number, not the frequency. (For non-periodic signals (inverse Fourier) - the integral is just a limit of a sum, with infinite number of harmonic numbers.)

Dr. Alipov - yes, I am of the same opinion. The artificial introduction of "negative" frequencies just simplifies all the calculations involved with FOURIER analyses. That`s all.

However, another question arises (with regard to your sentence about negative numbers): Do positive numbers (not only integer!) really "exist"?

How such an "existence" could be prooved? Or is it just a philosophical question?

I think: Either both of them (pos and neg. numbers) "exist" or they do not really "exist"

(whatever this word may mean).

Spacibo for your answer.

Regards

Lutz vW

Thank you Lutz,

I rather meant not the existence of "numbers" per se, but the presence of some material world origin (object or relation) reflected by an idea of a "number". I think these are two different questions, both philosophical for sure.

If we consider the existence of any numbers (real, complex etc.) per se, then, of course they all "exist", but they exist only as ideal objects, so far as the conscience exists that reasons them. If there is no human (or perhaps some other form of conscience, I don't know) who thinks numbers, there will be no numbers. And, of course, numbers per se do not exist as material objects - it is only some material phenomena reflected by our notion of a "number" that may (or may not) exist.

If we then consider these phenomena, then, for the natural numbers, there is a clear origin. The appearance of the very notion of a "number" in the primeval times was associated with this - with the count of objects. Different natural numbers just reflect the varying multitude of objects. For the number "zero", I think, it just reflects the phenomenon of the "absence" of an object.

When we come to positive rational numbers, the introduction of which was associated with the need for measurements, they reflect not objects, but rather relations - namely, ratios.

When we come to irrational numbers, here, I think, we have no more direct imprint by the material world phenomena on the human conscience, no direct analogy. An irrational number appeared purely as an abstraction, as a tool. It does reflect the material world, but not by itself as an idea, rather by serving as a tool of description and analysis. The same stands for imaginary/complex numbers, and (coming to the beginning of the discussion), I think - for any negative number, since there is no direct origin of an idea of a "negative number" in the material world, its introduction was consequence of introduction of arithmetic subtraction.

Regards,

Anton

Anton - thank you for clarification.

But it`s better to come back to the initial question regarding negative frequencies.

Perhaps some other RG members have another opinion?

Regards

Lutz vW

Dear Lutz,

To directly answer your question; no, there is no such physical thing as negative frequency. Frequency is the rate at which something happens, so by defintion it must be a positive real number.

Likewise a quantity of physical objects can also only be a positive real number, e.g. I have 4 and a half apples. I can't have minus four apples.

Obviously, both negative and imaginary numbers are very useful in the process of calculating the value of real quantities, but they have no physical reality of their own.

Regards,

Nicholas Lee

Nicholas - thank you. Full agreement from my side.

Negative frequencies arise in the Complex Math world. A complex number can be represented in a 2D-Cartesian plane.

Basically the first example was indeed a good one. Positive frequencies occur when the complex vector represented in a bidimensional real-world rotates counter-clockwise. Negative frequencies when it rotates clockwise.

If we think real world with only real numbers, these frequencies does not exist.

As mentioned already - for my opinion the use of the term "frequency" in the first example is not in agreement with the DEFINITION of "frequency".

If we consider the rotation of a complex vector clockwise, it is the "pi" phase shift of its imaginary component that causes it to rotate clockwise, not the "negative frequency" of rotation.

Regarding the example with a motor that Eligiusz suggests (I think it is quite different in its essense from the complex vector rotation discussed by Olivier), suppose that we consider rotating right a "positive" frequency and rotating left - a "negative" frequency. But then an idea of approaching the motor from the opposite side might occur to us - where we will find left and right changed places, so that the prior positive frequency becomes negative, and vice versa. Where is the reference, then?

http://en.wikipedia.org/wiki/Negative_frequency

The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way. A signed value of frequency can indicate both the rate and direction of rotation. The rate is expressed in units such as revolutions (aka cycles) per second (hertz) or radian/second (where 1 cycle corresponds to 2π radians).

The attached document and other answers in the links given therein regarding the same question asked earlier by Mohammad and a couple of several answers received may throw some light on the subject.

In a lighter vein(members not to mistake me ) let me quote a joke given in the Book," Engineering Electromagnetics" by William Hayt Jr.,published by Mc.GrawHill Publications

"If you want further Illumination on the topic hold it nearer to the Lamp"

P.S.

Prof. P. Subramanyam - thank you for your contribution and the link to wikipedia.

This wikipedia text was the real background for me to open this discussion.

I never blindly follow the wikipedia statements - and also in this particular case I have some doubts.

Initially, my question was if the term "negative frequency" would have any physical meaning. But now I think it is appropriate to be more specific:

"Frequencies" do physically not exist - neither positive nor negative.

What really exists is a physical quantity (let`s call it "signal") that changes its value periodically.

And this time dependent signal can be described by three parameters:

Shape (form), max. value (amplitude), repetition rate (frequency f).

(By the way: I think the symbol f may be used only for sinusoidal signals).

If we discuss the third characteristic parameter (frequency f) we have to refer to the definition: The number of periodic events per second. This is just a number, which cannot be negative. And also the reciprocal value (time for 1 period) T=1/f cannot be negative.

I think the introduction of "negative frequencies" has a mathematical background only - and also the wikipedia article speaks about the "concept" of negative frequencies. (Something that has a physical meaning, is not a "concept").

Of course, one can try to visualize this "artificial" quantity - for example, using a rotating wheel. But it is not correct to use such a mechanical device for introducing a new definition of the term "frequency" by allocating a direction to it.

Lutz vW

Very interesting, I was not acquainted with this Wikipedia article. I confess that I at first considered the original question by Lutz a bit idle, but now I see that it is not.

Although Wikipedia by all means holds it certain niche, I never use Wikipedia as a scientific or engineering reference. Moreover, I am strictly against referring to Wikipedia in academic publications (such as tutorials or textbooks), which practice is expanding in the last years. The key criterion for Wikipedia (resulting from the very essence of this resource) is consensus. This does not align with the criteria for scientific Truth, which are, in the first place, proof and confirmation.

Please excuse me for this offtopic.

Anton, as you have criticized Wikipedia, would you have a look at the Wikipedia article about Miller theorem? Is it worth to read it?

http://en.wikipedia.org/wiki/Miller_theorem

Dr. Alipov - thank you for clarification.

Here comes another argument against the "wheel"-example:

As far as I know, the concept of negative frequencies was introduced (and used) first in conjunction with the FOURIER transform in its complex form using EULER`s identity. And in this context it is to be stated that each negative frequency component always has a positive counterpart. That means: Both parts (positive/negative) appear as a pair only. This is in contrast to the attempt to visualize both parts using a rotating wheel.

Before going deep into the topic, I just thought that there can be no further Explanations and I quoted William Hayt Jr. "If you want further Illumination on the topic hold the book nearer to the Lamp" in a lighter vein as there seemed to be nothing more to discuss.

Later, on seeing Prof.Lutz's assertive objection, I feel that Prof.Lutz. has now brought the Lamp nearer to the topic and I could get more illumination on the subject and I can shred some more light on the subject.

Prof.Lutz's assertion that "the Frequency being number of periodic events per second cannot be negative as it is just a number" is correct.

The Fourier Series Expansion in trigonometric form is applicable for any non sinusoidal Periodic wave form. Sine wave is also a Cosine Wave displaced by 90 degrees.

The use of exponential form for Cosine wave as half the sum of two equal phasors rotating in opposite directions was treated by the earlier respondents to indicate the presence of negative frequency. Here lies the fallacy. The negative sign is applicable to the ' j' Operator and not to to ' wt '. Simply ( -j ) indicates measurement of the angle in the opposite direction and ( e ^ ( -jwt ) ) indicates rotation of the phasor in the opposite direction by an angle ( wt ).

If we consider t < 0 we find that the frequency is still positive and the wave form simply gets inverted for t < 0. Here time t is treated negative for past occurrence.

The wave form is continuous on both sides of time axis.

Sin( ( - wt ) ) is -Sin( wt ) indicates change in sign of instantaneous values not indicating frequency becoming negative and Cos ( -wt ) is Cos( wt ). Then also there is no question of Frequency becoming negative.

Thus on all counts there is no question of existence of Negative Frequency.

P.S.

Dear Lutz, Dear Colleagues of the RG,

It is a matter of definition. We have to precise our technical terms. I understand under frequency without any adjective or additives that it is the number of cycles per second in a periodic waveform. Accordingly, it is always a positive number. We never said that the the frequency is - 10 KHz for example.

There is the angular frequency omiga, which is termed also the angular velocity. It describes the angular velocity of the rotating objects. The angular frequency is related to frequency by omiga = 2 pi f in radian per seconds. Since rotation has two directions then the angular frequency may be positive or negative. Accordingly angular frequency may be positive or negative.

Real signals as cos wt = .5( e^jwt + e^-jwt) is the sum of two complex exponential signals having two opposite angular velocities w and -w. One has a positive angular frequency and the other has a negative angular frequency. These two components when represented in the angular frequency scale they appear on the opposite side around the the zero angular frequency. Dividing the angular frequency by 2 pi we get the frequency in cycle per seconds. For reduction and simplicity it is said double sided frequency representation of the frequency spectrum. That is positive and negative frequencies. But it must be said positive and negative angular frequencies.

I hope i shed some light on precising the definition of frequency.

I don't think this is a satisfactory construct from the methodological point of view. Suppose we consider that omega is negative. Since omega is 2*pi*f, and "2" and "pi" are positive, then we must admit that f is also negative.

It is not the negativity of omega that causes the complex vector to "rotate" in the opposite direction, but the minus sign before it, which relates to the phase shift of the imaginary part.

***

@Cyril: Sorry, I have not yet had time to read the wiki article.

i think the convention of negative frequency arise simply from introducing Euler's theorem for mathematical simplicity. And for the directional approach, if we start defining the frequency as a vector quantity including direction, like a wheel rotating in anti clockwise direction or clockwise direction, we can think of negative frequency. My idea is that the occurrence of negative frequency will be only in the mathematical world.

Yes - "we can think of negative frequency".

Agreed. It is a very helpful tool, which helps to simplify mathematical expressions (Euler). That`s all. They do not physically "exist".

What i said is precise and exact, The angular frequency which is the angular velocity may be either negative or positive according to the direction of rotation. This is a well established fact.Anton you speak about the absolute value of omiga which is always positive. The sign belongs to the omiga. The angular frequency is signed while the frequency is only a positive value.So in the Euler expression the sign belongs to omiga and not t for sure. It is more exact to say postive and negative angular frequencies rather than positive and negative frequency.This is my fixed believe.

Well, in mechanics angular velocity is not "positive" or "negative", but is a vector collinear to the axis of rotation.

@Lutz. Your question can be deconstructed into two separate issues:

a) First it is the wrong question. Because when it comes to maths, nothing is "real." We invent maths as we go along. It is a function of our imaginations. The real question is if can we can attach an interpretation to negative frequency or not.

b) Abdelhalim's answer is correct, ie. that a sinusoid can be thought of in terms of a sum of positive and negative phasors. Can we attach an interpretation to the sign? Eligiusz's answer at the beginning was close. We can think of one phasor rotating in one direction and one in the other direction.

My approach applies to any such question. For example if you ask if it is "real" that we can have an imaginary value for impedance, I will answer that is the wrong question. I will say what matters is the interpretation you give to an imaginary value for impedance.

This happens all the time in engineering. Whenever we come up with strange numbers, provided they are useful, we find a way to make an interpretation and then refine the definitions.

When negative numbers were first invented, people worried that it wasn't "real" to have negative cows. But that problem vanished once people realized it was a matter of interpretation and we can think of these negative cows as cows that are owed to a farmer.

All your problems vanish once you stop believing maths is "real" and start believing that it is merely a human construct that we make up as we go along. We are constantly making up the rules as we go.

Derek - thank you for the detailed and informative answer. I can assure you that I fully agree with your view (see my question at the top of this thread and my statement ("negative frequencies are just "mathematical" frequencies"). The background of my question was a document claiming that negative frequencies would exist in reality - and I wanted to start a discussion about this subject.

Regards

Lutz

Thank you Derec. In fact, i waited this judgement from Lutz.

Thank you Derec Sir for such a nice piece of information and explanation.

Quantum filed theory (quantum electrodynamics) has solutions with negative frequency. Dirac's relativistic electron equation has two solutions, one with positive frequency and one with negative frequency.  The absolute values of these frequencies are identical and satisfy E = Mc^2 = hf for the electron. The negative frequency means a state (a hole) in the sea of negative energies thus, according t Dirac, when an electron falls into such a hole, then annihilation happens and  2E = 2Mc^2 energy is radiated (two gamma photons with E=0.51 MEV energy in the opposite directions). Dirac called the "hole" positron and this was the theoretical discovery of antimatter.

There are obvious integrity problems with the picture (for example, negative frequency would mean negative energy but the energy is obviously positive at the annihilation, etc etc) but the math works. I think, this is an example for the situation many others described as the math has negative frequencies but is actual physical meaning is something else.

Note: I am not an expert; can't answer any question; just remembered my quantum electrodynamics class 35 years ago :-)