I might suggest that we consider the whole problem of negatives in math, and perhaps an even bigger context.  Involve your students in this analysis.

Fundamentally all symbols mean what we tell them to mean.  Often a negative value means what we decide it means.  For example, in optics, it means that something about the value is unexpected.  Ie. The height of the image is upside down.  The image is on the same side of the lens as the object. . . .  And then we build our equations to match.

Sometimes the relationship we developed with another data set constrains us to give -requires - a particular interpretation.  In kinematics a negative time simply means that something occurred before the instant we consider zero time.

Sometimes negatives are not meaningful.  In thin-film interference a negative thickness generally doesn't have meaning.

Stirring up this conversation involves them in the conversation about the judgments, and thus meaning behind calling something negative.

A warning though.  Your willingness to address this with students means that this conversation will occur many times, as each of them experiences the disequilibrium that their current thinking is inadequate at different times.  The blessing of this is that you actually stand the chance of seeing students change their thinking.  We often fail to recognize how rare that is.

I run into this problem when teaching about the constant (and its meaning) when the IV never actually takes on the value of 0 (like, birthweight predicting early Bailey score - no babies are born at 0 pounds).  What seems to help my students is to see the graph of the relationship starting at the actual base point, then extending the line artificially backwards.  If they can visualize it, they are better able to accomodate it.

For example, we can consider a sinusoid with frequency 2 Hz;

x1=sin(2*pi*2*t)

and another sinusoid with frequency -2 Hz;

x2=sin(2*pi*(-2)*t)

We know that they are not equal. So, negative frequency makes a difference.

But, on the other hand, we can easily show that

x2=-x1

or we can obtain x2 from x1 if we phase shift it - 180 or + 180 degrees.

As a result, x2 is nothing but a  phase shifted (or amplitude reversed) version of x1.

But, these are simple mathematical operations and they, by no means, have any physical significance. Or, do they???

I might suggest that we consider the whole problem of negatives in math, and perhaps an even bigger context.  Involve your students in this analysis.

Fundamentally all symbols mean what we tell them to mean.  Often a negative value means what we decide it means.  For example, in optics, it means that something about the value is unexpected.  Ie. The height of the image is upside down.  The image is on the same side of the lens as the object. . . .  And then we build our equations to match.

Sometimes the relationship we developed with another data set constrains us to give -requires - a particular interpretation.  In kinematics a negative time simply means that something occurred before the instant we consider zero time.

Sometimes negatives are not meaningful.  In thin-film interference a negative thickness generally doesn't have meaning.

Stirring up this conversation involves them in the conversation about the judgments, and thus meaning behind calling something negative.

A warning though.  Your willingness to address this with students means that this conversation will occur many times, as each of them experiences the disequilibrium that their current thinking is inadequate at different times.  The blessing of this is that you actually stand the chance of seeing students change their thinking.  We often fail to recognize how rare that is.

In my opinion, the key is to realise that mathematics allows us to create a model of physical reality, but is separate from it. In a mathematical model, mathematical rules apply; in a physical system, physical rules apply. It should not be taken for granted that both sets of rules are equivalent!

Modelling is a process whereby we consciously omit details of the system being modelled in order to make it more tractable. We may, for instance, suppose mathematical infinitude because border conditions are annoying, but physical systems do have borders! Therefore it is very important to remain aware of where a model applies and where it is too much of a distortion (again, because model and object belong to different domains with different rules).

I believe that a linguistic simil might help your students: think of the word 'house' and think of a proper house. You can say " 'house' is a short word", and also "that house is tall", but implying that " 'house' is a tall word" would make little sense (I may get my example wrong because of the language, but I guess you get the idea). 

In that sense, the appearence of negative frequencies in mathematical expressiones is due only to the rules of mathematics itself. It does not necessarily carry any meaning about the physical reality, nor does this fact imply that physical oscillations may support negative frequencies.

To sum up, a symbol that we, humans, use to refer to an object of reality and to reason about it, is distinct and separate from that object, thus different rules apply.

Hope that helps.

Cheers,

Fernando

https://en.wikipedia.org/wiki/Sign_(semiotics)

I am not mathematician but I may understand the negative frequency as follows:

For a vector in x-y plan rotates with angular velocity (w = 2*pi*frequency)

the frequency is positive if the rotation is counter-clockwise and negative if the rotation is clockwise.

Talib answer is the one we teach in Signal Processing, but we must join something to see the importance of negative frequencies: the modulation property of Fourier transform where 2 upper and 2 lower bands appear. This shows how they are really important

Frequency is derivative (velocity) of the instantaneous phase p of the sinusoid x=cos(p(t)). Therefore frequency w(t)=dp(t)/dt. Negative frequency means that instantaneous phase changes backwards. I would treat it as time reversal i.e. non-causal system, where consequence occur before the cause.

Very interesting discussion.

Basically, if you take a single frequency signal, a sine, sin(2.pi.f0.t), and want a frequency based representation, you simply put 1 for f=f0 and 0 elsewhere, or, in a more mathematically correct way, you put a delta (Dirac) impulse on f=f0. But when you perform a Fourier transform, you do not develop on sinus, but on complex exponentials and it is well known that you can get your sine from the difference of 2 exponentials, one with f=f0 frequency, and another one with f=-f0. If you see it this way, negative frequency have no physical meaning at all, but you need them to get your math correctly.

But all the answers above enlighten the problem.

negative frequency do not have any physical significance.It is all about convention.negative frequency means clockwise rotation and positive frequency means anticlockwise rotation or viseversa.

Dear Akay

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.

Best wishes

Whenever you express frequency in complex form ,the concept of negative frequencies arise. This can be explained using s-plane.For example Coswt=(ejwt + e-jwt)/2.

The concept of frequency has two meanings

1. The ratio of the number of actual to possible occurrences of an event.

2.The rate at which something happens in unit time (say)

The first is found in Statistics. The second in Engineering.

A good staring point to explain frequency as per 2 would be to use the notion of periodicity starting from a point moving on a circle . This immediately leads to how many radians in one second and the need to say whether it is clock wise or anticlockwise anticlockwise ie + or - .Subsequently the x / y component as sinusoids  become clear with connection to odd and even functions. Also as a bonus we get the idea of instantaneous  frequency which is quite often another difficulty experienced by many students.

Cheers

We could do (discrete) Fourier analysis (of real signals) using just cos and sin functions. Instead, we choose to use complex exponentials. I guess the main reason for this is that it allows us to simply apply a phase shift by multiplying by a complex factor (very useful!). As a 'side effect', this also allows us to analyse complex signals (not as useful in practice). To form a complete orthonormal basis set using these complex exponentials we need these negative frequencies. A basis function at a -ve freq is the same as the basis fcn at the +ve freq, but the imag part is shifted by 180 deg (i.e. they are complex conjugates of each other). So I would say negative frequencies are just a part of this mathematical 'apparatus'.     

 I think "get used to it" is a little weak with respect to didactics. ;-) 

I once read a reasoning on that issue, which somewhat stated: "the basis of fourier transform are infinitely extending sinusoidal waves... Who can tell in which direction they run? Left or right? Only our convention of time tells us that they are running from now to there, from left to right.... there is no difference for the actual wave when the time-arrow would run the other direction..."

I am citing from memory, but the concept of "time is not defining a physical representation", so both ways are needed, usually is accepted by my students.... 

Sorry, I should look up the citation on it.... 

I would like to thank all the contributors for their well thought-out and immensly enlightening answers. I am greatly informed on this topic. And I feel like, I will be well armed with possible answers next time I am encountered with this question in future Signals classes that I teach.  

plz go over a very good web link on basics of communication at http://complextoreal.com/

negative frequencies are only once we go in complex exponential domain as rightly highlighted by many in the preceding comments.

The person who down voted my answer has nothing to do with science and moral.

I am sorry for him. It is really one of the best answers without exaggeration.

The physical significance +ve and -ve quantities can usually be explained in a way that makes sense, but when somebody asks "What is the physical significance of this imaginary quantity?", the answer will usually be that there is none, or "Oops, I have made a math error!". Complex numbers are a mathematical tool of our creation with no physical significance. We start with real numbers/measurements, use complex numbers for analysis and computation, then return to the real domain. So I think in this case, "get used to it" is a reasonable response ;-). Get the students working (designing digital filters etc) in the frequency domain using real (cos and sin) functions (+ve freq only), then introduce them to complex math (with +ve & -ve freq), and I think they will love the utility and won't care about meaning anymore.