Randy Yates wrote:> > Glen Herrmannsfeldt wrote: > > [...] > > It seems obvious to me, though, that if you replace each complex number with > > an ordered pair of real numbers, and supply the appropriate operations to > > those ordered pairs such that you get the same results, that you have shown > > that you can do everything with (ordered pairs of) real numbers. > > Then why do we need complex numbers to solve polynomials with real > coefficients??? ...Glen, let me retract this. You are right, these two mathematical beasts (the complex and "ordered pairs of real numbers with the appropriate operations") are equivalent. However, this brings us to the heart of the problem: in an arithmetic system, there are only two operations: addition and multiplication, and there is only one set (the reals in this case). Therefore, YOU DON'T GET TO DEFINE NEW OPERATIONS AND NEW SETS (e.g., ordered pairs) from real numbers and call it the same mathematical system! The system is defined in terms of its set and the two operations. -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr
Negative Frequencies
Started by ●July 15, 2003
Reply by ●July 22, 20032003-07-22
Reply by ●July 22, 20032003-07-22
Piergiorgio Sartor wrote:> > Jerry Avins wrote: > > > The point please? I wrote -- anyway, intended to write, that for any a, > > there is one and only one cos(a). Do you contradict that? > > The equation cos(a)=1 has infinite solutions, not > only one. > > So, for each "a" there is one cos(a), but for each > cos(a) there are infinite "a"...What I wrote is, I thought, that cos(a) has only one value, whatever 'a' may be. It is the same as cos(-a) and cos(a + 2npi), but I didn't address that.> > Usually the form "one and only one" means that the > expression is invertible (one by one), which, of course > it is not true for the cos() operation.Not this time. For any a, there is one and only one cos(a).> > So, saying (or writing) "for any a there is one and > only one cos(a)" can lead to some confusion in the > reader (or listener). > Since I'm sure there is no mistake in the sentence, > than probably this needs just some clarification.I apologize for confusing you. I seem to be doing a lot of that lately.> > > P.S. "Contradict" means "speak against". When writing, should we use > > "contrascribe"? > > I guess the latin root (contra-dicere) has lost its > original meaning, to reach a much broader one, but > I can support "contrascribe" too! > > It's interesting that, in some countries, it is common > to use something like "see you again", when leaving > someone, also at the phone. In some other countries, > there is a different form, like "hear you again", for > the phone and "see you again" in case of face to face. > > bye, > > -- > Piergiorgio SartorI like language almost as much as technical things. Read you later. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 22, 20032003-07-22
Jerry Avins wrote:> What I wrote is, I thought, that cos(a) has only one value, whatever 'a' > may be. It is the same as cos(-a) and cos(a + 2npi), but I didn't > address that.OK. Clear.> I apologize for confusing you. I seem to be doing a lot of that lately.It's OK, I like to learn, without being confused, I cannot learn anything...> I like language almost as much as technical things. Read you later.See you later... ;-) bye, -- Piergiorgio Sartor
Reply by ●July 22, 20032003-07-22
Jerry Avins wrote:>...> > What I wrote is, I thought, that cos(a) has only one value, whatever 'a' > may be.That's not what I had written! It's not what I meant, either. The truth is, that if 'a' changes, cos(a) probably will too. I summed up what I mean later:> > ... For any a, there is one and only one cos(a). > >Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 22, 20032003-07-22
Jerry Avins wrote:> Stan Pawlukiewicz wrote: > >>Jerry Avins wrote: >> >>>Kari Pesonen wrote: >>> >>> >>>>"Bhanu Prakash Reddy" <itsbhanu@yahoo.com> wrote in message news:28192a4d.0307142216.4c6ee88@posting.google.com... >>>> >>>> >>>>>Hi, >>>>>Can anyone explain the concept of Negative frequencies clearly. Do >>>>>they really exist? >>>> >>>>About 30 years ago I wondered the same question (do negative frequencies >>>>exist?) >>>>Here you have one example from acoustics: >>>>Let's assume a sound wave propagating from a source at speed c to certain >>>>direction. We sample this sound wave by a microphone moving at speed >>>>2c to the same direction. What we get? The exact time domain wave form >>>>but the signal represented in time reversed. >>>>We can define the instantaneous frequency as the phase change in time >>>>unit (f = d phi/dt). In the microphone signal this change is negative >>>>when compared with the original signal. The frequency is negative by >>>>definition. >>>> >>>>In real life we can easily listen and analyze sounds "by negative frequencies" >>>>playing recorded signals backwards. Although the signal played forwards >>>>and backwards have same spectrum they sounds differently (at least when >>>>listened by humans) >>>> >>>>Kari Pesonen >>> >>> >>>There is linear or rotary motion involved in playing a tape or record >>>backward, so direction reversal makes sense. What about the loudspeaker >>>cone? Does that vibrate backwards? >> >>In active noise control, a speaker can be used to absorb sound energy. >> >> >>>The examples above are a sort of time reversal. Does that really negate >>>the frequencies? If so, how? >>> >>>Jerry > > > I see. Would you say that the speaker emits negative frequencies that > cancel the positive ones by combining with them? An interesting idea! > Come to think of it, no. That would imply that a negative frequency is a > positive one with a 180� phase reversal. So what _did_ you mean? > > JerryI was picking at your loud speaker vibration question. Instead of the vibration radiating, some cleaver folks discovered that a speaker can sink significant acoustic energy in some cases. It's an interesting phenomena. There was an IEEE Signal Processing magazine article on it a long time ago. I originally thought that sound control was essentially a method dependent on far field interference ( 180 degree reversal between a pair of radiators) but it is a lot more interesting in the near field. As far as the original thread goes, IMHO, an equivalent question would be "Are abstractions real?" As an engineer, I think that some abstractions have more utility than others.
Reply by ●July 22, 20032003-07-22
Jerry Avins wrote:> That's not what I had written! It's not what I meant, either. The truth > is, that if 'a' changes, cos(a) probably will too. I summed up what I > mean later:Uhm, Jerry, this is not OK, anymore... If "a" changes, cos(a) may change or not. We can introduce the concept of "small changes" to cope with this. bye, -- piergiorgio
Reply by ●July 22, 20032003-07-22
Piergiorgio Sartor wrote:> > Jerry Avins wrote: > > > ... probably ... > > Uhm, Jerry, this is not OK, anymore... > > If "a" changes, cos(a) may change or not. >Gimme a break! Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 23, 20032003-07-23
Jerry Avins wrote:> Gimme a break!You know that I'm quite "strict" on these issues... ;-) bye, -- Piergiorgio Sartor
Reply by ●July 23, 20032003-07-23
Piergiorgio Sartor wrote:> > Jerry Avins wrote: > > > Gimme a break! > > You know that I'm quite "strict" on these issues... ;-) > > bye, > > -- > Piergiorgio SartorWell, strictly then, "probably" implies "possibly not". Would you want to see the probability calculated? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 23, 20032003-07-23
Jerry Avins wrote:> Well, strictly then, "probably" implies "possibly not". Would you want > to see the probability calculated?Actually you will not believe me, but... yes, I was trying to calculate the probabilty that cos(a+x) = cos(a) (or not equal)... Or something like that... bye, -- Piergiorgio Sartor
