Eric Jacobsen wrote:>...> > ... As soon as the phase reference is in place the > phase vs time of the sinusoid can be determined and then the sign of > the frequency is relevant with respect to that reference. This > effectively makes the sinusoid representable as a rotating phasor.... That's the nub of it. The uses we put trigonometry to predisposes us to hang more on our analogies than they can support. Trig functions are one-to-one mappings of one scalar to another. That's not enough scaffolding for a solid link to something that rotates. It would be fatuous to pretend that I don't use understand the 'what ifs' and 'sorta likes', but that doesn't entitle me -- or anyone else -- to say, "Sure they're real, and when you grow up, you'll know that". A minus times a minus makes a plus for the convenience of doing math. The strongest one can say about that is that it's consistent. It's not true that it must be that way, just that it's more convenient. Some brilliant people -- my first wife was one -- are put off math by being told that something is a logical necessity when in fact it isn't. "If I can't see this logical necessity, the subject must be beyond me." That can be crippling. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Negative Frequencies
Started by ●July 15, 2003
Reply by ●July 19, 20032003-07-19
Reply by ●July 19, 20032003-07-19
On Sat, 19 Jul 2003 16:20:12 -0400, Jerry Avins <jya@ieee.org> wrote:>Eric Jacobsen wrote: >> > ... >> >> ... As soon as the phase reference is in place the >> phase vs time of the sinusoid can be determined and then the sign of >> the frequency is relevant with respect to that reference. This >> effectively makes the sinusoid representable as a rotating phasor. > > ... > >That's the nub of it. The uses we put trigonometry to predisposes us to >hang more on our analogies than they can support. Trig functions are >one-to-one mappings of one scalar to another. That's not enough >scaffolding for a solid link to something that rotates. It would be >fatuous to pretend that I don't use understand the 'what ifs' and 'sorta >likes', but that doesn't entitle me -- or anyone else -- to say, "Sure >they're real, and when you grow up, you'll know that". A minus times a >minus makes a plus for the convenience of doing math. The strongest one >can say about that is that it's consistent. It's not true that it must >be that way, just that it's more convenient. Some brilliant people -- my >first wife was one -- are put off math by being told that something is a >logical necessity when in fact it isn't. "If I can't see this logical >necessity, the subject must be beyond me." That can be crippling. > >JerryVery much agreed. I figgered we was on essentially the same page... Eric Jacobsen Minister of Algorithms, Intel Corp. My opinions may not be Intel's opinions. http://www.ericjacobsen.org
Reply by ●July 19, 20032003-07-19
Jerry Avins wrote:> What if I claim that w is the frequency, always positive, and that I can > form the cosine by using it to replace the underscore in either cos(_t) > or [exp(j_t) + exp(-j_t)]/2? It's interesting to note that making w a > negative number changes nothing at all. Hmmm... :-)Well, also the output of abs(x) does not change if I take abs(-x)... Any correlation with the above problem? bye, -- piergiorgio
Reply by ●July 19, 20032003-07-19
Jerry, Eric: Ponderables... Frequency modulation... and associated operations... Consider the case of frequency modulation of a zero frequency carrier. Is this possible? How? Why? Has it ever been done? What exactly are the bandwidth characteristics of a zero frequency carrier modulated by a negative frequency modulation waveform? What exactly is single sideband FM? Can full fidelity modulation waveforms be recovered from a single sideband FM signal? Filter out a conventional FM signal at high frequency, then mix it down to baseband [zero frequency] with a complex IQ demodulator, what do we have supported on the negative frequency domain? Food for thought! Thoughts, comments? -- Peter Consultant Indialantic By-the-Sea, FL. "Jerry Avins" <jya@ieee.org> wrote in message news:3F19A7FC.1640B45@ieee.org...> Eric Jacobsen wrote: > > > ... > > > > ... As soon as the phase reference is in place the > > phase vs time of the sinusoid can be determined and then the sign of > > the frequency is relevant with respect to that reference. This > > effectively makes the sinusoid representable as a rotating phasor. > > ... > > That's the nub of it. The uses we put trigonometry to predisposes us to > hang more on our analogies than they can support. Trig functions are > one-to-one mappings of one scalar to another. That's not enough > scaffolding for a solid link to something that rotates. It would be > fatuous to pretend that I don't use understand the 'what ifs' and 'sorta > likes', but that doesn't entitle me -- or anyone else -- to say, "Sure > they're real, and when you grow up, you'll know that". A minus times a > minus makes a plus for the convenience of doing math. The strongest one > can say about that is that it's consistent. It's not true that it must > be that way, just that it's more convenient. Some brilliant people -- my > first wife was one -- are put off math by being told that something is a > logical necessity when in fact it isn't. "If I can't see this logical > necessity, the subject must be beyond me." That can be crippling. > > Jerry > -- > Engineering is the art of making what you want from things you can get. > �����������������������������������������������������������������������
Reply by ●July 20, 20032003-07-20
Piergiorgio Sartor wrote:> > Jerry Avins wrote: > > > What if I claim that w is the frequency, always positive, and that I can > > form the cosine by using it to replace the underscore in either cos(_t) > > or [exp(j_t) + exp(-j_t)]/2? It's interesting to note that making w a > > negative number changes nothing at all. Hmmm... :-) > > Well, also the output of abs(x) does not > change if I take abs(-x)... > > Any correlation with the above problem? > > bye, > > -- > > piergiorgioHmmm... :-) -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 20, 20032003-07-20
Jerry Avins <jya@ieee.org> wrote in message news:<3F19A7FC.1640B45@ieee.org>...> Some brilliant people -- my > first wife was one -- are put off math by being told that something is a > logical necessity when in fact it isn't. "If I can't see this logical > necessity, the subject must be beyond me." That can be crippling.Quite true. I think much of the problem is that mathematicians want to condence their results into an as compact formula or proof as possible. That's OK for *doing* maths, but not necessarily for *teaching* maths or engineering. I do some occational teaching on underwater acoustics, and there are a set of formulas regarding Normal Modes that usually are presented in a condensed form. My impression is that students are a bit concerned when they come to this particular lesson, that they don't really see why those formulas pop out that way. So, after presenting the theory, I make a point of writing out the resulting formula in a "physical" way, so that each physical effect gets "its own" easily understood and identified term, in the formula. After having done that, it's a mere reordering of terms to get to the usual formulas. Not a big deal, but you need to see the detailed explanation to see why things pop out the way they do. Mathematicians like to see their craft as some sort of universial truth. Without going into the philosophical aspects of that claim, Man appears to have done his fair share of errors over the ages. For instance, the Romans, while ruling most of Europe by means of a huge army, building structures that lasted for thousands of years, could not do arithmetic tasks we take for granted. Try, for instance, to do this little computation, Subtract MCMXCIX from MIM and you will see two of the main problems with the roman number system. In arabic numbers the task says (provided I haven't messed up the roman numbers) Subtract 1999 from 1999 Of course, having more than one way of representing the same number is a problem: one can't devlop efficient arithmetics. And the romans did not have a way to treat the number 0 (zero). Only after the crusades did the europeans learn (from the arabs) the decimal positional number system, that included the number 0. Arithmetics became much easier after that. Now, even after the crusades there is still some 500 years to go by before Euler establishes most of what we regards as standard basic mathematical notation in the 18th century. In the mean time maths went through various stages of "alchemy and magic". There is, for instance, an hillareous tale about two guys that attend a public "duel" over maths, in Italy some time around the 15th century. At that time equations like x^3 + ax^2 + bx + c = 0 (I think there may be a restriction that a = 0, but I can't find the tale in my books) were the ultimate mathematical challenge. While solutions were known for particular sets of coefficients, no general way of solving for the roots was known. Furthermore, the cases x^3 = ax^2 + bx + c x^3+ax^2 = bx +c etc were considered as *different* equations. They wanted all coefficents a,b,c to be positive. So here comes this duel, where one guy have challenged some other guy (again, I don't remember the details and have lost the book) to solve some fourty equations of the given type. And to disclose his answers in a public event in a city square somewhere (Firenze?). Not totally dissimilar from the present-day public duels between politicians two days before election day. And the challengee has, incidentially, found the method of solution, though kept it secret. Instead of gloating over the chalengee's incapacity to solve the stated problems, the challeneger suffers the disgrace of not being able to state a problem too difficult for the challengee. And there was a huge fuzz some years later, when an apprentice of the discoverer of the solution discloses the method in a book. Not entirely dissimilar to the present day rages about patents and trade secrets. My point is that maths is hard work. The presentation of maths and mathematical concepts as "obvious" is, at best, uneducated. When you look at the time line of discoveries or the introduction of basic concepts, it goes on for decades, centuries and millennia. It appears that some 90% of all mathematical knowledge has been produced during the last two hundred years or so. Before that, people struggeled for centuries or even millennia just to get a working number system in place. Once that was done, they struggled for a few centuries more to get concepts like "negative numbers" in place. The time line alone should indicate that quite a few of those details we take for granted may not be as obvious as we like to think they are. Rune
Reply by ●July 20, 20032003-07-20
Rune Allnor wrote:>...> > My point is that maths is hard work. The presentation of maths and > mathematical concepts as "obvious" is, at best, uneducated. When you > look at the time line of discoveries or the introduction of basic > concepts, it goes on for decades, centuries and millennia. It appears > that some 90% of all mathematical knowledge has been produced during > the last two hundred years or so. Before that, people struggeled for > centuries or even millennia just to get a working number system in place. > Once that was done, they struggled for a few centuries more to get > concepts like "negative numbers" in place. The time line alone should > indicate that quite a few of those details we take for granted may > not be as obvious as we like to think they are. > > RuneWell said! Even further, many of the concepts that some take as necessary and fundamental -- as Euclid's fifth postulate once was -- are in fact merely postulates of convenience. Labeling them "obvious" or "true" is a disservice to those most inclined to ask "why". Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 20, 20032003-07-20
Jerry Avins wrote:> [...] > Trig functions are > one-to-one mappings of one scalar to another.y=cos(a) is not a one-to-one mapping from a to y. a1 != a2 does not imply that cos(a1) != cos(a2). -- % 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
Reply by ●July 20, 20032003-07-20
Randy Yates wrote:> > Jerry Avins wrote: > > [...] > > Trig functions are > > one-to-one mappings of one scalar to another. > > y=cos(a) is not a one-to-one mapping from a to y. a1 != a2 does > not imply that cos(a1) != cos(a2). > -- > % 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/~yatescrMaybe I said it wrong. What I mean is that for every a, there is one and only one cos(a), and that both a and cos(a) are scalars. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 21, 20032003-07-21
Jerry Avins <jya@ieee.org> wrote in message news:<3F1AC7E6.9A18EFC7@ieee.org>...> many of the concepts that some take as > necessary and fundamental -- as Euclid's fifth postulate once was -- are > in fact merely postulates of convenience. Labeling them "obvious" or > "true" is a disservice to those most inclined to ask "why". > > JerryMathematics is supposed to be based on just a few axioms. I have a vague recollection there are some eleven (or thereabouts) such axioms. Does anyone know where to find those axioms? Rune
