bulegoge@columbus.rr.com wrote:> On Jan 21, 9:37 pm, Jerry Avins <j...@ieee.org> wrote: >> buleg...@columbus.rr.com wrote: >>> On Jan 21, 2:03 pm, Darol Klawetter <darol.klawet...@l-3com.com> >>> wrote: >>>> On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote: >>>>> Hi, >>>>> I have begun a complex number tutorial which shows how e^jwt works. It >>>>> is located: >>>>> http://fourier-series.com/fourierseries2/complex_tutorial.html >>>>> I have more work to do on this, but come take a look. I created these >>>>> programs for complex numbers in order to explain the complex >>>>> representaion of the fourier series. >>>>> Brent >>>> Cute, but you should explain why Euler needed "j" in his identity. >>>> Explain why it's used as the vertical axis of the phasor plane. Many >>>> engineers want a physical intuition that is defied by a reference to >>>> the square root of -1. Most introductory texts just state Euler's >>>> identity and leave it at that. >>> I intend to add some more explanation in the near future. However, >>> this whole j thing is difficult because getting that physical >>> intuition has not really come to me yet. I wonder if you must simply >>> accept, as an article of faith that Eulers identity works and that >>> (e^jwt+e^-jwt)/2 = cos(wt). That is, the staring point is accepting >>> the identity. The equivelent of an axiom in geometry, then build off >>> of the axiom. >> No. At the very least, you can add the Taylor series of cos(x) to the >> Taylor series of i*sin(x) and observe that the sum is identical to the >> Taylor series of exp(i*x). A technicality makes that fall short of being >> a proof, but it sure is a good indication. >> > > That relationship is cool, It points to complex numbers as having more > meaning than might otherwise have been thought. It made complex > numbers a real "head scratcher", but, for me, this relationship does > not really bring forth any more insight into what imaginary numbers > are. To me, imaginary numbers are like a hidden world that works > behind the scene , but always turns up invisible to us. > > For me the real beauty is that the e^jwt is more intuitive as a > solution to a differential equation than say, sinx and cos x is. > Sure, it is observed that the fourth derivative of a sine wave is a > sine wave, but the e^jwt really brings the soultion to a differential > equation to life. > > We can intuitively understand solutions to differential equations that > have x, x^2, x^3 as solutions. > We can intuitively understand solutions that have e^-x or e^-t as > solutions. > > To understand rotational/sinusoidal solutions, however, we need to > understand complex numbers, which (perhaps) nobody really intuitively > understands. I guess when it all boils down, the only way to get a > complete solution to a differential equation (one that includes not > just the correct frequeny, but the correct phasing), you need the > hidden magic of complex numbers going on "behind the scenes". > > I know this sounds like rambling jibberish, and probably blows my > credibility, but, oh well.Imaginary numbers are just a compact bookkeeping scheme for computing relations that can be arrived at -- albeit with greater effort -- in other ways. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Complex Number tutorial
Started by ●January 21, 2008
Reply by ●January 21, 20082008-01-21
Reply by ●January 21, 20082008-01-21
On Jan 21, 10:10�pm, Jerry Avins <j...@ieee.org> wrote:> buleg...@columbus.rr.com wrote: > > On Jan 21, 9:37 pm, Jerry Avins <j...@ieee.org> wrote: > >> buleg...@columbus.rr.com wrote: > >>> On Jan 21, 2:03 pm, Darol Klawetter <darol.klawet...@l-3com.com> > >>> wrote: > >>>> On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote: > >>>>> Hi, > >>>>> I have begun a complex number tutorial which shows how e^jwt works. It > >>>>> is located: > >>>>>http://fourier-series.com/fourierseries2/complex_tutorial.html > >>>>> I have more work to do on this, but come take a look. �I created these > >>>>> programs for complex numbers in order to explain the complex > >>>>> representaion of the fourier series. > >>>>> Brent > >>>> Cute, but you should explain why Euler needed "j" in his identity. > >>>> Explain why it's used as the vertical axis of the phasor plane. Many > >>>> engineers want a physical intuition that is defied by a reference to > >>>> the square root of -1. Most introductory texts just state Euler's > >>>> identity and leave it at that. > >>> I intend to add some more explanation in the near future. �However, > >>> this whole j thing is difficult because getting that physical > >>> intuition has not really come to me yet. �I wonder if you must simply > >>> accept, as an article of faith that Eulers identity works and that > >>> (e^jwt+e^-jwt)/2 = cos(wt). �That is, the staring point is accepting > >>> the identity. �The equivelent of an axiom in geometry, then build off > >>> of the axiom. > >> No. At the very least, you can add the Taylor series of cos(x) to the > >> Taylor series of i*sin(x) and observe that the sum is identical to the > >> Taylor series of exp(i*x). A technicality makes that fall short of being > >> a proof, but it sure is a good indication. > > > That relationship is cool, It points to complex numbers as having more > > meaning than might otherwise have been thought. �It made complex > > numbers a real "head scratcher", but, for me, this relationship does > > not really bring forth any more insight into what imaginary numbers > > are. �To me, imaginary numbers are like a hidden world that works > > behind the scene , but always turns up invisible to us. > > > For me the real beauty is that the e^jwt is more intuitive as a > > solution to a differential equation than say, sinx and cos x is. > > Sure, it is observed that the fourth derivative of a sine wave is a > > sine wave, but the e^jwt really brings the soultion to a differential > > equation to life. > > > We can intuitively understand solutions to differential equations that > > have x, x^2, x^3 as solutions. > > We can intuitively understand solutions that have e^-x or e^-t as > > solutions. > > > �To understand rotational/sinusoidal solutions, however, we need to > > understand complex numbers, which (perhaps) nobody really intuitively > > understands. �I guess when it all boils down, the only way to get a > > complete solution to a differential equation (one that includes not > > just the correct frequeny, but the correct phasing), you need the > > hidden magic of complex numbers going on "behind the scenes". > > > I know this sounds like rambling jibberish, and probably blows my > > credibility, but, oh well. > > Imaginary numbers are just a compact bookkeeping scheme for computing > relations that can be arrived at -- albeit with greater effort -- in > other ways. > > Jerry > -- > Engineering is the art of making what you want from things you can get. > �����������������������������������������������������������������������- Hide quoted text - > > - Show quoted text -One of issues about complex numbers and book keeping has to do with my intro elctronics class many years ago. We were taught to multiply signals by A Le^jwt which was nice, but I don't think I ever fully appreciated that this trick assumes that the other side Le^-jwt automatically comes along for the ride and if you don't understand the implied conjugate, then you don't really get the shorthand notation (which I didn't , and didn't in that class)
Reply by ●January 22, 20082008-01-22
On Jan 21, 7:02 am, c...@claysturner.com wrote:> You may wish to look at Paul > Nahin's book "Dr. Euler's Fabulous Formula Cures Many Mathematical > Ills."Nahin's earlier book: 'An Imaginary Tale: The Story of "i"' is also a good read on the subject of complex numbers. It includes a historical perspective to the understanding of such. How humans in general came to understand some "complex" ideas might be a good personal approach as well. IMHO. YMMV. -- rhn A.T nicholson d.0.t C-o-M
Reply by ●January 22, 20082008-01-22
On Mon, 21 Jan 2008 12:39:49 -0800, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:>On Mon, 21 Jan 2008 11:41:55 -0800 (PST), bulegoge@columbus.rr.com >wrote: > >>On Jan 21, 2:03�pm, Darol Klawetter <darol.klawet...@l-3com.com> >>wrote: >>> On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote: >>> >>> > Hi, >>> >>> > I have begun a complex number tutorial which shows how e^jwt works. It >>> > is located: >>> >>> >http://fourier-series.com/fourierseries2/complex_tutorial.html >>> >>> > I have more work to do on this, but come take a look. �I created these >>> > programs for complex numbers in order to explain the complex >>> > representaion of the fourier series. >>> >>> > Brent >>> >>> Cute, but you should explain why Euler needed "j" in his identity. >>> Explain why it's used as the vertical axis of the phasor plane. Many >>> engineers want a physical intuition that is defied by a reference to >>> the square root of -1. Most introductory texts just state Euler's >>> identity and leave it at that. >> >>I intend to add some more explanation in the near future. However, >>this whole j thing is difficult because getting that physical >>intuition has not really come to me yet. I wonder if you must simply >>accept, as an article of faith that Eulers identity works and that >>(e^jwt+e^-jwt)/2 = cos(wt). That is, the staring point is accepting >>the identity. The equivelent of an axiom in geometry, then build off >>of the axiom. > >Hello bulegoge, > > I like your "complex numbers" demos. > >I'm not sure if it would benefit you but >you have a look at > >http://www.dspguru.com/info/tutor/QuadSignals.pdf > >Nice work bulegoge. > >[-Rick-]Hi Rick, Could you please fix the spelling of 'Cartisian' in that document for me? Thanks, Allan.
Reply by ●January 22, 20082008-01-22
bulegoge@columbus.rr.com wrote: ..> > One of issues about complex numbers and book keeping has to do with my > intro elctronics class many years ago. We were taught to multiply > signals by A Le^jwt which was nice, but I don't think I ever fully > appreciated that this trick assumes that the other side Le^-jwt > automatically comes along for the ride and if you don't understand the > implied conjugate, then you don't really get the shorthand notation > (which I didn't , and didn't in that class)I am a real maths dimwit so I have to breath in before sending this, so: My way of explaining a complex number to myself (and to patient musical friends) is to describe observing a pin on a rotating wheel viewed monoscopically on its edge. You need two viewing positions (at right-angles) to determine not only the position of the pin, but also the direction of rotation. From just one position, it is either moving up/down or side/side. So for me, it is not a case of the diagrams explaining complex numbers, but the numbers explaining the diagrams. (then I say "the rest is just algebra"...) It is also the key to proper reporting of a UFO. From one position, you cannot be sure if it is coming towards you but getting smaller, or v. versa, or... Richard Dobson
Reply by ●January 22, 20082008-01-22
On Jan 22, 12:41�am, "Ron N." <rhnlo...@yahoo.com> wrote:> On Jan 21, 7:02 am, c...@claysturner.com wrote: > > > You may wish to look at Paul > > Nahin's book "Dr. Euler's Fabulous Formula Cures Many Mathematical > > Ills." > > Nahin's earlier book: 'An Imaginary Tale: The Story of "i"' > is also a good read on the subject of complex numbers. > It includes a historical perspective to the understanding > of such. �How humans in general came to understand some > "complex" ideas might be a good personal approach as well. > > IMHO. YMMV. > -- > rhn A.T nicholson d.0.t C-o-MI currently have three of Nahin's books - they are all very good. He says his Dr. Euler's .... book is a continuation of the imaginary tale book. But he has put in more mathematical detail into the later book. His thesis (which he supports quite well) is imaginary numbers were actually needed to solve cubic equations and not for solving quadratic equations as is commonly taught. The 3rd book that I have by him is the story of Oliver Heaviside. I recommend this book for anyone who is interested in E-Mag theory as Maxwell died at a young age, and it was Heaviside who carried forward the fruits of Maxwell's work. Clay
Reply by ●January 22, 20082008-01-22
On Jan 21, 10:58�pm, buleg...@columbus.rr.com wrote:> On Jan 21, 10:10�pm, Jerry Avins <j...@ieee.org> wrote: > > > > > > > buleg...@columbus.rr.com wrote: > > > On Jan 21, 9:37 pm, Jerry Avins <j...@ieee.org> wrote: > > >> buleg...@columbus.rr.com wrote: > > >>> On Jan 21, 2:03 pm, Darol Klawetter <darol.klawet...@l-3com.com> > > >>> wrote: > > >>>> On Jan 21, 6:48 am, buleg...@columbus.rr.com wrote: > > >>>>> Hi, > > >>>>> I have begun a complex number tutorial which shows how e^jwt works. It > > >>>>> is located: > > >>>>>http://fourier-series.com/fourierseries2/complex_tutorial.html > > >>>>> I have more work to do on this, but come take a look. �I created these > > >>>>> programs for complex numbers in order to explain the complex > > >>>>> representaion of the fourier series. > > >>>>> Brent > > >>>> Cute, but you should explain why Euler needed "j" in his identity. > > >>>> Explain why it's used as the vertical axis of the phasor plane. Many > > >>>> engineers want a physical intuition that is defied by a reference to > > >>>> the square root of -1. Most introductory texts just state Euler's > > >>>> identity and leave it at that. > > >>> I intend to add some more explanation in the near future. �However, > > >>> this whole j thing is difficult because getting that physical > > >>> intuition has not really come to me yet. �I wonder if you must simply > > >>> accept, as an article of faith that Eulers identity works and that > > >>> (e^jwt+e^-jwt)/2 = cos(wt). �That is, the staring point is accepting > > >>> the identity. �The equivelent of an axiom in geometry, then build off > > >>> of the axiom. > > >> No. At the very least, you can add the Taylor series of cos(x) to the > > >> Taylor series of i*sin(x) and observe that the sum is identical to the > > >> Taylor series of exp(i*x). A technicality makes that fall short of being > > >> a proof, but it sure is a good indication. > > > > That relationship is cool, It points to complex numbers as having more > > > meaning than might otherwise have been thought. �It made complex > > > numbers a real "head scratcher", but, for me, this relationship does > > > not really bring forth any more insight into what imaginary numbers > > > are. �To me, imaginary numbers are like a hidden world that works > > > behind the scene , but always turns up invisible to us. > > > > For me the real beauty is that the e^jwt is more intuitive as a > > > solution to a differential equation than say, sinx and cos x is. > > > Sure, it is observed that the fourth derivative of a sine wave is a > > > sine wave, but the e^jwt really brings the soultion to a differential > > > equation to life. > > > > We can intuitively understand solutions to differential equations that > > > have x, x^2, x^3 as solutions. > > > We can intuitively understand solutions that have e^-x or e^-t as > > > solutions. > > > > �To understand rotational/sinusoidal solutions, however, we need to > > > understand complex numbers, which (perhaps) nobody really intuitively > > > understands. �I guess when it all boils down, the only way to get a > > > complete solution to a differential equation (one that includes not > > > just the correct frequeny, but the correct phasing), you need the > > > hidden magic of complex numbers going on "behind the scenes". > > > > I know this sounds like rambling jibberish, and probably blows my > > > credibility, but, oh well. > > > Imaginary numbers are just a compact bookkeeping scheme for computing > > relations that can be arrived at -- albeit with greater effort -- in > > other ways. > > > Jerry > > -- > > Engineering is the art of making what you want from things you can get. > > �����������������������������������������������������������������������- Hide quoted text - > > > - Show quoted text - > > One of issues about complex numbers and book keeping has to do with my > intro elctronics class many years ago. �We were taught to multiply > signals by A Le^jwt which was nice, but I don't think I ever fully > appreciated that this trick assumes that the other side Le^-jwt > automatically comes along for the ride and if you don't understand the > implied conjugate, then you don't really get the shorthand notation > (which I didn't , and didn't in that class)- Hide quoted text - > > - Show quoted text -It is more than just a trick. Look up Wronskians. If you are solving 2nd order differential equations, you know you will have two linear independent solutions to the homogenous part of the equations. The initial conditions will determine the appropriate combination. A Wronskian is a neat way to show that a set of functions are linearly independent. Or course with just two functions all you require is for their ratio to be nonconstant to show linear independence. For some types of 2nd order equations, the Wronskian is not only nonzero for two linearly independent functions, but sometimes it is constant. This shows up in some applications of Schrodinger's equation. So if you have a solution in part of a domain, the Wronskian can help you find the solution in another part of the domain. I hope this peaks your interests Clay p.s. You will find solutions to Hermitian systems will admit the conjugate of a function if the function itself is a solution. In DSP you will find a link between functions, their Fourier transforms, and Hermitian properties. In Physics, this shows up in Quantum Mechanics, where the position and momentum wave representations are Fourier transforms of each other.
Reply by ●January 22, 20082008-01-22
On Jan 21, 9:41�pm, buleg...@columbus.rr.com wrote:> I intend to add some more explanation in the near future. �However, > this whole j thing is difficult because getting that physical > intuition has not really come to me yet. �I wonder if you must simply > accept, as an article of faith that Eulers identity works and that > (e^jwt+e^-jwt)/2 = cos(wt). �That is, the staring point is accepting > the identity. �The equivelent of an axiom in geometry, then build off > of the axiom.- Hide quoted text - >The description that is intuitive for me is that multiplying a number on the x-axis by integer powers of -1 cause rotation by 180 degrees. Logically, to rotate by 90 degrees one should multiply by a number that you multiply by it again will give 180 degrees of rotation. This number is the square root of -1. Rocky
Reply by ●January 22, 20082008-01-22
On Mon, 21 Jan 2008 18:56:34 -0800 (PST), bulegoge@columbus.rr.com wrote: (snipped)> > To understand rotational/sinusoidal solutions, however, we need to >understand complex numbers, which (perhaps) nobody really intuitively >understands. I guess when it all boils down, the only way to get a >complete solution to a differential equation (one that includes not >just the correct frequeny, but the correct phasing), you need the >hidden magic of complex numbers going on "behind the scenes". > >I know this sounds like rambling jibberish, and probably blows my >credibility, but, oh well. > >BrentHi Brent, Don't worry. Reaching some sort of comfortable understanding of the true meaning (whatever that is) of complex numbers is not at all easy. I make no claim that I understand their meaning. I merely understand a little bit about their behavior. Remember, if complex numbers were easy to understand, the great mathematician Karl Gauss would *NOT* have called the j-operator "the shadow of shadows". [-Rick-]
Reply by ●January 22, 20082008-01-22
bulegoge@columbus.rr.com wrote: (snip)> I intend to add some more explanation in the near future. However, > this whole j thing is difficult because getting that physical > intuition has not really come to me yet. I wonder if you must simply > accept, as an article of faith that Eulers identity works and that > (e^jwt+e^-jwt)/2 = cos(wt). That is, the staring point is accepting > the identity. The equivelent of an axiom in geometry, then build off > of the axiom.Feynman has an interesting way to do it in "Lectures on Physics" volume 1. He starts out with fractional powers through successive square roots of 10. (For 10**0.5, 10**0.25, 10**0.125, etc.) That allows one to compute 10**(x/1024) for integer x. Then he computes 10**(i/1024) using complex multiplication, and uses powers of that to compute 10**(i x/1024) for integer x. The result, surprisingly (if you don't know the answer) is sine-like curves with an unusual period. He then goes back to show that if you change the base from 10 such that the period is 2 pi then the base must be e. -- glen
