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

# Complex Number tutorial

Started by ●January 21, 2008

Reply by ●January 21, 20082008-01-21

On Jan 21, 7:48�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. > > BrentHello Brent, It appears you are having fun with this. You may wish to look at Paul Nahin's book "Dr. Euler's Fabulous Formula Cures Many Mathematical Ills." I think you will find a lot of neat thinks to graphically demostrate. And yes the world appreciates your tutorials. Clay - a guy who had to learn things the old fashioned way before the advent of the internet.

Reply by ●January 21, 20082008-01-21

On Jan 21, 10:02�am, c...@claysturner.com wrote:> On Jan 21, 7:48�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 > > Hello Brent, > > It appears you are having fun with this. You may wish to look at Paul > Nahin's book "Dr. Euler's Fabulous Formula Cures Many Mathematical > Ills." I think you will find a lot of neat thinks to graphically > demostrate. > > And yes the world appreciates your tutorials. > > Clay - a guy who had to learn things the old fashioned way before the > advent of the internet.I think the last two of these programs are not loading correctly in some computers. My home computers load fine , but I am seeing other probs. If the last two programs don't load, I am sorry. I am trying to figure out why.

Reply by ●January 21, 20082008-01-21

On Jan 21, 10:26�am, buleg...@columbus.rr.com wrote:> On Jan 21, 10:02�am, c...@claysturner.com wrote: > > > > > > > On Jan 21, 7:48�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 > > > Hello Brent, > > > It appears you are having fun with this. You may wish to look at Paul > > Nahin's book "Dr. Euler's Fabulous Formula Cures Many Mathematical > > Ills." I think you will find a lot of neat thinks to graphically > > demostrate. > > > And yes the world appreciates your tutorials. > > > Clay - a guy who had to learn things the old fashioned way before the > > advent of the internet. > > I think the last two of these programs are not loading correctly in > some computers. �My home computers load fine , but I am �seeing other > probs. �If the last two programs don't load, I am sorry. �I am trying > to figure out why.- Hide quoted text - > > - Show quoted text -On the last two programs, click to download and then close out the program, then reclick , and it comes up correctly. Don't know why , yet

Reply by ●January 21, 20082008-01-21

On Mon, 21 Jan 2008 04:48:59 -0800, bulegoge 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. > > BrentLooks cool. At this moment it's telling me that 2.5 * cos 2 * pi = 2.53. Perhaps you need to make your rounding more consistent. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

Reply by ●January 21, 20082008-01-21

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. > > BrentCute, 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.

Reply by ●January 21, 20082008-01-21

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.

Reply by ●January 21, 20082008-01-21

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-]

Reply by ●January 21, 20082008-01-21

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.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. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������

Reply by ●January 21, 20082008-01-21

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. > > Jerry > -- > Engineering is the art of making what you want from things you can get. > �����������������������������������������������������������������������- Hide quoted text - > > - Show quoted text -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. Brent