Complex Numbers Meet Dynamic Systems – Lesson 2


In Lesson 1 we used complex numbers to represent a static characteristic- the size of a photograph.

For real world applications however, the characteristics of interest are often dynamic in nature, such as the mechanical vibration of a bearing or the alternating current through a device.

We’ll now show how complex numbers are used to represent these types of dynamic states. We will use rowers in a crew boat to illustrate the technique.

Let’s Get Moving

Meet our first row crew, Stan and Calvin, rowing in a smooth harmonic motion. Stan exhibits a perfect stroke and Calvin tries to follow. Our challenge is to precisely describe Calvin’s stroke relative to Stan’s using a single complex number.

Follow the exercise in the "PicoViewer" (left) to learn interactively about this challenge, then press on below to dig in.

Are You Real, or Imaginary?

Before trying to describe Calvin with a complex number, we introduce the Purist rowers, Rita and Ivan. Rita has a "Purely Real" dynamic response and Ivan has a "Purely Imaginary" dynamic response. Together, they will act as a component pair to build Calvin’s complex number description. Sound confusing? No worries, just jump into the PicoViewer below to get an interactive picture, starting with Rita.

Putting it Together

Now let’s assemble the Purist components to construct a complex personality, one having both real and imaginary parts. We show Calvin’s complex personality below, represented by an open circle with a cross on the complex plane. Jump into the Picoviewer below to see how the Real and Imaginary components compose his personality.

Lesson 2 Summary

We illustrated using complex numbers to describe dynamic output relative to input. The Real part of the the complex output represents the motion component which is in perfect timing with the input. The Imaginary part represents the motion component which is a quarter cycle ahead of the input.

In Lesson 3, we’ll use these concepts to characterize the vibration system from our intro. This application opens the secret into how imaginary numbers are actually used in real world situations.

On to Lesson 2

Comments Welcome!

3 + = six

38 Responses to “Complex Numbers Meet Dynamic Systems – Lesson 2”

  1. quote Says:


    Applications of Imaginary Numbers – Lesson 2

  2. Ivana Says:

    same question again: Why not use simple algebra in variables say x and y? or +-y ???

  3. Mike Hunt Says:

    totally my fav web search eva

  4. hi Says:


    also, did you notice thew were rowing BACKWARDS?


  5. steve Says:

    this was cool

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  9. frank jones Says:

    WOW these interactive devices just tickle my fancy!

  10. Amanto Love Says:

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  19. _IRIS Says:


  20. Sophia Goram Says:


  21. Cthulhu Says:

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  22. David Gonzalez Says:

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  23. Cthulhu Says:

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  27. Keegan and David aka. Rap masters Says:

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  28. Gabby Chenoweth Says:


  29. De'Andre Black aka Logan Klaers Says:

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  30. bchung Says:

    Doc MO,

    In the hypothetical case represented, Ivan represents and is defined as the imaginary component, which means that he is 90 degrees ahead of Stan. So by definition, you cannot change his phase – he is totally defined at 90 degrees. You can only change his amplitude. Rita is defined as the real component, which means that she is 0 degrees (exactly in sync) ahead of Stan. Also by definition, you cannot change her phase, but you can change her amplitude. Ivan and Rita (imaginary and real parts) can sum together as waves to make Calvin, or looked at the other way, you can take any output from Calvin and represent it as a sum of Ivan and Rita. To answer your question directly, Ivan’s phase does not change by definition, and yes, you can change is amplitude legally.

    I am wondering if you are confusing imaginary to be the same as phase, and real to be the same as amplitude as this is not correct. Is that what you were thinking? Generally speaking:

    Real is not equal to Amplitude
    Imaginary is not equal to phase.

    In order to translate real and imaginary components to amplitude and phase, the following equations are used


    where Real=magnitude of real component and Imag=magnitude of imaginary component

    Hope this helps.

  31. Doc MO Says:

    Is the purist rower Ivan correctly set up? It seems like when I change the imaginary component of his stroke, it should change his phase with respect to Stan. Actually when I drag the slider up and down it changes the amplitude of his stroke – seems like the real part is changing?

    Let me know if I misunderstand.

    Seems like a great idea!

  32. bchung Says:

    Hi Sandy,

    Thanks for the comment. This is a tough one, because I originally designed this site for what I thought would be useful for high school level, but folks that comment on getting a good intuitive understanding seem to generally be at the college level, or if they are in high school, are at a pretty advanced level.

    I’ll have to do some thinking on how to make this more accessible at the high school level – very tough one. Honestly I likely won’t be adding much to the imaginary numbers section in the near future. I’ve been working at a crawl pace on doing some electrical circuits tutorials, so that will come out first. Your comments are very helpful though to give me perspective on how to approach the circuits tutorial which is coming up next. Thanks for your comments!


  33. Sandy Says:

    Brad, thank you so much for making this site!!! I am a high school math teacher and took three semesters of calculus, differential equations and I think 12 or 15 more hours of math beyond that. Dif E is where I used imaginary numbers the most, though. I made an A in dif e and had not a clue what I was doing. I could do the math, but did not understand it. I asked my prof, but he wasn’t much help. All I knew was that it had something to do with electricity. I didn’t even know about the other examples. Thank you so much for this site!

    I do have a request, though. I currently homeschool and am tutoring an Algebra II class. We just covered imaginary numbers for the first time today. I showed the two girls your site and they were totally lost. Is there any way that you could conceptually explain on a more vague level what imaginary numbers are and their purpose? The girls got lost in the details as they were simply not following them. The photograph example made no sense to them whatsoever, mainly as you have said, insisting that using i was not necessary for that. The rowing example is fairly complex. Is there any way you could take say the vibrations of a cell phone and present an example in a way that the math and the application of it sort of wash over the viewer? Then I would put that example after the inro and before the rowing example. As a high school teacher, I would really, really appreciate that!!! Please let me know if you ever add anything to this web site!

  34. Wren Says:

    Can’t thank you enough for putting up such detailed explanations!

    Please keep up the good work :)

  35. Matt Says:

    Again, amazing!

  36. Bonita Says:

    ohhhhhh noooooo

  37. Hoyt Balmbergarg Says:

    Hi wasssp? I do not like green eggs in ham

  38. bchung Says:

    I’ ve been considering swapping Lesson 2 with Lesson 3 so that the real application is shown earlier, and possibly also shortening the number of rower exercises since they may be a little tedious. Any comments? Looking for some guidance from the users.