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.

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.

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.

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.

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.

0. Intro

1. Mapping Imaginary to Physical – Lesson 1

2. Complex Numbers and Dynamic Systems – Lesson 2

3. An Imaginary Number Application – Lesson 3

B. The Physical Link of the Imaginary Unit – Bonus Lesson

September 29th, 2013 at 6:36 am

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

January 4th, 2013 at 8:07 am

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November 13th, 2012 at 8:41 am

Helpful!

also, did you notice thew were rowing BACKWARDS?

starnge…

September 11th, 2012 at 9:44 am

this was cool

December 1st, 2011 at 7:44 am

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December 1st, 2011 at 7:26 am

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December 1st, 2011 at 5:19 am

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September 25th, 2011 at 6:42 pm

[...] learned, when confronted with the description of many problems in the physical world (diffraction, dynamic systems, or even quadratic or higher order polynomial equations requiring solution), it can be in practice [...]

May 31st, 2011 at 8:07 am

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May 31st, 2011 at 7:53 am

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April 26th, 2011 at 6:41 am

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April 25th, 2011 at 10:42 am

someone who portages

April 25th, 2011 at 10:41 am

…sigh*

April 25th, 2011 at 10:36 am

WHAT’S A PORTAGER, JOHN?

April 25th, 2011 at 10:31 am

Shut up, fool! The name of my prophet is HP Lovecraft, not witchcraft! I will destroy all that you hold dear!

P.S.

Drake is really Nyarlathotep.

.

April 25th, 2011 at 10:29 am

Hello haven’t you ever read any Witchcraft?

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Dear Gabby, hi.

Dear Logan, sh.

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April 25th, 2011 at 10:26 am

Stupid fools! Your juvenile stupidity will mean nothing when I awaken from my underwater city of R’yleh and wreak havoc on the world!

BTW, Logan should stop flexing or I will swallow his soul.

April 25th, 2011 at 10:22 am

Logan is a tool bag. He sits next to me and smells like dirty socks. Pishy pish. And he flex when he erases his oh-so-common mistakes. LOGAN RULES…bnr.

April 25th, 2011 at 6:21 am

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April 25th, 2011 at 6:19 am

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April 25th, 2011 at 6:13 am

Trig really crazy; you be lazy.

Can’t find the equation cause you’re not asian! (Uhhhhhhhhhh.)

April 25th, 2011 at 6:12 am

LOGAN! STOP FLEXING!

April 25th, 2011 at 6:08 am

What up my dawgs? I be chillin’

November 22nd, 2010 at 6:51 am

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

amplitude=sqrt(Real^2+Imaginary^2)

phase=inverse_tangent(Imaginary/Real).

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

Hope this helps.

November 19th, 2010 at 11:47 am

Brad-

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!

November 2nd, 2010 at 10:28 pm

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!

Brad

October 28th, 2010 at 10:44 pm

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!

September 26th, 2010 at 5:58 pm

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

Please keep up the good work

November 20th, 2008 at 6:21 pm

Again, amazing!

November 6th, 2008 at 5:07 pm

ohhhhhh noooooo

November 5th, 2008 at 9:21 pm

Hi wasssp? I do not like green eggs in ham

October 29th, 2008 at 9:37 pm

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.

picomonster