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The Physical Link of the Imaginary Unit - Bonus Lesson
Full Circle, Back to Theory
So far we’ve used i to represent a quarter cycle dynamic lead, but why? Seems arbitrary.
A physical interpretation of i reveals that its definition provides the link between real and imaginary dimensions, and a quarter cycle lead is simply a physical realization of this link. We’ll demonstrate this by mapping our dynamic rowers back into the abstract realm.
Note: this is a short but slightly tedious lesson. Just stick it out. It’s worth it!
The Block Diagram
We’ll start by recasting our rowers from lesson 2 into block diagram form (left).
As per previous examples, Stan leads the way by providing an Input stroke, Is for Calvin. Calvin operates on Stan’s input via his own Personality, Pc. This results in Calvin’s Output stroke, Oc, equal to Stan’s input multiplied by Calvin’s personality:
Rollover the block diagram twice to see two examples.
Cloning Calvin
Now let’s add another rower that has exactly the same personality as Calvin (Calvin’s clone). This means Stan’s stroke is operated on consecutively, first by Calvin with the resulting output O1=IsPc, then by Calvin’s clone with the output O2=O1Pc. Combining these operations, we get the final output based on Stan’s stroke and Calvin’s personality:
Example 1 (rollover once) demonstrates this result. A more interesting case however is if we are given the input and final output, and are asked to determine Calvin’s personality. In this case we can solve equation 2 for his personality:
Example 2 (rollover again) demonstrates this case, where the final output is 4.0 meters and the input is 1.0 meters, implying by equation 3 that Calvin’s personality must be 2.0. This means that Calvin’s personality is to double what he sees, his clone repeats that, and the result is 4 times what you started with.
Finally, a Physical Interpretation of i
OK, picture this. You are on a river bank on a sunny day watching your three friends rowing whom happen to be Stan, Calvin, and Calvin’s clone. Stan sets the pace with a stroke magnitude of exactly 1 meter. Calvin’s clone is at the opposite end, rowing with a stroke of -1 meters relative to Stan, moving his oars in exactly the opposite direction. You can’t quite make out what Calvin is doing in the middle because of the glare off the water.
In short, you know that the input stroke (Stan) is 1 meter and the final output stroke (Calvin’s clone) is -1 meter, but Calvin’s personality, Pc, is unknown. You can solve for this from equation 3. Plugging in:
This result means that Calvin’s personality is to look at the rower in front of him and row one quarter cycle ahead with the same size stroke. This operation is done twice, once by Calvin and then once by his clone, resulting in a final output that is half cycle ahead of Stan, effectively an exactly opposing stroke represented by -1 meters.
So, if a physical operation which is mathematically represented by multiplication (ie a rower operating on the rower’s stroke in front of him to yield an output stroke) is done twice identically, and the final output is opposite in sign but of equal magnitude relative to the input, we can say that this physical operation can be quantified by the imaginary unit i. The quarter stroke lead operation fits this imaginary unit description perfectly, which is why it turns out to be a physical realization of the imaginary unit.
Summary
We just demonstrated a physical realization of the imaginary unit, showing how the definition of the imaginary unit provides the connection between the real and the imaginary dimensions in physical systems. Understanding this physical realization of i is key, because it opens a giant toolbox of mathematical methods for describing the large class of two dimensional phenomena that follow the definition of i precisely. Applications abound in electrical engineering, vibration engineering, polymer science, navigation, and more. If you are in a profession that uses imaginary numbers, we’d love to hear from you - leave a comment! This is some powerful stuff. OK, ‘nuf said.
If you liked this web site and would like to see more of this type of learning, you can help us out:
1. Comment on how we may improve our site - EXTREMELY valuable to us.
2. Link to us.
3. Checkout our IMAGINARY STORE.
Thanks for visiting,
Bradley Chung (picomonster author)
Lesson Navigation
0. Home
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 or Browse Store
October 3rd, 2008 at 9:54 am
This website about was extremely helpful. I really like how you described i in physical terms, with the rowers. That really helped me understand the concept. This is a great website!
October 10th, 2008 at 1:07 pm
Thanks for the comment Eller. I’ve been wondering whether the interactive exercises were too tedious to be helpful, so it’s great to know that folks are getting something out of the interactive exercises.
Brad
October 17th, 2008 at 4:58 pm
Thank you for this useful site.
Can you please add more lessons.
for example the use of imaginary numbers in euler equation and how it is used in signal processing and power factors.
can you show us the most butiful equation in mathmatics ( euler identity) ei?=-1.
October 22nd, 2008 at 3:26 pm
Alex,
Thanks for the suggestion. I seem to see this topic (euler identity) a lot in imaginary number sites since it’s such an elegant form. I’ll have to scratch my head on that challenge of making this an easy to grasp concept. What I may try is to make it a short article in the blog. I’ll email you once I get it done (It will take a little while as I just got a new kid in the family!).
As for signal processing, I’m currently working on a basic circuits analysis tutorial which will incorporate imaginary numbers too.
Thanks again for the suggestion. It’s always helpful.
Brad
October 30th, 2008 at 2:27 pm
I really liked this example, it helped me a lot in my learning experiance, I understand imaginary numbers and complex numbers a lot better now with all those real examples… I really like calvin and I’m gonna miss him… Rita and that russian Ivan were also pretty cool, I liked Rita a lot more though because of her slim body and cute personality, she is a real hardworker, Ivan on the other hand, all he does is drink russian vodka and smoke ciggarettes on the job… He is a huge slacker and should go back to the Soviet Union… peace out Ivan if that is your real name… Oh yeah and is’nt Stan tired by now? Give him a break I mean sheesh
November 1st, 2008 at 6:39 pm
M.O.B.
Yes, many dimensions to all. Would be a great party. Love the comment.
November 5th, 2008 at 9:25 pm
I do not like green eggs and ham
November 18th, 2008 at 11:27 pm
Thank-you for your hard work and hard thinking to make such a clear explanation.
November 19th, 2008 at 4:07 am
You’re very welcome!
February 15th, 2009 at 3:59 pm
Thanks, this is enormously helpful. The interactive lessons are great. You sounded concerned that they might be tedious, but I didn’t find them so at all.
February 15th, 2009 at 9:46 pm
Hi EM; very useful feedback, as it helps me set direction for future work. I’m glad that you didn’t find it tedious, so I’ll shy away from changing the level of detail for this lesson, or for future work. Thanks for taking the time to comment.
March 21st, 2009 at 1:56 pm
Thank you Bradley.
I was struggling to visualize the use of imaginary numbers in electrical circuits. It helped me immensely.
It would be nice if you could also explain Laplace and Fourier transforms in a similar manner. It will help…
March 23rd, 2009 at 7:10 am
Hi Sudarshan, I remember turning in homework in my basic circuits engineering class, with a whole host of complex number answers, and absolutely NO idea what that meant, even though the answers were correct and I could show my work. My goal in doing this tutorial, was to see if I could make a difference in those situations, and I’m extremely excited to hear that you benefited from this. As for the Laplace Xform and Fourier Xforms, I’m first going to try to do more explanations of circuit elements (like capacitors/inductors), so I may have to put that off until my kids are in college, since they’re keeping me really busy! (They’re 2 and 5). Thanks for the comment.
Brad
March 23rd, 2009 at 10:58 am
Hi Brad,
Nice work and very helpful! I don’t see any tedium in your examples; I think they do a fine job of clarifying a topic that is not at all intuitive (to most of us anyway!) the way it’s usually presented.
Thanks for the resource!
April 5th, 2009 at 10:32 pm
Great examples. I’ve taken several years of classes in complex numbers and am now in a class on complex variables and I still didn’t have that good of an understanding of the physical implications of imaginary numbers. Your tutorial definitely helped me with that! Thanks!
April 12th, 2009 at 9:50 pm
Thanks. I’m covering complex analysis in my circuits class. This website helped me to understand the concept.
July 31st, 2009 at 4:41 am
Thanks for the excellent tutorial! Just what I’ve always wanted to know. How about an intuitive explanation for the link between complex numbers and sinusoids? Why, when you raise i to successively higher powers the result is cyclic instead of exponentially increasing?
For an intuitive explanation of Fourier theory, see my web page
http://sharp.bu.edu/~slehar/fourier/fourier.html
August 2nd, 2009 at 10:05 pm
Steven,
I think that the Euler identity may be one of the mathematical links to sinusoids, although it’s not exactly the case you are referring to. A constant (e) raised to an imaginary number is equivalent to a sinusoidal function. I’m not sure that I can make the bridge in my mind of the case you’re talking about though where i raised to a real power can by itself be directly related to a sinusoid.
I would love to checkout your Fourier theory explanation, but the link appears to be broken from my end. I even tried to get there by searching on Google under sharp.bu.edu, but the link there seemed broken too. Is it missing something? Would love to check it out.
Thanks for the comment!
August 3rd, 2009 at 5:41 am
Just checked the intuitive Fourier website again - it’s up! Looks very interesting. Can’t wait to go through it.
August 3rd, 2009 at 7:04 pm
Steven,
Love the Fourier website
http://sharp.bu.edu/~slehar/fourier/fourier.html
- very elegant. I thought I had the idea of Fourier transforms ok, but this imagery is a great tool and makes it crystal clear. I especially liked the way that you show how the images can be transformed back and forth between domains, totally reversibly. Seeing that with real pictures is amazing! More interesting (to me) than the dry equations.
December 9th, 2009 at 7:57 am
Hi, I love this website and the elegant simplicity of your examples. Very well done. One suggestion you might consider would be to prepare an example based upon the synchronization of the phases in three phase power delivery systems. We used imaginary numbers a lot in those problems because the second phase lags the first phase by 120 degrees and the third phase lags by 240 degrees, making your clone example a perfect representation of these type of systems as the phase shifts are captured using imaginary number calculations. Might be something to consider
December 9th, 2009 at 8:01 pm
Hi John,
This suggestion fits really well with my interest in extending these tutorials into real world practical examples, and I’m very interested in your suggestion. I’m starting to envision some animation examples but need a little more education myself in the specifics. Could you point me to a specific problem that you guys may typically solve in class, maybe a homework problem/test problem if you are a teacher, or maybe a scenario if you are in industry?
I’ll also try you by email as it may be easier to communicate.
Thanks for the comment!
August 7th, 2010 at 2:14 am
Fantastic!