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Mapping Imaginary to Physical - Lesson 1
Contrary to its name, the Imaginary Number exists in a similar fashion to the Real number, often describing physical characteristics that we can detect, observe, and measure. Common applications are circuit analysis in electrical engineering, and vibration analysis in mechanical engineering.
In Lesson 1, we demonstrate this link to the physical with a very simple example - quantifying the size of a photograph. We build on this fundamental concept later to demonstrate utility in real world applications.
Got Milk?
OK, first the dry stuff from school; we’ll be very brief. The imaginary unit, i, is shown right. Rollover once for implications. Rollover again for the imaginary number, defined as the imaginary unit multiplied by a real number; some examples of imaginary numbers: 4i, or -2.7i.
This definition gives the imaginary number special properties, putting it on a separate dimension, or number line, from the real number. This second dimension is managed by using the complex number.
The 2 Dimensional Complex Number
The complex number, represented here as z, packages the real and imaginary number into a single variable. It is defined as z=a+bi where a and b are real numbers.
The complex number is useful for describing two dimensional variables. The real part of the complex number quantifies one dimension, and the imaginary part quantifies another. Rollover image right for an example.
Complex Plane Imaging
OK. So the complex number can be used for two dimensional things, but we need a more visual representation - the complex plane. The complex plane is simply a graph onto which we plot the Real and Imaginary parts of our complex number. We have placed a System Variable, S (red dot) in quadrant I, to represent an initial physical state (the photograph on the left).
You can play on the complex plane above by dragging on either the red, purple, or orange handles. As you play around, the photograph will change size and proportion in accordance with the complex system state. This represents the intimate link between a complex variable and a physical system. In up coming lessons we will apply this simple concept to the more real world vibration example you saw on the home page.
Lesson 1 Summary
Imaginary numbers are used in real world applications to quantify physical characteristics. They are typically paired with real numbers to form a complex number, which is useful for describing two dimensional characteristics. Engineers like to plot two dimensional variables on the complex plane to get a visual snapshot of their system.
In lesson 2 we will showcase the power of complex numbers in describing dynamic systems. Press on to learn more…
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
June 9th, 2008 at 7:01 am
Love the 3D animations. First site I’ve seen that really looks interactively at HOW imaginary numbers are used. Thanks!!
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October 29th, 2008 at 9:35 pm
Glad it provided some inspiration. You must mean a “new” wife because the web was not around 50 years ago!
November 6th, 2008 at 4:57 pm
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November 6th, 2008 at 5:00 pm
awesome dawg! i love math.
imaginary numbers still don’t make sense to me though….
November 20th, 2008 at 6:18 pm
This is just great!
March 10th, 2009 at 5:15 pm
Hey I was looking around to find some ways to inspire my weak algebra students on the importance of the imaginary unit. However your site made me question it… I understand needing a way to put two dimensions in one equation but what does sizing a picture have to do with the square root of negative 1?
March 11th, 2009 at 8:52 am
Ms. Math Teacher: Sizing a photograph interactively is simply an intermediate step to illustrate that a variable can contain two dimensions. Internalizing the difference between one dimensional variables and two dimensional variables is a critical step to understanding complex numbers, and if we were to start with the actual application (electrical circuit, vibration motion, etc), getting that fundamental dimension concept across may be a non-starter. So the photograph concept is simply a way to get the idea across simply and easily via analogy.
Once this concept is solid, we move through lessons 2,3, then the bonus lesson to show that in real world applications, the imaginary unit is the transformation that mathematically describes the link between the real and imaginary axes (see the bonus lesson). This last bonus lesson completes the picture assembling the two dimensional concept with how the imaginary unit links both dimensions mathematically. Hope this helps.
August 3rd, 2009 at 3:34 am
This site is great. Excellent job on the illustrations & interactive diagrams, everything is well explained, keep up the good work.
Thank you
August 3rd, 2009 at 5:51 am
Thanks Ukash,
I’m still working ever so slowly but steadily on a highly interactive tutorial for an intuitive understanding of circuit analysis. I’m excited about it but of course it has it’s tedious parts to get through. This kind of positive feedback keeps me going!
February 17th, 2010 at 3:33 pm
Awesome! This website makes math cool and fun to learn. I get off the bus and go on this site!
February 18th, 2010 at 1:47 pm
Ur amazing whoever u r–keep up the work> i really appreaciate it
May 24th, 2010 at 9:06 am
Heyy yall, my name is Raul. I’m sitting in math class just having a swell time learning about how imaginary numbers change how this little cutie pie looks in this picture. Love yall