[ExI] imaginary numbers

[John Clark]

On Nov 27, 2010, at 2:15 AM, Mike Dougherty wrote:

> Imaginary numbers really do require imagination to grok i^2 = -1.  I'd love to see that represented in a visually intuitive way*.

Make the Real numbers be the horizontal axis of a graph and the imaginary numbers be the vertical axis, now whenever you multiply a Real or Imaginary number by i you can intuitively think about it as rotating it by 90 degrees in a counterclockwise direction. 

Look at i, it sits one unit above the real horizontal axis so draw a line from the real numbers to i, so if you multiply i by i (i^2)  it rotates to become −1, multiply it by i again(i^3) and it becomes −i, multiply it by i again (i^4) and it becomes 1, multiply it by i again (i^5) and you've rotated it a complete 360 degrees and you're right back where you started at i.

It is this property of rotation that makes i so valuable, the best example may be electromagnetism where Maxwell used it to describe how electric and magnetic fields change in the X and Y direction (that is to say in the Real and Imaginary direction) as the wave propagates in the Z direction.

 John K Clark  

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