[ExI] imaginary numbers

[Mike Dougherty]

2010/11/27 John Clark <jonkc at bellsouth.net>

> 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.
>
>
if you put real numbers on X and real numbers on Y then the product is the
number of unit squares that cover the area.  So a 5 x 5 square is literally
a 25 unit square.  A 5i x 5i square is a negative 25 unit square?  What does
the negative mean in that sense?  Am I missing something fundamental about i
?  (I think I am)  This is one of the rare occasions where I'm not being
facetious or tongue-in-cheek.
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