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Re: In "square root of -1", should we say "minus 1" or "negative 1"?
Posted:
Dec 3, 2012 12:47 PM
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On Mon, Dec 3, 2012 at 6:36 AM, Joe Niederberger
<niederberger@comcast.net> wrote:
>>Then, we can define 'complex numbers' as 'entities' consisting of a combination of real numbers and real numbers (x)i. And also: just do away with the '+' sign in a complex number.
>
> I'm not proposing this a a fully thought-out approach, but rather is just a nascent idea.
>
> *If* negative numbers can be successfully taught as "composite numbers" - a magnitude and a sign, where the sign is constrained to relate to directions along a line,
> (and that composite nature is consistently emphasized), then why not just introduce complex numbers as numbers where the directional component is allowed into the 2D plane.
>
> It would be fun and non-trivial to try to motivate the rest of the story from that starting point.
>
> Cheers,
> Joe N
Complex numbers may be introduced earlier but in my assessment should
be explored using mechanical aids, namely computers with interactive
ways of doing them. Fractals and their generation need to be
introduced, talked about. Programs should be written and run, even if
only simple ones.
The curriculum I talk about in one of my slide shows features
"Tractors" which plow a "field" which is no more than an n x n array
of ASCII characters. The tractors move across this field and compute
whether the ASCII character should change based on Mandelbrot's z =
z*z + c where c is the current position on that plane and z starts at
0. After a few iterations, the tractor will know what to plop down:
http://www.flickr.com/photos/kirbyurner/5645244292/in/photostream/
http://www.visualphotos.com/image/1x7580552/crop_formation_in_form_of_mandelbrot_set
http://www.4dsolutions.net/ocn/python/OST/mandelbrot_eclipse.png
>>> c = -0.3 + 0.3j
>>> z= 0
>>> for _ in range(10):
... z = z * z + c
...
>>> z
(-0.2646679898049498+0.1936749768223369j)
Numbers that "blow up" (spiral out) get "color coded" for being
outside the set, based on how fast they diverge.
Here's a direct quote from the program, taking the absolute value of
the number resulting from the above process, with "depth" the number
of iterations.
# "color coding, worth fine tuning"
if abs( v ) <= 1:
self.marker = "#"
elif abs( v ) <= 10:
self.marker = "@"
elif abs( v ) <= 100000:
self.marker = "."
http://4dsolutions.net/ocn/fractals.html
I'll do this with adults, but I see no reason to keep this material
away from teens. I've explored it with teens many times, including
this year at University of Portland, where a self selected group and I
watched Youtubes (good projector, good sound):
http://www.youtube.com/watch?v=93akxnQ1xxw
(re the above, some teachers of teens might be scarred away by the
title 'Seduction' and showing of cleavage at 32 seconds, but if you
can get passed that... if we're old enough to appreciate Shakespeare
in any way, then we might be mature enough to not be bozos
(fortunately, in my adult ed workshops, I don't have to worry about
keeping it all G-rated)).
Kirby
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