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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


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 thoughtout 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 nontrivial 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 Grated)).
Kirby



