# More cogitations on group theory...

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## More cogitations on group theory...

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## Re: More cogitations on group theory...

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## Re: More cogitations on group theory...

 > Groupoids, categories, rings (clock time), fields (modular > arithmetic), vector spaces, and algebras require a bit more thought, > but I am sure that they can be done. > That's perfect Ed.  Good to hear for another die-hard "group theory for children" dude, a vanishing breed perhaps. My intended audience might actually be older people, including so-called "retirement community" students who have grand kids and want to have inter-generational topics.  Many learned BASIC as kids (the Bill Gates generation). Another group you can massage into a field are integers multiplying modulo N, except not just any integers, only N's totatives.  Back to permworld and Guido's exceedingly simple implementation of Euclid's Algorithm. >>> def gcd(a,b):         while b:                 a , b = b, a % b         return a >>> gcd(12, 4) 4 >>> gcd(12, 5) 1 >>> totatives12 = [m for m in range(12) if gcd(m, 12) == 1 ] >>> totatives12 [1, 5, 7, 11] >>> from random import choice >>> (choice(totatives12) * choice(totatives12)) % 12 11 >>> (choice(totatives12) * choice(totatives12)) % 12 1 >>> (choice(totatives12) * choice(totatives12)) % 12 5 Asserting closer (group property): >>> if (choice(totatives12) * choice(totatives12)) % 12 in totatives12: print (True) True >>> if (choice(totatives12) * choice(totatives12)) % 12 in totatives12: print (True) True >>> if (choice(totatives12) * choice(totatives12)) % 12 in totatives12: print (True) True If the target number is prime instead of composite (e.g. 23 instead of 12), then you have field properties, not just group properties i.e. + is closed as much as * is. You'll find me ranting on mathfuture how high schools bleep over any opportunity to introduce "totative" or "totient" in favor an an exclusive "factor tree" based approach to gcd.  That made more sense before RSA was in every web browser.  In a "how things work" curriculum, one would wish for more computer literacy. http://groups.google.com/group/mathfuture/msg/11005d0c9dc9eba2(I've gotten more correspondence from Milo -- he wants to make sure we all know that Turing at Bletchley Park did *not* solve the German U-boat 5-rotor puzzle, doesn't like how much credit Turing gets). I'd like want to use John Zelle's graphics.py in the module where we draw some Wolfram checkerboard of black and orange rectangles ala New Kind of Science (NKS). We were doing that back in February 2007. http://mail.python.org/pipermail/edu-sig/2007-February/007736.htmlThe new version gets Conway's Game of Life from the same "turtle" (called a "tractor" in farmworld, but the same idea, transferred to all-ASCII waves of grain). Even Mandelbrot is rendered in ASCII "tractor art": http://mybizmo.blogspot.com/2011/05/lesson-planning.htmlThese are ancient threads as far as edu-sig is concerned. We've always been trendy around here. :) > Of course, in every case I am talking about extracting and presenting > the fundamental ideas, and leaving proofs, notations, and all but the > simplest calculations for later. > Of course.  I've got an older bunch but this isn't a course about Group Theory, it's a course about learning to program in the Python computer language, with a backdrop of standard Computer Science courses (Euclid's Algorithm chief among them, at least here on edu-sig). Kirby _______________________________________________ Edu-sig mailing list [hidden email] http://mail.python.org/mailman/listinfo/edu-sig