[Edu-sig] Re: Idle play
Kirby Urner
urnerk at qwest.net
Fri Apr 29 15:47:54 CEST 2005
> To break no rules - PyGeo is written in Python. In my case, necessarily.
>
>
> Art
Good theorem. In a newbie geometry class, I wish they'd run through a lot
more like Pascal's (mention his age when he hit on it), using something a
lot like PyGeo (or just use PyGeo why not?).
A big thing for me, before getting into proofs, is to just make clear what a
theorem *asserts* (pretty clear in this case, not always so obvious).
I think too often we're in too big a rush to deal with the proof (which is
likely pre-supplied, if the theorem is old and/or interesting). But with
newbies, let's just linger on the wealth of assertions in our treasure chest
(like this one from Pascal).
That'd built up interest and appreciation for what the millennia have
bequeathed to us. Proofs later (e.g. I like Karl von Staudt's re V + F = E
+ 2).
Another example: Fermat's Little Theorem (not Last, although that's a good
one too), itself a special case of one of Euler's:
Fermat: if gcd(b,p) == 1 and p prime: assert pow(b, p-1, p) == 1.
Or Euler's: if gcd(b,n) == 1: assert pow(b, tot(n), n) == 1, where tot(n) =
len([x in range(1, n) if gcd(x, n)==1]) -- see 'totatives' at MathWorld.
Note that I'm deliberately using Python as my language of expression for
these proofs. In the geometry/programming course I'm imagining, that'd make
oh-so-much sense.
Kirby
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