[Edu-sig] Pythonic Math must include...

[kirby urner]

Yes, note that my Pascal's includes it, with an embedded zip.  Another place
list comprehension comes up is in our naive definition of totatives as:

def totative(n):
    return [ t for t in range(1, n) if gcd(t, n) == 1]

i.e. all 0 < t < n such that (t, n) have no factors in common (are
relatively prime).  Then our totient function is simply the len of this
list, giving us a quick way to assert (test) Euler's theorem:  b **
totient(d) % d == 1 where gcd(b,d)==1.  There's an easy enough proof in
'Number' by Midhat Gazale, a must have on our syllabus (I'm suggesting).

Kirby


On Thu, Jan 15, 2009 at 6:35 AM, michel paul <mpaul213 at gmail.com> wrote:

> I like this.
>
> I think another 'must include' for math classes would be list comprehension
> syntax.  Not an algorithm in itself, but an important way of thinking.  It's
> what we try to get them to do using set notation, but in math classes it
> seems simply like a formality for describing domains, nothing more.  In
> Python, it DOES stuff.
>
> - Michel
>
> 2009/1/14 kirby urner <kirby.urner at gmail.com>
>
>> Candidates:
>>
>> "Must include" would be like an intersection of many sets in a Venn
>> Diagram, where we all gave our favorite movies and a very few, such as
>> 'Bagdad Cafe' or 'Wendy and Lucy' or... proved common to us all (no
>> suggesting those'd be -- just personal favorites).
>>
>> In this category, three candidates come to mind right away:
>>
>> Guido's for the gcd:
>>
>> def gcd(a,b):
>>     while b:
>>         a, b = b, a % b
>>     return a
>>
>> Then two generics we've all seen many times, generators for Pascal's
>> Triangle and Fibonacci Sequence respectively:
>>
>> def pascal():
>>     """
>>     Pascal's Triangle **
>>     """
>>     row = [1]
>>     while True:
>>         yield row
>>         row = [ a + b for a, b in zip( [0] + row, row + [0] ) ]
>>
>> and:
>>
>> def fibonacci(a=0, b=1):
>>     while True:
>>         yield a
>>         a, b = a + b, a
>>
>> IDLE 1.2.1
>> >>> from first_steps import *
>> >>> gcd(51, 34)
>> 17
>> >>> g = pascal()
>> >>> g.next()
>> [1]
>> >>> g.next()
>> [1, 1]
>> >>> g.next()
>> [1, 2, 1]
>> >>> g.next()
>> [1, 3, 3, 1]
>> >>> f = fibonacci()
>> >>> f.next()
>> 0
>> >>> f.next()
>> 1
>> >>> f.next()
>> 1
>> >>> f.next()
>> 2
>> >>> f.next()
>> 3
>> >>> f.next()
>> 5
>>
>> Check 'em out in kid-friendly Akbar font (derives from Matt Groening of
>> Simpsons fame): http://www.wobblymusic.com/groening/akbar.html
>>
>> http://www.flickr.com/photos/17157315@N00/3197681869/sizes/o/
>>
>> ( feel free to link or embed in your gnu math website )
>>
>> I'm not claiming these are the only ways to write these.  I do think it's
>> a feature, not a bug, that I'm eschewing recursion in all three.  Will get
>> to that later, maybe in Scheme just like the Scheme folks would prefer (big
>> lambda instead of little, which latter I saw put to good use at PPUG last
>> night, well attended (about 30)).
>>
>> http://mybizmo.blogspot.com/2009/01/ppug-2009113.html
>>
>> Rationale:
>>
>> In terms of curriculum, these belong together for a host of reasons, not
>> just that we want students to use generators to explore On-Line Encyclopedia
>> of Integer Sequences type sequences.  Pascal's Triangle actually contains
>> Fibonaccis along successive diagonals but more important we're laying the
>> foundation for figurate and polyhedral ball packings ala The Book of
>> Numbers, Synergetics, other late 20th century distillations (of math and
>> philosophy respectively).  Fibonaccis converge to Phi i.e. (1 + math.sqrt(5)
>> )/2.  gcd will be critical in our relative primality checks, leading up to
>> Euler's Theorem thence RSA, per the review below (a literature search from
>> my cube at CubeSpace on Grand Ave):
>>
>> http://cubespacepdx.com/
>> http://mathforum.org/kb/thread.jspa?threadID=1885121&tstart=0
>> http://www.flickr.com/photos/17157315@N00/3195148912/
>>
>> Remember, every browser has SSL, using RSA for handshaking, so it's not
>> like we're giving them irrelevant info.  Number theory goes into every
>> DirecTV box thanks to NDS, other companies making use of this powerful
>> public method.^^
>>
>> You should understand, as a supermarket manager or museum administrator,
>> something about encryption, security, what's tough to the crack and what's
>> not.  The battle to make RSA public property was hard won, so it's not like
>> our public school system is eager to surrender it back to obscurity.
>> Student geek wannabes perk up at the thought of getting how this works, not
>> hard to show in Javascript and/or Python.  Makes school more interesting, to
>> be getting the low-down.
>>
>> By the same token, corporate trainers not having the luxury of doing the
>> whole nine yards in our revamped grades 8-12, have the ability to excerpt
>> specific juicy parts for the walk of life they're in.
>>
>> Our maths have a biological flavor, thanks to Spore, thanks to Sims.  We
>> do a Biotum class almost right away ("Hello World" is maybe part of it's
>> __repr__ ?).  I'm definitely tilting this towards the health professions,
>> just as I did our First Person Physics campaign (Dr. Bob Fuller or leader,
>> University of Nebraska emeritus).
>>
>> The reason for using bizarre charactersets in the group theory piece is we
>> want to get their attention off numbers and onto something more generic,
>> could be pictograms, icons, pictures of vegetables...
>>
>> Feedback welcome,
>>
>> Kirby
>>
>>
>> ** http://www.flickr.com/photos/17157315@N00/3198473850/sizes/l/
>> ^^
>> http://www.allbusiness.com/media-telecommunications/telecommunications/5923555-1.html
>>
>>
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