Another example of Pythonic mathematics, two posts so far at: http://mathforum.org/kb/message.jspa?messageID=4019537&tstart=0 Here's a new edition, better formatting, typo fixed: ======== I've got a gig coming up, so thought I'd do some notes here. It's a Python class, and I'll be using math topics to motivate problem solving. For example, there's no factorial built-in = an opportunity to program: from operator import mul def factorial(n): return reduce(mul, range(1,n+1))
factorial(10) 3628800 factorial(100) 93326215443944152681699238856266700490715968264381621468592963 89521759999322991560894146397615651828625369792082722375825118 5210916864000000000000000000000000L
The course starts out with data structures, a foundation on which to build. Data structures are for storing and later accessing information. In Python, this might involve pickling, i.e. serialization of objects. Or we might write out to text files, use a SQL engine, save to the ZODB, or pipe out through sockets to some unspecified receiver with a TCP/IP address (an IP number). We might use Twisted for this latter purpose. On the math side, the data structure we call a dictionary is a collection type known as a mapping, in that it associates or pairs keys with values. We may pick our keys and values from the same domain and codomain, in which case we have a mapping of maybe integers into integers, or floats into floats, or characters into characters. This last mapping (characters into characters) is a jumping off point to simple encryption by means of letter substitution (we treat the space as just one more letter):
import string string.uppercase 'ABCDEFGHIJKLMNOPQRSTUVWXYZ\x8a\x8c\x8e\x9f\xc0\xc1\xc2\xc3\xc4\xc5\xc6\ xc7\xc8\xc9\xca\xcb\xcc\xcd\xce\xcf\xd0\xd1\xd2\xd3\xd4\xd5\xd6\xd8\xd9\ xda\xdb\xdc\xdd\xde' myset = string.uppercase[:26]+' ' myset 'ABCDEFGHIJKLMNOPQRSTUVWXYZ ' from random import shuffle domain = list(myset) shuffle(myset)
Traceback (most recent call last): File "<pyshell#15>", line 1, in -toplevel- shuffle(myset) File "c:\python24\lib\random.py", line 269, in shuffle x[i], x[j] = x[j], x[i] TypeError: object does not support item assignment
shuffle(list(myset)) myset 'ABCDEFGHIJKLMNOPQRSTUVWXYZ ' outputs = domain[:] shuffle(outputs) myfunc = dict(zip(domain,outputs)) myfunc {' ': 'C', 'A': 'R', 'C': 'V', 'B': 'P', 'E': 'N', 'D': 'I', 'G': 'K', 'F' : 'A', 'I': 'Q', 'H': 'Y', 'K': 'X', 'J': 'G', 'M': 'D', 'L': 'O', 'O': 'W', 'N': 'F', 'Q': 'S', 'P': 'J', 'S': ' ', 'R': 'E', 'U': 'B', 'T': 'U', 'W': 'Z', 'V': 'L', 'Y': 'H', 'X': 'M', 'Z': 'T'} p = 'THE RAIN IN SPAIN STAYS MAINLY IN THE PLAIN' def encrypt(p, subdict): e = '' for c in p: e = e + subdict[c] return e
encrypt(p)
Traceback (most recent call last): File "<pyshell#29>", line 1, in -toplevel- encrypt(p) TypeError: encrypt() takes exactly 2 arguments (1 given)
encrypt(p, myfunc) 'UYNCERQFCQFC JRQFC URH CDRQFOHCQFCUYNCJORQF'
Where we're going with this is towards the relations and functions concepts. Relations pair members in two sets, perhaps the same key (domain element) with two or more different values (range element), while functions guarantee a unique and determined value for a given key. OK for Relation, but not Function: R(r) --> b R(r) --> c OK for Function, and for Relation: f(r) --> b f(s) --> b In addition to defining functions by roster (explicit pairing), we'll be defining functions according to rules, which is where we transition from exploring data structures to writing short top-level functions. Or functions will often be about sequences, e.g. figurate and polyhedral, Fibonacci, maybe chaotic.[1] ===== ADDENDUM: Note to the Teacher ====== [2] The Python community is in a sense an ethnicity, in that there's a lot of shared jargon and concepts. This is true in mathematics as well (sub-ethnicities within ethnicities, e.g. topologists). Per an earlier thread here at the Math Forum: all math is ethnomath. This math/CS hybrid involves mixing heroes from disparate domains. We learn about Descartes and Fermat, sure, but also about Von Neumann, Dijkstra, Ada Byron. http://ei.cs.vt.edu/~history/VonNeumann.html http://en.wikipedia.org/wiki/Edsger_Dijkstra http://www.google.com/search?hl=en&q=Ada+Byron&btnG=Google+Search We will also explore the historical beginnings of the World Wide Web: e.g. Vannevar Bush (MEMEX), Ted Nelson (Xanadu), and Tim Berners-Lee (W3).[3] http://en.wikipedia.org/wiki/Vannevar_Bush http://xanadu.com/ http://www.w3.org/People/Berners-Lee/ I'm suggesting this as an alternative to precalc/calc, i.e. a discrete math track in K-12, perhaps leading to a different mix of majors, but still college prep and still an on ramp into engineering and the sciences for the most part, although these skills also transfer to art/design majors in other departments as well. Kirby [1] added paragraph for edu-sig edition [2] this was a separate post, originally. [3] http://worldgame.blogspot.com/2004/12/dj-vu.html