Pedagogy_of_LLL_Play.sagews

Some casual thoughts on structured play and learning

My daughter is now a toddler. Her central activity is play. In the course of her play she figures out practical boundary conditions (e.g. climbing, falling, and balance), patterns (e.g. if I uncover my eyes and look surprised mom and dad will say "peek-a-boo" and smile), and surprising facts (e.g. that small box in the kitchen has very thin metal which makes fantastic crumply noises, what joy!).

Play does not provide the principle which underlies these observations. Reflection and analysis are needed to connect the practical to the abstract.

However, it is in play that we build our hypotheses. Intuition makes study more robust (when it is available). Without a "feel" for the facts we stumble around for longer (IMHO).

This school of thought is rooted in Praxis, the intermingling of the theory and the realization of the theory.

x = 2
print x*23

46

x.class_number?

File: /projects/sage/sage-6.9/src/sage/rings/integer.pyx
Signature : x.class_number(self, proof=True)
Docstring :
Returns the class number of the quadratic order with this
discriminant.

INPUT:

* "self" -- an integer congruent to 0 or 1mod4 which is not a
  square

* "proof" (boolean, default "True") -- if "False" then for
  negative disscriminants a faster algorithm is used by the PARI
  library which is known to give incorrect results when the class
  group has many cyclic factors.

OUTPUT:

(integer) the class number of the quadratic order with this
discriminant.

Note: This is not always equal to the number of classes of
  primitive binary quadratic forms of discriminant D, which is
  equal to the narrow class number. The two notions are the same
  when D<0, or D>0 and the fundamental unit of the order has
  negative norm; otherwise the number of classes of forms is twice
  this class number.

EXAMPLES:

   sage: (-163).class_number()
   1
   sage: (-104).class_number()
   6
   sage: [((4*n+1),(4*n+1).class_number()) for n in [21..29]]
   [(85, 2),
   (89, 1),
   (93, 1),
   (97, 1),
   (101, 1),
   (105, 2),
   (109, 1),
   (113, 1),
   (117, 1)]

TESTS:

The integer must not be a square or an error is raised:

   sage: 100.class_number()
   Traceback (most recent call last):
   ...
   ValueError: class_number not defined for square integers

The integer must be 0 or 1 mod 4 or an error is raised:

   sage: 10.class_number()
   Traceback (most recent call last):
   ...
   ValueError: class_number only defined for integers congruent to 0 or 1 modulo 4
   sage: 3.class_number()
   Traceback (most recent call last):
   ...
   ValueError: class_number only defined for integers congruent to 0 or 1 modulo 4

This is a title

This is a subtitle (f(x) = x^2)

M = matrix(ZZ, 10, 10)

M

[0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 0 0 0]

latex(M)

\left(\begin{array}{rrrrrrrrrr} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right)

$$\left(\begin{array}{rrrrrrrrrr} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right)$$

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