We started from the Euclidean algorithm, that was designed to find
the 
of two integers 
, and we have described the
change of understanding introduced circa 1800 : the main result of
that algorithm are the Bezout's coefficients 
that satisfy
. Thereafter, we have described
how the initial Bezout's algorithm, designed for integers, can be
used for polynomials, and to approximate irrationals numbers, as well
as holomorphic functions.
From a syntactical point of view, the whole thing can be described by an unique main procedure, the ``details'' being encapsulated into auxiliary functions, as summarized in Algorithm 7.
The matricial writing of the algorithm leads to a better understanding
of what happens, since it gives versatile meanings to the many objects
that occur during computations. Homographies can be introduced in
a more natural way, and relations like 
can be interpreted together as a Bezout's formula, a determinant,
or the fact that 
is really an homography, and not a constant.
By the way, we have described the algorithm using centered remainders
(in 
) and characterized the convergents that are skipped by
that acceleration. The same ideas can be applied to Padé expansion,
using two terms quotients and giving Jacobi or Stieltjes fractions.
Continued fractions are an efficient computing method together with a powerful theoretical tool. It is useful to understand how they work, how to program them and how to show them.
In a second place, the computations can be followed more easily, the applicability to teaching purpose being plain.
