id: 06408322
dt: j
an: 2015b.00643
au: Griffiths, Martin
ti: Continued fractions: rational approximations in the classroom. II.
so: Math. Sch. (Leicester) 39, No. 4, 15-17 (2010).
py: 2010
pu: Mathematical Association (MA), Leicester
la: EN
cc: F60 F50 N20
ut: finite continued fractions; infinite continued fractions; Euclid’s
algorithm; limits; periodic infinite continued fractions; irrational
numbers; irrational roots of quadratic equations with rational
coefficients; surds; roots; decimal fractions; number representations;
infinite continued fraction expansions; rational approximation
ci: ME 2011b.01005
li:
ab: From the introduction: In [the author, ibid. 39, No. 2, 16‒17 (2010; ME
2011b.01005)], we looked at simple ways of obtaining ever-better
rational approximations to numbers of the form $\sqrt m$, where $m\in
\mathbb{N}$ is not a perfect square. We also showed how to derive
rational approximations to the transcendental number $e$. In both cases
the approximations were calculated by way of recursive procedures.
While perfectly acceptable, these methods may have appeared to be
rather ad hoc. As mathematicians, rather than having to resort to ad
hoc procedures in this way, we would much prefer to develop a general
method for obtaining these approximations. Thus, in the second article
of this series, we start to piece together a coherent mathematical
framework for dealing with the problem of approximating irrational
numbers by rational ones. It is hoped that teachers might wish to
create their own classroom activities based on the ensuing mathematical
ideas.
rv: