id: 02313394
dt: b
an: 1996f.03725
au: Lang, S.
ti: Introduction to diophantine approximations.
so: Springer, New York, NY (ISBN 0-387-94456-7). 140 p. (1995).
py: 1995
pu: Springer, New York, NY
la: EN
cc: F65
ut: approximation to algebraic numbers; diophantine inequalities; algebraic
numbers; transcendental numbers; asymptotic approximation; quadratic
irrationalities
ci:
li:
ab: The book gives an introduction to continued fractions and diophantine
approximations, readable by undergraduates but also of interest at the
research level because the theory leads immediately into unsolved
problems. Emphasis is placed on classical numbers, and also phenomena
valid for almost all numbers. For instance, the continued fraction for
e is computed. Tables of computations done with W. Adams and H. Trotter
have been added to the original edition to see experimental data
concerning possible conjectures about the behavior of algebraic numbers
with respect to their continued fractions and approximations by
rational numbers. The subject is particularly interesting for
undergraduates who can be put in contact with deep mathematics without
a very extensive building of theories. One general idea is that
algebraic numbers will exhibit a behaviour that is the same as almost
all numbers in a probabilistic sense, except under very specific
structural conditions, namely quadratic numbers. Results for almost all
numbers (due to Khintchine) show an interplax between calculus and
number theory, which will also show undergraduates how analysis mixes
with number theory. (orig)
rv: