id: 06655907
dt: j
an: 2016f.00942
au: Lyons, Christopher
ti: The secret life of $1/n$: a journey far beyond the decimal point.
so: Math. Enthus. 13, No. 3, 189-216 (2016).
py: 2016
pu: Information Age Publishing (IAP), Charlotte, NC; University of Montana,
Department of Mathematical Sciences, Missoula, MT
la: EN
cc: F60 F40
ut: unit fractions; decimal expansions; digits; length; period; rational
numbers; primitive roots; class numbers; number theory; magic and
mystery; divisibility; Euler’s theorem; random look; digit frequency;
binary quadratic forms; binary expansion; abstract algebra; ideal class
group; imaginary quadratic fields
ci: Zbl 0827.11004; Zbl 0839.11049
li: http://scholarworks.umt.edu/tme/vol13/iss3/3
ab: Summary: The decimal expansions of the numbers $1/n$ (such as
$1/3=0.3333\dots$, $1/7=0.142857\dots$) are most often viewed as tools
for approximating quantities to a desired degree of accuracy. The aim
of this exposition is to show how these modest expressions in fact
deserve have much more to offer, particularly in the case when the
expansions are infinitely long. First we discuss how simply asking
about the period (that is, the length of the repeating sequence of
digits) of the decimal expansion of $1/n$ naturally leads to more
sophisticated ideas from elementary number theory, as well as to
unsolved mathematical problems. Then we describe a surprising theorem
of {\it K. Girstmair} [Acta Arith. 67, No. 4, 381‒386 (1994; Zbl
0827.11004); Am. Math. Mon. 101, No. 10, 997‒1001 (1994; Zbl
0839.11049)] showing that the digits of the decimal expansion of $1/p$,
for certain primes $p$, secretly contain deep facts that have long
delighted algebraic number theorists.
rv: