
06655907
j
2016f.00942
Lyons, Christopher
The secret life of $1/n$: a journey far beyond the decimal point.
Math. Enthus. 13, No. 3, 189216 (2016).
2016
Information Age Publishing (IAP), Charlotte, NC; University of Montana, Department of Mathematical Sciences, Missoula, MT
EN
F60
F40
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
Zbl 0827.11004
Zbl 0839.11049
http://scholarworks.umt.edu/tme/vol13/iss3/3
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, 381386 (1994; Zbl 0827.11004); Am. Math. Mon. 101, No. 10, 9971001 (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.