CHAPTER VI. 
PERIODIC DECIMAL FRACTIONS; PERIODIC FRACTIONS: FACTORS 
OF I0"±1. 
Ibn-el-Banna^ (Albanna) in the thirteenth century factored 10" — 1 for 
small values of n. The Arab Sibt el-Maridini^" in the fifteenth century 
noted that in the sexagesimal division of 47° 50' by 1° 25' the quotient 
has a period of eight terms. 
G. W. Leibniz^ in 1677 noted that \/n gives rise to a purely periodic 
fraction to any base h, later adding the correction that n and h must be 
relatively prime. The length of the period of the decimal fraction for 1/n, 
where n is prime to 10, is a divisor of n — 1 [erroneous for n = 21 ; cf . Wallis^] . 
John Wallis^ noted that, if N has a prime factor other than 2 and 5, the 
reduced fraction M/N equals an unending decimal fraction with a repetend 
of at most A^ — 1 digits. If N is not divisible by 2 or 5, the period has two 
digits if N divides 99, but not 9; three digits if A^ divides 999, but not 99. 
The period of 1/21 has six digits and 6 is not a divisor of 21 — 1. The 
length of the period for the reciprocal of a product equals the 1. c. m. of 
the lengths of the periods of the reciprocals of the factors [cf. Bernoulli^]. 
Similar results hold for base 60 in place of 10. 
J. H. Lambert^ noted that all periodic decimal fractions arise from 
rational fractions; if the period p has n digits and is preceded by a decimal 
with m digits, we have 
lO'" ' lO'^lO" lO'^lO^" lO'^ClO^-l) 
John Robertson^ noted that a pure periodic decimal with a period P of 
k digits equals P/9 ... 9, where there are k digits 9. 
J. H. Lambert^ concluded from Fermat's theorem that, if a is a prime 
other than 2 and 5, the number of terms in the period of \/a is a divisor 
of a — 1. If S' is odd and \/g has a period oi g — 1 terms, then ^ is a prime. 
If \/g has a period of m terms, but ^ — 1 is not divisible by m, g is composite. 
Let 1/a have a period of 2m terms; if a is prime, A; = lO'^+l is divisible by 
a; if a is composite, k and a have a common factor; if k is divisible by a 
and if m is prime, each factor other than 2^5^ of a is of period 2m. 
Let a be a composite number not divisible by 2, 3 or 5. If 1/a has a 
period of m terms, where w is a prime, each factor of a produces a period 
'Cf. E. Lucas, Arithm^tique amusante, 1895, 63-9; Brocard.'o^ 
i«Carra de Vaux, Bibliotheca Matb., (2), 13, 1899, 33-4. 
^Manuscript in Bibliothek Hannover, vol. Ill, 24; XII, 2, Blatt 4; also. III, 25, Blatt 1, seq., 
10, Jan., 1687. Cf. D. Mahnke, BibUotheca Math., (3), 13, 1912-3, 45-48. 
^Treatise of Algebra both historical & practical, London, 1685, ch. 89, 326-8 (in manuscript, 
1676). 
*Acta Helvetica, 3, 1758, 128-132. 
»Phil. Trans., London, 58, 1768, 207-213. 
"Nova Acta Eruditorum, Lipsiae, 1769, 107-128. 
159 
