CHAPTER VI.
PERIODIC DECIMAL FRACTIONS; PERIODIC FRACTIONS; FACTORS
OF 10n=fcl.
Ibn-el-Banna1 (Albanna) in the thirteenth century factored 10n — 1 for small values of n. The Arab Sibt el-M£ridinila 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. Leibniz2 in 1677 noted that 1/n gives rise to a purely periodic fraction to any base 6, later adding the correction that n and b 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 . Wallis3] .
John Wallis3 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 N— 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 N 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. Bernoulli8]. Similar results hold for base 60 in place of 10.
J. H. Lambert4 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
I
n
-.--
10m ' 10m10n 10w102n 10m(10n-l)
John Robertson5 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. Lambert6 concluded from Fermat's theorem that, if a is a prime other than 2 and 5, the number of terms in the period of I/a is a divisor of a— 1. If g is odd and l/g has a period of g — 1 terms, then g is a prime. If l/g has a period of wterrns, but g — 1 is not divisible by m, g is composite. Let I/a have a period of 2m terms; if a is prime, k — 10™ +1 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 2P6Q of a is of period 2m.
Let a be a composite number not divisible by 2, 3 or 5. If I/a has a period of m terms, where m is a prime, each factor of a produces a period
»Cf. E. Lucas, Arithme'tique amusante, 1895, 63-9; Brocard.103
»«Carra de Vaux, Bibliotheca Math., (2), 13, 1899, 33-4.
2Manuscript in Bibliothek Hannover, vol. Ill, 24; XII, 2, Blatt 4; also, III, 25, Blatt 1, seq.,
10, Jan., 1687. Cf. D. Mahnke, Bibliotheca Math., (3), 13, 1912-3, 45-48. 3Treatise of Algebra both historical & practical, London, 1685, ch. 89, 326-S (in manuscript,
1676).
«Acta Helvetica, 3, 1758, 128-132. 6Phil. Trans., London, 58, 1768, 207-213.