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Some Properties of Pure Recurring Decimals

Published online by Cambridge University Press:  03 November 2016

J.S. Batty
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Extract

The earlier sections of this account are concerned with some standard properties of pure recurring decimals. These include the cyclic property, illustrated for example by the decimals representing the six proper fractions with denominator 7, each of which has a period using a cyclic permutation of the digits 142857 It was a question concerning this property, raised by a student, which led to the writer's interest in the properties considered in the later sections. References are given in Dickson's History of the Theory of Numbers, Vol. I, Ch. 6.

Type
Research Article
Information
The Mathematical Gazette , Volume 38 , Issue 324 , May 1954 , pp. 90 - 95
Copyright
Copyright © The Mathematical Association 1954

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References

page 90 note * Decimals to the base 10 are used throughout; the changes necessary for a general base, prime to q, are trivial.

page 90 note † Hardy and Wright, Ch. 9.

page 91 note * Dickson, p. 166.

page 91 note † For example, Hardy and Wright, Ch. 9; Ore, n’umber Theory and its History, Ch. 13.

page 91 note ‡ Dickson, p. 166.

page 93 note * This is the only with period 9, the smallest odd composite period.

page 95 note * If (5.3) is satisfied this result is trivial, the multiple being unity