10

Recurring Decimals

In the previous chapter, attention was drawn to the peculiar
properties of the number 142857. There are, in addition to
this, many numbers—usually much greater—with similar or
related properties and they are all of one type.

The number 142857 is not just a haphazard stringing
together of digits but represents the cycle of recurring deci-
mals which results from the division of unity by the number
7. Thus, 1=-14285% where the dots over the first and last
digits mean that this set of six digits is repeated over and
over again always in the same order. In the same way, the
number 076923 (to which reference has also already been
made) is derived from the decimal equivalent of the fraction
Ts5(=076923).

These sets of digits which recur over and over again are
referred to here as recurring decimals, and these always re-
sult from the division of unity by a prime number (other than
2 or 5) or by any number containing such a prime number
as one of its factors. The division of unity by 2 or 5 or any
power of these numbers will not give rise to recurring deci-
mals, but division by any number having 2 or 5 as one of its
factors will nevertheless give recurring decimals if the divisor
has any other prime as another factor.

In some cases digits recur immediately (4=-3) or follow-
ing one other digit which is itself not repeated (1=-16). In
most cases, however, there results a cycle of decimals in which
a group of digits (as distinct from a single digit) recurs in the
same order.

If the two numbers 142857 and 076923 are re-examined,
the following two facts emerge:

7 97

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