Number times Recurring Part of Reciprocal gives 9-Repdigit/Mistake/First Edition

Source Work

1986: David Wells: Curious and Interesting Numbers:

The Dictionary

$142,857$

Mistake

This is a property of all the periods of repeating decimals. If the period of $n$ is multiplied by $n$, the result is as many $9$s as there are digits in $n$.

Correction

This should read:

This is a property of all the periods of reciprocals of (strictly) positive integers. If the period of $1 / n$ is multiplied by $n$, the result is as many $9$s as there are digits in the period of $1 / n$.

In Curious and Interesting Numbers, 2nd ed. of $1997$, this has been partially corrected to:

This is a property of all the periods of repeating decimals. If the period of $n$ is multiplied by $n$, the result is as many $9$s as there are digits in the period of $1 / n$.

But even then, this appears not to be true.

The number $6$, for example, does not have this property.

$\dfrac 1 6 = 0 \cdotp 1 \dot 6$

but:

$6 \times 6 = 36$

which is not equal to $9$ as the theorem states ought to be the case.

Research is ongoing as to what the theorem should actually say.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $142,857$