Number times Recurring Part of Reciprocal gives 9-Repdigit/Mistake/First Edition
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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): $142,857$