13
$\begingroup$

There appears to be an interesting pattern in the decimal expansion of $\dfrac1{243}$:

$\frac1{243}=0.\overline{004115226337448559670781893}$

I was wondering if anyone could clarify how this comes about?

  • 0
    By the way, $243=3^5$. I wonder if this is the only fifth power with this property...2018-05-25

1 Answers 1

36

$\frac{1}{243}=\frac{1}{333}+\frac{10}{8991}$

$\frac{1}{333}=.\overline{003}$

$\frac{1}{8991}=.\overline{000111222333444555666777889}=\frac{111}{998001}=\frac{111}{10^6-2\cdot10^3+1}$

  • 0
    @wim: fixed. Thanks.2011-12-12