Reciprocal of 27

Theorem

The decimal expansion of the reciprocal of $27$ is as follows:

$\dfrac 1 {27} = 0 \cdotp \dot 03 \dot 7$

This sequence is A021027 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

Performing the calculation using long division:

   0.037...
  --------
27)1.00000
     81
     --
     190
     189
     ---
       100
        81
       ---
        ...

It is to be noted that this is because $999 = 27 \times 37$.

$\blacksquare$