Reciprocal of 23

Theorem

The decimal expansion of the reciprocal of $23$ is as follows:

$\dfrac 1 {23} = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3$

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:

   0.043478260869565217391304...
  -----------------------------
23)1.00000000000000000000000000
     92     184     23
     --     ---     --
      80     160    170
      69     138    161
      --     ---    ---
      110     220     90
       92     207     69
      ---     ---     --
       180     130    210
       161     115    207
       ---     ---    ---
        190     150     30
        184     138     23
        ---     ---     --
          60     120     70
          46     115     69
          --     ---     --
          140      50     100
          138      46      92
          ---      --     ---
            200     40    ...
            184     23

$\blacksquare$