Reciprocal of 83

Theorem

The decimal expansion of the reciprocal of $83$ is as follows:

$\dfrac 1 {83} = 0 \cdotp \dot 01204 \, 81927 \, 71084 \, 33734 \, 93975 \, 90361 \, 44578 \, 31325 \, \dot 3$

This sequence is A021087 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

Performing the calculation using long division:

   0.0120481927710843373493975903614457831325301...
  -------------------------------------------------
83)1.0000000000000000000000000000000000000000000...
     83
   ----
     170    640    310   810    120   260   100
     166    581    249   747     83   249    83
     ---    ---    ---   ---    ---   ---   ---
       400   590    610   630    370   110 
       332   581    581   581    332    83
       ---   ---    ---   ---    ---   ---
        680    90    290   490    380   270
        664    83    249   415    332   249
        ---    --    ---   ---    ---   ---
         160    700   410   750    480   210
          83    664   332   747    415   166
         ---    ---   ---   ---    ---   ---
          770    360   780    300   650   440
          747    332   747    249   581   415
          ---    ---   ---    ---   ---   ---
           230    280   330    510   690   250
           166    249   249    498   664   249
           ---    ---   ---    ---   ---   ---
            640    310   810    120   260    100

$\blacksquare$