Reciprocal of 59

Theorem

The decimal expansion of the reciprocal of $59$ is as follows:

$\dfrac 1 {59} = 0 \cdotp \dot 01694 \, 91525 \, 42372 \, 88135 \, 59322 \, 03389 \, 83050 \, 84745 \, 76271 \, 18644 \, 06779 \, 66 \dot 1$

This sequence is A021063 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division: 

   0.016949152542372881355932203389830508474576271186440677966101...
  ------------------------------------------------------------------
59)1.000000000000000000000000000000000000000000000000000000000000...
     59
   ----
     410   150   170   350    230     500   370   380    570
     354   118   118   295    177     472   354   354    531
     ---   ---   ---   ---    ---     ---   ---   ---    ---
      560   320   520   550    530     280   160   260    390
      531   295   472   531    472     236   118   236    354
      ---   ---   ---   ---    ---     ---   ---   ---    ---
       290   250   480   190    580     440   420   240    360
       236   236   472   177    531     413   413   236    354
       ---   ---   ---   ---    ---     ---   ---   ---    ---
        540   140    80   130    490     270    70    400    60
        531   118    59   118    472     236    59    354    59
        ---   ---    --   ---    ---     ---    --    ---    --
          90   220   210   120    180     340   110    460    100
          59   177   177   118    177     295    59    413     59
          --   ---   ---   ---    ---     ---   ---    ---    ---
          310   430   330    200    300    450   510    470   ...
          295   413   295    177    295    413   472    413
          ---   ---   ---    ---    ---    ---   ---    ---
           150   170   350    230     500   370   380    570

$\blacksquare$

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