Reciprocal of 47

Theorem

The decimal expansion of the reciprocal of $47$ is as follows:

$\dfrac 1 {47} = 0 \cdotp \dot 02127 \, 65957 \, 44680 \, 85106 \, 38297 \, 87234 \, 04255 \, 31914 \, 89361 \, \dot 7$

This sequence is A021051 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division: 

   0.021276595744680851063829787234042553191489361702...
  ------------------------------------------------------
47)1.000000000000000000000000000000000000000000000000...
     94            
   ----    
      60   270    400    140   160    150   440
      47   235    376     94   141    141   423
      --   ---    ---    ---   ---    ---   ---
      130   350    240    460   190     90   170
       94   329    235    423   188     47   141
      ---   ---    ---    ---   ---     --   ---
       360   210     50    370    200   430   290
       329   188     47    329    188   423   282
       ---   ---     --    ---    ---   ---   ---
        310   220     300   410    120    70    80
        282   188     282   376     94    47    47
        ---   ---     ---   ---    ---    --    --
         280   320     180   340    260   230   330
         235   282     141   329    235   188   329
         ---   ---     ---   ---    ---   ---   ---
          450   380     390   110    250   420    100
          423   376     376    94    235   376     94
          ---   ---     ---   ---    ---   ---    ---
           270    40     140   160    150   440   ...

$\blacksquare$

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