Reciprocal of 43

Theorem

The decimal expansion of the reciprocal of $43$ is as follows:

$\dfrac 1 {43} = 0 \cdotp \dot 02325 \, 58139 \, 53488 \, 37209 \, \dot 3$

This sequence is A021047 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Performing the calculation using long division: 

   0.02325581395348837209302...
  -----------------------------
43)1.00000000000000000000000...
     86        
   ----
     140    410    310
     129    387    301
     ---    ---    ---
      110    230     90
       86    215     86
      ---    ---     --
       240    150     400
       215    129     387
       ---    ---     ---
        250    210     130 
        215    172     129
        ---    ---     ---
         350    380      100
         344    344       86
         ---    ---      ---
           60    360     ...
           43    344
           --    ---
           170    160
           129    129
           ---    ---
            410    310

$\blacksquare$

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