Reciprocal of 13

Theorem

The decimal expansion of the reciprocal of $13$ is as follows:

$\dfrac 1 {13} = 0 \cdotp \dot 07692 \dot 3 \ldots$

This sequence is A021017 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

Performing the calculation using long division:

   0.07692307...
  -------------
13)1.00000000...
     91            
   ----    
      90
      78
      --
      120
      117
      ---
        30
        26
        --
         40
         39
         --
          100
           91
          ---
          ...

$\blacksquare$