Reciprocal of 9091

Theorem

The reciprocal of $9091$ is:

$\dfrac 1 {9091} = 0 \cdotp \dot 00010 \, 9998 \dot 9$

Proof

Performing the calculation using long division:

     0.00010999890001...
    --------------------------------------------
9091)1.00000000000000...
       9091
       ----
        90900
        81819
        -----
         90810
         81819
         -----
          89910
          81819
          -----
           80910
           72728
           -----
            81820
            81819
            -----
                10000
                 9091
                -----
                  ...

$\blacksquare$