Reciprocal of 1089

Theorem

The reciprocal of $1089$ is:

$\dfrac 1 {1089} = 0 \cdotp \dot 00091 \, 82736 \, 45546 \, 37281 \, 9 \dot 1$

This sequence is A113657 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Note that it consists mainly of the unit multiples of $9$ from $1$ to $9$ concatenated.

Proof

Performing the calculation using long division:

     0.00091827364554637281910009...
    --------------------------------------------
1089)1.00000000000000000000000000...
       9801    5445    1089
       ----    ----    ----
        1990    5950    9910
        1089    5445    9801
        ----    ----    ----
         9010    5050    1090
         8712    4356    1089
         ----    ----    ----
          2980    6940      10000
          2178    6534       9801
          ----    ----       ----
           8020    4060      ....
           7623    3267
           ----    ----
            3970    7930
            3267    7623
            ----    ----
             7030    3070
             6534    2178
             ----    ----
              4960    8920
              4356    8712
              ----    ----
               6040    2080
               5445    1089        

$\blacksquare$

Historical Note

According to David Wells in his $1997$ work Curious and Interesting Numbers, 2nd ed., this result can be attributed to N. Goddwin, but no corroboration or expansion of this has yet been found.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $1089$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $1089$