Reciprocal of 81

Theorem

The decimal expansion of the reciprocal of $81$ has a particularly interesting pattern:

$\dfrac 1 {81} = 0 \cdotp \dot 01234 \, 567 \dot 9$

This sequence is A021085 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

Performing the calculation using long division:

   0.0123456790...
  ----------------
81)1.0000000000000
     81
     --
     190
     162
     ---
      280
      243
      ---
       370
       324
       ---
        460
        405
        ---
         550
         486
         ---
          640
          567
           --
           730
           729
           ---
             100
              81
             ---
             ...

$\blacksquare$

Also see

• Reciprocal of Square of 1 Less than Number Base

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $81$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $81$