998001 Redux

I posted in an earlier article about how 1 / 998001 yields the decimal:-

0.000001002003004005006007008009 ... 995996997998999 ...

where the recurring element shows all of the three digit numbers, in sequence, from 000 to 999.

I commented on how 1 / 99980001 similarly yields:-

0.00000001000200030004000500060007 ... 999499959996999799989999

and 1 / 9801 yields:-

0.000102030405060708091011121314 ... 919293949596979899

I commented, further, how

9999 x 9999 = 99980001
999 x 999 = 998001
99 x 99 = 9801

What about the most trivial case, 9 x 9?

9 x 9 = 81

By this right, 1 / 81 should have a very specific pattern: the recurring portion has to be 0123456789 ...

A quick check confirms -

1 / 81 = 0.012345678901234567890123456789 ...

That would make 80 / 81

80 / 81 = 0.98765432109876543210 ...

Things get interesting when you look at 10/81:-

10/81 = 0.1234567890 ...

Just to confirm ...

(10 / 81) - (1 / 81) = 9 / 81 = 1 / 9

0.12345678901234567890 ...
0.01234567890123456789 ...
0.11111111111111111111 ...

which is, of course, the decimal expression of 1 / 9.