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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.


4 COMMENTS

  1. The Integral Calculator site looks as if it relates to integral calculus: taking a formula and finding out what function it is derived from, for example such as finding out that the formula 2πr for the circumference of a circle is the derivative of the formula of the area of a circle, πr^2.

    It looks as if it's specifically aimed at integral calculus, so I don't know if it can help with more general resolution of formulas, and it's not strictly relevant to this particular post above.

    Also, the site exhibits some distinctly dodgy behaviour, redirecting me immediately to another site – which suggests spamming.

  2. 1/81 actually is .0123456790 and not .01234567890. This can be proved as follows :
    (10/81)-(1/81) =(9/81)
    .123456790123456790
    (-) .012345679012345679
    _________________________
    .11111111111111111
    _________________________

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