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# Cool Fractions

Preamble: I (S.W.) have been told that some of you are expecting something really cool from the PoW for the 1000th problem of the program. Here is one that is a little odd, but nevertheless, very unusual, with a truly surprising answer.

The decimal representation of 10/81 is cool:

0.123456790 123456790 123456790 123456790 and so on forever.

But the missing 8 is a bit of a flaw. The easiest way to remedy this is to subtract 1/10^9,

(10/81) - 1/1000000000 = 9999999919/ 81000000000 =
0.123456789 123456790 123456790 123456790 and so on forever.

But this too is less than perfectly cool since the first 9 is not followed by a 0 or a 10.

Show that one can turn 10/81 into a very cool number by subtracting (m/n)/10^9 from it so that the result looks like

0.1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 . . .

for a LONG way. The idea is to continue the integer sequence for as long as you can using no more than 4-digit numbers for m and n.

Source: Cool Irrational Numbers and Their Rather Cool Rational Approximations, by Francesco Calogero, The Mathematical Intelligencer, Fall 2003, p. 72-76.