An interesting fraction

[soroban]

\(\displaystyle \dfrac{1}{998,001} \;=\;0.\overline{000\,001\,002\,003\,004\,005\, \cdots\,996\,997\,999}\, \cdots\)

The decimal representation contains all the 3-digit numbers except 998
. . and the 2997-digit cycle repeats forever.

This is just one of a family of such fractions.
Can you determine the underlying characteristic?

 

[daon2]

I 'discovered' the similar value 1/81 while playing with a calculator in my freshman nutrition class. My mind was blown.

I suspect it has to do with the remainders:

10^2/81 has remainder 19

10^4/9801 has remainder 199

10^6/998001 has remainder 19999

And I suspect 10^(2k)/(99..9800..01) with exactly (k-1) 9's and 0's, will also exhibit the pattern.

And actually, since 1/81=1/9^2, 1/9801=1/99^2, there's probably something there too. I'll think about it more later if no one else chimes in.

 

[pappus]

I 'discovered' the similar value 1/81 while playing with a calculator

...

This fraction can be used to do a "trick" which impresses sometimes even 4th-graders:

1. Write the ciphers

12345679

(you certainely have noticed that the 8 is missing)

2. Ask which cipher looks really ugly. (The wittier in class - those with the big mouth - will surely call 8).

3. Let them do by hand!

12345679 * 72

(If you want to get only 3s as result the factor is 27. In genaral: You'll get only ciphers a if the factor is 9a)