A Little NUmber Curiosity

[DrDave]

Recently, I was doing some number crunching that involved 9 and multiples thereof. Just for fun, look at the digits for the INVERSE of each of these numbers. The "special properties" (whatever that might mean) of 9 have been studied and used since ancient times.



In the numbers below, they mostly show insertion of zeros. But the set begining with 81 runs a bit differently by inserting various digits. And it appears 81*9 =729 contiunes with the insertion of digits. I didn't pursue this one further.



I'm sure those of you that are pattern finders will see also that the sum of the digits reduce to 9.



Also, why don't ALL numbers that reduce to 9 give these "unusual" sequences? For example, 27 doesn't produce  a very "pleasing" sequences of digits.



9, 99, 999, 9999, ...



18, 1818, 181818, ...



45, 4545, 454545, ...



36, 3636, 363636, ...



81, 8181, 818181, 81818181, ...



108, 108108, 108108108, ...



81*9 = 729



DrDave

[cormullion]

I think there's a book about number 9. 9 is a bit odd in my view.

[DrDave]

9 is a bit odd in my view.

You are so right. And it's also a little square.

[Kazimir Majorinc]

When you speak about problems with 27, that means that 1/27 doesn't make nice output, right?

[DrDave]

When you speak about problems with 27, that means that 1/27 doesn't make nice output, right?

Essentially, yes. When you compare the inverse of 27, 27227, 27272, etc. with any of the others that I listed above, it doesn't seem to be inserting either zeros or other digits as the above numbers do.