I found these simple scaling factors between 1 and 2 which do not have any repeating decimals even when inverted: 1.000 ... 1.000 0.800 ... 1.250 0.625 ... 1.600 0.500 ... 2.000 Are there others? Proof / disproof?

0.25,0.1,0.2 .. ?? First specify the number of decimal places you will allow (obviously a finite number) then, er, do something. 0.0000000000000000001 Edit - between 1 and 2 - forget the above.

Yes, between 1 and 2. I don't think there are any others, but I'm not sure why, or how to prove it. It seems a little strange that there would only be 1.250 and 1.600 but I guess it is a feature of base 10, rather than a bug, lol. If we're not limiting it to between 1 and 2, we can extend the pattern indefinitely, and then eventually get to all of the other numbers that you suggested.