# Ratios that are invertible, yet remain rational

Discussion in 'Physics & Math' started by Neddy Bate, Aug 4, 2020.

1. ### Neddy BateValued Senior Member

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2,121
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?

Last edited: Aug 4, 2020

3. ### Confused2Registered Senior Member

Messages:
607
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.

5. ### Neddy BateValued Senior Member

Messages:
2,121
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.