How to find the position of a given rational number. Create a recursive sequence beginning with the number and ending with 1 using the following rule:
If the number is less than 1, create its reciprocal;
if the number is greater than 1, subtract 1;
stop when 1 is reached.
(keep track of the order of these steps.)
Example beginning with 7/16 (step in parentheses) last column explained below
7/16 (r) 101
16/7 (1) 100
9/7 (1) 50
2/7 (r) 25
7/2 (1) 24
5/2 (1) 12
3/2 (1) 6
1/2 (r) 3
2 (1) 2
1
The last column is the value of n for each value in the first column and is found by starting with 1 at the bottom and working up by the following rule:
If there is a (1) then multiply by 2;
if there is a (r) then add 1.
So the number a_{101}=7/16.
It seems clear that each rational number should reach 1 in a unique way and that the reversal process should reach a unique value. I have not actually proven this.
Edit was to fix a typo. I made a different change than the one Charlie suggested.
Edited on August 29, 2016, 8:42 am

Posted by Jer
on 20160828 17:09:33 