(I love sevenths in decimal. One seventh is 0.1̅4̅2̅8̅5̅7̅, two sevenths 0.2̅8̅5̅7̅1̅4̅, three sevenths 0.4̅2̅8̅5̅7̅1̅, &c. Just remember 7 × 2 = 14, 14 × 2 = 28, 28 × 2 = 56 and change the six to a seven for some reason and start again with that, and rotate the whole lot as necessary. It’s just beautiful.)
Firstly, why are your numbers wearing hats? (I know it represents a repeating number, but how do you type numbers with hats? I assume you're not copy-pasting 5̅7̅1̅ whenever you want to write it out. Is there a shorthand on MacOS?)
Regarding the 7's, I don't understand the pattern from 0.142857 to 0.285714 to 0.428571. Are you saying that 0.428571 is easy to remember, somehow? Or derivable from 0.285714? In fact, did you write out these numbers without using a calculator?
I see what you're saying with 7 x 2 = 14, x 2 = 28, x 2 = 56. And I see that if you change 56 to 57, then yes indeed it does match the ending of 0.1̅4̅2̅8̅5̅7̅, and then you can repeat the pattern of 7 x 2 = 14, which matches the ending of your 0.2̅8̅5̅7̅1̅4̅ number. But what's the connection to 0.4̅2̅8̅5̅7̅1̅? It's unclear how to use your trick to go from 7̅1̅4̅ to 5̅7̅1̅. Or even why that's helpful in general, or what the overall connection is between all of these seemingly unrelated things.
Thanks for sharing!
for 2/7, take the second lowest number (2), and rotate it to the front: 285714 for 3/7, third lowest (4): 428571 for 4/7 it then becomes 571428 and so on
This seems to hold up with my calculator. Pretty cool.
The 7x2, 14x2, 28x2 is jus a trick for remembering what these numbers are and their order. (142856, +1).
<Multi_key> <asciicircum> <underscore> : "̅" U0305 # COMBINING OVERLINE